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Quantum Groups
Algebra has moved well beyond the topics discussed in standard
undergraduate texts on \modern algebra." Those books typically
dealt with algebraic structures such as groups, rings and fields:
still very important concepts! However, Quantum Groups: A Path
to Current Algebra is written for the reader at ease with at least one
such structure and keen to learn the latest algebraic concepts and
techniques.
A key to understanding these new developments is categorical
duality. A quantum group is a vector space with structure. Part of
the structure is standard: a multiplication making it an \algebra."
Another part is not in those standard books at all: a comultiplication,
which is dual to multiplication in the precise sense of category
theory, making it a \coalgebra." While coalgebras, bialgebras and
Hopf algebras have been around for half a century, the term
\quantum group," along with revolutionary new examples, was
launched by Drinfel'd in 1986.
AUSTRALIAN MATHEMATICAL SOCIETY LECTURE SERIES
Editor-in-chief: Professor Michael Murray, University of Adelaide, SA 5005, Australia
Editors:
Professor P. Broadbridge AMSI, The University of Melbourne,
Victoria 3010, Australia
Professor C. C. Heyde, School of Mathematical Sciences,
Australian National University, Canberra, ACT 0200, Australia
Professor C. E. M. Pearce, Department of Applied Mathematics,
University of Adelaide, SA 5005, Australia
Professor C. Praeger, Department of Mathematics and Statistics,
University of Western Australia, Crawley, WA 6009, Australia
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
Introduction to Linear and Convex Programming, N . C A M E R O N
Manifolds and Mechanics, A . J O N E S , A . G R A Y & R . H U T T O N
Introduction to the Analysis of Metric Spaces, J . R . G I L E S
An Introduction to Mathematical Physiology and Biology, J . M A Z U M D A R
2-Knots and their Groups, J . H I L L M A N
The Mathematics of Projectiles in Sport, N . D E M E S T R E
The Peterson Graph, D . A . H O L T O N & J . S H E E H A N
Low Rank Representations and Graphs for Sporadic Groups,
C. PRAEGER & L. SOICHER
Algebraic Groups and Lie Groups, G . L E H R E R ( e d . )
Modelling with Differential and Difference Equations,
G. FULFORD, P. FORRESTER & A. JONES
Geometric Analysis and Lie Theory in Mathematics and Physics,
A. L. CAREY & M. K. MURRAY (eds)
Foundations of Convex Geometry, W . A . C O P P E L
Introduction to the Analysis of Normed Linear Spaces, J . R . G I L E S
The Integral: An Easy Approach after Kurzweil and Henstock,
L. P. YEE & R. VYBORNY
Geometric Approaches to Differential Equations,
P. J. VASSILIOU & I. G. LISLE
Industrial Mathematics, G . F U L F O R D & P . B R O A D B R I D G E
A Course in Modern Analysis and its Applications, G . L . C O H E N
Chaos: A Mathematical Introduction, J . B A N K S , V . D R A G A N & A . J O N E S
Quantum Groups
A Path to Current Algebra
ROSS STREET
Technical Editor:
ROSS MOORE
CAMBRIDGE UNIVERSITY PRESS
Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo
Cambridge University Press
The Edinburgh Building, Cambridge CB2 8RU, UK
Published in the United States of America by Cambridge University Press, New York
www.cambridge.org
Information on this title: www.cambridge.org/9780521695244
© R. Street 2007
This publication is in copyright. Subject to statutory exception and to the provision of
relevant collective licensing agreements, no reproduction of any part may take place
without the written permission of Cambridge University Press.
First published in print format 2007
ISBN-13
ISBN-10
978-0-511-26900-4 eBook (EBL)
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ISBN-13
ISBN-10
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Cambridge University Press has no responsibility for the persistence or accuracy of urls
for external or third-party internet websites referred to in this publication, and does not
guarantee that any content on such websites is, or will remain, accurate or appropriate.
To Oscar and Jack
Contents
Introduction
1
Revision of basic structures
page ix
1
2
3
Duality between geometry and algebra
The quantum general linear group
5
9
4
5
Modules and tensor products
Cauchy modules
13
21
6
7
8
Algebras
Coalgebras and bialgebras
Dual coalgebras of algebras
27
37
47
9
10
Hopf algebras
Representations of quantum groups
51
59
11
12
Tensor categories
Internal homs and duals
67
77
13
14
Tensor functors and Yang–Baxter operators
A tortile Yang–Baxter operator for each
85
15
16
finite-dimensional vector space
Monoids in tensor categories
Tannaka duality
93
97
109
17
18
Adjoining an antipode to a bialgebra
The quantum general linear group again
117
119
19 Solutions to Exercises
References
121
133
Index
135
vii
viii
Bradshaw: “Ceremonial Figure”, Tassel Bradshaw Group, [Wal94, Plate 20].
Introduction
Algebra has moved well beyond the topics discussed in standard undergraduate texts on “modern algebra”. Those books typically dealt with algebraic
structures such as groups, rings and fields: still very important concepts!
However, Quantum Groups: A Path to Current Algebra is written for the
reader at ease with at least one such structure and keen to learn the latest
algebraic concepts and techniques.
A key to understanding these new developments is categorical duality.
A quantum group is a vector space with structure. Part of the structure
is standard: a multiplication making it an “algebra”. Another part is not
in those standard books at all: a comultiplication, which is dual to multiplication in the precise sense of category theory, making it a “coalgebra”.
While coalgebras, bialgebras and Hopf algebras have been around for half
a century, the term “quantum group”, along with revolutionary new examples, was unleashed on the mathematical community by Drinfeld [Dri87] at
the International Congress in 1986. Before launching into an explanation
of the duality required, I should mention here that an ordinary group gives
rise to a quantum group by taking the vector space with the group as basis.
When pushed to provide formal proofs of our claims, mathematicians
generally resort to set theory. We build our structures on sets and feel
satisfied when we can be explicit about the elements of our constructed
objects. Up to the mid twentieth century, algebraic objects were sets with
selected operations which assigned elements to lists of elements. Typically,
we would have binary operations which might be called addition, multiplication or Lie bracket respectively assigning a sum, product or formal
commutator to each pair of elements.
In those days, the importance was recognized of dealing with the homomorphisms between algebraic structures: these were the functions which
preserved the operations involved in the kind of structure at hand. The existence of a bijective homomorphism (isomorphism) between two algebraic
objects meant that the two objects played the same role. So how could
the literal elements be the defining ingredient? The important issue was
the way the algebraic object related to others of its own kind by means
of homomorphisms into it or out of it. Quite often the elements could be
recaptured as homomorphisms from a particular object into the one of interest. For example, the elements of a vector space were in bijection with
the linear functions from a selected one-dimensional space.
ix
x
Introduction
Homomorphisms into an object might therefore be called “generalized
elements” of the object. However, this notion of element of the object will
depend on the kind of structure we are studying since that will determine
what a homomorphism is (a group homomorphism, a linear function, a ring
homomorphism, or whatever).
We quite often wish to add more elements to our sets to improve the
properties of the operations: as when we construct the integers from the
natural numbers to obtain subtraction; or when we construct the rational
numbers from the integers to obtain division; or when we construct the real
polynomials from the real numbers to obtain an indeterminate. These constructions can be described explicitly as sets with operations that include
the original ones. More importantly, each such construction is unique up to
/ C out
isomorphism with a universal property: each homomorphism X
of the original structure X, into a set C with the extra structure, extends
/ C out of the constructed object X̂ .
to a homomorphism X̂
In this way it was realized that knowing the homomorphisms out of
objects determined the objects just as surely as knowing the homomorphisms into them did. It is natural then to call homomorphisms out of
an object “generalized co-elements”. Once this kind of duality principle is
acknowledged, interesting facts appear.
Let us take a simple example purely using sets. Consider two sets X
and Y . Their cartesian product X × Y is the set whose elements are pairs
(x, y) where x lies in X and y lies in Y . We are not studying any structure
on these sets except for the property of being a set. So homomorphisms in
/ X ×Y
this case are merely functions. It is clear that functions f : T
into X × Y from a test object T are in bijection with pairs of functions
/ X and f2 : T
/ Y . In other words, T -elements
(f1 , f2 ) where f1 : T
of X × Y are in bijection with pairs consisting of a T -element of X and a
T -element of Y . All that is fairly straightforward.
Now suppose that our sets X and Y have no common elements; if they
are not disjoint, replace them by isomorphic sets which are. Write X + Y
for the union; we write X + Y rather than X ∪ Y to emphasize that it
is the disjoint union (if X and Y were finite, the number of elements of
X + Y would be the sum of the number of elements in X and the number
/ T is determined by its restriction to X
in Y ). A function f : X + Y
and its restriction to Y . In other words, the co-T -elements of X + Y are in
bijection with pairs consisting of a co-T -element of X and a co-T -element
of Y .
We conclude that the constructions X × Y and X + Y are duals of one
another. This is not something that was stressed when we were taught the
more abstract multiplication and addition of numbers in infants’ school.
If we now look at vector spaces or groups X and Y , the cartesian
product X × Y as sets becomes a vector space or group by means of coordinatewise operations from X and Y ; again this has pairs as the generalized
Introduction
xi
/ X × Y . However, to obtain the dual constructions in these
elements T
cases is quite different from the disjoint union of sets: in the case of vector
spaces, we have that X × Y is self-dual (called direct sum and denoted by
X ⊕ Y ); in the case of groups, the dual notion is rather complicated (called
the free product by group theorists).
In order to formalize the way in which constructions such as these
can be dual, we can use the notion of category. I intend to give a definition of this concept in this introduction. Before doing so, I would like to
draw an analogy. It was noticed in projective plane geometry that theorems occurred in pairs: one such pair consists of Pascal’s Mystic Hexagram
Theorem and Brianchion’s Theorem; both are about conics. Given one theorem in a pair, the other is obtained by interchanging the role of points and
lines, reversing the incidence relation (“lies on” becomes “goes through”).
To formally explain this duality, we abstract the notion of projective plane.
Here is the essence of the definition. A projective plane P consists of
two sorts of elements: one sort called points, the other called lines. It also
consists of a relation between these elements, called incidence (this is a rule
telling when a point is incident with a line). There are some axioms which
include:
1.
for distinct points P and Q , there is a unique line L such that P
and Q are both incident with L ; and,
2.
for distinct lines L and M , there is a unique point P such that P is
incident with both L and M .
Any system satisfying this is a projective plane! The points do not need
to look like points and the lines do not need to look like lines in any sense.
Of course, we still draw pictures to help our intuition.
Now we are ready to formalize duality. Given a projective plane P , we
obtain a projective plane P rev whose points are the lines of P , whose lines
are the points of P , and whose incidence relation is the reverse of that of
P . Notice that axioms (1) and (2) for P rev are respectively axioms (2)
and (1) for P . This means that, if we prove a theorem about all projective
planes, then the dual theorem is automatically true by applying the original
theorem to P rev .
It turns out that there are not too many interesting theorems assuming
only axioms (1) and (2). A further axiom based on a theorem of Pappus can
be added and the system remains self-dual. In fact, conics can be defined
using an idea of Steiner, and Pascal’s Theorem can be proved. Let us now
discontinue discussion of this analogy and return to the formalization of
the duality at hand.
A category A consists of two sorts of elements: one sort called objects,
the other called morphisms (or arrows). It also consists of three functions.
The first function assigns to each morphism f a pair (A, B) of objects in
xii
Introduction
which case A is called the domain (or source) of f while B is called the
/ B and A f / B are used.
codomain (or target) of f ; the notations f : A
/A
The second function assigns to each object A a morphism 1A : A
called the identity morphism of A . A pair (f, g) of morphisms is called
composable when the codomain of f is equal to the domain of g . The third
function assigns to each composable pair (f, g) of morphisms, a morphism
g ◦ f , called the composite of f and g , whose domain is that of f and whose
codomain is that of g . There are two axioms:
1.
if (f, g) and (g, h) are composable pairs of morphisms then
(h ◦ g) ◦ f = h ◦ (g ◦ f ) ; and,
2.
if f : A
/ B is a morphism then f ◦ 1 = f = 1 ◦ f .
A
B
The standard argument shows that identity morphisms are unique. The
notation A(A, B) (or HomA (A, B)) is used for the set of all morphisms in
A from A to B .
There is a category Set whose objects are sets, morphisms are functions, and composition is the usual composition of functions. There is a
category Vectk whose objects are vector spaces over a fixed field k and
morphisms are linear functions; composition is as usual. Similarly we have
a category whose objects are groups and a category whose objects are rings.
However, there are categories whose objects do not look like sets and
whose morphisms do not look like functions. For example, there is a category whose objects are integers, whose morphisms are pairs (m, n) of integers such that the domain of (m, n) is m and the codomain is the product
mn ; a pair ((m, n), (r, s)) of morphisms is composable when mn = r and
the pair’s composite is (m, ns) .
Now to duality. Given a category A , there is a category Aop whose
objects are the objects of A , and morphisms are the morphisms of A ;
however, the domain of a morphism is its codomain in A while its codomain
in Aop is its domain in A . A pair (g, f ) of morphisms is composable in Aop
if and only if (f, g) is composable in A ; its composite f ◦ g in Aop is the
composite g ◦ f in A . We call Aop the dual or opposite of the category A .
Perhaps it helps to say that Aop is the category obtained from A by
/ B in A is precisely a morphism
reversing arrows: a morphism f : A
op
/
A in A . Admittedly, if the objects of A look like sets (that is,
f : B
are sets with some structure), the same is true of Aop ; but the same cannot
be said for morphisms that are functions, since formally reversed functions
can scarcely be thought of as functions.
The duality between cartesian product and disjoint union can now
be made precise. In a category A , a product for two objects A and B
/ A , p : A×
consists of an object A×B and two morphisms p1 : A×B
2
/
B (called projections) with the following “universal” property: for all
B
Introduction
xiii
/ A, b : T
/ B , there exists a unique
objects T and morphisms a : T
/ A × B, denoted by (a, b) , such that p ◦ (a, b) = a and
morphism T
1
p2 ◦ (a, b) = b . This means that T -elements of A × B are in bijection with
pairs consisting of a T -element of A and a T -element of B .
/ D in a category A is called a right inverse for a
A morphism h : C
/
morphism k : D
C when k ◦ h = 1C ; we also say that k is a left inverse
for h . A morphism h is invertible (or an isomorphism) when it has both a
left and right inverse; in this case, a familiar argument shows that the left
and right inverse agree and this common morphism is unique, being called
the inverse of h and denoted by h−1 . If there exists an invertible morphism
/ D then we say C and D are isomorphic and write C ∼
C
= D . In a
category, we think of isomorphic objects as being essentially the same. Any
two products of two objects A and B can be proved, by an easy argument,
to be isomorphic.
Now we have our duality between cartesian product and disjoint union
of sets: cartesian product is the product in the category Set while disjoint
union is the product in the category Setop.
We can give an even simpler example. An object K of a category A is
called terminal when, for all objects A of A , there is precisely one morphism
/ K . The singleton set 1 is terminal in the category Set while the
A
empty set ∅ is terminal in Setop.
Any concept defined for all categories A has a dual concept which is
the same concept translated to Aop : the prefix “co-” is used. So a product
in Aop is called a coproduct in A . A terminal object in Aop , under this
system, would be called a coterminal object in A ; but it is also called an
initial object of A .
In the spirit of category theory itself, we should consider appropriate
/X
morphisms of categories. These are called functors. A functor F : A
between categories A and X consists of two functions. The first assigns to
each object A of A an object F A of X . The second function assigns to each
/ F B of X . There
/ B of A a morphism F f : F A
morphism f : A
are two axioms:
1.
F 1A = 1F A for all objects A of A ; and,
2.
F (g ◦ f ) = F g ◦ F f for all composable pairs (f, g) in A .
It is easy to see that functors preserve invertibility of morphisms: in fact
they take inverses to inverses.
Let us look at a couple of examples of functors.
• Each object T of a category A determines a functor RT = A(T, ) :
/ Set called the functor represented by T ; the elements of
A
/ A in A (that is, the
RT A = A(T, A) are the morphisms a : T
/ R B takes
T -elements of A), while the function RT f : RT A
T
such an a to f ◦ a .
xiv
Introduction
• Suppose K is an object of A for which a product K ×A exists (and is
/A
chosen) for all objects A . There is a functor F = K × : A
defined on objects by F A = K ×A and on morphisms by F f = K ×f
/ K ×B.
where K × f = (p1 , f ◦ p2 ) : K × A
Categories were invented not only to formalize duality but to formalize
the concept of “naturality” in mathematics. The idea was that a natural
transformation should be one that involves no ad hoc choices. For example,
if V is a vector space and V ∗ is the vector space of linear functions from V
/ V ∗∗ which
into the base field k , there is a natural linear function V
∗
/
k defined by evaluation
takes v ∈ V to the linear function ev : V
/ V that depends on a choice of
at v . However, any linear function V ∗∗
basis for V should not be natural.
/ X and G : A
/ X are functors between the same
Suppose F : A
/ G is a function. The funccategories. A natural transformation θ : F
/ GA of X .
tion assigns to each object A of A a morphism θA : F A
/
There is a single axiom: for each morphism f : A
B,
Gf ◦ θA = θB ◦ F f .
There is an obvious componentwise
composition of natural transformations.
This defines a category A , X , called a functor category, where the objects
/ X and the morphisms are natural transformations.
are functors F : A
A natural isomorphism is an invertible morphism in the functor category.
/ Set is called representable when it is isomorphic to
A functor F : A
RT for some object T ; such a T is called a representing object for F . For
/ Set , which takes each vector space to
example, the functor U : Vectk
its underlying set and each linear function to that function, is representable:
we have U ∼
= Rk since the linear functions from the field k to a vector
space V are in natural bijection with elements of V . Many constructions
in mathematics are designed to provide representing objects for interesting
functors.
Let us look at a couple of examples of natural transformations:
/ Set is a functor and T is an object of A . Each
• Suppose F : A
/F
element x of F T determines a natural transformation x̂ : RT
defined by x̂A (a) = (F a)(x)
.
The
Yoneda
Lemma
states
that
this
defines a bijection F T ∼
= A , Set (RT , F ) . The inverse bijection
/ F to
is even easier: it takes the natural transformation θ : RT
the element θT (1T ) of F T .
/ L is a morphism of a category A in which prod• Suppose h : K
ucts of pairs of objects exist. Then we obtain a natural transforma/L×
whose value at the object A is the
tion f × : K ×
/ L×A .
morphism f × A = (f ◦ p1 , p2 ) : K × A
Introduction
xv
Modern algebra in the sense of the first half of the twentieth century dealt with sets equipped with operations. Soon after, the idea of
co-operations crept into mathematics. The notion of coalgebra is dual to
algebra. This is the main concept in this book.
Now I turn to the book’s contents. Chapter 1 gives precise definitions
of monoids and groups; the axioms are expressed in terms of diagrams
ready to be imported to a general category. This importation is carried
out in Chapter 2 where we provide the important example of 2 × 2 matrices in readiness for the quantum version. A duality between geometry
and algebra is explained. In Chapter 3, we describe the quantum general
linear group of 2 × 2 matrices as a coalgebra. This comes from lectures
by Manin in Montréal. Chapter 4 is about modules over rings; we find it
natural to take a 2-sided point of view so that our basic module M has a
left action by a ring R and a right action by a ring S which compatibly
interact. Chapter 5 concerns finitely generated, projective modules under
the mysterious name of “Cauchy modules”. It turns out that F. W. Lawvere
noticed a concept in enriched category theory which has Cauchy sequences
as an example; when interpreted for additive categories, it leads to modules
that are finitely generated and projective. Chapter 6 discusses algebras,
Lie algebras and the Poincaré–Birkhoff–Witt Theorem. Chapter 7 is about
coalgebras and bialgebras. A coalgebra is a vector space with a comultiplication. A bialgebra is an algebra which is also a coalgebra subject to a
compatibility condition. The dual vector space of a coalgebra is an algebra,
however, the usual dual of an algebra need not naturally be a coalgebra. In
Chapter 8 we describe Sweedler’s modification (see [Swe69] and [Abe80])
of the dual of an algebra which is a coalgebra.
In Chapter 9, we look at Hopf algebras. These should be thought of
as generalized groups. An important part of group theory is the theory of
their representations: these are modules over the group ring. In Chapter 10
we look at modules over bialgebras. Then, in Chapter 11, we move to use
categories more seriously. We discuss categories equipped with an abstract
tensor product: monoidal or tensor categories. We discuss examples involving braids. A deep example, not treated here, is the subject of the
paper [JS95]. An important property of the tensor product U ⊗V of vector
spaces is that it represents the vector space V , W of linear functions from
V to W :
Vectk (U ⊗V , W ) ∼
= Vectk U, V , W .
In Chapter 12, this idea is lifted to arbitrary tensor categories. Examples
from knot theory are provided.
The Yang–Baxter equation, from the branch of physics called statistical
mechanics, had a major influence on the new examples of Hopf algebras
called quantum groups. In Chapter 13, an algebraic concept of Yang–
Baxter operator, which makes sense in any tensor category, is explained.
A family of examples from linear algebra is provided in Chapter 14.
xvi
Introduction
In Chapter 15, the notion of monoid is lifted to the level of generality
at which “algebras” and “coalgebras” become precise categorical duals.
For the first time, I believe, in a text at this level, emphasis is placed on
“2-cells” between monoid morphisms, providing the student with a gentle
introduction to higher-dimensional category theory.
Each bialgebra has a tensor category of representations. This correspondence is a modern formulation of Tannaka–Krein duality. The treatment of this topic in Chapter 16 makes use of the 2-dimensional structure
of monoids from Chapter 15. There is by now a vast literature on Tannaka
duality. We satisfy ourselves with a sketch in Chapter 17 of an application
to construct universally a Hopf algebra from a bialgebra.
Finally, in Chapter 18, the example of Chapter 3 is revisited in the
light of what has been learned. There are exercises at the end of several
chapters. Solutions to most of these are provided in Chapter 19.
Acknowledgements ...
Many ideas presented here are my version of joint research with André
Joyal. I consider myself very fortunate to have experienced such exciting
collaboration.
I would like to thank the students and staff who attended the original
course in the first half of 1990 at Macquarie University. I am very grateful to Paddy McCrudden (as a Vacation Scholar under my supervision in
January–February 1995) for his careful reading of my handwritten notes,
and particularly for his systematic checking of the exercises.
It is a pleasure to acknowledge the support of grants from the Australian Research Council during the preparation of this work.
... the typing process
Many of the chapters were carefully typed by Elaine Vaughan. Technology
moved ahead incredibly from that word-processor available in our Department in 1990. Typesetting with TEX and LATEX was begun by Ross Moore
and continued by several post-doc and graduate students; namely Sam
Williams, Ross Talent (now deceased) and Simon Byrne. Each added further exercises and solutions, as these were encountered in lecture courses.
Most important in this was the use of XY-pic1 to produce the commutative diagrams that appear throughout this book, and which play such an
integral part in the visualization and understanding of much of the material. None of these diagrams has been imported as an image built using
other software. All, including the braids, tangles, and 2-cell diagrams, are
1 Originally written by Kristoffer Rose; extended and enhanced by Rose and Moore
for mathematical applications and higher quality output. The XY-pic package and documentation is now included with all TEX and LATEX distributions.
Introduction
xvii
specified within the LATEX source using the XY-pic package syntax. Indeed
the syntax and coding to handle curves and 2-cells was written in 1993–94
by Ross Moore, specifically for use with this book. Since then the XY-pic
package has become a useful tool for presenting diagrammatic material in
Category Theory and other branches of mathematics, computer science and
linguistics.
As an application of Ross Moore’s work with the LATEX2HTML translation software, an earlier version of the present manuscript was made available via the “world-wide web”, now known as the internet. In that form it
was used as a source of lecture notes for courses at Macquarie.
A great deal of credit is also due to Simon Byrne (as a Vacation Scholar
in January–February 2005) for finishing off the typing of exercises and for
assembling the manuscript into a form ready to submit as a proposal for
the Australian Mathematical Society Lecture Series. With this go-ahead,
the final version of the manuscript, complete with up-to-date Bibliography,
Index, front-matter and filler images was prepared by Ross Moore, who is
acknowledged here as being the Technical Editor for this monograph.
... the illustrations
The illustrations appearing at the end of some chapters are reproduced from
Grahame Walsh, Bradshaws [Wal94]. I am very grateful to the Bradshaw
Foundation and Edition Limitée for consenting to their inclusion. The
original coloured rock paintings, which the silhouettes trace, are the work
of Australian people living as much as 50 millennia before our time.
xviii
Introduction
These paintings have been mentioned already in the mathematico-scientific
literature, in connection with knots and braids; viz.
How old are knots? It has been suggested that the Stone Age
should be called the Age of String. The extraordinary tasselled figures photographed and described by G. L. Walsh in
Bradshaws: Ancient Rock Paintings of Western Australia (Edition Limitée, 1994) have been suggested to be 50,000 years
old. Knots have been intimately linked with the development
of humans, through weapons, fishing, hunting, clothing, housing, boating and a myriad of other ways.
The metaphor of knots is found throughout literature, and
knots and interlacing are featured in many forms of art.
Ronald Brown, review of: “The Knot Book:
An Elementary Introduction to the Mathematical Theory
of Knots” by Colin C. Adams (W. H. Freeman 1994),
appeared in Nature, Vol. 371 (13 October 1994).
Suggested Further Reading
[JS91b] André Joyal and Ross Street. An introduction to Tannaka duality
and quantum groups. In Category Theory (Como, 1990), volume
1488 of Lecture Notes in Mathematics, pages 413–492. Springer,
MR1173027
Berlin, 1991.
[Kas95] Christian Kassel. Quantum Groups, volume 155 of Graduate Texts
in Mathematics. Springer, New York, 1995.
MR1321145
[Maj95] Shahn Majid. Foundations of Quantum Group Theory. Cambridge
University Press, Cambridge, 1995, (paperback, 2000). MR1381692
[SS93]
Steven Shnider and Shlomo Sternberg. Quantum Groups: From
CoAlgebas to Drinfel d Algebras. Graduate Texts in Mathematical
Physics, II. International Press, Cambridge, MA, 1993. MR1287162
[Yet01]
David N. Yetter. Functorial Knot Theory, volume 26 of Series on
Knots and Everything. World Scientific Publishing Co. Inc., River
MR1834675
Edge, NJ, 2001.
1
Revision of basic structures
The cartesian product of n sets X1 , . . . , Xn is the set
X1 × · · · × Xn = {(x1 , . . . , xn ) | xi ∈ Xi } .
There is a canonical bijection
(X1 × · · · × Xm ) × (Xm+1 × · · · × Xn ) ∼
= X1 × · · · × Xn
given by deleting the inside brackets. The diagonal function
δ:X
/ X × ···× X
is given by δ(x) = (x, . . . , x) .
The cartesian product of no sets is the special set 1, with precisely one
element, which should technically be denoted by empty parentheses ( ) .
Particular cases of the canonical bijections are
X ×1 ∼
= X ∼
= 1×X .
/ 1 will be denoted by ε rather than δ ; it is the only
The diagonal X
/ Y1 , . . . , f : Xn
/ Yn induce
/
1 . Functions f1 : X1
function X
n
a function
/ Y1 × · · · × Yn
f1 × · · · × fn : X1 × · · · × Xn
given by f1 × · · · × fn (x1 , . . . , xn ) = f1 (x1 ), . . . , fn (xn ) .
/ X on a set X is given by 1 (x) = x .
The identity function 1X : X
X
/
1 is uniquely determined. Similarly the diagonal
We noted that ε : X
/ X × X is unique, determined by commutativity of the diagram
δ: X
X
∼
=
(Identity)
∼
=
δ
1×X
ε×1X
X ×X
1
1X ×ε
X ×1.
2
Quantum Groups: A Path to Current Algebra
Furthermore, the following diagram commutes
(Associativity)
δ
X
X ×X
δ×1X
X ×X ×X .
1X ×δ
/ X × X × X so determined is none other than the
The function X
ternary diagonal.
A monoid is a set M together with special purpose functions η : 1
/ M such that the following diagrams commute.
µ : M ×M
/M,
M
∼
=
(Id)
∼
=
µ
1×M
(Assoc)
µ
M
M ×M
η×1M
M ×M
1M ×η
µ×1M
1M ×µ
M ×1
M ×M ×M
If we write 1 for the value of η at the only element of 1 and we write x y
for µ(x , y) then the above diagrams translate to the equations
1x = x = x1
(x y) z = x (y z)
for all x , y , z ∈ M .
This time functions η and µ are not uniquely determined by the set M .
However given µ , condition (Id) uniquely determines η while the condition
/M
(Assoc) gives an unambiguous ternary operation µ : M × M × M
which we write as µ(x , y , z) = x y z . Generally there is an unambigu/ M determined by the
ous multiple product function µ : M × · · · × M
binary µ .
An element x ∈ M is called invertible when there exist y , z ∈ M such that
y x = 1 and x z = 1 . Notice that
y = y 1 = y(x z) = (y x)z = 1 z = z
so each invertible element x determines uniquely an element, denoted x−1 ,
satisfying x−1 x = 1 = x x−1 .
A group is a monoid in which each element is invertible. Then we have a
/ M such that this next diagram commutes.
function ι : M
M
δ
M ×M
(Invertibility)
ι×1M
1M ×ι
ε
M ×M
η
I
µ
M
Revision of basic structures
3
Note carefully the dependence of this axiom on the diagonal structure of
cartesian product.
For a set X, the n-fold cartesian product X × · · · × X is denoted by X n .
Each permutation ξ on {1, . . . , n} induces a bijection
/ Xn
σξ : X n
given by σξ (x1 , . . . , xn ) = (xξ(1) , . . . , xξ(n) ) . In particular, we have the
switch coming from the non-identity permutation of {1 , 2} :
/ X ×X
σ : X ×X
,
σ(x , y) = (y , x) .
Each σξ is a composite of bijections of the form 1X × · · · × σ × · · · × 1X .
Notice that the following diagram commutes.
X
(Commutativity)
δ
δ
σ
X ×X
X ×X
A monoid M , η , µ is called commutative when the following diagram
commutes.
M
µ
µ
σ
M ×M
It follows that the composite M n
permutation ξ .
σξ
M ×M
Mn
µ
M is independent of the
Suppose M and N are monoids. A monoid morphism (or homomorphism
/ N such that the following diagrams
of monoids) is a function f : M
commute.
1
η
M ×M
η
f ×f
µ
M
f
N
M
N ×N
µ
f
N
Expressed in terms of elements, these diagrams merely say that f (1) = 1
and f (x y) = f (x)f (y) . If N has left-cancellation (i.e., ab = ac implies
b = c ; e.g., if N is a group) then f (1) = 1 is redundant.
4
Quantum Groups: A Path to Current Algebra
Monoid morphisms preserve invertibility : if x ∈ M is invertible, f (x−1 ) =
f (x)−1 . So for groups M and N we have commutativity of the square
M
f
ι
M
N
ι
f
N.
A rig is a set R enriched with two monoid structures, a commutative one
written additively and the other written multiplicatively, such that the
following equations hold:
(Distributive)
a0 = 0 = 0a
a(b + c) = a b + a c , (a + b)c = a c + b c .
The natural numbers N = {0 , 1 , 2 , . . . } provide an example of a rig.
A ring is a rig for which the additive monoid is a group. The integers Z
provide an example.
A rig is commutative when the multiplicative monoid is commutative.
A field is a commutative ring for which each element is either 0 or has a
multiplicative inverse.
/ S is a function which is a
For rigs R and S a rig morphism f : R
monoid morphism for both the additive and multiplicative structures.
Let k denote a field. A k-algebra is a ring A together with a ring morphism
/ A . Notice that either A is trivial (i.e., 1 = 0) , or η is injective
η: k
[ κ = κ ⇒ κ − κ = 0 ⇒ κ − κ is invertible ⇒ η(κ − κ ) is invertible
1=0
==⇒ η(κ − κ ) = 0 ⇒ η(κ) = η(κ ) ] . We can define scalar multiplication
/ A by κ a = η(κ) a .
k×A
/ B is a ring
For k-algebras A and B, a k-algebra morphism f : A
morphism such that the next diagram commutes;
k
η
η
A
f
B
that is, f (κ a) = κ f (a) . We write Algk A , B for the set of k-algebra
/ B.
morphisms f : A
An isomorphism is a bijective morphism; automatically its inverse function
is also a morphism.
2
Duality between geometry and
algebra
The purpose of this section is to convince you that commutative algebras
are really spaces seen from the other side of your brain.
For a compact hausdorff space X, we have the algebra C(X) of continuous,
/ C . The addition and multiplication
complex-valued functions a : X
are obtained pointwise from C .
/ Y gives rise to an algebra morphism
A continuous function f : X
/
C(f ) : C(Y )
C(X) (note the reversal of direction!) via C(f )(b) = a ,
/ 1 gives the algebra
where a(x) = b(f (x)) . In particular, the unique X
/ X of the
/
C(X) , while each point x : 1
morphism η : C = C(1)
/
space gives an algebra morphism C(X)
C.
Actually C(X) is more than just a C-algebra; it is what is called a
commutative C ∗ -algebra (there is a norm and an involution ( )∗ coming
from conjugation). With this extra structure the duality becomes precise:
Each commutative C ∗ -algebra A is isomorphic to C(X) for
some compact hausdorff space X; each C ∗ -algebra morphism
/ C(X) has the form C(f ) for a unique continuous
C(Y )
/Y.
function f : X
This result is commonly referred to as Gelfand duality.
Algebraic geometry is the study of spaces called varieties : the solutions to
polynomial equations in several variables. In studying the variety given by
x2 + 2y 3 = z 4 over the field k, we pass to the k-algebra
A = k[ x , y , z ] / (x2 + 2y 3 = z 4 ) .
By k[ x , y , z ] we mean the k-algebra of polynomials in three commuting
indeterminates x , y , z ; the elements are expressions
αijk xi y j z k
i,j,k
5
6
Quantum Groups: A Path to Current Algebra
where αijk ∈ k and (i , j , k) runs over a finite subset of N3 . The quotient
algebra A is obtained from k[ x , y , z ] by identifying elements when they
may be transformed one into another by means of the equation x2 +2y 3 = z 4
and the algebra axioms.
/ B is
Given a k-algebra B , a k-algebra morphism f : k[ x , y , z ]
uniquely determined by its values on x , y , z . In fact we have a bijection
Algk k[ x , y , z ] , B ∼
= B3 .
Similarly, we have a bijection
Algk A , B ∼
= (u , v , w) ∈ B 3 | u2 + 2v 3 = w4
/B
where A is as above. Again we see that a k-algebra morphism A
corresponds to a map of varieties in the reverse direction.
For general k-algebras A and B, it is suggestive to call a morphism f :
/ B a B-point of A . A point of (the space corresponding to) A is a
A
k-point, not to be confused with an element of the algebra A itself.
commutative
k-algebras
op
spectrum
spaces
coordinate algebra
Let X denote a category. I am thinking of the objects of X as spaces X
/ Y as the maps appropriate to that kind
and Y say, and the arrows
X
of space. Write X X , Y for the set of arrows from X to Y in X .
Let X1 , . . . , Xn be arbitrary objects of X . A product for this list of
objects consists of an object X1 × · · · × Xn together with arrows
pi : X1 × · · · × Xn
/ Xi
for i = 1, . . . , n
such that, given any other object K and arrows
/ Xi
f :K
there exists a unique arrow f : K
for i = 1, . . . , n
/ X1 × · · · × Xn with p ◦ f = f .
i
i
pi
X1 × · · · × Xn
f
Xi
fi
K
This means we have a bijection
X K , X1 × · · · × Xn ∼
= X K, X1 × · · · × X K, Xn .
Duality between geometry and algebra
7
In particular, the empty product is called a terminal object, denoted by 1 .
We have
X K, 1 ∼
= 1.
Products are unique up to isomorphism (if they exist).
/ X × · · · × X is defined by p ◦ δ = 1 for all i . The
The diagonal δ : X
i
X
canonical isomorphisms f1 × · · · × fn and isomorphisms σξ can be defined
as for sets.
The diagrammatic definition of monoid and group can be carried into the
category X (provided the products exist;
1 and
M × M are enough). If M
is a monoid (group) in X then each X K, M becomes a monoid (group)
using the multiplication ∗ given by
f ∗ g = µ ◦ (f × g) ◦ δ
K
δ
K ×K
f ×g
M ×M
µ
M.
A group in the category of topological spaces and continuous maps is called
a topological group. A group in the category of smooth manifolds and
smooth maps is called a Lie group.
We are more interested here in groups in the category (Comm Algk )op
of commutative k-algebras and reversed morphisms; these are called affine
groups over k . This is the variety point of view. On the algebraic side they
are called commutative Hopf algebras over k . Product of varieties becomes
tensor product A⊗k B of k-algebras (more on this later). A commutative
Hopf algebra H thus has structure given by the k-algebra morphisms
ε:H
/k ,
δ:H
/H⊗ H ,
k
ν:H
/H
called counit , comultiplication , antipode (corresponding respectively to the
unit, multiplication, inversion for the group).
Now for each commutative
k-algebra A , we obtain a group Algk H, A of A-points in H.
It will also be necessary to consider the algebraic version of affine
monoids over k . These are called commutative bialgebras over k . They
have a counit and comultiplication, but generally no antipode.
Example 2.1 Let M(2) denote k[ a , b , c , d ] as a commutative k-algebra.
/ k is defined by ε(a) = ε(d) = 1 , ε(b) = ε(c) = 0 .
A counit ε : M(2)
Clearly
k[ a , b , c , d ] ⊗k k[ a , b , c , d ] ∼
= k[ a , b , c , d , a , b , c , d ]
8
Quantum Groups: A Path to Current Algebra
with the coprojections
k[ a , b , c , d , a , b , c , d ]
k[ a , b , c , d ]
a , b , c , d and a , b , c , d
a,b,c,d .
/
The comultiplication δ : M(2)
M(2)⊗k M(2) is given by
k[ a , b , c , d ]
a,b,c,d
a,b,c,d
a a + b c , a b + b d , c a + d c , c b + d d .
This makes M(2) into a commutative k-bialgebra. Notice that we have a
monoid isomorphism
Algk M(2) , A ∼
= Mat 2 , A
where on the right we have the multiplicative monoid of 2 × 2 matrices with
entries in A . Thus M(2) is the coordinate k-algebra of the variety of 2×2
matrices.
To obtain the coordinate k-algebra of the general linear group, we take
GL(2) = k[ a , b , c , d , x ] / (x a d − x b c = 1) .
/ GL(2) which induces
There is an epimorphic k-algebra morphism M(2)
a bialgebra structure on GL(2) from that on M(2). The antipode
ν : GL(2)
a,b,c,d,x
GL(2)
x d , −x b , −x c , x a , a d − b c
makes GL(2) into a commutative Hopf algebra.
Bradshaw: Tassel Bradshaw Group, [Wal94, Plate 22].
3
The quantum general linear group
The passage from quantum to classical mechanics is quite well defined by
taking the limit as Planck’s constant tends to 0. The passage in the other
direction is not so clear cut, and may not be uniquely determined. On the
algebraic side, “quantization” involves deforming commutative algebras to
non-commutative ones:
e.g.,
xy = yx
becomes
x y = e y x .
Usually we deal with q = e rather than , so classical results correspond
to the case q = 1 . Quantum spaces correspond to more general k-algebras,
not necessarily commutative.
Let k be a fixed field and fix q ∈ k with q = 0 . Write k
x1 , . . . , xn for
the k-algebra of polynomials in non-commuting indeterminates x1 , . . . , xn .
As a vector space over k , a basis is given by those elements
m
m
m
1
2
r
xξ(2)
· · · xξ(r)
xξ(1)
for which r ∈ N and m1 , . . . , mr ∈ Z+ and ξ : {1, . . . , r}
any function. Notice that
/ {1, . . . , n} is
k[ x , y ] = k
x , y /( x y = y x ) .
The coordinate algebra of the space of quantum 2 × 2 matrices is defined by
Mq (2) = k
a , b , c , d /R
where R is the system of equations
ab = q −1 ba , ac = q −1 ca , cd = q −1 dc , bd = q −1 db
bc = cb , ad − da = (q −1 − q) bc .
a
b
c
d
(mnemonic)
9
10
Quantum Groups: A Path to Current Algebra
The monomials am1 bm2 cm3 dm4 form a basis for the algebra, as a vector
space over k .
a b
Algk Mq (2) , A ∼
∈ Mat (2 , A) | R holds .
=
c d
a b
a b
be two A-points of Mq (2) such
Theorem 3.1 Let
and
c d
c d
that each entry of the first commutes with each entry of the second.
a b
a b
(as matrices) is an A-point of Mq (2) .
(i) The product
c d
c d
a b
= (ad − q −1 bc) commutes with
(ii) The “q-determinant” detq
c d
each of a , b , c , d and
a b
a b
a b
a b
× det q .
det q
= detq
c d
c d
c d
c d
a b
is invertible in A then
(iii) If det q
c d
−1 a b −1
a b
d
−qb
= det q
−1
c d
a
c d
−q c
is an A-point of Mq−1 (2) .
The above result can be proved by direct calculation, but this gives
little insight into the special nature of the relations R . Examples such as
this arose in work of L. D. Faddeev [FRT88] and his school on the quantum
inverse scattering transform (QIST) method. The version I present here
comes from some lectures of Yu. Manin [Man88] given at Université de
Montréal in June 1988. The following “explanation” of Theorem 3.1 is due
to Yu. Kobyzev (Moscow, winter 1986–87).
Introduce the quantum plane, as defined by the k-algebra
= k
x , y /(xy = q −1 yx) .
A2|0
q
The monomials xm y n with m , n ∈ N form a basis for this as a vector space.
We also need to consider a quantized version of a Grassmannian algebra in
two variables:
= k
ξ , η /(ξ 2 = η 2 = 0 = ξη + q ηξ) .
A0|2
q
The monomials ξ m η n with m , n ∈ {0, 1} form a basis for this algebra. The
reason for the funny superscripts 2|0 and 0|2 comes from “supergeometry”
where dimensions are represented by pairs d | d of numbers. This A0|2
q is a
quantum superplane.
/A
An A-point of B is called generic when the algebra morphism B
is injective.
The quantum general linear group
11
Theorem 3.2 Suppose (x, y) is a generic A-point of A2|0
q and (ξ , η)
is a generic A-point of A0|2
q . Suppose a , b , c , d ∈ A all commute with
x , y , ξ , η . Put
x
a b
x
x
a c
x
ξ
a b
ξ
=
,
=
,
=
.
y
y
c d
y
b d
η
y
c d
η
If q 2 = −1 , the following conditions are equivalent :
(i) (x , y ) and (x, y ) are points of A2|0
q ;
2|0
(ii) (x , y ) is a point of Aq and (ξ , η ) is a point of A0|2
q ;
a c
(iii)
is a point of Mq (2) .
b d
[For q 2 = −1 we only have (iii) ⇒ (i) & (ii).]
−1 Proof. (i) ⇔ (iii). (x , y ) is a point of A2|0
q iff x y = q y x ; that is, iff
−1
(a x + b y)(c x + d y) = q (c x + d y)(a x + b y) . Multiply out the products
using the fact that a , b , c , d each commute with x and y ; since (x , y) is
generic, we can equate coefficients of x2 , y 2 , x y . So the single equation is
in fact equivalent to the following set of three equations:
(∗)
ac = q −1 ca
,
bd = q −1 db
,
ad − da = q −1 cb − q bc .
Interchanging b and c we see that (x , y ) is a point of A2|0
q iff
(∗∗)
ab = q −1 ba
,
cd = q −1 dc
,
ad − da = q −1 bc − q cb .
Taking the last equations in (∗) & (∗∗) we get q −1 cb − q bc = q −1 bc − q cb ;
that is, (q + q −1 )(bc − cb) = 0 hence bc = cb, provided q 2 = −1 .
So (iii) ⇔ (∗) & (∗∗), which together are equivalent to (i).
2
2
(ii) ⇔ (iii). (ξ , η ) is a point of A0|2
q iff 0 = (a ξ + b η) = (c ξ + d η) =
(a ξ +b η)(c ξ +d η)+q (c ξ +d η)(a ξ +b η) . Using ξ 2 = η 2 = 0 these become
ab ξη+ba ηξ = 0 and cd ξη+dc ηξ = 0 and ab ξη+bc ηξ+q (cb ξη+da ηξ) = 0 .
Using ξη = −q ηξ and the linear independence of η and ξ in A, we get that
−q ab+ba = 0 and that −q cd+dc = 0 and also −q (ad+q cb)+bc+q da = 0 .
These are equivalent to (∗∗). So (ii) ⇔ (∗) and (∗∗) ⇔ (i).
a b
c d
and its transpose to both transform the quantum plane into itself; or for
a b
to transform both the plane and superplane into themselves.
c d
In other words, the relations R are precisely what is needed for
Proof of Theorem 3.1. (i) Let B be the free k-algebra containing the
indeterminates a , b , c , d , a , b , c , d , x , y subject to the relations on these
variables in the hypotheses of Theorems 3.1 and 3.2. Then (x , y) is generic;
12
a
c
Quantum Groups: A Path to Current Algebra
b
a b and
are B-points of Mq (2) . By Theorem 3.2, we have
d
c d
x
a c
x
a b
are B-points of A2|0
that
and q . Each coordinate
y
c d
b d
y
in the first of these commutes with all of a , b , c , d while coordinates in the
a c
x
second commute with a , b , c , d . Also
is generic since when
b d
y
2|0
/ Aq for which (a , b , c , d , x , y) / (1 , 0 , 0 , 1 , x , y)
composed with B
a b
x
we get (x, y), which is generic. Similarly
is generic. So by
y
c d
a c
x
a c
x
a b
a b
Theorem 3.2 we have
and
b d
c d
y
b d
y
c d
a b
a b
2|0
both being B-points of Aq . Again by Theorem 3.2,
is a
c d
c d
B-point of Mq (2).
/ A for which
To obtain the result for the given A apply the morphism B
/ (a , b, . . . , d , 0 , 0) .
(a , b , . . . , d , x , y)
(ii) We now get a natural definition of the quantum determinant which
immediately yields its multiplicativity: in the notation of Theorem 3.2,
ξ η = (a ξ + b η)(c ξ + d η) = det q
a
c
(iii) This is left as an exercise for the reader.
b
d
ξη .
The quantum general linear group is defined from 2×2 matrices by inverting
the determinant:
GLq (2) = Mq (2)[t]/(t a = a t , t b = b t , t c = c t , d t = t d , t det q = 1) .
Similarly, the quantum special linear group is defined by requiring that the
determinant be equal to 1:
SLq (2) = Mq (2)/(det q = 1) .
Theorem 3.2 describes the representations of these “groups” on quantum
0|2
spaces A2|0
q and Aq .
Exercise 3.1 Give a direct proof of Theorem 3.1 applied to quantum 2×2
matrices.
4
Modules and tensor products
Let R be a ring (not necessarily commutative). We write Rop for the ring
with opposite multiplication
σ
R×R
R×R
µ
R.
(To say R is commutative is to say Rop = R .)
A left R-module is an abelian group M (written additively) together
with a function
R×M
/M
whereby (r , m) / rm
called scalar multiplication, such that
1m = m
,
(r s) m = r (s m)
(r + r ) m = r m + r m
,
r (m + m ) = r m + r m .
/M.
A right R-module is defined similarly, with multiplication M × R
op
A left R -module structure on an abelian group M “is the same” as
/ M is a scalar
a right R-module structure. More precisely, µ : R × M
σ
µ
multiplication for a left Rop -module iff M × R
R×M
M is one
for a right R-module. In this way, we can deal only with left R-modules
and omit “left”, unless we explicitly stipulate otherwise.
If R is commutative, R = Rop and there is no need to distinguish
left and right modules. If R is a field, an R-module is precisely a vector
space over R . Furthermore, Z-modules are precisely abelian groups since
each abelian group A admits a unique Z-scalar multiplication
given by
n a = a + · · · + a (n terms) for n ≥ 0 and n a = − (−n)a for n < 0 .
A subset X of an R-module M is said to generate M (or span M ) when,
for each m ∈ M , there exist r1 , . . . , rn ∈ R and x1 , . . . , xn ∈ X such that
(∗)
m = r1 x1 + · · · + rn xn .
Call M finitely generated when it is generated by some finite subset.
13
14
Quantum Groups: A Path to Current Algebra
A (not necessarily finite) subset X of M is linearly independent when for
x1 , . . . , xn ∈ X distinct elements, having a relation of the form r1 x1 +
· · · + rn xn = 0 with r1 , . . . , rn ∈ R implies that r1 = · · · = rn = 0 .
Then each expression (∗) is unique up to order of factors (with x1 , . . . , xn
distinct).
An R-module F is said to be free when it is generated by some linearly
independent subset. Every vector space is free, but this is peculiar to R
being a field. It is easy to see that Z/(2) is not a free abelian group.
Each set X determines an R-module
FR (X) = {r1 x1 + · · · + rn xn | ri ∈ R , xi ∈ X , n ∈ N}
with addition and scalar multiplication defined in the obvious way. We
can identify x ∈ X with 1 x ∈ FR (X) and see easily that X is linearly
independent and generates FR (X) . So FR (X) is free.
/ N is (left)R-linear (or an
For R-modules M and N , a function f : M
R-module morphism) when f (m+m ) = f (m)+f (m ) and f (r m) = r f (m)
for all m , m ∈ M and r ∈ R . Write HomR (M, N ) for the abelian group
/ N ; the addition is given by (f + g)(m) =
of R-linear functions f : M
f (m) + g(m) .
Warning: You may think HomR (M, N ) becomes an R-module by defining
(rf )(m) = r f (m) . But this rf does not preserve scalar multiplication
when R is non-commutative.
/ Y . An RFor sets X and Y , write Y X for the set of all functions f : X
/ M is uniquely determined by its restriction
linear function f : FR (X)
to X . Indeed, this gives an isomorphism of abelian groups
HomR (FR (X) , M ) ∼
= MX
where the addition on M X is pointwise.
A submodule H of an R-module M is a subset which is closed under addition
and scalar multiplication. This gives an equivalence relation ≡H on M
whereby
m ≡H m
if and only if
m − m ∈ H .
The equivalence class containing m ∈ M is m + H = {m + h | h ∈ H} ,
called the H-coset containing m . The set M/H of H-cosets becomes an
R-module via
(m + H) + (n + H) = (m + n) + H
,
r(m + H) = r m + H .
/ M/H for which ρ(m) =
We have a surjective R-linear function ρ : M
/
N with g(m) = 0 for all m ∈ H,
m + H . For each R-linear g : M
Modules and tensor products
15
/ N with ĝ ◦ ρ = g . The kernel
there exists a unique R-linear ĝ : M/H
/ N is a submodule
ker f = {m ∈ M | f (m) = 0} of any R-linear f : M
of M ; we have a commutative diagram
f
M
N
ρ
M/ ker f
∼
=
im f
of R-modules, where im f = {f (m) | m ∈ M } is the image of f , the bottom
arrow is an R-linear isomorphism, and the right arrow is an inclusion of a
submodule.
The submodule (X) generated by a subset X of an R-module M is the
smallest submodule of M which contains X . As such it is the image of
/ M whose restriction to X is the inclusion
the R-linear function FR (X)
/
X
M . Of course (X) is generated by X, but in general not freely.
Suppose that M is a right R-module and N is a left R-module. A function
/ A into an abelian group A is R-bilinear when it satisfies
f : M ×N
f (m , n + n )
= f (m , n) + f (m , n )
f (m + m , n) = f (m , n) + f (m , n)
f (m r , n) = f (m , r n) .
Write BilR (M, N ; A) for the abelian group, which is a subgroup of AM×N ,
/ A . Our main goal is to construct a
of R-bilinear functions f : M × N
/ M⊗ N .
“universal” bilinear function λ : M × N
R
Let B denote the subset of the abelian group FZ (M × N ) consisting of
all elements of the form
(m + m , n) − (m , n) − (m , n) ,
(m, n + n ) − (m , n) − (m , n ) ,
(m r , n) − (m , r n)
for m , m ∈ M with n , n ∈ N and r ∈ R . Put
M ⊗R N = FZ (M × N )/(B) .
Then we have abelian group isomorphisms
HomZ (M ⊗R N , A) = HomZ FZ (M × N )/(B) , A
∼ {g ∈ Hom F (M × N ) , A | g is zero on B}
=
∼
=
=
Z
Z
{f ∈ AM×N | f is R-bilinear}
BilR (M, N ; A) .
16
Quantum Groups: A Path to Current Algebra
/A
In particular by taking A = M ⊗R N we get the identity morphism A
corresponding, under the composite of the above string of isomorphisms,
/ M ⊗ N . Then we easily see that
to a bilinear morphism λ : M × N
R
/ A uniquely determines an abelian group
each R-bilinear f : M × N
/ A with g ◦ λ = f .
morphism g : M ⊗R N
For (m , n) ∈ M × N , we put m ⊗ n = λ(m , n) . A typical element of
M ⊗R N then has the form
k
mi ⊗ n i
i=1
where m1 , . . ., mk ∈ M and n1 , . . ., nk ∈ N . These elements satisfy
(m + m ) ⊗ n
= m ⊗ n + m ⊗ n
m ⊗ (n + n )
m r ⊗n
= m ⊗ n + m ⊗ n
= m ⊗r n .
/
With R and S rings, a module M from R to S, written M : R
S , is
an abelian group M enriched with a left R-module structure and a right
S-module structure related by
(r m)s = r(m s)
for all r ∈ R , m ∈ M and s ∈ S . (In the literature this structure is also
known as a left R-/right S-bimodule.) In this notation, tensor product can
be viewed as a kind of “composition of modules”.
S
M
R
N
M⊗S N
T
For M and N as above, M ⊗S N becomes a module from R to T by defining
r(m ⊗ n)t = (r m) ⊗ (n t) .
This composition of modules is not strictly associative, but is associative
up to canonical isomorphisms much like cartesian product of sets. This can
be seen by defining a multiple tensor product as we now proceed to do.
For rings R and S and any set X , there is a free module from R to S
generated by X . It is denoted by FRS(X) and its elements have the form
r1 x1 s1 + · · · + rn xn sn
for ri ∈ R , si ∈ S , xi ∈ X , n ∈ N .
Modules and tensor products
17
/
S we have
HomRS FRS(X) , M ∼
= MX
For each module M : R
where HomRS (N, M ) is the abelian group which has as elements the left R/M.
/right S-module morphisms N
Given rings and modules as in the diagram
M2
M1
R2
M3
...
R1
..
.
Mn
L
R0
a function f : M1 × · · · × Mn
the equations
Rn
/ L is called multilinear when it satisfies
f (m1 , . . . , mi + mi , . . . , mn ) = f (m1 , . . . , mi , . . . , mn )
+ f (m1 , . . . , mi , . . . , mn )
r0 f (m1 , . . . , mn ) = f (r0 m1 , m2 , . . . , mn )
f (m1 , . . . , mi ri , mi+1 , . . . , mn ) = f (m1 , . . . , mi , ri mi+1 , . . . , mn )
f (m1 , . . . , mn ) rn
= f (m1 , . . . , mn−1 , mn rn )
for ri ∈ Ri and mi , mi ∈ Mi . Write
Mult (M1 , . . . , Mn ; L)
for the abelian group of such functions f . It should now be clear how to
construct a module
M1 ⊗R1 M2 ⊗R2 · · · ⊗Rn−1 Mn : R0
/
Rn
and multilinear function
λ : M1 × · · · × Mn
/ M ⊗ ··· ⊗
1 R1
Rn−1 Mn
having the universal property that, for each multilinear function f : M1 ×
/ L , there exists a unique left R -/right R -module morphism
· · · × Mn
0
n
/ L for which g ◦ λ = f . This describes an
g : M1 ⊗R1 · · · ⊗Rn−1 Mn
abelian group isomorphism
n
Mult (M1 , . . . , Mn ; L) ∼
= HomR
R0 (M1 ⊗R1 · · · ⊗Rn−1 Mn , L)
(where HomRS (M, N ) = Mult (M, N ) is the abelian group of left R-/right
/ N ). When there is no ambiguity about the
S-module morphisms M
18
Quantum Groups: A Path to Current Algebra
rings, we write M1 ⊗ · · · ⊗Mn instead of M1 ⊗R1 · · · ⊗Rn−1 Mn . As with
cartesian product we have canonical isomorphisms
(M1 ⊗ · · · ⊗Mk )⊗(Mk+1 ⊗ · · · ⊗Mn ) ∼
= M1 ⊗ · · · ⊗Mn .
/ M ⊗M in which m / m ⊗ m , does not preHowever, the diagonal M
serve addition. The empty tensor product M1 ⊗ · · · ⊗Mn for n = 0 is just
/
R0 as a module R0
R0 , using multiplication in R as scalar multiplication on both sides. We have canonical isomorphisms
R⊗R M ∼
= M ∼
= M ⊗S S .
Given M, M : R
/
S , we write
M
/
S
f : M ⇒ M : R
or
R
f
S
M
/ M is a left R- and right S-module morphism. Given
to mean f : M
the data
M1
R0
f1
M1
M2
R1
f2
Mn
...
R2
Rn−1
M2
fn
Rn
Mn
we obtain f1 ⊗ · · · ⊗ fn : M1 ⊗R1 · · · ⊗Rn−1 Mn ⇒ M1 ⊗R1 · · · ⊗Rn−1 Mn :
/
R0
Rn given by (f1 ⊗ · · · ⊗ fn ) ◦ λ = λ ◦ (f1 × · · · × fn ) .
We have seen that tensor products allow us to represent bilinear functions
as module morphisms. Another way of doing this uses Hom instead of
tensor. Given a triangle of modules
M
R
S
N
L
T
we can enrich the abelian group HomR (M, L) (resp. HomT(N , L)) of left R(resp. right T -) module morphisms with a module structure
/
HomR (M, L) : S
T
/
T
S)
(resp. Hom (N , L) : R
Modules and tensor products
19
using the scalar multiplications
(s f t)(m) = f (ms) t
(resp. (r g s)(n) = r g(sn) ) .
We then have abelian group isomorphisms
HomTS (N , HomR (M, L)) ∼
= Mult (M, N ; L)
∼
= HomSR (M , HomT(N , L))
induced by the canonical bijections
M N
M
∼
L
= LM×N ∼
= LN .
Combining these with the earlier results, we have
HomTS (N , HomR (M, L))
∼
=
∼
=
HomTR (M ⊗S N , L)
HomSR (M, HomT(N , L)) .
These isomorphisms are determined by the evaluation morphisms
evM : M ⊗S HomR (M, L)
T
evN : Hom (N , L)⊗S N
/L
/L
,
m⊗f ,
g ⊗n Explicitly, the first isomorphism takes any u : N
composite
M ⊗S N
1M ⊗u
M ⊗S HomR (M, L)
/ f (m)
/ g(n) .
/ Hom (M, L) to the
R
evM
L.
Exercise 4.1 For rings R , S , T and any sets X , Y prove that
T
= FRS(X)⊗S FST(Y )
FR (X × Y ) ∼
(x , y )
x⊗y .
Hint:
Look at left R-/right T -module morphisms into M : R
/
T .
Exercise 4.2 Describe Z/(2)⊗Z Z/(5) .
Exercise 4.3 (a) If R, S are rings, describe a canonical ring structure on
R⊗Z S .
(b) Is the function from R to R⊗Z S taking R to r ⊗ 1 a ring morphism?
Why?
(c) Show that R⊗Z S is the coproduct of R, S in the category of commutative rings.
20
Quantum Groups: A Path to Current Algebra
Exercise 4.4 Show that a module M from R to S amounts to the same
thing as a left R⊗Z S op -module.
Exercise 4.5 Describe explicitly the construction of M ⊗S N ⊗T L .
Bradshaw: Tassel Bradshaw Group, [Wal94, Plate 23].
5
Cauchy modules
/
/
A module M : R
S gives rise to a module M ∗ = HomR (M, R) : S
R
called the left dual of M . There is a canonical module morphism
/ Hom (M, L)
R
∗
ρM
L : M ⊗R L
given by ρM
L (u ⊗ l)(m) = u(m)l , for each left R-module L .
/
Call an M : R
S Cauchy when ρM
L is an isomorphism for all left
R-modules L . Our goal in this section is to characterize Cauchy modules
more intrinsically.
A module P is called projective when, for all surjective module morphisms
/ L and all module morphisms f : P
/ L , there exists some
e: L
/
module morphism g : P
L with f = e ◦ g .
P
f
g
L
e
L
/ N is said to be a retraction when there exists a
A morphism r : M
/ M with r ◦ i = 1 . When a retraction exists from M
morphism i : N
N
to N , we call N a retract of M .
Proposition 5.1 A module P is projective iff P is a retract of some free
module F .
Proof. (1) A retract Q of a projective P is projective. To see this take
/ P and r : P
/ Q with r ◦ i = 1 . Suppose e : L
/ / L is a
i:Q
Q
/
/
L . Then f ◦ r : P
L , and since
surjective morphism and f : Q
/ L with e ◦ h = f ◦ r. But
P is projective, there is a morphism h : P
then e ◦ (h ◦ i) = (e ◦ h) ◦ i = f ◦ r ◦ i = f ◦ 1Q = f , so g = h ◦ i has e ◦ g = f .
/ L surjective and
(2) Free modules F (X) are projective. Take e : L
/
L . Then we can choose (using the axiom of choice)
any f : F (X)
21
22
Quantum Groups: A Path to Current Algebra
an element g(x) ∈ L for each x ∈ X such that e(g(x)) = f (x) . Since
/ L;
F (X) is free, we can extend g uniquely to a morphism g : F (X)
and furthermore e ◦ g = f since they agree on X .
(3) For each module M there is a free module F and a surjective morphism
/ M . Just take F to be the free module F (M ) on the underlying
e: F
/ M we only have to give it on M ,
set of M . To give a morphism e : F
so we take e(m) = m . Clearly this e is surjective.
/ P is surjective and P projective then e is a retraction.
(4) If e : F
For we have i as in:
P
1P
i
e
F
P .
This brings us to the fundamental theorem of “Morita theory”.
/
Theorem 5.2 The following conditions on a module M : R
S are
equivalent.
(i) M is Cauchy.
/
S such that both the
(ii) There exists a morphism d : S ⇒ M ∗ ⊗R M : S
following two composites are identity morphisms
M ∼
= M ⊗S S
1M ⊗d
M∗ ∼
= S⊗S M ∗
d⊗1M ∗
M ⊗S M ∗ ⊗R M
M ∗ ⊗R M ⊗S M ∗
(iii) There exists a module N : S
e : M ⊗S N
/R
evM ⊗1M
R⊗R M ∼
= M
1M ∗ ⊗evM
M ∗ ⊗R R ∼
= M∗ .
/
R and morphisms
,
d:S
/ N⊗ M
R
such that the following composite is the identity morphism
M ∼
= M ⊗S S
1M ⊗d
M ⊗S N ⊗R M
e⊗1M
R⊗R M ∼
= M.
(iv) M is a finitely generated projective left R-module.
Proof. (i) ⇒ (ii). Since ρM
M is an isomorphism, there is an element of
∗
M
/ M . This element of M ∗ ⊗ M now
M ⊗R M taken by ρM to 1M : M
R
/ M ∗ ⊗ M whose value at 1 ∈ S
determines a unique morphism
d
:
S
R
is the element.
Write d(1) = i ui ⊗ mi . The condition ρM
M (d(1))(m) = m
becomes i ui (m) mi = m for all m ∈ M . This immediately gives that the
Cauchy modules
23
∗
first composite of (ii) takesm to m . To see
that the second takes u ∈ M
to itself we use u(m) = u( ui (m) mi ) = ui (m) u(mi ) .
(ii) ⇒ (iii). Just take N = M ∗ , e = evM and d as in (ii).
k
(iii) ⇒ (iv). Just put d(1) = i=1 ni ⊗ mi ∈ N ⊗
R M . From the fact that
the composite in (iii) is the identity, we have
i e(m ⊗ ni ) mi = m for
all m ∈ M . So M is generated by m1 , . . . , mk . It remains to see that
/ L surjective and f : M
/ L . Then
M is projective. Take s : L
/L
g: M
we can choose
l1 , . . . , lk ∈ L with s(li ) = f (mi ) . Define
by g(m) =
i ) li and we get s(g(m)) =
i e(m ⊗ n
i e(m ⊗ ni ) s(li ) =
⊗
⊗
e(m
n
)
f
(m
)
=
f
(
e(m
n
)m
)
=
f
(m)
,
as
required.
i
i
i
i
i
i
(iv) ⇒ (i). It is easy to see that a retract of a Cauchy module is Cauchy
(Exercise 5.3). So it suffices to show that M = FR (X) is Cauchy for X
a finite set {x1 , . . . , xk }. But then M ∗ = HomR (F (X) , R) ∼
= Rk and
k
∼
HomR (M , L) = HomR (F (X) , L) = L . Under these isomorphisms ρM
L
/ Lk with (r , . . . , r ) ⊗ l /
carries across to the morphism Rk ⊗R L
k
k 1
/
(r1 l , . . . , rk l) which has inverse (l1 , . . . , lk )
i=1 ui ⊗ li , in which
ui ∈ Rk projects to 0 in all components except the i-th where it projects
to 1 . So ρM
L is an isomorphism.
/ R we obtain two
Given rings R and S , from any ring morphism f : S
/
/
modules f R : S
R and Rf : R
S , which have R as underlying
abelian group. They have scalar multiplicatons
/
S × fR
Rf × S
fR
,
,
/ Rf
×R
R × Rf
/ fR
/R
f
(r , r ) (r , r) / r r
/ r r .
fR
given by, respectively
(s , r) (r , s) For any module L : R
/ f (s) r
/ r f (s)
,
,
/
T we have canonical isomorphisms
f R ⊗R L
∼
=
r ⊗l
L
∼
=
HomR (Rf , L)
l
u(1)
u.
It follows easily from this that
(Rf )∗ ∼
=
and that Rf is Cauchy.
fR
24
Quantum Groups: A Path to Current Algebra
/
A module M : R
S is called convergent when there exists a ring
/ R and a module isomorphism M ∼
morphism f : S
= Rf .
/
/
The product i∈I Mi : R
S of a family of modules Mi : R
S with
i ∈ I , has as elements the families m = (mi )i∈I with mi ∈ Mi ; addition
and scalar multiplication are given by
m + m = (mi + mi )i∈I
,
r m s = (r mi s)i∈I .
There are projections
prj :
/M
j
Mi
for each j ∈ I
i∈I
given by prj (m) = mj . There are also injective module morphisms
/
inj : Mj
for each j ∈ I
Mi
i∈I
given by inj (m) = m where mj = m and mi = 0 for
all i = j ; we can
use these to identify each Mj with the submodule of i∈I Mi consisting of
those m with mi = 0 for all i = j .
/
The direct sum
S is the submodule of i∈I Mi which
i∈I Mi : R
consists of those m for which mi = 0 for all but finitely many i ∈ I .
This is the submodule generated by the union ∪i∈I Mi , hence we can write
i∈I miinstead of m ∈
i∈I Mi . Of course the injections inj actually
land in i∈I Mi .
Proposition 5.3 There are module isomorphisms:
(a)
HomR
i∈I
f
(b)
i∈I
/
(f ◦ ini )i∈I ,
∼
=
o
HomR (Mi , L)
i∈I
o
Mi ⊗S N
i mi ⊗ n
∼
=
Mi , L
Mi ⊗S N
/
i∈I
i (mi ⊗ n)
.
Proof. (a) Injectivity.
If f ◦ ini = 0 for all i ∈ I then f is zero on each Mi
and hence on
Mi .
Cauchy modules
25
/ L for all i ∈ I , define f : M
/ L by
Surjectivity.
Given fi : Mi
i
fi (mi ) .
f ( mi ) =
∼
(b)
HomR
Mi ⊗S N , L
Mi , HomR (N, L)
= HomS
i
∼
=
i
Hom Mi , HomR (N, L)
S
i
∼
=
HomR (Mi ⊗S N , L)
i
∼
=
HomR
(Mi ⊗S N ) , L
i
and the composite isomorphism is induced by the given map in (b). This
proves it. (Why?)
When I is finite, notice that i∈I Mi = i∈I Mi . This is also frequently
written ⊕i∈I Mi . So M ⊕ N = M × N = M + N .
Exercise 5.1 Show that a module P is finitely generated and projective if
and only if P is a retract of a free module on a finite set.
Hint: In part (3) of the proof of Proposition 5.1 we did not need F (M );
only F (X) for any X generating M .
Exercise 5.2 Suppose M is a finitely generated projective module over a
commutative ring R. Show that M ∗ is a finitely generated projective module
and that the canonical morphism M → M ∗∗ is bijective.
Exercise 5.3 Prove directly from the definition of “Cauchy module” that
a retract of a Cauchy module is Cauchy.
Exercise 5.4 Re-examine the proof of Theorem 5.2 to show that a module
/
M :R
S is Cauchy if and only if ρM
M is surjective.
26
Bradshaw: Tassel Bradshaw Group, [Wal94, Plate 30].
6
Algebras
Let R be any ring. An algebra over R (or R-algebra) is a module A : R
together with module morphisms
µ : A⊗R A
/A
,
η:R
/
R
/A
such that
A⊗R A⊗R A
(Associativity)
(Identity)
A
η ⊗1A
1A ⊗η
µ⊗1A
µ
A⊗R A
1A ⊗µ
A
µ
A⊗R A
A.
1A
Notice that A becomes a ring with multiplication a b = µ(a ⊗ b) and identity
1 = η(1) .
/
For R-algebras A , B : R
R an algebra morphism f : A
module morphism satisfying
A⊗R A
f ⊗f
R
B⊗R B
η
η
µ
A
/ B is a
µ
f
A
B
f
B.
We write AlgR (A , B) for the set of algebra morphisms from A to B .
27
28
Quantum Groups: A Path to Current Algebra
/
Example 6.1 For any module M : R
S , the endomorphism algebras, over S and R respectively, are given by
/
S
EndR (M ) = HomR (M, M ) : S
/
S
S
R .
End (M ) = Hom (M, M ) : R
In each case the multiplication is given by composition.
A module morphism
µ̂ : A ⇒ EndS(M ) : R
/
R
corresponds to a module morphism
/
S.
µ : A⊗R M ⇒ M : R
To say that µ̂ is an algebra morphism is precisely to say that µ is a scalar
multiplication enriching M with the structure of left A-module.
Example 6.2 For any module M : R
/
R , write
M ⊗n = M ⊗R · · · ⊗R M
(n terms) .
The tensor algebra on M is defined by the “geometric series”
T (M ) =
∞
M ⊗n
n=0
/ T (M ) induced by the canonical
with multiplication µ : T (M )⊗R T (M )
isomorphisms
M ⊗p ⊗R M ⊗q
and unit η : R
∼
=
M ⊗(p+q)
/ T (M ) equal to the injection
in0 : R = M ⊗0
/
∞
M ⊗n .
n=0
Composition with the injection in1 : M
/ T (M ) gives a bijection
AlgR (T (M ) , A) ∼
= HomR
R (M, A)
/ A to g : T (M )
/A
for any algebra A . The inverse takes f : M
given by g(m1 ⊗ · · · ⊗ mr ) = f (m1 ) · · · f (mr ) . In particular, if we take
M = A and f = 1A , we obtain an algebra morphism
µ : T (A)
/A
with
µ(a1 ⊗ · · ·
⊗ ar )
= a1 · · · ar .
Algebras
29
Example 6.3 Let G be any monoid. There is an R-algebra R(G) which
is just the free module FRR(G) on the underlying set of G together with the
multiplication µ which extends that of G in the sense that
R
R
R(G × G) ∼
(G)⊗R FR
(G)
= FR
FR
G×G
µ
µ
R
(G) .
FR
G
This R(G) is called the monoid R-algebra of G ; or when G is a group,
/ A into the
the group R-algebra of G . Each monoid morphism G
multiplicative monoid of A extends uniquely to an R-algebra morphism
/ A.
R(G)
A representation of G on M is an R(G)-module. Scalar multiplication
/ M can be viewed as a monoid morphism
R(G)⊗R M
G
/ End (M ) .
R
The subset of M given by {gm−m | g ∈ G , m ∈ M } generates a submodule
(gm − m | g ∈ G , m ∈ M ) and we write M/G for the quotient module
M/(gm − m | g ∈ G , m ∈ M ) .
An ideal in an algebra A is a submodule I such that a x b ∈ I for all x ∈ I
and a , b ∈ A . There is a unique structure of algebra on the quotient module
/ A/I is an algebra morphism. The
A/I for which the canonical ρ : A
/ B is an ideal in A .
kernel of any algebra morphism f : A
If X is a subset of an algebra A , we write (X) for the smallest ideal of A
containing X . This should not cause confusion with the module notation;
the ideal (X) is precisely the submodule (A X A) generated by the subset
A X A = {a x b | a , b ∈ A , x ∈ X} of A . Now given any algebra morphism
/ B satisfying g(x) = 0 for all x ∈ X , then an algebra morphism
g: A
/ B is uniquely determined via the equation f ◦ ρ = g .
f : A/(X)
Now suppose that R is a commutative ring. Then left R-modules are
“the same thing” as right R-modules. Moreover, each left R-module M can
/
R by defining
be naturally regarded as a module M : R
r m s = (r s) m
for all r , s ∈ R and m ∈ M .
In dealing with modules over a commutative ring, we happily regard left
modules as two-sided via this process. Thus for R-modules M1 , . . . , Mn
we have a tensor product R-module
M1 ⊗R · · · ⊗R Mn .
30
Quantum Groups: A Path to Current Algebra
Furthermore every permutation ξ on the set {1 , . . . , n} induces a canonical
module isomorphism
∼
=
σξ : M1 ⊗R · · · ⊗R Mn
m1 ⊗ · · ·
Mξ(1) ⊗R · · · ⊗R Mξ(n)
mξ(1) ⊗ · · ·
⊗ mn
⊗m
ξ(n)
.
Given an algebra A over R with multiplication µ and unit η , we obtain an
opposite algebra Aop on the same module A , with multiplication
σ
µop : A⊗R A
A⊗R A
µ
A
/ A . Call A commutative when Aop = A
and with the same unit η : R
as algebras. It follows that the composite
A⊗R · · · ⊗R A
σξ
A⊗R · · · ⊗R A
µ
A
is independent of the permutation ξ .
Example 6.4 For any set X, the set RX of all functions from X into
the commutative ring R becomes a commutative R-algebra after defining
addition, scalar multiplication and multiplication as acting pointwise. The
/ RX is given by η(r)(x) = r for all r ∈ R and x ∈ X .
unit η : R
Example 6.5 Let M be any module over the commutative ring R . There
is a natural representation of the symmetric group Sn on M ⊗n given by
/ End (M ⊗n ) ; that is, ξ · (m ⊗ · · · ⊗ m ) = m
σ : Sn
1
n
R
ξ(1) ⊗ · · · ⊗ mξ(n) .
The symmetric R-algebra on M is given by the “exponential series”
S(M ) =
∞
M ⊗n /Sn .
n=0
Another way of constructing this is as follows. For any R-algebra A we
can form a commutative R-algebra by taking the quotient of A by the ideal
(a b − b a | a , b ∈ A ) . Applying this construction to the tensor algebra
T (M ) gives S(M ) .
For every commutative R-algebra A , we have that
AlgR (S(M ) , A) ∼
= HomR (M , A) .
In particular, corresponding to the identity map 1A : A
/ A.
R-algebra morphism µ : S(A)
/ A there is an
Algebras
31
The following diagram of “forgetful” and “free” constructions summarizes
some of the above.
Comm AlgR
S
AlgR
R
T
forget
multiplication
forget
module
ModulesR
Monoids
word monoid
forget
multiplication
Sets
FR
forget
module structure
Skew commutativity a b + b a = 0 for an R-algebra is too strong as a
requirement for all a , b ∈ A . For example taking b = 1, it would give
(1 + 1)a = 0 . Hence if R is a field of characteristic other than 2 (meaning
1 + 1 = 0 in R), we would get a = 0 , and so A = {0} .
An R-algebra A is said to be skew commutative when for all a ∈ A
either a2 = 0 or a ∈ η(R) . Then, provided none of a , b and a + b is in the
/ A , we have
image of η : R
a b + b a = (a + b)2 − a2 − b2 = 0 .
Example 6.6 For any R-algebra A we can form the quotient by the ideal
(a2 | a ∈
/ η(R)) to obtain a skew commutative algebra. If we do this to the
tensor algebra T (M ) we obtain the exterior algebra Λ(M ) . Alternatively,
let Λn (M ) be the quotient module of M ⊗n by the submodule generated by
the elements m1 ⊗ · · · ⊗ mn with mi = mj for some i = j (this submodule
is {0} when n = 0 or 1); then
Λ(M ) =
∞
Λn (M ) .
n=0
We write m1 ∧ · · · ∧ mn for the image of m1 ⊗ · · · ⊗ mn in Λ(M ) . For all
x , y , z ∈ M we have
x∧x
(rx + sy) ∧ z
= 0
therefore x ∧ y = − y ∧ x
= r(x ∧ z) + s(y ∧ z) .
32
Quantum Groups: A Path to Current Algebra
If M = FR {x1 , . . . , xk } is a free module on a k-element set then Λn (M ) is
a free module on a nk -element set; so Λ(M ) is a free module on a set with
2k -elements. In particular Λk (M ) is free on the singleton set {x1 ∧· · ·∧xk },
so if
k
rij xj
yi =
j=1
then y1 ∧ · · · ∧ yk must be a unique scalar multiple
y1 ∧ · · · ∧ yk = det(rij ) x1 ∧ · · · ∧ xk
of x1 ∧ · · · ∧ xk . This can be taken as a definition of the determinant of
(rij ) ∈ Mat (k , R) .
If A is a skew commutative algebra then we have a bijection
AlgR (Λ(M ) , A) ∼
= HomR (M, A) .
An R-Lie algebra is an R-module L together with a module morphism
/ L satisfying the conditions
β : L⊗R L
(Jacobi identity)
β(x , x) = 0
β β(x , y) , z + β β(z , x) , y + β β(y , z) , x = 0 .
Call such a β a Lie bracket on the module L .
Example 6.7 For any R-algebra A the commutator [ a , b ] = a b − b a
defines a Lie bracket on the underlying R-module of A :
[[a,b],c]+ [[c,a],b]+ [[b,c],a]
= [a,b]c − c[a,b] + [c,a]b− b[c,a] + [b,c]a− a[b,c]
= (a b c − b a c) − (c a b − c b a) + (c a b − a c b) − (b c a − b a c)
+ (b c a − c b a) − (a b c − a c b) = 0 .
So A becomes a Lie algebra, denoted by AL . It turns out (at least when R
is a field) that every Lie algebra is a submodule, closed under commutator,
of such an example.
/
Example 6.8 Let A be an R-algebra and M : A
A a module. Then
/ M is an R-module morphism satisfying
a derivation D : A
(Leibniz rule)
D(a b) = D(a) b + a D(b) .
Algebras
33
Notice that a = b = 1 gives D(1) = 2 D(1) , so D(1) = 0 . Let DerR (A , M )
denote the submodule of HomR (A, M ) consisting of the derivations. We
write DerR (A) for DerR (A , A) . It is easy to check that DerR (A) is closed
/A
under commutator in the algebra EndR (A) ; that is, if D1 , D2 : A
are derivations then so is [ D1 , D2 ] = D1 ◦ D2 − D2 ◦ D1 .
Example 6.9 The tangent space at the identity of each Lie group is a Lie
algebra. The pioneering work of Sophus Lie and Eli Cartan showed how
much information about the Lie group is obtainable from the Lie algebra
(especially in the compact case).
The Lie groups GL (n , R) and SL (n , R) and O(n , R) consist of those
matrices x ∈ Mat(n , R) for which respectively x is invertible , det x = 1
and x xt = 1 . They have associated Lie algebras
gl(n , R) =
sl(n , R) =
o(n , R)
=
Mat(n , R)
x ∈ gl(n , R) | trace(x) = 0
x ∈ gl(n , R) | xt = −x .
(We shall not stop to prove this here.) The Lie bracket is [ x , y ] = x y−y x
in each case. As an exercise the reader should check that sl(n , R) and
o(n , R) are closed under commutator.
/ L is an R-module
Suppose L and L are R-Lie algebras and f : L
morphism. Then f is a Lie algebra morphism when it satisfies
f β(x , y) = β f (x) , f (y) .
Write LieR (L , L ) for the set of Lie algebra morphisms f : L
/ L .
We saw in Example 6.7 above that each R-algebra A gives rise to an R-Lie
algebra AL using the commutator. We shall describe an “adjoint” for this
process: for each R-Lie algebra L we obtain an R-algebra U(L) , called the
universal enveloping algebra of L , such that there is a natural bijection
AlgR U(L) , A ∼
(∗)
= Lie(L , AL ) .
For this we use the tensor algebra T (L) on the underlying R-module of L ,
and take the quotient by the appropriate ideal:
U(L) = T (L)/ x ⊗ y − y ⊗ x − β(x , y) | x , y ∈ L .
34
Quantum Groups: A Path to Current Algebra
We have a Lie algebra morphism i as in:
i
L
in1
U(L)L
ρ
T (L)
and it is composition with i that induces the bijection (∗) .
The direct sum L1 ⊕L2 of Lie algebras L1 , L2 is their direct sum as modules
together with the Lie bracket
β (x1 , x2 ) , (y1 , y2 ) = β(x1 , y1 ) , β(x2 , y2 ) .
Proposition 6.10 There is an algebra isomorphism
U(L1 ⊕ L2 ) ∼
= U(L1 ) ⊗R U(L2 )
whose composite with i : L1 ⊕ L2
to x1 ⊗ 1 + 1 ⊗ x2 .
/ U(L ⊕ L ) takes the pair (x , x )
1
2
1
2
Proof. It is left to the reader to check that
/ U(L ) ⊗ U(L )
/ x1 ⊗ 1 + 1 ⊗ x2
L1 ⊕ L2
(x1 , x2 ) 1
2 L
R
/
/ (x1 , 0)
L1
U(L1 ⊕ L2 )L
x1
/
/ (0 , x2 )
L2
U(L1 ⊕ L2 )L
x2
are Lie algebra morphisms. These three must therefore be composites with i
of algebra morphisms
φ : U(L1 ⊕ L2 )
ψ1 : U(L1 )
ψ2 : U(L2 )
/ U(L ) ⊗ U(L )
1
2
R
/ U(L ⊕ L )
1
2
/ U(L ⊕ L ) .
1
2
/ U(L ⊕L ) by ψ(a ⊗ b) = ψ (a) ψ (b) . Then
Define ψ : U(L1 ) ⊗R U(L2 )
1
2
1
2
we have that
ψ φ(x1 , x2 ) = ψ(x1 ⊗ 1 + 1 ⊗ x2 ) = (x1 , 0) + (0 , x2 ) = (x1 , x2 )
= φ(x1 , 0) = x1 ⊗ 1
φ ψ1 (x1 )
= φ(0 , x2 ) = 1 ⊗x2 .
φ ψ2 (x2 )
Hence φ and ψ are mutually inverse.
Algebras
35
A deeper result which we shall not prove here is:
Proposition 6.11 (Poincaré–Birkhoff–Witt) If the R-Lie algebra L is
/ U(L)L is injective.
free as an R-module then i : L
A Lie algebra L is called commutative when β(x , y) = 0 for all x , y ∈ L .
So an algebra A is commutative iff AL is commutative.
Notice that, for any module M , we can make M into a commutative Lie
algebra. Then the universal enveloping algebra of M is precisely the same
as the symmetric algebra of M ; that is, U(M ) = S(M ) . In particular, we
have (Proposition 6.10):
S(M ⊕ M ) ∼
= S(M ) ⊗R S(M ) .
Exercise 6.1 Let R be a commutative ring and G be a group. Consider
left modules M, N, L over the group algebra R(G).
(a) Show that M ⊗R N becomes an R(G)-module on defining:
g(m ⊗ n) = (gm) ⊗ (gn) for g ∈ G, m ∈ M, n ∈ N .
(b) Show that HomR (M, L) becomes an R(G)-module on defining:
(gu)(n) = gu(g −1 m) for g ∈ G, u ∈ HomR (M, L), m ∈ M .
(c) Show that evaluation evM : M ⊗R HomR (M, L)
R(G)-module morphism.
/ L is an
(d) Prove that evaluation induces an isomorphism of R-modules
HomR(G) N, HomR (M, L) ∼
= HomR(G) M ⊗R N, L .
36
Bradshaw: Stylized Bradshaw Group, [Wal94, Plate 40].
7
Coalgebras and bialgebras
Let R be any ring. By a coalgebra over R (or R-coalgebra) we mean a
/
module C : R
R together with module morphisms
δ:C
/ C⊗ C
R
and
ε:C
/R
such that
C
C
δ
δ
δ ⊗1X
C⊗R C
C⊗R C⊗R C
1X ⊗δ
C⊗R C
ε⊗1X
1X ⊗ε
C .
1C
We call δ the comultiplication and ε the counit. This structure provides a
module with “formal diagonals”. There is a uniquely determined
/ C⊗ C⊗ · · · ⊗ C
R
R
R
where for each c ∈ C we have δ(c) = c1i ⊗ · · · ⊗ cni . The notation
δ:C
i
δ(c) =
c(1) ⊗ · · ·
⊗ c(n)
(c)
is sometimes used even though the representation of δ(c) in the tensor
product is not uniquely determined — we act as though a choice of this
representation has been made for each c ∈ C . Given a multilinear function
/ A we also write
f : C × ···× C
f (c(1) , . . . , c(n) ) .
f δ(c) =
(c)
37
38
Quantum Groups: A Path to Current Algebra
In terms of this notation the axioms can be rewritten as
δ(c(1) ) ⊗ c(2) =
c(1) ⊗ c(2) ⊗ c(3) =
c(1) ⊗ δ(c(2) )
(c)
c =
(c)
ε(c(1) ) ⊗ c(2) =
(c)
(c)
c(1) ⊗ ε(c(2) ) .
(c)
Suppose C and D are coalgebras. A coalgebra morphism f : C
module morphism such that
f
C
D
δ
f
C
/ D is a
D;
δ
ε
C⊗R C
f ⊗f
ε
D⊗R D
R
that is,
f (c(1) ) ⊗ f (c(2) )
=
(c)
f (c)(1) ⊗ f (c)(2)
(f (c))
ε(f (c))
=
ε(c) .
We write CogR (C , D ) for the set of coalgebra morphisms from C to D .
Suppose R is commutative. A coalgebra C over R is cocommutative when
C
δ
δ
C⊗R C
σ
C⊗R C .
Return now to a general ring R . Suppose that A is an R-algebra and C
is an R-coalgebra. Then HomR (C , A ) becomes an R-algebra under the
following convolution structure:
HomR (C , A )⊗R HomR (C , A )
⊗
HomR (C⊗R C , A⊗R A )
µ◦ ◦δ
HomR (C , A )
R
∼
=
HomR (R , R )
η◦ ◦ε
HomR (C , A ) .
Coalgebras and bialgebras
39
In terms of elements, for left R-module morphisms f , g : C
convolution product is given by
f ∗ g = µ ◦ (f ⊗ g) ◦ δ
and
/ A their
1 = η◦ε.
Using the notation for comultiplication this becomes the formula
f (c(1) ) g(c(2) ) .
(f ∗ g)(c) =
(c)
In particular, with A = R each R-coalgebra C gives rise to a convolution
R-algebra structure on the dual C ∗ = HomR (C , R ) . However, we prefer
/ C∗
to regard C ∗ as an R-algebra via the multiplication µ : C ∗ ⊗R C ∗
defined by the following diagram.
µ⊗1
C ∗ ⊗R C ∗ ⊗R C
C ∗ ⊗R C
1⊗1⊗δ
e
C ∗ ⊗R C ∗ ⊗R C⊗R C
R
e
1⊗e⊗1
C ∗ ⊗R C
(This works even for non-commutative R.)
Example 7.1 Suppose X is any category which admits finite products.
/ ModR is a functor into the category of modules from
Suppose F : X
R
R to R . Suppose there are natural module morphisms
φX
1 , ... , Xn
: F (X1 × · · · × Xn )
/ FX ⊗ FX ⊗ ··· ⊗ FX
1 R
2 R
n
R
compatible with the canonical associativity isomorphisms for product and
tensor product. Then for each object X of X we obtain a coalgebra F X,
with comultiplication and counit
FX
FX
Fδ
Fε
F (X × X)
F1
φX,X
φ0
F X⊗R F X
R.
If furthermore R is commutative and F is compatible with the switches,
then this coalgebra is cocommutative.
40
Quantum Groups: A Path to Current Algebra
Sub-example (a). (R commutative) The free R-module construction
gives a functor
/ Mod
FR : Set
R
from the category of sets to ModR . We have isomorphisms
φ : FR (X1 × · · · × Xn )
∼
=
FR X1 ⊗R · · · ⊗R FR Xn
x1 ⊗ · · · ⊗ xn .
(x1 , . . . , xn )
Proof. (n = 2)
∼
HomR FR (X × Y ) , M
= M X×Y ∼
= (M X )Y
∼
= HomR (FR Y , M X )
∼
= HomR FR Y , Hom(FR X , M )
∼
= Hom (F X⊗F Y , M ) .
R
R
R
So each FR X becomes an R-coalgebra.
Sub-example (b). The universal enveloping algebra provides a functor
U : LieR
/ Mod
R
and we have already observed
U (L ⊕ L ) ∼
=
(x, x )
U (L)⊗R U (L )
x ⊗ 1 + 1 ⊗ x .
and
U ({0}) ∼
= R
Since direct sum is product in LieR we have another standard example.
Thus each universal enveloping algebra U (L) becomes an R-coalgebra. The
comultiplication here is determined by
L
x
i
U (L)
δ
U (L)⊗ U (L)
x ⊗ 1+1 ⊗x .
Sub-examples (a), (b) suggest two definitions that we can make for any
coalgebra C .
• Say that c ∈ C is set-like when δ(c) = c ⊗ c and ε(c) = 1 . (In the
case of C = FR (X) the set-like elements are precisely the elements
of X .) Write D(C) for the set of set-like elements of C .
Coalgebras and bialgebras
41
• Say that c ∈ C is primitive when
δ(c) = c ⊗ 1 + 1 ⊗ c .
(In the case of C = U (L) each element of L is primitive.)
Write P(C) for the submodule of primitive elements of C .
Proposition 7.2 (with R a field.) The set-like elements of any coalgebra
C form a linearly independent subset D(C) .
Proof. Suppose D(C) is linearly dependent. Let n + 1 be the first natural
number for which there is a linearly dependent subset of D(C) with that
many elements. Then any set of n elements of D(C) must necessarily be
linearly independent, while there exist distinct g, g1 , . . . , gn ∈ D(C) which
are linearly dependent. Then we can write
g = λ1 g1 + · · · + λn gn
with the λi ∈ R all non-zero. Then
n
λi gi ⊗ gi
=
i=1
=
n
λi δ(gi ) = δ(g) = g ⊗ g
i=1
n
λi λj gi ⊗ gj .
i,j=1
Since {g1 , . . . , gn } is linearly independent in C then { gi ⊗ gj } is linearly
independent in C⊗C , so we can equate coefficients: λi λj = 0 for i = j and
λi = λ2i . Since λi = 0 this means n = 1 and λi = 1 . But g = g1 was not
allowed.
We shall come back to set-like and primitive elements in the context of
bialgebras.
Example 7.3 (with R commutative.) Let C = FR N be the free R-module
on the countable set N . Define
1 for n = 0
δ(n) =
p ⊗q
and
ε(n) =
0 for n > 0 .
p+q=n
This defines a cocommutative coalgebra structure on C .
Take an R-algebra A and look at an example of convolution with this
coalgebra C . The convolution structure transports across the R-module
isomorphism
( sequences in A )
HomR (C , A ) ∼
= AN
42
Quantum Groups: A Path to Current Algebra
to give the multiplication
ab =
ap b q
p+q=n
for sequences a = (an ) = (a0 , a1 , . . . ) and b = (bn ) = (b0 , b1 , . . . ) in A .
The unit sequence is (1 , 0 , 0 , 0 , . . . ) . A precise definition of indeterminate
can be taken to mean the sequence
x = (0 , 1 , 0 , 0 , 0 , 0 , . . . ) ∈ AN
in A . Each u ∈ A is identified with u 1 = (u , 0 , 0 , 0 , . . . ) ∈ AN . Then
each a ∈ AN can be written as a formal (no convergence requirements!)
power series
∞
an xn .
a =
n=0
N
Write A[[x]] for A with this algebra structure. It is the R-algebra of formal
power series in A . If A is commutative so is A[[x]] . In particular when
A = R we obtain the commutative R-algebra C ∗ = R[[x]] .
Example 7.4 Let n = {1, 2, . . . , n} and put C = FR (n × n) . Then C
becomes an R-coalgebra on defining
δ(i , j) =
n
(i , k) ⊗ (k , j)
and
ε(i , j) =
k=1
1
0
for i = j
otherwise .
Given any R-algebra A , the convolution structure simply transports across
the R-module isomorphism
HomR (C , A ) ∼
= An×n
( n × n matrices in A )
to give the usual matrix multiplication
(aij ) (bij ) =
n
aik bkj .
k=1
In this way we obtain the R-algebra Mat(n , A ) ∼
= EndR (An ) of n × n
matrices with entries in A .
Suppose that R is a commutative ring. An R-bialgebra is an R-module B
together with algebra and coalgebra structures
/B
µ : B⊗R B
/ B⊗ B
δ: B
R
and
η: R
/B
and
ε: B
/R
Coalgebras and bialgebras
43
satisfying the conditions
µ
B⊗R B
B
δ ⊗δ
δ
∼
=
1⊗σ⊗1
B⊗R B⊗R B⊗R B
B⊗B
ε⊗ε
η ⊗η
R⊗R
ε
B⊗B
B⊗B
∼
=
µ
B
µ⊗µ
B⊗R B⊗R B⊗R B
δ
η
R
B
B
η
ε
1R
R
R.
Notice the complete duality between (µ , η ) and (δ , ε ) . When expressed
in terms of elements the duality is not so apparent:
δ(x , y) =
x(1) y(1) ⊗ x(2) y(2)
(x) (y)
ε(x y) = ε(x) ε(y) and δ(1) = 1 ⊗ 1
and ε(1) = 1 .
For R commutative, the tensor product A⊗R A of R-algebras A and A
becomes an R-algebra via the multiplication
(A⊗R A )⊗R (A⊗R A )
∼
=
1⊗σ⊗1
σ1324
(A⊗R A)⊗R (A ⊗R A )
µ⊗µ
A⊗R A
and unit
R ∼
= R⊗R R
η ⊗η
A⊗R A .
Also the tensor product C⊗R C , of R-coalgebras C and C , becomes an
R-coalgebra via the comultiplication
C⊗R C δ ⊗δ
(C⊗R C)⊗R (C ⊗R C )
and counit
C⊗R C ε⊗ε
∼
=
1⊗σ⊗1
(C⊗R C )⊗R (C⊗R C )
R⊗R R ∼
= R.
44
Quantum Groups: A Path to Current Algebra
With this, we can make the observation:
Proposition 7.5 Suppose (µ , η ) and (δ , ε ) are respectively, algebra and
coalgebra structures on the R-module B . Then the following conditions
are equivalent:
(i) B is a bialgebra;
(ii) µ : B⊗R B
(iii) δ : B
/ B and η : R
/ B⊗ B and ε : B
R
/ B are coalgebra morphisms;
/ R are algebra morphisms.
/ B is a function
For bialgebras B and B a bialgebra morphism f : B
which is both an algebra and coalgebra morphism. Write BigR (B , B ) for
the set of such functions f .
Before giving examples of bialgebras we prove some extra results on the
set-like and primitive elements for the bialgebra case.
Proposition 7.6 If B is a bialgebra then the set-like elements are closed
under multiplication: so D(B) becomes a monoid.
Proof. δ(bb ) = δ(b) δ(b ) = (b ⊗ b)(b ⊗ b ) = bb ⊗ bb for b , b ∈ D(B) ; also
ε(bb ) = ε(b) ε(b ) = 1 · 1 = 1 .
Proposition 7.7 If B is a bialgebra then the set of primitive elements is
closed under commutator, so P(B) becomes a Lie algebra. Also ε(x) = 0
for all x ∈ P(B) .
Proof. For x , y ∈ P(B) we have
δ([ x , y ])
= δ(x) δ(y) − δ(y) δ(x)
= (x ⊗ 1 + 1 ⊗ x)(y ⊗ 1 + 1 ⊗ y) − (y ⊗ 1 + 1 ⊗ y)(x ⊗ 1 + 1 ⊗ x)
= xy ⊗ 1 + x ⊗ y + y ⊗ x + 1 ⊗ xy − (yx ⊗ 1 + y ⊗ x + x ⊗ y + 1 ⊗ yx)
= [ x , y ]⊗1 + 1 ⊗[ x , y ]
so that [ x , y ] ∈ P(B) . Also x = (1 ⊗ ε) δ(x) = (1 ⊗ ε) (x ⊗ 1 + 1 ⊗ x) =
x + ε(x) ; hence ε(x) = 0 .
Example 7.8 Return to the situation of coalgebras in Example 7.1. There
/ Mod which give rise to
are two conditions on the functor F : X
R
bialgebras F X .
Coalgebras and bialgebras
45
(a) When the morphisms φ are all invertible.
Then F takes each monoid G in X to a bialgebra F G . The multiplication
and unit for G give an algebra structure
F G ⊗R F G ∼
= F (G × G)
Fµ
FG
and
R ∼
= F1
Fη
FG
on F G . These are coalgebra morphisms since all arrows in X “commute
with diagonals”. By Proposition 7.5, each F G becomes a bialgebra. This is
/ Mod , so for each monoid G
the situation for the functor FR : Set
R
the monoid
algebra
R(G)
is
a
cocommutative
bialgebra. Notice here that
G∼
= D R(G) as monoids (see Proposition 7.6).
/ Alg .
(b) When F lifts to F : X
R
In this case each F X is clearly a bialgebra since the comultiplication and
counit are algebra morphisms (Proposition 7.5). In particular, for the func/ Alg this is indeed the situation. Thus we have that:
tor U : LieR
R
each universal enveloping algebra U (L) is a cocommutative bialgebra.
Example 7.9 Return to Example 7.3 of a coalgebra. This time, to use the
symbol N to denote our countable set would be confusing. Instead we denote
it by E = {e0 , e1 , e2 , e3 , . . . } . Then the coalgebra structure on FR (E) is
0 for n > 0 ,
δ(en ) =
ep ⊗ eq
and
ε(en ) =
1
for n = 0 .
p+q=n
We now make FR (E) into an algebra via
ep eq =
(p + q)!
ep+q
p! q!
with
e0 = 1 .
(The binomial coefficient is an integer and so “lives” in any ring R .) Then
FR (E) is a bialgebra. If R is a field of characteristic 0 (i.e., 1 + · · ·+ 1 = 0
in R for any non-zero number of terms) put x = e1 so one easily sees that
1 n
x . Hence, as an algebra, FR (E) is isomorphic to the polynomial
en =
n!
algebra R[x] in one variable. For general R we can think of FR (E) as the
algebra of Hurwitz polynomials in one indeterminate:
k
an xn
n!
n=0
with each an ∈ R .
Example 7.10 Return to Example 7.4 and form the symmetric algebra
S(C) of the coalgebra C = FR (n × n) . Since using n × n can be confusing
we replace it by any set X = {xij | i, j ∈ n} of cardinality n2 . Then we
46
Quantum Groups: A Path to Current Algebra
identify S(FR (X)) with the polynomial R-algebra R[(xij )] in n2 commuting
indeterminates xij for i , j ∈ n . In Example 7.1 we saw that this becomes
a bialgebra by virtue of the fact that it is the universal enveloping algebra of
a commutative Lie algebra FR (X), but this is not the structure of interest
here. The coalgebra C induces the bialgebra structure
1 for i = j
δ(xij ) =
xik ⊗ xkj
and
ε(xij ) =
0 for i = j
k
which we call the matrix bialgebra M(n) over R . This must not be confused
with the matrix algebra
Mat(n , R) ∼
= AlgR (M(n) , R )
(which is the algebra of “points” of M(n) ).
Exercise 7.1 For any R-coalgebra C, prove the following identities:
(a) δ(c) =
ε(c(2) ) ⊗ δ(c(1) )
=
(c)
δ(c(2) ) ⊗ ε(c(1) )
(c)
(b)
(c)
(c)
ε(c(1) ) ⊗ c(3) ⊗ c(2) =
c(1) ⊗ ε(c(3) ) ⊗ c(2)
(c)
c(2) ⊗ c(1)
(c)
ε(c(1) ) ⊗ ε(c(3) ) ⊗ c(2) = c .
(c)
Bradshaw: “Transitionary Figures”, Clothes Peg Figure Period, [Wal94, Plate 66].
8
Dual coalgebras of algebras
We have seen that the dual C ∗ of a coalgebra has a natural structure of an
algebra. One might expect the dual A∗ of an algebra to be a coalgebra in
an obvious way, but this is not true because of the failure of the canonical
morphism
/ (M ⊗ N )∗
M ∗ ⊗R N ∗
R
to be always invertible. If M is Cauchy the morphism is invertible since
(M ⊗R N )∗
= HomR (M ⊗R N, R )
∼
= HomR (M, N ∗ ) ∼
= M ∗ ⊗R N ∗ .
So for an algebra A which is Cauchy (as a module) we obtain a coalgebra,
denoted by A∗ , via
δ : A∗
ε : A∗
µ
η
∗
/ (A⊗ A )∗ ∼
= A∗ ⊗R A∗
R
/ R∗ ∼
=R.
However, instead of restricting A, which is unsatisfactory since many of the
examples are not Cauchy, we modify the definition of the dual A∗ .
Let’s call an ideal I of an algebra A coCauchy when the quotient algebra
A/I is Cauchy (as a module). Define
A0 = { u ∈ A∗ | u is zero on some coCauchy ideal of A } .
Proposition 8.1 (with R a field.)
(a) A0 is a submodule of A∗ .
(b) If f ∈ AlgR (A , B ) then f ∗ : B ∗
with f , takes B 0 into A0 .
/ A∗ , given by composition
(c) For any R-algebra B the canonical morphism A∗ ⊗B ∗
∼
induces an isomorphism A0 ⊗B 0 = / (A⊗B)0 .
47
/ (A⊗B)∗
48
Quantum Groups: A Path to Current Algebra
/ A0 ⊗A0 satisfying
(d) There exists a unique δ : A0
δ
A0
A∗
∼
=
A0 ⊗A0
µ∗
(A⊗A)0
(A⊗A)∗ .
Proof. (a) If u ∈ A0 and r ∈ R then ker(r u) ⊇ ker u , so that r u ∈ A0 .
Take u , v ∈ A0 zero on coCauchy ideals I and J respectively. We can
find subspaces U , V and W of A with A = (I ∩ J) ⊕ U ⊕ V ⊕ W and
I = (I ∩ J) ⊕ U and J = (I ∩ J) ⊕ V . So A/I ∼
= V ⊕ W and A/J ∼
= U ⊕W
∼
are finite dimensional. Thus A/I ∩ J = U ⊕ V ⊕ W is finite dimensional.
Hence I ∩ J is a coCauchy ideal on which u + v is zero.
(b) Take v ∈ B 0 zero on coCauchy J in B . Then f −1 (J) ⊆ ker(vf ) =
f
/ B/J so
ker f ∗ (v) is an ideal of A ; but f −1 (J) is the kernel of A
B
−1
that A/f (J) is isomorphic to a subspace of B/J . So f −1 (J) is coCauchy.
(c) We shall use Exercise 8.1.
Before beginning the proof of Proposition 8.1(c) notice that for any coCauchy ideal K in A⊗B we have coCauchy ideals
I = {a ∈ A | a ⊗ 1 ∈ K} = ( A
J = {b ∈ B | 1⊗b ∈ K} = ( B
A⊗R J + I⊗R B = ker A⊗R B
A⊗B )∗ (K)
⊗1
A⊗B )∗ (K)
/ (A/I)⊗ (B/J) R
1⊗
of A and B and A⊗B respectively.
Now take w ∈ (A⊗B)0 which is zero on some coCauchy K as above.
Then w is zero on A⊗J + I⊗B ⊆ K . However, A⊗B/(A⊗J + I⊗B) ∼
=
A/I ⊗ B/J so there exists a unique w̄ :
w
A⊗B
R
w̄
ρ⊗ρ
(A/I)⊗(B/J)
Furthermore since A/I and B/J are finite dimensional, so that we have
∼
∗
⊗(B/J)∗ =
(A/I⊗B/J)∗ is invertible, there is some element
(A/I)
h̄i ⊗k̄i ∈ (A/I)∗ ⊗(B/J)∗ corresponding to w̄ . In particular, for x ∈ A/I
and y ∈ B/J we have that
w̄(x⊗y) =
i
h̄i (x) k̄i (y) .
(∗)
Dual coalgebras of algebras
49
/ A/I
/ R and similarly define
Define the composite hi = h̄i ◦ ρ : A
/ B/J
/ R . These are in A0 and B 0 since they are
ki = k̄i ◦ ρ : B
zero on I and J respectively. Hence we have
hi ⊗ ki ∈ A0 ⊗B 0
i
which is the image of w ∈ (A⊗B)0 because of (∗) .
Conversely, if h ∈ A0 and k ∈ B 0 vanish on coCauchy I and J (ideals
of A and B) then h ⊗ k vanishes on A⊗J + I⊗B which is a coCauchy ideal
of A⊗B .
(d) Suppose u ∈ A0 vanishes on a coCauchy ideal I . Then µ∗ (u)(a ⊗ b) =
(uµ)(a ⊗ b) = u(ab) , so µ∗ (u) vanishes on A⊗I + I⊗A which is a coCauchy
ideal of A⊗A . Hence µ∗ takes A0 into (A⊗A)0 and δ exists as desired. Corollary 8.2 (with R a field.) For each algebra A a coalgebra structure
∗

/ A∗ η
on A0 is given by the δ in 8.1(d) and ε = (A0 R∗ ∼
= R).
/ B induces a coalgebra morphism
Also each algebra morphism f : A
/ A0 given by restriction of f ∗ (see Proposition 8.1(b) ).
f 0 : B0
Proof. Draw the diagrams expressing the axioms on µ , η and f . Simply
apply ( )∗ then restrict to ( )0 .
For each algebra A we obtain a left and right A-module structure on A∗
given as follows, for a ∈ A and u ∈ A∗ :
(au)(x) = u(xa)
In fact A∗ : A
,
(ua)(x) = u(ax) .
/
A . For any f ∈ A∗ write
Af
and f A and Af A
for the R-submodules of A∗ consisting of those elements of the form af
and f a and af b respectively with a , b ∈ A .
Proposition 8.3 (with R a field.) For f ∈ A∗ these are equivalent:
(1)
(2)
(3)
(4)
f ∈ A0 ;
/ (A⊗A )∗ ;
µ∗ (f ) is in the image of A∗ ⊗A∗

/ (A⊗A )∗ ;
µ∗ (f ) is in the image of (A⊗A )0 Af is Cauchy;
(5) f A is Cauchy;
(6) Af A is Cauchy.
Proof. (1) ⇒ (3) by Proposition 8.1(d). Also (3) ⇒ (2) is trivial.
(2) ⇒ (4). Letµ∗ (f ) be the image of i ui ⊗ vi ∈ A∗ ⊗A∗ . Then we have
that f (ab) = i ui (a) vi (b) so bf = i vi (b) ui ∈ A∗ . Thus bf is in the
subspace of A∗ spanned by the ui . Hence Af is finite dimensional.
50
Quantum Groups: A Path to Current Algebra
(4) ⇒ (1). Suppose Af is finite dimensional. Then also EndR (Af ) is finite/ End (Af ) given by
dimensional, so the kernel I of the morphism A
R
/
/
(bf
bf a) , is a coCauchy ideal of A . But a ∈ I implies 1f a = 0 ,
a
so f (a) = 0 . Hence f is zero on I so that f ∈ A0 .
(5) ⇒ (1) is similar to (4) ⇒ (1) and (6) ⇒ (5) is trivial.
(1) ⇒ (6). Take f ∈ A0 zero on the cofinite ideal I . Then for c ∈ I we have
(a f b)(c) = f (b c a) = 0 . Thus A f A ⊆ I ⊥ = {u ∈ A∗ | u(I) = 0} ∼
= (A/I)∗
which is finite-dimensional since A/I is so.
Corollary 8.4 For any coalgebra C the canonical injection d : C
given by d(c)(u) = u(c) , has image in (C ∗ )0 .
/ C ∗∗
Proof. Take c ∈ C . Then C ∗ d(c) = {u d(c) | u ∈ C ∗ }⊆ C ∗∗ . Now
(u d(c))(v) = d(c)(v ∗ u) = (v ∗ u)(c) = (v ⊗ u)δ(c) = (v ⊗ u) (c) c(1) ⊗ c(2) =
) u(c(2) ) using the definition of multiplication v ∗ u in C . Thus
(c) v(c(1)
u d(c) = (c) u(c(2) ) d(c(1) ) , which is in the subspace of C ∗∗ spanned by
the d(c(1) ) . Hence C ∗ d(c) has finite dimension and by using (4)⇔(1) of
Proposition 8.3 it follows that d(c) ∈ (C ∗ )0 .
Theorem 8.5 (with R a field.) For all algebras A and coalgebras C there
is a bijection
AlgR (A , C ∗ )
f
A
C∗
∼
=
/
CogR (C , A0 )
d
C
(C ∗ )0
f0
A0
.
/ A∗ induces i∗ : A∗∗
/ A0 ∗ , while the
Proof. The inclusion i : A0
/ f 0 ◦ d takes g ∈ Cog (C, A0 ) to the composite
inverse to f
R
A
d
A∗∗
i∗
A0
∗
g∗
C∗ .
The remaining details are left to the reader.
/ N is an injective module
Exercise 8.1 (with R a field.) When f : M
/
morphism then f ⊗R 1 : M ⊗R L
N ⊗R L is injective. Furthermore we
/ (M ⊗ N )∗ is injective. Show this.
have that M ∗ ⊗R N ∗
R
Exercise 8.2 Show that an algebra morphism f : A
element of A0 .
/ R is a set-like
/ R which factor (in
Exercise 8.3 Show that A0 consists of those u : A
the category of R-modules) as u = w ◦ f , where f is surjective with Cauchy
codomain. Use this to simplify some proofs in this chapter.
9
Hopf algebras
Our base ring R will always be assumed commutative, and whenever ( )0
appears we happily suppose it to be a field.
An R-Hopf algebra is an R-bialgebra H together with R-module morphism
ν:H
/H
called the antipode, which satisfies the following diagram.
H
δ
ν ⊗1
H⊗H
1⊗ν
H⊗H
ε
µ
H
η
R
For any Hopf algebra H let H op denote the Hopf algebra obtained by
µ /
σ /
replacing µ with µ ◦ σ : H⊗H
H⊗H
H and replacing δ with
δ /
σ /
σ◦δ : H
H⊗H
H⊗H while keeping the same η , ε and ν.
There is also a bialgebra H obtained more simply by just replacing δ
δ /
σ /
with σ ◦ δ : H
H⊗H
H⊗H while keeping the same µ , η , ε and ν.
In general however, this H is not a Hopf algebra.
Proposition 9.1 Let H be a Hopf algebra. Then
(a) the antipode ν is uniquely determined;
(b) ν : H op
/ H is a bialgebra morphism;
(c) H is a Hopf algebra if and only if ν is bijective (moreover the antipode
for H is the inverse for ν );
(d) if H is commutative or cocommutative then ν ◦ ν = 1H (that is, ν is
an involution).
Proof. (a) Since H is a coalgebra and an algebra, we have the convolution
algebra structure on HomR (H, H) . An antipode is precisely an inverse for
1H ∈ HomR (H, H) under convolution. For any monoid, inverses are unique.
51
52
Quantum Groups: A Path to Current Algebra
(b) To show ν : H op
µ
H⊗H
ν
H
/ H preserves multiplication we must show that
H
=
σ
H⊗H
/ H⊗H
ν ⊗ν
H⊗H
/ H .
µ
We do this by showing that, under convolution, the left-hand side is a left
inverse for µ ∈ HomR (H⊗H, H) while the right-hand side is a right inverse.
H⊗H
δ ⊗δ
(ν◦µ)∗µ
H
µ
1⊗σ⊗1
⊗4
H
⊗4
bialgebra
axiom
ε⊗ε
bialgebra
axiom
δ
H
ε
definition of
convolution
µ⊗µ
H ⊗2
ν ⊗1
H
antipode axiom
η
R
δ ⊗δ
H ⊗2
µ
H ⊗2
H ⊗4
1⊗σ⊗1
H ⊗4
µ⊗1⊗1
H ⊗3
δ ⊗1
1⊗σ
H ⊗3
1⊗ε
1⊗σ
H ⊗3
1⊗1⊗ε
H
δ
1⊗δ ⊗1
H ⊗4
1⊗ε⊗1
H ⊗4
1⊗η ⊗1
1⊗1
H ⊗3
1⊗1⊗ν ⊗1
H ⊗2
ε⊗ε
µ⊗1⊗1
1⊗ν ⊗1
µ⊗1⊗1
H ⊗3
1⊗µ⊗1
H ⊗3
1⊗1⊗ν
µ⊗1
H ⊗3
µ⊗1
µ⊗1
ε
H ⊗2
1⊗ν
1⊗µ
H ⊗2
µ
R
η
H
While the second commutativity is perhaps more easily seen by looking at
elements, the bonus we get on using diagrams is that, formally reversing
all the arrows and replacing µ and η by δ and ε , we have the proof that
Hopf algebras
53
/ H preserves comultiplication. The following diagram proves
ν : H op
ν preserves unit, while the dual diagram proves ν preserves counit.
η
R
ν
H
H
1⊗η
η ⊗η
η
H
1
δ
ν ⊗1
H ⊗2
H ⊗2
1
ε
µ
η
R
H.
(c) ν is a (composition) right inverse for ν ⇔ ν ◦ ν = 1H ⇔ ν ◦ ν is a
convolution left inverse for ν (since 1H is the convolution inverse for ν )
⇔
H
δ
H⊗H
1⊗ν H⊗H
ν ⊗ν
ε
µ
H⊗H
H
η
(using (b))
R
⇔
H
δ
H⊗H
1⊗ν H⊗H
σ
µ
H⊗H
H
ν
H
η
ε
R
⇔
H
δ
H⊗H
σ
H⊗H
ε
ν ⊗1
H⊗H
µ
H
ν
H .
η
R
The last condition is the condition that ν should be a left convolution
inverse for 1H in Hom(H , H ) , except that ν is applied to the condition.
Similarly, we get that ν is a left (composition) inverse for ν if and only
if ν satisfies the condition to be a right convolution inverse for 1H in
Hom(H , H ) with ν applied to the condition. It follows then that ν and ν are mutually (composition) inverse precisely when ν and 1H are mutually
convolution inverse; that is, if and only if ν is an antipode for H .
(d) If H is cocommutative then H = H so that H is a Hopf algebra with
antipode ν = ν . So ν is its own (composition) inverse; that is, ν ◦ ν = 1H .
For the commutative case replace H by H op .
Remark. Proposition 9.1(d) can also be seen from the observation that
commutative Hopf algebras are groups in the opposite of the category of
commutative algebras, while cocommutative Hopf algebras are groups in
the category of cocommutative coalgebras; the antipode is inversion so is
clearly involutory.
54
Quantum Groups: A Path to Current Algebra
Proposition 9.2 Let H and K be any Hopf algebras. Then each bialgebra
/ K preserves antipode.
morphism f : H
f
H
K
ν
ν
f
H
K
/ B are coalgebra and algebra
/ C and g : A
Proof. Clearly if f : D
morphisms respectively, then
Hom( C , A )
/ Hom(D , B )
whereby u / g◦u◦f
is a monoid morphism for the convolution structures. In particular, here
we have two monoid morphisms
◦f
and f ◦
: Hom(H, H)
/ Hom(H, K)
that both take 1H to f . Monoid morphisms take inverses to inverses. So
ν◦f =
convolution inverse of f in Hom(H, K)
= f ◦ν .
Using other fancier words, the category HopfR of Hopf algebras is a full
subcategory of the category BigR of bialgebras.
For any algebra H we have seen that H 0 becomes a coalgebra. If H is a
bialgebra then H 0 becomes a bialgebra using the multiplication
H 0 ⊗H 0 ∼
= (H⊗H)0
and unit
R ∼
= R0
ε0
δ0
H0
H0
(recall Proposition 8.1). Furthermore, if H is a Hopf algebra then so is H 0
with antipode
/ H0 .
ν0 : H 0
What we have here is a contravariant “self-adjoint” functor
( )0 : HopfRop
/ Hopf .
R
What “self-adjoint” means in this context is that
BigR (H, K 0 ) ∼
= BigR (K , H 0 ) .
Hopf algebras
55
Proposition 9.3 If H is any Hopf algebra then the monoid D(H) of setlike elements is a group.
Proof. For g ∈ D(H) we have
ν(g) g = µ ◦ (ν ⊗ 1) (g ⊗ g)
= µ ◦ (ν ⊗ 1) δ(g)
= µ ◦ (ν ⊗ 1) ◦ δ (g) = η ε(g) = η(1) = 1 .
An A-point of a Hopf algebra H is an algebra morphism f : H
/ A.
/ A are commuting A-points of H
Proposition 9.4 (a) If f , g : H
/ A is
(meaning that [ f (h) , g(k) ] = 0 for all h , k ∈ H ) then f ∗ g : H
an A-point of H.
/ A is an A-point of H then f has a convolution inverse
(b) If f : H
/ A which is an A-point of H op .
f ◦ ν : H op
Proof. (a) The commuting property yields that
H⊗H
f ⊗g
A⊗A
µ
A
/ H⊗H is an algebra morphism
is an algebra morphism. But δ : H
since H is a bialgebra. So f ∗ g ∈ Alg(H, A ) .
(b) Clear from Proposition 9.1(b).
Example 9.5 For a monoid G , we have seen that the monoid algebra
R(G) is a bialgebra. If G is a group then the group algebra R(G) becomes
/ R(G) given by ν(g) = g −1 .
a Hopf algebra with antipode ν : R(G)
−1
/ G expressed diagrammatically in Set are
(The axioms for ( ) : G
/ Mod into the axioms which define
taken by the functor FR : Set
R
the antipode.)
Example 9.6 For a Lie algebra L , write Lop for the Lie algebra with the
same module L but with Lie bracket β op given by β op (x , y ) = β( y , x ) .
For any algebra A we have (Aop )L = (AL )op . It follows (why?) that we
have a canonical algebra isomorphism
U(Lop ) ∼
= U(L)op .
/ Lop taking x to −x (note that
We have a Lie algebra isomorphism L
/ U(L)op by
[ −x , −y ] = [ x , y ] = −[ y, x ] ). So we define ν : U(L)
56
Quantum Groups: A Path to Current Algebra
L
−
Lop
i
i
U(L)
U(L)op ∼
= U(Lop ) .
ν
One easily checks that for x1 , . . . , xn ∈ L
ν(i(x1 ) · · · i(xn )) = (−1)n i(x1 ) · · · i(xn ) .
With this antipode U(L) becomes a Hopf algebra.
Example 9.7 The matrix bialgebra M(n) (Example 7.10 of a bialgebra) is
not a Hopf algebra. We need to
“adjoin
an inverse for the determinant”.
Recall that M(n) = R[ X ] = S FR (X) where X = {xij | i , j = 1, . . . , n}
has cardinality n2 . Define
(−1)|ξ| x1 ξ(1) x2 ξ(2) . . . xn ξ(n)
det(X) =
ξ∈Sn
where |ξ| is the least number of simple transpositions required to obtain the
permutation ξ . Form the following
commutative
polynomial R-algebra:
R[ X ∪ {t} ] = R[ (xij ) , t ] = S FR (X ∪ {t}) , in n2 + 1 (commuting)
indeterminates t and xij with (1 ≤ i , j ≤ n) . Put
GL(n) = R[ X ∪ {t} ]/( t det(X) − 1 )
as a commutative R-algebra. We make GL(n) into a bialgebra by defining
δ(xij ) =
n
xik ⊗ xkj
δ(t) = t ⊗ t
k=1
ε(xij ) = δij
ε(t) = 1
modulo ( t det(X) − 1 ) . Put Xij = {xrs | r = i , s = j } . Now define the
/ GL(n) by
morphism ν : GL(n)
ν(xij ) =
ν(t)
=
t det(Xji )
det(X)
modulo ( t det(X) − 1) . Then GL(n) becomes a Hopf algebra.
For any commutative R-algebra A we have a canonical isomorphism
of groups
AlgR (GL(n) , A) ∼
= GL(n , A) .
Examples 9.5 and 9.6 above exhibit cocommutative Hopf algebras R(G)
and U(L) , while Example 9.7 is a commutative Hopf algebra GL(n) . It
is only recently that the importance of Hopf algebras which are neither
commutative nor cocommutative has been properly understood.
Hopf algebras
57
Example 9.8 We now describe a “quantum deformation” of Example 9.7.
This is a generalization to n×n, from the 2×2 case discussed in Chapter 3.
Take X = {xij | i , j = 1 , . . . , n} as in Example 9.7. First we form the
free algebra R
X = T FR (X) on the (non-commuting) indeterminates
xij . Let Mq (n) denote the quotient of R
X by the ideal generated by the
following elements:
xir xjk − xjk xir
xir xjk − xjk xir − (q − q
xik xjk − q xjk xik
for i < j and k < r
−1
) xik xjr
for i < j and r < k
for i < j
xik xir − q xir xik
for k < r .
This becomes a coalgebra with comultiplication
δ(xij ) =
n
xir ⊗ xrj
( modulo the ideal )
r=1
and counit
ε(xij ) = δij
( Kronecker delta ) .
Define the “quantum determinant” by
det q (X) =
(−q)|ξ| x1 ξ(1) x2 ξ(2) · · · xn ξ(n)
ξ∈Sn
which is a central element of Mq (n) (that is, it commutes with all other
elements). The quantum general linear group is defined by
GL q (n) = Mq (n)[ t ]/( t det q (X) − 1 ) .
We adjust the comultiplication and counit of Mq (n) by defining δ(t) = t ⊗ t
and ε(t) = 1 . Then we have a bialgebra epimorphism
ρ : Mq (n)
Define ν : GL q (n)
/ GL (n) .
q
/ GL (n) by
q
ν(xij ) = t det q (Xji )
and
ν(t) = det q (X) .
Then GL q (n) becomes a Hopf algebra. Notice that GL q (n)op = GL q-1(n) .
Many claims have been made in this section. For n = 2 the calculations
in Theorem 3.1 prove them all. (This should be compared with Proposition 9.4 in the present section.) The general case can be verified similarly,
but will follow from later work.
58
Quantum Groups: A Path to Current Algebra
Exercise 9.1 [Swe69] Assume our base ring R is a field and write ⊗ for
⊗R . An ideal in an algebra A is a submodule I such that µ(I⊗A + A⊗I)
⊆ I . We know that A/I becomes an algebra. A coideal of a coalgebra C is
a submodule I such that δ(I) ⊆ I⊗C + C⊗I and ε(I) ⊆ 0 .
(a) If I is a coideal of a coalgebra C, describe a coalgebra structure on
/ C/I becomes a coalgebra morphism. If C is
C/I for which ρ : C
a bialgebra and I is also an ideal, show that C/I is a bialgebra. What
condition on I ensures C/I has an antipode if C has? I is called a
Hopf ideal when this holds.
(b) Verify that the polynomial R-algebra B = R
x, y, z on three noncommuting indeterminates becomes a bialgebra with
δ(x) = x⊗x ,
δ(y) = y⊗y ,
ε(x) = ε(y) = 1 ,
δ(z) = 1⊗z + z⊗x
ε(z) = 0 .
(c) Verify that the ideal (xy − 1 , yx − 1) is a coideal in B. Let H denote
the quotient bialgebra.
(d) Show that H is a Hopf algebra with antipode ν given by
ν(x) = y , ν(y) = x , ν(z) = −zy (modulo the ideal of (c) ).
Show further that ν 2n (z) = xn zy n , ν 2n+1 (z) = −xn zy n+1 . Hence, this
antipode has infinite order.
(e)
i. Show that the ideals In = (xn zy n − z), Jn = (xn − 1) are Hopf
ideals in H.
ii. Show that the antipodes of both H/In and H/Jn have order 2n .
Bradshaw: Elegant Action Figure Group, [Wal94, Plate 84].
10
Representations of quantum groups
We mentioned in Example 6.3 that a representation of a group G was an
R(G)-module. One kind of representation for a Hopf algebra H therefore
suggests itself: a module over H . We begin by discussing modules over
bialgebras.
/ A is a ring morphism then each (left)
First note that if f : E
A-module M becomes a (left) E-module via the action
for e ∈ E , m ∈ M .
e m = f (e)m
This is called restriction of scalars along f .
Let A be an R-algebra. Then each module is automatically an R/ A . Alternamodule via restriction of scalars along the unit η : R
tively, we can view an A-module as an R-module M with a ring morphism
/ End (M ) . Later, we want to look at “comodules”, and so we
µ̂ : A
R
want a definition of A-module which dualizes. The good version is: an R/ M , called the action
module M with a module morphism µ : A⊗R M
of A on M , satisfying
A⊗R A⊗R M
µ⊗1
A⊗R M
1⊗µ
µ
M
A⊗R M
η ⊗1
M
µ
1
M.
We write ModR (A) for ModA just to emphasize that we build it up from
ModR .
Suppose M and N are (left) modules over the R-algebra A. Re / op
/ op
/
R and N : R
A . We see that M ⊗R N : A
A ,
gard M : A
which means M ⊗R N becomes an A⊗A-module. If A is a bialgebra then we
59
60
Quantum Groups: A Path to Current Algebra
/ A⊗ A to obtain an A-module structure
can restrict scalars along δ : A
R
on M ⊗R N . Explicitly, the action is the composite
A⊗R M ⊗R N
δ ⊗1⊗1
A⊗R A⊗R M ⊗R N
1⊗σ⊗1
A⊗R M ⊗R A⊗R N
µ ⊗µ
M ⊗R N .
This generalizes to multiple tensor products (over R) of A-modules. In
particular, the empty tensor product R becomes an A-module by restricting
/ R.
scalars along the counit ε : A
/ op
A
With M and N left A-modules as before, we can regard M : R
/ op
op /
op
and N : R
A , so that HomR (M, N ) : A
A ; or in other words
HomR (M, N ) becomes an Aop ⊗A-module. Thus if A = H is a Hopf algebra, we can restrict scalars along the R-algebra morphism
δ
H
H⊗R H
ν ⊗1
H op ⊗R H
to make HomR (M, N ) into an H-module. Explicitly, the action of H on
HomR (M, N ) is the composite
H⊗R HomR (M,N )
δ ⊗1
H⊗R HomR (M,N )⊗R H
1⊗σ
H⊗R H⊗R HomR (M,N )
µ1 ⊗ν
HomR (M,N )⊗R H
µ2
HomR (M,N )
where µ1 and µ2 are the left and right actions:
H⊗R HomR (M,N )
µ̂⊗1
HomR (N,N )⊗R HomR (M,N )
◦
µ1 : h ⊗ f
HomR (M,N )⊗R H
1⊗µ̂
HomR (M,N )⊗R HomR (M,M )
µ2 : f ⊗ h
◦
HomR (M,N )
/ h(f m))
(m HomR (M,N )
/ f (h m)) .
(m Proposition 10.1 For left modules M and N over the Hopf algebra H,
the canonical R-module morphisms
/N
e : HomR (M, N )⊗R M
/ Hom (N, M ⊗ N )
d:M
R
R
where
where
/ f (m)
f ⊗m / (n / m ⊗ n)
m are left H-module morphisms.
Proof. Omitting ⊗R and HomR from the notation, we obtain the first of
Representations of quantum groups
61
these from the following diagram. The second we leave to the reader.
δ11
H(M N )M
HH(M N )M
1δ11
1σ1
δ111
δ111
1ε11
11σ1
HHH(M N )M
1
H(M N )HM
HH(M N )HM
1ν111
H(M N )M
HHH(M N )M
1η11
11η1
1σ11
11σ1
HH(M N )HM
H(M N )HHM
1σ11
1µ11
HH(M N )M
11ν11
H(M N )HHM
11µ1
1σ1
1µ2 11
1
H(M N )HM
H(M N )HM
1µ2 1
1µ2 1
H(M N )M
µ1 1
1e
HN
(M N )M
µ1 11
1µ
(M N )HM
e
µ
N
Corollary 10.2 For modules M , N and L over a Hopf algebra H, the
canonical isomorphism
HomR (M ⊗R N , L) ∼
= HomR (M , HomR (N, L))
restricts to an isomorphism
HomH (M ⊗R N , L) ∼
= HomH (M , HomR (N, L)) .
Proof. The canonical isomorphism is obtained from the evaluation e and
the canonical d of Proposition 10.1.
In other words, we have a nice tensor–hom situation for the category
ModR (H) of (left) H-modules. Both the tensor and the hom are preserved
by the functor
/ Mod
ModR (H)
R
given by ignoring the H-action.
62
Quantum Groups: A Path to Current Algebra
Although modules over the group algebra are representations of the group,
so that the study of modules over a Hopf algebra does suggest itself, the
point of view of Chapter 2 (i.e. space–algebra duality) leads more naturally
/ H⊗H of the
to “comodules”. For here, it is the comultiplication δ : H
Hopf algebra which corresponds to the spatial multiplication.
Suppose C is an R-coalgebra. A (left) C-comodule is an R-module
/ C⊗ M , called the coaction of C
M with a module morphism δ : M
R
on M , satisfying
M
δ
δ ⊗1
C⊗R M
1⊗δ
C⊗R C⊗R M
C⊗R M
ε⊗1
δ
1
M
M.
We write ComR (C) for the category whose objects are C-comodules and
whose arrows are C-comodule morphisms; that is, R-module morphisms
/ N such that
f : M
M
δ
C⊗R M
1⊗f
f
N
δ
C⊗R N .
Each C-comodule M becomes a C ∗ -module with the action
C ∗ ⊗R M
1⊗δ
C ∗ ⊗R C⊗R M
e⊗1
M.
See Chapter 7 for the algebra structure on C ∗ .
By the fundamental theorem of Morita theory (Theorem 5.2), if C is
Cauchy (as an R-module) then this gives a bijection between C-coactions δ
and C ∗ -actions µ on each R-module M : recover δ as the composite
M
d ⊗1
C⊗R C ∗ ⊗R M
1⊗µ
C⊗R M .
So for C Cauchy, we have an isomorphism of categories
ComR (C) ∼
= ModR (C ∗ ) .
If C is an R-bialgebra not necessarily Cauchy we obtain, in a manner
dual to that for modules, a coaction on the tensor product (over R) of
Representations of quantum groups
63
C-comodules. Explicitly for C-modules M and N , the coaction for M ⊗R N
is given by the composite
M ⊗R N
δ ⊗δ
C⊗R M ⊗R C⊗R N
σ1324
C⊗R C⊗R M ⊗R N
The empty tensor product R has the coaction η : R
µ⊗1⊗1
C⊗R M ⊗R N .
/ C⊗ R .
R
When it comes to Hom our formal duality fails: in reversing arrows we have
maintained ⊗, yet Hom does not maintain its universal property. However, if M is Cauchy, HomR (M, N ) does have the reverse-arrow universal
property: there is a bijection between R-module morphisms
HomR (M, N )
L
and R-module morphisms
M ⊗L
N
since HomR (M, N ) ∼
= M ∗ ⊗R N and M ⊗R L ∼
= HomR (M ∗ , L).
Proposition 10.3 Each Cauchy R-module M gives rise to an R-coalgebra
/ R and comultiplication
M ⊗R M ∗ with counit e : M ⊗R M ∗
/ M ⊗ M ∗⊗ M ⊗ M ∗
R
R
R
1 ⊗ d ⊗ 1 : M ⊗R M ∗
(see Theorem 5.2). For any R-coalgebra C, the assignment
δ̂ = (M ⊗R M ∗
δ ⊗1
C⊗R M ⊗R M ∗
1⊗e
C)
determines a bijection between coactions
δ:M
/ C⊗ M
R
of C on M and coalgebra morphisms
δ̂ : M ⊗R M ∗
/C .
M ∗ ⊗R M HomR (M, M ) has the universal
Proof. M ⊗R M ∗ σ
ρ
property of Hom under reversal of arrows; so the diagrammatic proof that
EndR (M ) is an algebra and that an action is an algebra morphism of the
/ End (M ) , dualizes.
kind A
R
Take M = Rn in the above proposition and let e1 , . . . , en be the standard basis. Now let e∗1 , . . . , e∗n be the dual basis for Rn ∗ so e∗i (ej ) = δij
64
Quantum Groups: A Path to Current Algebra
(Kronecker-δ). A coaction of C on Rn thus amounts to a coalgebra morph/ C , and this is determined by its values on the basis
ism δ̂ : Rn ⊗R Rn ∗
∗
n
elements ei ⊗ ej of R ⊗R Rn ∗ :
δ̂(ei ⊗ e∗j ) = xij ∈ C .
So C-comodule structures on Rn are in bijection with multiplicative matrices in C; that is, matrices x = (xij ) in C satisfying
δ(xij ) =
xik ⊗ xkj , ε(xij ) = δij .
k
Following Manin [Man88], we write the last two equations as
ε(x) = i
where i is the identity matrix and x ⊗y = ( k xik ⊗ ykj ) is not the usual
tensor product of matrices.
x 0
Example 10.4 In the situation of Example 9.8, x = (xij ) and
0 t
are multiplicative matrices for Mq (n) and GL q (n), respectively.
δ(x) = x ⊗x ,
Now suppose C = H is a Hopf algebra and M is a Cauchy R-module. By
applying Proposition 10.1 to M ∗ (and using the canonical M ∗∗ ∼
= M ), we
see that M ∗ ⊗R M becomes a coalgebra with counit
σ
M ∗ ⊗R M
e
M ⊗R M ∗
R
and comultiplication
M ∗ ⊗R M
1⊗d⊗1
M ∗ ⊗R M ∗ ⊗R M ⊗R M
σ
M ∗ ⊗R M ⊗R M ∗ ⊗R M .
/ H is a coalgebra morphism if and
Proposition 10.5 (a) δ̂ : M ⊗R M ∗
only if the composite
M ∗ ⊗R M
1⊗σ⊗1
δ̂
H op
νk
H .
M ⊗R M ∗
is a coalgebra morphism.
(b) Suppose M is an H-comodule and put
δ̂k = M ⊗R M ∗
δ̂
H
For k even, δ̂k is a coalgebra morphism and has convolution inverse δ̂k+1
in HomR (M ⊗R M ∗ , H) .
Representations of quantum groups
65
Proof. (a)
M ∗M
σ
1⊗d⊗1
M ∗M ∗M M
δ̂
1⊗d⊗1
σ4231
M M ∗M M ∗
σ ⊗σ
M M ∗M M ∗
H
δ
δ̂ ⊗δ̂
σ3412
1⊗σ⊗1
M ∗M M ∗M
MM∗
H⊗H
σ
δ̂ ⊗δ̂
H⊗H
M ∗M
σ
MM∗
δ̂
H
σ
MM∗
e
ε
e
R
/ H op is a coalgebra morphism, by Proposition 9.1(b);
(b) Since ν : H
/ H is a coalgebra morphism for k even;
then it is also true that ν k : H
k
so δ̂k = ν ◦ δ̂ is a coalgebra morphism, as required. This also means
that right composition with δ̂k preserves convolution. Since 1H and ν are
convolution inverses, so are 1H ◦ δ̂k and ν ◦ δ̂k ; that is, so are δ̂k and δ̂k+1 .
Proposition 10.6 Suppose that M is a comodule over the Hopf algebra H
and that M is Cauchy as an R-module. Then M ∗ becomes an H-comodule
via
δ̂
σ
ν
H .
δ̂ = M ∗ ⊗R M
M ⊗R M ∗
H
Moreover, the R-module morphisms
/R
e : M ⊗R M ∗
/ M ∗⊗ M
d:R
R
become H-comodule morphisms. One might therefore say that M becomes
a Cauchy H-comodule.
Proof. By Propositions 10.5(a) and 9.1(b), the stated δ̂ is a coalgebra
morphism. So by Proposition 10.5, it determines a coaction of H on M ∗ .
Tracing through, one sees that this is dual to the situation for HomR (M, R)
as in Proposition 10.1; so the proof dualizes, but it can also be shown
directly that e and d are comodule morphisms.
Remark. To obtain results as in Proposition 10.5(b) for k odd, apply
Proposition 10.5(b) to the M ∗ of Proposition 10.6; compare with [Man88,
p.14].
Exercise 10.1 Give a direct proof of Proposition 10.3, concerning the
coalgebra structure on M ⊗R M ∗ where M is a Cauchy R-module.
66
Bradshaw: Sash Bradshaw Group, [Wal94, Plate 52].
11
Tensor categories
It is clear that specific categories have entered explicitly into the above
discussion, but we have made little use of them as categories apart from
diagrams and duality. For what follows it is hard to imagine how to express
the results without categories.
A tensor category (also called “monoidal category” [EK66]) is a category V
/ V called tensor product, an object I
together with functor ⊗ : V × V
of V called the unit object , and natural families of isomorphisms
/ A⊗(B⊗C)
aA,B,C : (A⊗B)⊗C
/ A , l : I⊗A
/A
rA : A⊗I
A
called respectively the associativity constraint, the right unit constraint
and the left unit constraint, subject to the two conditions:
(A⊗B)⊗(C⊗D)
aA⊗B,C,D
aA,B,C⊗D
((A⊗B)⊗C)⊗D
A⊗(B⊗(C⊗D))
1⊗aB,C,D
aA,B,C ⊗1
(A⊗(B⊗C))⊗D
A⊗((B⊗C)⊗D)
aA,B⊗C,D
(A⊗I)⊗C
aA,I,C
A⊗(I⊗C)
1⊗lC
rA ⊗1
A⊗C .
Define A1 ⊗ · · · ⊗An to be the object obtained by inserting brackets in some
chosen preassigned way, such as from the left ((· · · (A1 ⊗A2 )⊗ · · · )⊗An ) .
67
68
Quantum Groups: A Path to Current Algebra
It is an important fact (Mac Lane’s coherence theorem) that, in general,
the only automorphism which is a composite of isomorphisms of the form
1 ⊗ (x ⊗ 1) or (1 ⊗ x) ⊗ 1 , where x is a component of a, r, l or their inverses, is
the identity arrow of A1 ⊗ · · · ⊗An . This essentially allows one to work as if
the a, r, l are all identities. If all the a, r, l are indeed identities, then the
tensor category is called strict . The opposite V op of a tensor category V
consists of the opposite category of V (obtained by reversing the direction
of arrows of V) and the reverse tensor product, so that A⊗B in V op is just
B⊗A in V.
A braiding for a tensor category V is a natural family of isomorphisms
/ B ⊗A
: A⊗B
c
A,B
subject to the conditions
c
A⊗(B⊗C)
A,B ⊗ C
(B⊗C)⊗A
aB,C,A
aA,B,C
(A⊗B)⊗C
B⊗(C⊗A)
cA,B ⊗1
1⊗cA,C
(B⊗A)⊗C
(A⊗B)⊗C
aB,A,C
c ⊗
A B,C
B⊗(A⊗C)
C⊗(A⊗B)
−1
aA,B,C
−1
aC,A,B
A⊗(B⊗C)
(C⊗A)⊗B
cA,C ⊗1
1⊗cB,C
A⊗(C⊗B)
−1
aA,C,B
(A⊗C)⊗B .
A braided tensor category is a tensor category with a chosen braiding
(see [JS93]).
A symmetry for a tensor category is a braiding which satisfies the following
extra condition:
1
A⊗B
A⊗B
cA,B
cB,A
B ⊗A .
Tensor categories
69
A symmetric tensor category is a tensor category with a chosen symmetry.
Example 11.1 The braid category B has as objects the natural numbers
/ n the braids on n strings; there are no
0, 1, 2, . . . and as arrows α : n
/
n for m = n . A braid α
arrows m
5
•
•
•
•
•
5
•
•
•
•
•
α
on n strings can be regarded as an element of the Artin braid group Bn
with generators s1 , . . . , sn−1 subject to the relations
si sj
for j < i − 1
= sj si
si+1 si si+1
= si si+1 si
where si is the braid depicted as:
n
1
•
i−1
•
i
•
n
•
•
•
n
•
i+1 i+2
•
•
si
•
•
•
Composition of braids is just multiplication in this group, represented
diagrammatically by vertical stacking of braids with the same number of
strings.
•
•
•
3
β
3
•
•
•
3
•
•
•
γ
Tensor product of braids adds the number of strings by placing one braid
next to the other longitudinally.
•
•
•
•
•
•
⊗
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
•
=
•
•
70
Quantum Groups: A Path to Current Algebra
This makes B a strict tensor category. A braiding c m,n : m + n
is given by crossing the first m strings over the remaining n .
m
4+2
/ n+m
n
•
•
•
•
•
•
•
•
•
•
•
•
c4,2
2+4
The axioms that show B is braided are easily checked diagrammatically.
Example 11.2 The category ModR of modules over a commutative ring
R is a symmetric tensor category with tensor product ⊗R , with the canon/ B⊗ A .
ical constraints, and with symmetry σ : A⊗R B
R
Example 11.3 Let A be an R-bialgebra. If M and N are A-modules, we
have an A-module structure on M ⊗R N given by
a · (m ⊗ n) =
a(1) m ⊗ a(2) n
(a)
as seen in the last chapter. So ModR (A) becomes a tensor category with
tensor product ⊗R .
/ N⊗ M
If A is cocommutative, the switch morphism σ : M ⊗R N
R
is a symmetry for ModR (A) . However, as in the rest of this book, we are
more interested in non-cocommutative A .
We ask: what are the possible braidings on the tensor category ModR (A)?
/ N ⊗ M gives, for each A , a morphism
A braiding cM,N : M ⊗R N
R
/ A⊗ A which gives an element γ = c (1 ⊗ 1) ∈ A⊗A .
cA,A : A⊗R A
R
A,A
Conversely, each element γ = i ui ⊗ vi ∈ A⊗A determines a natural
/ N ⊗ M via the formula
morphism cM,N : M ⊗R N
R
cM,N (m ⊗ n) =
(ui n) ⊗ (vi m) .
i
This is a bijection, as can be seen from the following diagram in which m̂ :
/ M is the unique module morphism with m̂(1) = m .
A
R⊗R R
η ⊗η
A⊗R A
cA,A
m̂⊗n̂
M ⊗R N
A⊗R A
n̂⊗m̂
cM,N
N ⊗R M .
Tensor categories
71
In order for each cM,N to be an isomorphism it is necessary for γ ∈ A⊗R A
to be invertible. In order for each cM,N to be a module morphism we need
c a · (m ⊗ n) = c
(a(1) m) ⊗ (a(2) n)
(a)
=
(ui a(2) n) ⊗ (vi a(1) m)
i
(a)
to be equal to
a · c(m ⊗ n) = a ·
(ui n) ⊗ (vi m)
i
=
(a)
(a(1) ui n) ⊗ (a(2) vi m) .
i
This is equivalent to the requirement
(ui a(2) ) ⊗ (vi a(1) ) =
(a(1) ui ) ⊗ (a(2) vi ) .
i,(a)
i,(a)
/ A⊗ A whose value at
Regarding γ ∈ A⊗R A as a morphism γ : R
R
1 ∈ R is the given γ , we can express this condition diagrammatically as
(B0)
A
γ ⊗δ
A⊗4
σ1423
σ3142
A⊗4
µ⊗µ
A⊗2 .
For a braiding, we require two more conditions:
cM,N ⊗L (m ⊗ n ⊗ l) = (1N ⊗ cM,L )(cM,N ⊗ 1L )(m ⊗ n ⊗ l) ,
cM ⊗N,L (m ⊗ n ⊗ l) = (cM,L ⊗ 1N )(1M ⊗ cN,L )(m ⊗ n ⊗ l) ;
that is,
ui(1) n ⊗ ui(2) l ⊗ vi m =
ui n ⊗ uj l ⊗ vj vi m ,
i,j
i,(ui )
ui l ⊗ vi(1) m ⊗ vi(2) n =
uj ui l ⊗ vj m ⊗ vi n .
i,j
i,(vi )
These are equivalent to the two conditions
ui(1) ⊗ ui(2) ⊗ vi =
ui ⊗ uj ⊗ vj vi ,
i,j
i,(ui )
i,(vi )
ui ⊗ vi(1) ⊗ vi(2) =
i,j
uj ui ⊗ vj ⊗ vi .
72
Quantum Groups: A Path to Current Algebra
Diagrammatically, these conditions become:
(B1)
γ ⊗γ
R
A⊗4
σ1342
γ
1⊗1⊗µ
δ ⊗1
A⊗2
(B2)
A⊗4
γ ⊗γ
R
A⊗4
A⊗3
σ3142
γ
A⊗4
µ⊗1⊗1
1⊗δ
A⊗2
A⊗3 .
Hence, we define a braiding element for a bialgebra A to be an invertible
element γ ∈ A⊗R A which satisfies (B0), (B1), (B2). We have proved above
that braiding elements for A are in bijection with braidings on the tensor
category ModR (A) .
A braided bialgebra (also called “quasitriangular bialgebra”) is a bialgebra equipped with a braiding element γ ∈ A⊗R A. A braiding element γ
is called a symmetry element when γ 2 = 1 ∈ A⊗R A; these are in bijection
with symmetries on ModR (A). A symmetric bialgebra (also known as a
“triangular algebra”) is a bialgebra equipped with a symmetry element.
Before leaving this example, we point out that conditions (B1), (B2)
can be put in a more familiar form in thecase where A is Cauchy as an Rmodule. For in this case, elements γ = i ui ⊗ vi ∈ A⊗R A are in bijection
/ A via the formula
with R-module morphisms g : A∗
γ=
R
d
A∗ ⊗R A
g⊗1A
A⊗R A
.
Condition (B1) precisely says that g preserves comultiplication, while condition (B2) says that g reverses multiplication. In fact, if γ is a braiding
/ Aop is a bialgebra morphism; preservation of unit and
element, g : A∗
counit follows from cM,I = cI,M = 1M .
We shall just look at the translation of (B2) to g. Begin with the defining diagram
R
d
A∗ ⊗R A
A∗ ⊗R A∗ ⊗R A⊗R A
δ ∗ ⊗1⊗1
d
A∗ ⊗R A
1⊗d⊗1
1⊗δ
A∗ ⊗R A⊗R A
Tensor categories
73
for δ ∗ , which is the multiplication for A∗ . To prove g reverses multiplication is to prove
A∗ ⊗R A∗
g ⊗g
A⊗R A
σ
A⊗R A
d∗
µ
g
A∗
A.
This is equivalent to proving the legs are equal after applying
and composing with
R
d
A∗ ⊗R A∗
1⊗d⊗1
⊗R A⊗R A
A∗ ⊗R A∗ ⊗R A⊗R A .
From the defining diagram for δ ∗ , this amounts to
R
d
A∗ ⊗R A∗
1⊗d⊗1
A∗ ⊗R A∗ ⊗R A⊗R A
g⊗g⊗1⊗1
A⊗4
σ
A⊗4
µ⊗1⊗1
1⊗δ
g⊗1⊗1
A∗ ⊗R A⊗R A
A⊗3 .
Using γ = (g ⊗ 1A ) ◦ d , we easily see that this is equivalent to (B2).
Although a braiding is as useful as a symmetry for most purposes, there is
sometimes further structure on a braiding which makes it even more like a
symmetry without actually forcing it to be one.
Suppose V is a braided tensor category. A twist for V is a natural
family of isomorphisms
/A
θA : A
such that θI = 1I and
A⊗B
cA,B
B⊗A
θB ⊗θA
θA⊗B
A⊗B
cB,A
B⊗A .
A balanced tensor category is a braided tensor category with a chosen twist.
(A braiding is a symmetry if and only if the identity arrows provide a twist.)
74
Quantum Groups: A Path to Current Algebra
Example 11.4 The braid category B is canonically balanced. The twist
/ n is obtained by taking n vertical parallel strings with ends tied
θn : n
to two horizontal parallel rods, and rotating the bottom rod through a full
2π twist in the right-hand screw direction with thumb vertical. Then θ0 , θ1
are identities, while θ2 (which can be written as (s1 )2 using the notation
from Example 11.1) is:
2
•
•
2
•
•
θ2
Example 11.5 There is a tensor category B̃ which is defined similarly
to B, except that the arrows are braids on ribbons (instead of on strings)
and it is permissible to twist the ribbons through full 2π turns (as in the
following diagram).
3
α
3
The homsets B̃(n, n) = B̃n are groups under composition. A presentation of this group B̃n is given by generators s1 , . . . , sn where s1 , . . . , sn−1
satisfy the relations as for Bn . These are depicted by thickened versions of
the diagrams in Example 11.1, along with the extra relation
sn−1 sn sn−1 sn = sn sn−1 sn sn−1
where sn is depicted as follows
1
2
···
n
n−1
n
···
sn
n
Composition in B̃ is vertical stacking of diagrams, and tensor product
for B̃ is horizontal placement of diagrams, much as for B. The braiding
Tensor categories
75
/ n+m for B̃ is obtained by placing the first m ribbons over
c m,n : m+n
/n
the remaining n without introducing any twists. Now the twist θn : n
for B̃ is obtained by regarding the two boundary edges of the ribbons as extra
/ 2n in B. Then in B̃ we have
strings and taking θ2n : 2n
1
2
θ1
θ2
1
2
/ B be a
Example 11.6 Let A and B be abelian groups and f : A × A
bilinear function. There is a balanced strict tensor category Cf constructed
as follows. The objects are the elements of A. The homset Cf (x, y) is
empty unless x = y , in which case Cf (x, x) = B . The tensor product is
given by
β /
α+β
α /
(x
x) ⊗ (y
y) = (x + y
x + y) .
/
The braiding is c x,y = f (x, y) : x + y
y + x and the twist is given by
/
θ = f (x, x) : x
x.
X
Example 11.7 Let A be a braided R-bialgebra with braiding element γ =
i ui ⊗ vi ∈ A⊗R A . A twist element for A is an invertible central element
τ ∈ A such that ε(τ ) = 1 and
(ui τ vj ) ⊗ (vi τ uj ) .
δ(τ ) =
i,j
Diagrammatically the last equation becomes:
R
γ ⊗τ ⊗τ ⊗γ
A⊗6
σ136245
A⊗6
µ⊗1µ⊗1
A⊗4
τ
µ⊗µ
A
δ
A⊗2 .
Twist elements τ for A are in bijection with twists θ for the braided tensor
/ M means it has the form
category ModR (A) . Naturality of θM : M
76
Quantum Groups: A Path to Current Algebra
θM (m) = τ m for some τ ∈ A ; for θM to be an A-module morphism, τ
needs to be central (meaning τ · a = a · τ for all a ∈ A); for θM to be an
isomorphism, τ needs to be invertible; for θM = 1R , the condition ε(τ ) = 1
is needed; and of course the remaining twist conditions correspond.
A balanced bialgebra is a braided bialgebra with a twist.
Exercise 11.1
(a) In a braided tensor category V show that (ignoring the constraints
a, l, r) cA,I = cI,A = 1A and
A⊗C⊗B
c⊗1
C⊗A⊗B
1⊗c
1⊗c
A⊗B⊗C
C⊗B⊗A
c⊗1
c⊗1
B⊗A⊗C
1⊗c
B⊗C⊗A
(b) For a braided bialgebra A and V = ModR (A), interpret the properties
of c in (a) in terms of the braiding element γ ∈ A⊗R A.
(c) Draw diagrams of braids which express the hexagonal diagram of (a)
in the braid category B.
Exercise 11.2 Define the centre Z V of a tensor category V to be the cat/ ⊗A
egory whose objects are pairs (A, a) where A ∈ V and a : A⊗
is a natural isomorphism such that the following conditions hold:
• aI = 1 (more precisely, aI is the composite of the canonical isomorphisms A⊗I ∼
=A∼
= I⊗A).
• aX⊗Y = (1 ⊗ aY ) ◦ (aX ⊗ 1) for all X, Y ∈ V.
/ (B, b) in Z V is an arrow f : A
An arrow f : (A, a)
for all X ∈ V, we have bX ◦ (f ⊗ 1) = (1 ⊗ f ) ◦ aX .
/ B such that,
(a) Show that Z V becomes a tensor category with:
(A, a)⊗(B, b) = A⊗B, (a ⊗ 1) ◦ (1 ⊗ b) .
(b) Show that the tensor category Z V is braided via
c(A,a),(B,b) = aB : (A, a)⊗(B, b)
/ (B, b)⊗(A, a) .
Exercise 11.3 Show that the first dot-point condition on objects of Z V is
redundant.
12
Internal homs and duals
Suppose V is a tensor category with A and B being objects of V. A (left)
internal hom for A and B consists of an object [A , B ] of V together with
an arrow
/B
(called evaluation)
e : [A , B ]⊗A
A
/ B , there exists a unique arrow fˆ :
such that, for all arrows f : C⊗A
/ [A , B ] with
C
f
=
C⊗A
fˆ⊗1A
eA
[A , B ]⊗A
B
.
Thus we have a natural bijection
V(C , [A , B ])
∼
=
V(C⊗A , B)
eA ◦ (g ⊗ 1A ) .
g
A tensor category is called left-closed when each pair of objects has a left
/ D are arrows of V then,
/ A and g : B
internal hom. If f : C
provided the internal homs exist, there is a unique arrow
[f , g ] : [A , B ]
/[C ,D ]
such that
[f,g]⊗1C
[A , B ]⊗C
[ C , D ]⊗C
1[A,B] ⊗f
eC
[A , B ]⊗A
eA
B
g
D.
In this way, when V is left-closed the internal hom becomes a functor
[ , ] : V op × V
77
/ V.
78
Quantum Groups: A Path to Current Algebra
From the universal property, the internal hom for A, B is unique up to
isomorphism. An internal hom for I, A always exists; namely
[I ,A] = A
with
/A .
eA = rA : A⊗I
If B, C and A⊗B, C have internal homs then so do A, [B , C ]; namely,
/[B ,C ] .
[A , [ B , C ] ] = [A⊗B , C ] with eA = êA⊗B : [A⊗B , C ]⊗A
Usually the internal hom functor [ , ] is given a priori; in which case all
we have are canonical isomorphisms
[I ,A] ∼
=A
[A , [ B , C ] ] ∼
= [A⊗B , C ] .
and
There is a composition arrow
/ [A , C ]
[ B , C ]⊗[A , B ]
in V (whenever the internal homs exist) which corresponds, by using the
universal property of [A , C ] , to the composite
[B , C ]⊗[A , B ]⊗A
1⊗eA
[B , C ]⊗B
eB
C .
A right internal hom [A , B ] for A, B comes equipped with an arrow
eA : A⊗[A , B ]
/B
which induces a bijection
V(C , [A , B ] ) ∼
= V(A⊗C , B)
for all objects C. If V is braided, each left internal hom [A , B ] gives a
right internal hom via [A , B ] = [A , B ] and
eA =
A⊗[A , B ]
cA,[A,B]
[A , B ]⊗A
eA
B
.
A tensor category is called closed when all left- and right-internal homs
exist. (In the literature, “closed” is sometimes used for our left-closed,
while “biclosed” is used for our closed.) When the internal homs exist, we
have an arrow
/ [ [A , B ] , B ]
ωA : A
which corresponds to eA : [A , B ]⊗A
ωA : A
corresponding to eA .
/ B via e
[A,B] . Similarly, we have
/ [ [A , B ] , B ]
Internal homs and duals
79
Example 12.1 The symmetric tensor category ModR of modules over a
commutative ring R is closed with internal homs given by
[ M, N ] = [ M, N ] = HomR (M, N ) .
Example 12.2 Suppose H to be an R-Hopf algebra. Then the tensor category ModR (H) of H-modules (with tensor product ⊗R ) is left-closed with
internal hom given by
[ M, N ] = HomR (M, N ) ;
see Proposition 10.1.
Suppose the antipode ν for H is invertible. Using Proposition c(c),
(M, N ) for
H becomes a Hopf algebra having antipode ν −1 . Write HomR
HomR (M, N ) as a left H -module. Clearly, ModR (H) is right-closed with
[ M, N ] = HomR
(M, N ). Therefore ModR (H) becomes closed when ν is
invertible. The forgetful functor
ModR (H)
/ Mod
R
preserves tensor product, and both left and right internal homs.
Example 12.3 Let Ban denote the category of Banach spaces (over the
/ B are linear functions for
complex numbers), where the arrows f : A
which
f (a) ≤ a .
(The analysts in the audience will think these fairly uninteresting functions.) We make Ban into a symmetric tensor category by taking tensor
products as vector spaces, completing in the obvious way. The internal hom
[A , B ] exists for all Banach spaces A, B; it is the Banach space of bounded
linear functions from A to B with the usual norm. (These functions are
of more interest to the analyst.) Thus Ban is a closed symmetric tensor
category.
Suppose V is a tensor category. An object J is called left dualizing when,
for all objects A, internal homs [A, J ], [A, J ] exist, and the arrow
ωA : A
/ [ [A , J ] , J ]
is invertible. It follows that V is left-closed with
[A , B ] = [A⊗[B , J ] , J ] ,
since
V(C , [A⊗[B, J ] , J ] ) ∼
=
∼
=
V(C⊗A⊗[B, J ] , J)
V(C⊗A , [ [B, J ] , J ]) ∼
=
V(C⊗A , B) .
80
Quantum Groups: A Path to Current Algebra
The concept of right dualizing object is defined in the same way, with ω
replaced by ω . A dualizing object is one which is both left and right
dualizing.
Example 12.4 The category of finite-dimensional vector spaces over a
field has a dualizing object, namely the field itself. In this case, the dualizing
object is the unit for tensor product.
Example 12.5 Fix a field k . A quadratic algebra is a pair V , R
where V is a finite-dimensional vector
R is a subspace of V ⊗V .
space
and
/ W , S is a linear function
A quadratic algebra morphism f : V , R
/ W for which
f : V
(f ⊗ f )(R) ⊆ S .
Write
QA for the category of quadratic algebras. Each quadratic algebra
V , R determines an actual algebra
T (V )/R
where T (V ) is the tensor algebra on V (see Example 6.2). The category
QA has a symmetric tensor product given by
V , R ⊗ W , S = V ⊗W , σ1324 (R⊗S) .
The unit object is I = k , k ⊗k .
We claim that J = k , 0 is a dualizing object. It is easy to see that
[ V , R , J ] = V ∗ , R⊥
where V ∗ = Hom k (V, k) and R⊥ is the kernel of the composite surjection
V ∗ ⊗V ∗
with i : R
∼
=
(V ⊗V )∗
i∗
R∗
/ V ⊗V being inclusion. It follows that R is the kernel of
V ⊗V
ω ⊗ω
V ∗∗ ⊗V ∗∗
so we have an isomorphism
ω : V ,R
∼
=
(V ∗ ⊗V ∗ )∗
i∗
R⊥ ∗
/ V ∗∗ , R⊥⊥ .
The quadratic algebra which we identify with the quantum plane A2|0
q
(recall Chapter 3) is
= k2 , (y ⊗ x − q x ⊗ y)
A2|0
q
Internal homs and duals
81
where x = (1, 0), y = (0, 1) ∈ k2 . Let ξ , η ∈ k2 ∗ be the dual basis given by
ξ(x) = η(y) = 1, ξ(y) = η(x) = 0. Then
(y ⊗ x − q x ⊗ y)⊥ = ( ξ ⊗ ξ , η ⊗ η , ξ ⊗ η + q η ⊗ ξ )
as a subspace of k2 ∗⊗ k2 ∗ . Hence the quantum superplane arises as the
quadratic algebra
2∗
A0|2
=
k , ( ξ ⊗ξ , η ⊗ η , ξ ⊗ η + q η ⊗ ξ )
q
[ A2|0
q ,J ] .
=
Notice also that
2|0
A0|2
q ⊗ Aq
=
k2∗⊗ k2 , ( b ⊗ a − q a ⊗ b , d ⊗ c − q c ⊗ d ,
q −1 b ⊗ c − q c ⊗ b + d ⊗ a − a ⊗ d )
(which should be compared with equations (∗∗) in the proof of Theorem 3.2)
where
a = ξ ⊗x , b = ξ ⊗ y , c = η ⊗ x , d = η ⊗y
gives half the relations required for Mq (2).
Let V be a tensor category. Write d , e : B A , or briefly B A , for
objects A, B of V and arrows
/I ,
e : B⊗A
/ A⊗B
d:I
when the following diagrams commute:
A
d⊗1A
1A
B
A⊗B⊗A
1B ⊗d
1A⊗e
B⊗A⊗B
e⊗1B
1B
A
B.
We call B a left dual for A, and we call A a right dual for B. We call e the
counit and d the unit . Duals are uniquely determined up to isomorphism.
The tensor category is called left autonomous when each object A has a left
/ B determines a unique arrow f ∗ : B ∗
/ A∗
dual A∗. Each arrow f : A
given by the composite
B∗
1⊗d
B ∗ ⊗A⊗A∗
1⊗f ⊗1
B ∗ ⊗B⊗A∗
This makes left dual into a functor
( )∗ : V op
/ V.
e⊗1
A∗ .
82
Quantum Groups: A Path to Current Algebra
We also have (A⊗B)∗ ∼
= B ∗ ⊗A∗ and I ∗ ∼
= I. The tensor category is called
autonomous when each object A has both a left dual A∗ and right dual A∨ .
If V is a braided tensor category then each left dual A∗ is also a right
dual with counit and unit, respectively
A⊗A∗
I
cA,A∗
d
A∗⊗A
A⊗A∗
e
−1
cA,A
∗
I
A∗⊗A .
This implies A∗∗ ∼
=A.
A tortile tensor category is an autonomous balanced tensor category in
which the twist is related to the dual via the condition
θA∗ =
θA ∗ : A∗
/ A∗ .
A left dual A∗ for A gives a left internal hom [A , B ] = B⊗A∗ with “evaluation”:
/B
e = 1 ⊗ e : B⊗A∗⊗A
A
B
for all objects B of V. A right dual A∨ for A gives a right internal hom
[A , B ] = A∨ ⊗B. Hence a left/right autonomous tensor category is left/right-closed.
Example 12.6 For each commutative ring R, an object M of ModR has
a (left) dual if and only if it is Cauchy (Theorem 5.2). Write PrfR for the
full subcategory of ModR consisting of Cauchy R-modules. Since PrfR is
closed under tensor, it is an autonomous (symmetric) tensor category.
Example 12.7 Let H be an R-Hopf algebra. An object M of the tensor
category ModR (H) has a left dual precisely when it is Cauchy as an Rmodule; in this case, M ∗ = HomR (M, R) (Proposition 10.1). If H has an
(M, R).
invertible antipode then each such M has a right dual M ∨ = HomR
Write PrfR (H) for the full subcategory of ModR (H) consisting of those
H-modules M which are Cauchy when viewed as R-modules. For H with
invertible antipode, PrfR (H) is an autonomous tensor category.
Example 12.8 Tangles on strings. (This example was discovered by
Freyd–Yetter [FY89].) Let P be a Euclidean plane. A geometric tangle
T is a compact 1-dimensional oriented submanifold of [0, 1] × P which is
tamely embedded and whose boundary ∂T is equal to T ∩∂([0, 1]×P ). Thus
a geometric tangle T is a disjoint union of directed (topological) circles
contained in (0, 1) × P and of directed paths connecting two points on the
boundary ∂([0, 1] × P ). The target of T is the subset ∂T ∩ ({1} × P )
Internal homs and duals
83
as an oriented 0-dimensional manifold. The source of T is the subset
∂T ∩ ({0} × P ) but with orientation reversed. A geometric tangle can be
pictured as follows:
−
−
−
−
+
−
+
+
−
+
A tangle is an isotopy class of geometric tangles where isotopies keep the
boundaries fixed. The source and target of a tangle are regarded as signed
subsets of P . Let 1, 2, 3, . . . denote equally spaced collinear points on P .
Now we can define the autonomous braided tensor category T of tangles.
/ {+, −} for n ≥ 0 , called
The objects are functions A : {1, 2, . . . , n}
signed sets. The arrows of T are the tangles which have these signed sets
as sources and targets. Composition and tensor-product are as for braids.
The braiding is illustrated below.
+
−
+
+
+
−
+
−
+
−
+
+
∗
The left dual A of a signed set A is given by reversing the order and the
sign of the points; that is, A∗(i) and A(n − i + 1) are opposite signs for
1 ≤ i ≤ n . The counit and unit are illustrated below for A = {− + −} .
∗
e : A ⊗A
d:I
+
−
+
−
+
−
−
+
−
+
−
+
/I
/ A ⊗A∗
84
Quantum Groups: A Path to Current Algebra
The next diagram proves one triangle for e and d; the other is similar.
+
−
+
+
−
+
−
+
−
+
−
+
+
−
+
Example 12.9 Tangles on ribbons. (The full details of this example
appear in the thesis of Shum [Shu94].) The category T of tangles on ribbons
is obtained from T just as we obtained B from B in Example 11.5. The
directed strings of tangles are thickened into (directed) ribbons. Ribbons
obtained from strings with boundary may be twisted through complete turns.
Those which are thickenings of closed strings may have twists, as long as
they remain 2-sided 2-manifolds; the Möbius ribbon is not allowed.
Again we obtain an autonomous tensor category. The counits and units
look like this:
+
−
−
+
+
−
−
e
+
−
+
−
+
d
In fact, T is a tortile tensor category. The twist is as for B and the identity
θA∗ = θA∗ can be seen from the following diagram.
=
13
Tensor functors and Yang–Baxter
operators
/V
Suppose C and V are tensor categories. A tensor functor F : C
/ V (denoted by the same symbol) together
consists of a functor F : C
∼
with a natural isomorphism φA,B : FA⊗FB = / F (A⊗B) and another iso∼
morphism φ0 : I = / FI , such that
FA⊗FB⊗FC
φA,B ⊗1
F (A⊗B)⊗FC
1⊗φB,C
φA⊗B⊗C
FA⊗F (B⊗C)
φA,B⊗C
F (A⊗B⊗C)
and
FA
1⊗φ0
1
φ0 ⊗1
FI⊗FA
φI,A
FA⊗FI
φA,I
FA
(where we have suppressed the constraints a, r, l as usual). If the condition
that φ, φ0 be invertible is dropped, we have a weak tensor functor. If in
fact φ and φ0 are identities, then F is called a strict tensor functor. (Weak
tensor functors are also called “monoidal functors” and tensor functors
called “strong monoidal functors”.)
If C and V are braided, we describe a tensor functor as braided when
FA⊗FB
φA,B
F (cA,B )
cFA,FB
FB⊗FA
F (A⊗B)
φB,A
85
F (B⊗A) .
86
Quantum Groups: A Path to Current Algebra
If C and V are symmetric, we say F is symmetric instead of “braided”. If
C and V are balanced, we say F is balanced, when it is braided and
F (θA ) =
/ FA .
θFA : FA
/ V is a weak tensor functor and C and V are leftSuppose F : C
closed tensor categories. Then the composite
F [A , B ]⊗FA
φ[A,B],A
F eA
F ([A , B ]⊗A)
FB
corresponds, using the defining property of [FA , FB ] , to an arrow
/ [FA , FB ] .
φ̃
: F [A , B ]
A,B
We call F a left-closed tensor functor when each φ̃A,B is invertible; rightclosed and closed are now defined in the obvious way. (This differs from
the notion of “closed functor” in the literature.)
When it comes to duals, the situation is better: tensor functors preserve
/ V is a tensor functor and d , e : B A
duals. More precisely, if F : C
in C, then FB FA in V with unit
I
φ0
FI
Fd
F (B⊗A)
−1
φ[A,B],A
FA⊗FB
and counit
FB⊗FA
φ[B,A]
F (B⊗A)
Fe
FI
φ0−1
I .
Hence, if C is (left-) autonomous and V is (left-) closed, then each tensor
/ V is (left-) closed (since [A , B ] = B⊗A∗ in C).
functor C
Example 13.1 The category Set of (small) sets is a tensor category using
cartesian product as tensor product. For each commutative ring R, the “free
module functor” (see Chapter 4)
FR : Set
/ Mod
R
is a tensor functor. It is certainly not closed since we have
HomR (FR X, FR Y ) ∼
YX.
= (FR Y )X ∼
=
/ Set which takes each module to its underlying
The functor | | : ModR
set |M | is a good example of a weak tensor functor: we have functions
φM,N : |M | × |N |
(m , n)
φ0 = η :
which are not invertible.
1
/ |M ⊗N |
m ⊗n
/ |R|
Tensor functors and Yang–Baxter operators
87
Example 13.2 Universal algebra. The “universal enveloping algebra”
/ Alg is a tensor functor (see Proposition 6.10).
functor U : LieR
R
Example 13.3 Yang–Baxter operators. We want to examine what is
/ V from the braid category
involved in giving a tensor functor F : B
into an arbitrary tensor category V.
A Yang–Baxter (YB) operator on an object A of V is an invertible
/ A⊗A such that the following hexagon commutes.
arrow y : A⊗A
A⊗A⊗A
1⊗y
A⊗A⊗A
y ⊗1
y ⊗1
A⊗A⊗A
A⊗A⊗A
1⊗y
1⊗y
A⊗A⊗A
y ⊗1
A⊗A⊗A
For example, the object 1 of B admits the following YB-operator:
c1,1 = s1 : 1 + 1
/ 1+1
which is the element of the braid group B2 depicted by the next diagram.
s1 :
The YB-hexagon becomes the following simple identity.
=
/ V preserves tensor products “up to
Since any tensor functor F : B
coherent isomorphism”, we obtain a YB-operator y on F (1) = A ; namely,
φ
y : A⊗A
1,1
F (1 + 1)
F s1
φ−1
F (1 + 1)
1,1
A⊗A .
Conversely, given a YB-operator y on an object A of V, we can determine
/ V such that F (1) = A and y is the above coma tensor functor F : B
posite. In fact, F is unique up to isomorphism (arising from the different
88
Quantum Groups: A Path to Current Algebra
possible choices for the n-fold tensor product A⊗n ). Since F is to be a
tensor functor, we are forced to have
F (n) = F (1 + 1 + · · · + 1) ∼
= A⊗n .
Each generator si of Bn can be written as si = 1i−1 ⊗ s1 ⊗ 1n−i−1 in B , so
/ A⊗n is forced. We just need to check that
the definition of F si : A⊗n
this is compatible with the braid relations (Example 11.1); but this follows
from the YB-hexagon and the functoriality of tensor product. Details are
left as an exercise (which is worth doing).
Hence, up to the appropriate notion of isomorphism, tensor functors
/ V correspond to pairs (A, y) consisting of an object A of V and
F :B
a YB-operator y on A. We can express this by saying:
B , 1 , s1 is the free tensor category having an object equipped
with a YB-operator.
/ V also deterExample 13.4 A tensor functor from ribbons F : B
mines a YB-operator y on F 1 = A as in Example 13.3 (compare with
/ 1 in B gives an isomorphism
[Tur88]). This time the twist θ1 : 1
/
z = F θ1 : A
A.
Due to the equalities in B
,
=
=
this gives an example of the following concept.
A YB-operator y on an object A of V is said to be balanced when it is
/ A such that these commute:
equipped with an isomorphism z : A
A⊗A
y
1⊗z
A⊗A
A⊗A
z ⊗1
y
A⊗A
A⊗A
,
y
z ⊗1
A⊗A
A⊗A
1⊗z
y
A⊗A .
Each balanced YB-operator determines a unique (once the n-fold tensors
/ V from which A, y, z are recovA⊗n are chosen) tensor functor F : B
ered as above. So we have that
B , 1 , c1,1 , θ1 is the free tensor category containing an object
equipped with a balanced YB-operator.
Tensor functors and Yang–Baxter operators
89
The easier part of Example 13.3 can be obtained from two observations.
• Tensor functors take YB-operators into YB-operators.
/ V is a tensor functor and if y is a YBMore precisely, if F : C
/ F (X⊗X) carries across
operator on X in C, then F y : F (X⊗X)
the isomorphism φX,X to a YB-operator y on F X in V.
• A braiding on a tensor category gives, on each object X, a YB/ X⊗X .
operator: cX,X : X⊗X
Moreover, tensor functors take balanced YB-operators into balanced YBoperators. Furthermore, in a balanced tensor category there is a balanced
YB-operator (cX,X , θX ) on each object X. (See Example 13.4 .)
We now look at compatibility of YB-operators with duals.
A YB-operator y on A is called (left-) dualizable when A has a left dual
/ A∗ given by the composites:
A∗ and both the arrows u , v : A∗⊗A
1⊗1⊗d
A∗⊗A
A∗⊗A⊗A⊗A∗
1⊗y ⊗1
1⊗y
−1
A∗⊗A⊗A⊗A∗
⊗1
e⊗1⊗1
A⊗A∗
e⊗1⊗1
A∗⊗A∗
are invertible. It follows that the composite w given by
1⊗1⊗d
A∗⊗A∗
A∗⊗A∗⊗A⊗A∗
1⊗u⊗1
A∗⊗A⊗A∗⊗A∗
has inverse w−1 given by the composite:
A∗⊗A∗
1⊗1⊗d
A∗⊗A∗⊗A⊗A∗
1⊗v ⊗1
A∗⊗A⊗A∗⊗A∗
e⊗1⊗1
A∗⊗A∗ .
Proposition 13.5 In a braided tensor category, if an object A has a dual,
then y = cA,A is a dualizable YB-operator on A with
−1
u = cA,A
∗ ,
−1
v = cA
∗,A ,
−1
w = cA
∗,A∗ .
Proof. To prove cA,A∗ ◦ u = 1A∗⊗A , it suffices (by the property of duals) to
show that equality holds after applying A⊗ to both sides and composing
with d ⊗ 1A . Thus the following diagram gives the first equation.
A⊗A∗⊗A⊗A⊗A∗
1⊗1⊗1⊗d
A⊗A∗⊗A
d⊗1
A
1⊗1⊗cA,A ⊗1
d⊗1⊗1⊗1
A⊗A⊗A∗
A⊗A∗⊗A⊗A⊗A∗
d⊗1⊗1⊗1
cA,A ⊗1
A⊗A⊗A∗
cA,A⊗A∗
1⊗d
cA,I = 1
A
d ⊗1
1⊗e⊗1⊗1
1
A⊗A⊗A∗
1⊗cA,A∗
A⊗A∗⊗A
90
Quantum Groups: A Path to Current Algebra
−1
The result for v follows by using the braiding cB,A
in place of cA,B . For w,
consider A∗ in place of A.
/ V preserve dualizability. So if C is braided and
Tensor functors F : C
X has a dual in V, we obtain a dualizable YB-operator on F X in V.
A YB-operator on A is called tortile when it is balanced, dualizable, and
y ⊗1
A⊗A⊗A∗
1⊗v -1
A⊗A⊗A∗
A⊗A∗⊗A
1⊗d
z
A
1⊗e
z
A
A.
Proposition 13.6 In a balanced tensor category, if an object A has a dual
then the pair (cA,A , θA ) is a tortile YB-operator precisely when θA∗ = (θA )∗ .
Proof. The following diagram proves the equation
(θA∗ )∗ θA = (1 ⊗ e) ◦ (1 ⊗ v −1 ) ◦ (y ⊗ 1) ◦ (1 ⊗ d) .
θA
A
1⊗d
A
A⊗A⊗A∗
1⊗cA−1
∗,A
A⊗A∗⊗A
θA⊗1⊗1
1⊗d
1⊗cA−1
∗,A
A⊗A⊗A∗
A⊗A∗⊗A
cA−1
∗,A
θA⊗θA∗⊗1
1⊗θA∗⊗1
A⊗A∗⊗A
−1
cA,A
∗⊗1
cA−1
∗,A⊗1
A∗⊗A⊗A
y ⊗1
= cA,A⊗1
A∗⊗A⊗A
θA∗A⊗1
e⊗1
1⊗cA,A
e⊗1
A∗⊗A⊗A
A
cA∗A,A
cA−1
∗,AA
θI = I
A
cI,A = 1
1
cA∗,A⊗1
A⊗A⊗A∗
1⊗v −1 = 1⊗cA−1
∗,A
A⊗A∗⊗A
1⊗e
A
So the balanced YB-operator (y, z) is tortile if and only if (θA∗ )∗ θA = z 2 ;
that is, if and only if (θA∗ )∗ θA = (θA )2 ⇔ (θA∗ )∗ = θA ⇔ θA∗ = (θA )∗ .
Tensor functors and Yang–Baxter operators
91
It follows that, in a tortile tensor category, each object A is equipped with
a tortile YB-operator (cA,A , θA ).
Example 13.7 Since T is a tortile tensor category, we obtain a tortile
YB-operator (c+,+ , θ+ ) on the object + of T . Thus, each tensor functor
/ V yields a tortile YB-operator on F (+) in V.
F : T
In fact, T , + , c+,+ , θ+ is the free tensor category equipped with a
tortile YB-operator ( [Shu94] together with [JS91c]). This means that, given
a tortile YB-operator (y, z) on an object A in a tensor category V, there
/ V which
exists a (unique up
tensor functor F : T
to isomorphism)
takes + , c+,+ , θ+ to A , y , z . We do not intend to prove this here; after
all, our geometric description of T was incomplete. We hope the result
is believable. All we really need is that such a free T should exist; but the
description of the realistic model is too pretty to omit.
It should be clear how to define F in terms of A, y, z. For example,
F (+ − − − + −)
F (c+,− )
=
=
A⊗A∗⊗A∗⊗A∗⊗A⊗A∗
u−1
F (c−,− )
F (θ+ )
=
=
w
z
F (θ− )
=
z∗
and so on. Any tangle of ribbons can be decomposed, using composition
and tensor product in T , into single crossings (c+,+ , c+,− , c−,+ , c−,− , and
their inverses), turnings ( e and d) and twists ( θ+ , θ− , and their inverses).
So the value of F on the tangle is forced. The hard part, which we shall
not include in these notes, is to show that this value is independent of the
decomposition.
It is instructive to see in this example what is meant by the equation
z 2 = (1 ⊗ e) ◦ (1 ⊗ v −1 ) ◦ (y ⊗ 1) ◦ (1 ⊗ d). It is expressed by the following
diagram (which can be tested by taking off your belt).
=
92
Bradshaw: “Barred Variation”, Sash Bradshaw Group, [Wal94, Plate 58].
14
A tortile Yang–Baxter operator for
each finite-dimensional vector space
Let k be a field and let q ∈ k be a fixed non-zero element. Let V be a
vector space over k with basis ε1 , ε2 , . . . , εn . Define a linear function
y : V ⊗V
/ V ⊗V
on the basis elements εi⊗ εj of V ⊗V by


 εj ⊗ εi
y(εi⊗ εj ) =
εj ⊗ εi + (q − q −1 ) εi⊗ εj


q εi⊗ εi
for i > j
for i < j
for i = j .
In order to check the YB-hexagon for y, we look at (y ⊗ 1)(1 ⊗ y)(y ⊗ 1),
(1 ⊗ y)(y ⊗ 1)(1 ⊗ y) at each εi ⊗ εj ⊗ εk . There are thirteen of these cases to
check to account for all possible relative positions of i, j and k. We shall
only give three of these cases as an illustration: put ρ = q − q −1 and omit
the ε and ⊗ symbols from the notation.
i < j < k : (ijk)
y ⊗1
(jik) + ρ (ijk)
1⊗y
(jki) + ρ (jik) + ρ (ikj) + ρ2 (ijk)
y ⊗1
(kji) + ρ (jki) + ρ (ijk) + ρ (kij) + ρ2 (ikj)
+ ρ2 (jik) + ρ3 (ijk)
(ijk)
1⊗y
(ikj) + ρ (ijk)
y ⊗1
(kij) + ρ (ikj) + ρ (jik) + ρ2 (ijk)
1⊗y
(kji) + ρ (kij) + ρ (ijk) + ρ (jki) + ρ2 (jik)
+ ρ2 (ijk) + ρ3 (ijk) ,
93
94
Quantum Groups: A Path to Current Algebra
j < i < k : (ijk)
(ijk)
y ⊗1
1⊗y
(jki) + ρ (jik)
y ⊗1
(kji) + ρ (jki) + ρ (ijk) + ρ2 (jik)
1⊗y
(ikj) + ρ (ijk)
y ⊗1
1⊗y
i = j < k : (iik)
(iik)
(jik)
y ⊗1
(kij) + ρ (ikj) + ρ (jik)
(kji) + ρ (ijk) + ρ (jki) + ρ2 (jik) ,
q (iik)
1⊗y
q (iki) + ρq (iik)
y ⊗1
q (kii) + ρq (iki) + ρq 2 (iik)
1⊗y
(iki) + ρ (iik)
y ⊗1
1⊗y
(kii) + ρ (iki) + ρq (iik)
q (kii) + ρ (iik) + ρq (iki) + ρ2q (iik) .
(Note that q 2 ρ = ρ + q ρ2 since ρ = q − q −1 .)
Clearly y is invertible with inverse given by:


 εj ⊗ εi
−1
y (εi⊗ εj ) =
εj ⊗ εi + (q −1 − q) εi⊗ εj

 −1
q εi⊗ εi
for i < j
for i > j
for i = j .
Hence, y is a YB-operator on the object V of Modk .
It is now possible to calculate the operators u, v, w and their inverses
(see the definition of dualizable YB-operator in Chapter 13). For this, let
ε1∗ , . . . , εn∗ ∈ V ∗ be the dual basis for ε1 , . . . , εn ∈ V ; this means
εi∗ (εj ) = δij .
Tortile Yang–Baxter operator for vector spaces
Recall that e : V ∗ ⊗V
Now we obtain
u(εi∗⊗ εj ) =
=
/ k is the evaluation functor and d = ε ⊗ ε∗ .
k
k k
(e ⊗ 1 ⊗ 1)(1 ⊗ y ⊗ 1)(1 ⊗ 1 ⊗ d)(εi∗⊗ εj )
(e ⊗ 1 ⊗ 1)(1 ⊗ y ⊗ 1)(εi∗⊗ εj ⊗ εk⊗ εk∗ )
k
=
95
(e ⊗ 1 ⊗ 1) εi∗⊗ εk⊗ εj ⊗ εk∗ + (q − q −1 ) εi∗⊗ εj ⊗ εk⊗ εk∗
k>j
+ q (e ⊗ 1 ⊗ 1) (εi∗⊗ εj ⊗ εj ⊗ εj∗ ) +
(e ⊗ 1 ⊗ 1) (εi∗⊗ εk ⊗ εj ⊗ εk∗ )
k<j
=
δik εj ⊗ εk∗ + (q − q −1 ) δij εk ⊗ εk∗ + q δij (εj ⊗ εj∗ )
k>j
+
δik εj ⊗ εk∗
k<j
=

∗

 εj ⊗ εi
εj ⊗ εi∗

 −1
q εi⊗ εi∗ +
(q − q −1 ) εk ⊗ εk∗
for j < i
for i < j
for i = j .
k>i
The other operators are calculated similarly. We record the results below:
for i = j
εj ⊗ εi∗
∗⊗
u(εi εj )
=
−1
∗
∗
q εi⊗ εi +
(q − q ) εk ⊗ εk
for i = j
k>i
for i = j
εj∗⊗ εi
u−1 (εi⊗ εj∗ ) =
−1
(q − q)q −2(k−i) εk∗⊗ εk for i = j
q −1 εi∗⊗ εi +
k>i
∗
⊗
ε
for i = j
ε
i
j
=
v(εi∗⊗ εj )
−1
−1
∗
∗
q εi⊗ εi +
(q − q) εk⊗ εk
for i = j
k<i
for i = j
εj∗⊗ εi
v −1 (εi⊗ εj∗ ) =
−1
∗⊗
2(i−k) ∗⊗
(q − q )q
εk εk
for i = j
q εi εi +
k<i
 ∗ ∗
for i < j
 εj ⊗ εi
w(εi∗⊗ εj∗ ) =
εj∗⊗ εi∗ + (q − q −1 ) εi∗⊗ εj∗
for i > j
 ∗ ∗
q εi ⊗ εi
for i = j
 ∗ ∗
for i > j
 εj ⊗ εi
εj∗⊗ εi∗ + (q −1 − q) εi∗⊗ εj∗
for i < j
w−1 (εi∗⊗ εj∗ ) =
 −1 ∗ ∗
q εi ⊗ εi
for i = j .
96
Quantum Groups: A Path to Current Algebra
Hence y is dualizable. It enriches to a balanced YB-operator on defining
/ V simply to be the homothety
z: V
z(x) = q n x .
Proposition 14.1 The YB-operator (y, z) defined above is tortile.
Proof. First observe that, for i = j , the value of e v −1 at εi ⊗ εj∗ is 0 ; while
for i = j the value is
q +
(q − q −1 )q 2(i−k)
k<i
= q + (q − q −1 ) q 2(i−1) + q 2(i−2) + · · · + q 2(i−(i−1))
(q 2 )i−1 − 1
q2 − 1
2 i−1
− q = q 2i−1 .
= q + q(q )
= q + q(q 2 − 1)
Hence, we have the following remarkable calculation:
(1 ⊗ e)(1 ⊗ v −1 )(y ⊗ 1)(1 ⊗ d)(εi )
(1 ⊗ (e v −1 ))(y ⊗ 1)(εi ⊗ εj ⊗ εj∗ )
=
j
=
j<i
+
(1 ⊗ (e v −1 ))(εj ⊗ εi⊗ εj∗ ) + (1 ⊗ (e v −1 )) q (εi⊗ εi⊗ εi∗ )
(1 ⊗ (e v −1 )) (εj ⊗ εi⊗ εj∗ ) + (q − q −1 ) (εi ⊗ εj ⊗ εj∗ )
i<j
= 0 + q q 2i−1 εi + 0 +
(q − q −1 )q 2j−1 εi
i<j
=
=
=
=
2i
q + (q − q −1 )(q 2i+1 + q 2i+3 + · · · + q 2n−1 ) εi
2i
q + (q 2 − 1) q 2i (1 + q 2 + · · · + q 2(n−i−1) ) εi
2 n−i
−1
2i
2
2i (q )
q + (q − 1) q
εi
q2 − 1
2i
q + q 2i (q 2n−2i − 1) εi
= q 2n εi
= z z(εi ) .
15
Monoids in tensor categories
A monoid in a tensor category V consists of an object A and arrows
µ : A⊗A
/A
and
/A
η:I
which satisfy the usual identity and associativity conditions (see Chapters
/ B is an arrow in V which preserves
1 and 6). A monoid arrow f : A
µ and η , in the diagrammatically expressed sense.
It is also useful to consider “arrows between monoid arrows”. Suppose
/ B are monoid arrows. A 2-cell
that f , g : A
ξ : f ⇒g :A
is defined to be an arrow ξ : I
/ B in V such that
ξ ⊗f
A
g ⊗ξ
/B
B ⊗B
µ
B.
The more 2-dimensional notation
f
A
ξ
B
g
is also used. (In the case where V = Set , such a 2-cell amounts to an
element ξ ∈ B for which ξf (a) = g(a)ξ for all a ∈ A .)
There are two basic pasting operations for 2-cells. Given the situation
f
A
h
A
ξ
B
k
B
g
where the arrows are monoid arrows and ξ is a 2-cell, there is a 2-cell
kf h
A
kξ
kgh
97
B
98
Quantum Groups: A Path to Current Algebra
obtained since we have the following factorization:
ξ ⊗(f h)
ξ ⊗f
h
A
A
µ
B ⊗B
g ⊗ξ
(gh)⊗ξ
B
k⊗ k
k
µ
B ⊗B
B
This is called whiskering ξ by h and k .
The other basic pasting operation is vertical composition, which takes a
pair of 2-cells ξ and ζ , as in the following situation
f
ξ
A
B
g
ζ
h
to a 2-cell
f
A
B
ζ∗ξ
h
where ζ ∗ ξ =
I
ζ ⊗ξ
B ⊗B
B .
µ
The identity 2-cell η : f ⇒ f : A
vertical composition.
/ B is an identity for the operation of
When we write a diagram such as
f2
f1
A
f3
B
ξ
g1
g4
g2
g3
/ B . This allows us to
it is intended that ξ : f3 f2 f1 ⇒ g4 g3 g2 g1 : A
define a more general pasting operation, which assigns to a diagram like
Monoids in tensor categories
99
f3
f2
f4
f1
f5
f7
f8
f6
A
B
g4
g3
g1
g2
/ B , obtained (quite
a 2-cell f8 f7 f6 f5 f4 f3 f2 f1 ⇒ g4 f6 g3 g2 g1 : A
possibly in several different ways) by first whiskering the 2-cells in the
diagram to be of the form
B
A
as well as being vertically composable, and then composing vertically.
As an example of this pasting, consider the diagram
f
A
ξ
g
l
h
ζ
B
k
First whisker ξ and ζ appropriately, as in
f
A
gl
ξ
g
and
B
h
A
lh
ζ
k
to obtain two vertically composable 2-cells
gf
glk
A
gξ
B.
ζ
kh
Then vertically compose to obtain a 2-cell
gf
A
B
kh
B
100
Quantum Groups: A Path to Current Algebra
which is called the pasted composite of the original diagram. In this case
there was only one way of performing the pasting.
As another example, consider the diagram
f
A
h
ξ
ζ
g
B.
k
On the one hand we can whisker ξ and ζ as in
f
A
h
ξ
h
B
,
A
g
ζ
g
B
k
and then vertically compose; while on the other hand we can whisker ξ and
ζ as in
f
h
A
f
ζ
B
,
A
ξ
k
B
g
k
and then vertically compose. The reader should verify that the resultant
2-cells of the form
hf
A
B
kg
are actually equal.
It is a general fact that the result of pasting is independent of the way
it is broken down into basic pasting operations. In fact, all ambiguities in
the method can be traced back to instances of the last example. For the
particular diagrams we shall use here, it is easily shown that they have a
uniquely determined pasted composite.
Write Mon(V) to denote the category of monoids in V ; the arrows are
monoid arrows. With the extra structure of 2-cells, Mon(V) is an example
of a “2-category” (see [KS74]).
(For a commutative ring R , we have that Mon(ModR ) = AlgR and also
Mon(ModRop ) = CogRop where ModR and ModRop have the same tensor
product ⊗R . )
Monoids in tensor categories
101
• Weak tensor functors take monoids to monoids.
/ W is a weak tensor functor, each monoid A
More precisely, if F : V
in V gives a monoid F (A) in W with multiplication
φA,A
F (A)⊗F (A)
F (µ)
F (A⊗A)
F (A)
and unit
φ0
I
F (I)
F (η)
F (A) .
In fact, we obtain a functor
Mon(F ) : Mon(V)
/ Mon(W)
which preserves the basic pasting operations of 2-cells (and so is an example
of a “2-functor”).
For each monoid A in V there is a category, called ModV (A) , of (left)
A-modules. An A-module consists of an object M of V and an arrow
/ M , called the action, which satisfies all the usual defining
µ : A⊗M
/N
diagrams for a module (see Chapter 9). An A-module arrow µ : M
is an arrow in V such that
A⊗M
1⊗u
A⊗N
µ
µ
u
M
N .
/ V which simply forgets
There is a “forgetful” functor UA : Mod V (A)
the module action.
/ B determines a functor
Each monoid arrow f : A
Mod(f ) : Mod V (B)
/ Mod (A)
V
given by “restriction of scalars” along f . That is, for a B-module M , we
take Mod(f )(M ) to be M with A-action
A⊗M
f ⊗1
µ
B⊗M
M .
Each B-module arrow becomes an A-module arrow, thereby giving the
following commutative triangle of categories and functors:
Mod V (B)
UB
Mod(f )
V
Mod V (A)
UA
102
Quantum Groups: A Path to Current Algebra
/ B between monoid arrows
Furthermore, each 2-cell ξ : f ⇒ g : A
f and g in V determines a natural transformation
Mod(ξ) : Mod(f )
/ Mod(g)
whose component at the B-module M is the composite
ξ ⊗1
M
B⊗M
µ
M
which is an A-module arrow, as can be seen from the diagram
A⊗M
1⊗ξ ⊗1
A⊗B ⊗ M
f ⊗1
1⊗µ
A⊗M
g ⊗1
g⊗1⊗1
A⊗M
ξ ⊗1⊗1
µ
B⊗B⊗M
1⊗µ
ξ ⊗1
M
1⊗µ
B⊗M
µ
µ⊗1
µ
B⊗M
M
Naturality follows from the following diagram involving a B-module arrow
/N.
u: M
M
ξ ⊗1
u
N
B⊗M
µ
M
u
1⊗u
ξ ⊗1
B⊗N
µ
N
The assignment ξ / Mod(ξ) turns the two basic pasting operations into
corresponding familiar operations on natural transformations. This gives
an example of a “2-functor”
ModV : Mon(V)op
/ Cat
where Cat is some appropriate “2-category” of categories.
For our purposes, it is important to remember the forgetful functors
/ V . So, rather than Cat , we consider Cat/V, whose
UA : Mon V (A)
/ V, whose arrows
objects are functors F : C
/D ,G
T : C ,F
/ D such that G T = F , and for which the 2-cells α :
are functors T : C
/
T are arbitrary natural transformations from T to T (that is, no
T
Monoids in tensor categories
103
condition relating it to F and G ). Observe that ModV really lands in
Cat/V by taking A ∈ Mon(V)op to ModV (A) , UA . So we have that
/ Cat/V
ModV : Mon(V)op
is a 2-functor.
There is an obvious candidate for a tensor product on Cat/V, namely
C ,F ⊗ D,G = C × D, C × D
F ×G
V×V
⊗
V
.
This tensor product respects all of the pasting operations for natural transformations. Ignoring
the 2-cells,
Cat/V becomes a tensor category with
/ V .
unit given by 1 , I : 1
Suppose now that V is braided. For monoids A and B in V, we enrich
A⊗B with the multiplication
1⊗cB,A ⊗1
A⊗B⊗A⊗B
µ⊗µ
A⊗A⊗B⊗B
A⊗B
/ A⊗B ; the braiding properties imply (Exercise 15.1)
and unit η ⊗ η : I
that this makes A⊗B into a monoid. Thus Mon(V) becomes a tensor
category such that the forgetful functor
Mon(V)
/V
is a strict tensor functor. The tensor product on Mon(V) respects the
basic pasting operations of 2-cells; see Exercise 15.2.
We shall now see that
ModV : Mon(V)op
/ Cat/V
is essentially a weak tensor functor. For this, observe that ModV(I) = V
and we have arrows
I
1
V
1V
I
Φ
ModV(A) × ModV(B)
ModV(A⊗B)
UA ⊗UB
V
V×V
UA⊗B
⊗
V
in Cat/V , where Φ(M , N ) = M ⊗N , with action
A⊗B⊗M ⊗N
1⊗cB,M ⊗1
A⊗M ⊗B⊗N
µ⊗µ
M ⊗N .
104
Quantum Groups: A Path to Current Algebra
The reason for the word “essentially” is that the axioms for a weak tensor
functor (see the beginning of Chapter 12) hold only up to isomorphism
(instead of equality); in fact, the isomorphisms are precisely provided by
the associativity and unit constraints a, r, l for the tensor product.
Just as weak tensor functors take monoids to monoids, the 2-functor
ModV takes tensor objects in Mon(V)op to tensor objects in Cat/V .
• A tensor object is “essentially” a monoid.
• A tensor object in Cat is precisely a tensor category; whereas a monoid
in Cat is a strict tensor category. • A tensor object in Cat/V is a pair C , F consisting of a tensor category
/ V.
C and a strict tensor functor F : C
• A tensor object in Mon(V)op will be called a quasi-bimonoid in V . This
/ A⊗A , : A
/I,
consists of a monoid A in V , monoid arrows δ : A
and 2-cells:
δ
A
A
A⊗2
1A
δ
1A
δ
δ ⊗1
α
ρ
λ
⊗2
A
⊗3
A
A
1⊗δ
⊗1
A⊗2
A
1⊗
which are invertible under vertical composition and satisfy the following
equalities between pasted diagrams.
δ
A
A⊗2
δ
A
δ
A
⊗3
A
1⊗α
1⊗δ
1⊗δ ⊗1
A
A
δ
δ
A⊗2
1⊗δ
δ ⊗1
A⊗3
A⊗2
A
δ
δ ⊗1
α
δ ⊗1⊗1
A⊗3
=
δ
A⊗2
1
A⊗2
1⊗1⊗δ
δ ⊗1
ρ⊗1
A⊗3
1⊗λ
1
1⊗⊗1
A⊗2
A⊗3
1⊗δ
A⊗4
1⊗1⊗δ
1⊗δ
α
A⊗2
δ ⊗1⊗1
⊗3
A⊗2
δ
=
α⊗1
1⊗δ
⊗2
δ ⊗1
α
A⊗3
δ ⊗1
α
A⊗2
δ
δ ⊗1
1
1⊗⊗1
A⊗2
A⊗4
Monoids in tensor categories
105
A bimonoid in V is a strict quasi-bimonoid; that is, one for which the
2-cells α, λ, ρ are actually identity 2-cells. (A bimonoid in ModR is an
R-bialgebra; see Proposition 7.5.)
Hence we have that a (quasi-) bimonoid A in V determines the structure of a tensor category on ModV(A) as well as a tensor functor structure
/ V (see Chapter 10 for those cases when V = Mod
on UA : ModV(A)
R
op
and when V = ModR ).
A (quasi-) Hopf monoid in V is a (quasi-) bimonoid H together with
/ H , called the antipode, such that
an arrow ν : H
H
δ
ν ⊗1
H⊗H
1⊗ν
H⊗H
µ
H
η
I
Drinfeld [Dri89] obtained interesting examples of quasi-Hopf monoids in
Modk .
Proposition 15.1 Suppose V is a braided tensor category and H is a
Hopf monoid in V . If M is an H-module which has a left dual M ∗ as
an object of V then M ∗ becomes the left dual of M in ModV(H) via the
action
H⊗M ∗
ν ⊗1
H⊗M ∗
cH,M ∗
1⊗µ⊗1
M ∗ ⊗H
1⊗d
M ∗ ⊗M ⊗M ∗
M ∗ ⊗H⊗M ⊗M ∗
e⊗1
M∗ .
Proof. This is a matter of proving that e and d are module arrows. For
this, compare with Propositions 10.1 and 10.5.
A braiding for a (quasi-) bimonoid A in V is a 2-cell
A
δ
δ
γ
A⊗A
cA,A
A⊗A
which is invertible (with respect to vertical composition) and satisfies the
two equalities below.
106
Quantum Groups: A Path to Current Algebra
A
δ
δ
A⊗A
δ
A
δ
δ
A⊗A
A⊗A
α
γ
α
α
A⊗A
1⊗δ
cA,A
δ ⊗1
A⊗A
δ ⊗1
1⊗δ
=
A⊗A
δ
δ ⊗1
1⊗γ
1⊗δ
A⊗A⊗A
γ ⊗1
δ ⊗1
1⊗cA,A
1⊗δ
cA,A⊗1
A⊗A⊗A
A⊗A⊗A
cA⊗A,A
A
δ
δ
A⊗A
A⊗A⊗A
δ
A
δ
δ
A⊗A
A⊗A
α-1
γ
α-1
α-1
A⊗A
δ ⊗1
cA,A
1⊗δ
δ
A⊗A
1⊗δ
δ ⊗1
=
A⊗A
A⊗A⊗A
cA⊗A,A
δ ⊗1
1⊗δ
γ ⊗1
A⊗A⊗A
1⊗γ
1⊗δ
cA,A⊗1
δ ⊗1
1⊗cA,A
A⊗A⊗A
A⊗A⊗A
cA⊗A,A
A⊗A⊗A
cA,A⊗A
A⊗A⊗A
Now suppose V is symmetric. A twist for A is a 2-cell
1A
A
A
τ
1A
which is invertible, with respect to ∗ , and satisfies
δ
A
1
A
τ
A
δ
cA,A
1
A⊗A =
δ
1
A⊗A
γ
δ
γ
A⊗A
τ ⊗τ
A⊗A
1
cA,A
1
A⊗A
1
A
τ
1
A
I = A
1
I .
A (quasi-) bimonoid with a braiding and twist is called balanced.
Monoids in tensor categories
107
The reader should interpret the above pasting diagrams in the special case
where V = Modk to see that these definitions agree with the definitions
of braiding element and twist for bialgebras as in Examples 11.3 and 11.7 .
We now have a conceptual version of the calculations in those examples.
Proposition 15.2 Suppose V is a symmetric tensor category and A is
a bimonoid in V. There is a bijection between braidings γ for A and
braidings c for ModV(A) determined by cM,N as the composite:
M ⊗N
γ ⊗1⊗1
1⊗cA⊗M,N
A⊗A⊗M ⊗N
A⊗N ⊗A⊗M
µ⊗µ
N ⊗M .
There is a bijection between twists τ for A and twists θ for ModV(A)
determined by
θM =
τ ⊗1
M
µ
A⊗M
M
.
Proof. Apply the 2-functor ModV to the triangle containing γ , and paste
below it a square containing a natural isomorphism whose components are
the symmetry of V (we omit the subscripts V on maps in the diagram):
ModV(A)
Mod(δ)
Mod(δ)
Mod(γ)
ModV(A⊗A)
ModV(A⊗A)
Mod(cA,A )
Φ
Φ
c
ModV(A) × ModV(A)
σ
ModV(A) × ModV(A)
The result is a natural isomorphism, whose component at the pair (M , N ) ∈
ModV(A) × ModV(A) is the cM,N as stated in the proposition. The axioms
on γ convert to the braiding axioms for c .
Conversely, to recapture γ from c , take the composite
γ =
I
η ⊗η
A⊗A
cA,A
A⊗A
,
where we regard A as an object of ModV(A) with action µ . The proof of
the twist bijection is similar.
108
Quantum Groups: A Path to Current Algebra
A tortile bimonoid in V is a balanced Hopf monoid H such that
I
τ
H
τ
ν
H .
Proposition 15.3 Suppose that V is a symmetric autonomous tensor category. If H is a tortile bimonoid in V then ModV(H) is a tortile tensor
category.
Proof. By Propositions 15.1 and 15.2, ModV(H) becomes an autonomous
balanced tensor category. All that remains to see is that θM ∗ = (θM )∗ ,
which follows from ν ◦ τ = τ .
Consider the replacement of the tensor category V by its opposite tensor category V op . Monoids become comonoids, but bimonoids and Hopf
/ I on
monoids are unchanged. For a bimonoid H in V , a cotwist τ : H
op
/I
H in V is defined to be a twist on H in V ; a cobraiding γ : H⊗H
op
on H in V is defined to be a braiding on H in V .
Thus we have the corresponding notions of cobraided, cobalanced and
cotortile bimonoid in V .
Our proposal for a definition of a quantum group over R is that it should be
a cotortile bimonoid in ModR . In Chapter 18 we shall see that our main
example, the quantum general linear group, does indeed give an instance
of this concept.
Exercise 15.1 Check that the tensor product of monoids in a braided tensor category is a monoid.
Exercise 15.2 Show that the natural tensor product on Mon(V) preserves
whiskering and vertical composition of 2-cells in each variable.
Exercise 15.3 Verify that braidings and twists in Modk , in the sense of
this chapter, include those of Examples 11.3 and 11.7.
16
Tannaka duality
Given a compact group G, the set Rep G of isomorphism classes of appropriate representations admits various operations; for example direct sum
and tensor product. Tannaka’s duality theorem (1939) provided a recipe
for recovering a compact group Gp R from a structure R such as Rep G
whereby Gp Rep G ∼
= G.
For algebraic groups, Saavedra Rivano [SR72] considered the category
of appropriate representations together with the tensor structure and the
underlying functor into vector spaces. He gave criteria on a tensor functor
into vector spaces under which it should be equivalent to such an underlying functor. A non-commutative generalization of this was given by Ulbrich [Ulb89]. We shall lead into this Hopf algebra version by examining
the 2-functor Mod V of the previous chapter.
For simplicity of exposition we suppose our tensor category V is strict.
This loses no generality in fact since every tensor category is equivalent to
a strict one (Mac Lane’s coherence theorem). We also suppose that V is
symmetric (but we cannot suppose the tensor product is strictly commutative). A consequence of this simplification is that we really do have a weak
tensor functor
/ Cat/V .
Mod V : Monop
V
op
We are now interested in going back from Cat/V to Mon(V) . A possible
way to do this is via a left adjoint to ModV , if it exists. Under reasonable
conditions, a left adjoint EF ∈ Mon(V) does exist at an object C , F of
Cat/V. It is constructed as follows.
If each internal hom [ FX , FX ] exists in V and if V is suitably complete, we put
[ FX , FX ]
EF =
X∈C
where the integral sign denotes the end (see Mac Lane [Mac71]) of the
/ V taking (X , Y ) to [ FX, F Y ]; it is the equalizer of
functor C op × C
the obvious pair of arrows (see Exercise 16.1)
[ FX , F Y ] .
[ FX , FX ]
f :X→Y
X
109
110
Quantum Groups: A Path to Current Algebra
There are projection arrows
/ [ FX , FX ]
πX : EF
for each object X ∈ C. These correspond, using the definition of internal
hom, to arrows
/ FX .
µX : EF ⊗FX
The universal properties of end and internal hom show that there exists
/ E in V and natural families of
a bijection between the arrows f : A
F
/
FX , given by
arrows θX : A⊗FX
θX = µX ◦ (f ⊗ 1X ) .
The natural families
EF ⊗EF ⊗FX
1⊗µX
µX
EF ⊗FX
FX ,
FX
1FX
FX
induce, under such bijections, the monoid structure on EF :
/E
F
µ : EF ⊗EF
,
η:I
/E .
F
Example 16.1 Take V = ModR for some commutative ring R . Then
/ V, the
we have that Mon(V) = AlgR . Now for any functor F : C
algebra EF has as elements the natural families θ = (θX )X∈G of R-linear
/ FX ; addition and multiplication by scalars are
morphisms θX : FX
done componentwise, while multiplication is componentwise composition.
In particular, for any R-algebra A , if we have
/ Mod
R
F = UA : ModR (A)
then there is a natural isomorphism of algebras
EF ∼
= A.
To see this, notice that each element m of an A-module M determines a
/ M in Mod (A) with m̂(1) = m ; so for a natural θ :
unique m̂ : A
R
/ U , we have
UA
A
A
θA
m̂
M
A
m̂
θM
M
which implies θM (m) = θA (1) m ; so θ is determined by θA (1) ∈ A .
Tannaka duality
111
Suppose A is a monoid in V . The two axioms which are required for an
/ E to be in Mon(V) translate to the two conditions on
arrow f : A
F
/ FX which say that
the corresponding family of arrows θX : A⊗FX
each θX is an action of A on FX. This is precisely what is needed to lift
/ Mod (A) such that U T = F ; just put T X =
F to a functor T : C
V
A
FX , θX . This gives a natural bijection between hom sets
,
C , F , ModV (A) , UA
Mon(V)(A , EF ) ∼
= Cat/V
op
/ Cat/V .
whereby C , F / EF is left adjoint to ModV : Mon(V)
In fact, the above bijection becomes an isomorphism of categories, since it
extends to 2-cells:
T
C
f
A
ξ
f
ModV (A)
T
EF
F
UA
V
/ E is left 2-adjoint to Mod .
This is expressed by saying that C , F F
V
Taking A = EF in the above bijection and looking at the image of the
identity arrow, we obtain
N
C
F
ModV (EF )
UE
F
V
where N X = F X , µX . We obtain a (partial) 2-functor
/ (Mon(V))op
: Cat/V
/ D , G in Cat/V into the
by taking the 2-cell α : T ⇒ T : C , F
/ E in Mon(V) corresponding (under the
2-cell Eα : ET ⇒ ET : EG
F
2-adjunction) to the 2-cell in Cat/V :
/ Mod (E ) , U .
Nα : NT ⇒ NT : C , F
V
G
E
E
G
/ V are
Remark 16.2 The Formal Tannaka Duality criteria on F : C
/
ModV (EF ) should be faithful and that every “appropriate”
that N : C
EF -module should be isomorphic to some N X .
112
Quantum Groups: A Path to Current Algebra
We can equally well regard E
E
as a 2-functor
: (Cat/V)op
/ Mon(V)
whereupon (for general reasons as an adjoint to ModV ) it is a weak tensor
functor. It preserves the unit in the sense that EI ∼
= I, while we have a
canonical arrow φ such that
φF,G
EF ⊗FG
EF ⊗G
πX⊗Y
πX⊗πY
[ FX , FX]⊗[ GY, GY ]
⊗
[ FX⊗GY, FX⊗GY ]
where the bottom arrow corresponds to the composite
[FX, FX]⊗[ GY, GY ]⊗FX⊗GY
1⊗c⊗1
e⊗e
[FX, FX]⊗FX⊗[ GY, GY ]⊗GY
FX⊗GY .
So E takes monoids in (Cat/V)op to monoids in Mon(V), the latter
being the commutative monoids in V , but this is of no interest to us here.
Our real interest is in to what extent
E
: (Cat/V)
/ Mon(V)op
takes monoids to monoids. This will be true of those monoids C , F in
/E
Cat/V for which φF,F : EF ⊗EF
F ⊗F is invertible.
Is this a reasonable condition? At first glance, invertibility of
/
[ FX , FX ] ⊗
[ GY, GY ]
[ FX⊗GY, FX⊗GY ]
φF,G :
X
Y
X,Y
looks unlikely. It would be implied by the two conditions:
(a) each A⊗
: V
/ V preserves ends; and
⊗
(b) each [A , B ]⊗[ C , D ]
[ A⊗C , B⊗D ] is invertible,
for every A = FX and every C = GY .
However, these look unlikely too, if we think in terms of Example 16.1 .
We shall look at the conditions (a) and (b) more closely. If V is a closed
tensor category then A⊗ preserves colimits (since it has a right adjoint
Tannaka duality
113
[A , ] ; see Mac Lane [Mac71, Chapter V §5]. But end is a limit, not a
colimit. So (a) can be ensured by taking V to be the opposite of a complete
closed tensor category. We need to be careful here since we still need the
internal homs of the form [FX, FX ] in V , not in V op .
Condition (b) is true, for example for finite-dimensional vector spaces.
What is needed is that A and C should have duals; then we have canonical
isomorphisms
[ A , B ]⊗[ C , D ] ∼
= A∗⊗ B ⊗ C ∗ ⊗D
∼
= [ A⊗C , B⊗D ] .
= (A⊗C)∗ ⊗ (B⊗D) ∼
Hence, conditions (a) and (b) are not unreasonable after all. They are
satisfied when V is the opposite of a closed symmetric tensor category
which is cocomplete enough for coends over C to exist, and when each FX
and GY has a dual.
Suppose then that V op is a closed symmetric (strict) tensor category
which is (small) cocomplete. Suppose C is a left autonomous small (strict)
/
tensor category and F : C
V is a (strict) tensor functor. Then each FX
∗
has a dual FX . Since C , F is a monoid in Cat/V, we obtain a monoid
EF in Mon(V)op ; that is, a bimonoid EF in V. This gives a factorization
N
C
ModV (EF )
F
UEF
V
of our tensor functor F into tensor functors N and UE .
F
In fact, EF is a Hopf monoid. To see this, define F ∗ : C op
∗
F X = FX ∗ . We obtain a monoid arrow
/ V by
/ (C op , F ∗ )
( )∗ : (C, F )
in Cat/V. This induces a monoid arrow ν with
ν
EF ∗
πX ∗
EF
πX ∗
FX⊗FX ∗ .
It is easy to see that EF ∗ = EF op as bimonoids in V (that is, EF ∗ is just
EF with switched multiplication and switched comultiplication), and ν is
an antipode for the bimonoid EF .
114
Quantum Groups: A Path to Current Algebra
Now suppose C is braided. The braiding can be regarded as an invertible
2-cell:
C ,F ⊗ C ,F
σ
C ,F ⊗ C ,F
c
⊗
⊗
C ,F
in Cat/V. Applying E , we obtain an invertible 2-cell in Mon(V) :
c
EF ⊗EF
EF ⊗EF .
γ
δ
δ
EF
The braiding arrows for c on C carry over precisely to those for γ on EF .
/ Mod (E ) becomes a braided tensor functor.
Moreover, N : C
V
F
Next suppose C is balanced. The twist on C can be regarded as being an
invertible 2-cell:
C ,F
1C
θ
C ,F
1C
in Cat/V, and, applying E , we obtain a twist
1
EF
τ
EF
1
for the braided bimonoid EF . So EF becomes a balanced Hopf monoid and
/ Mod (E ) becomes a balanced tensor functor.
N : C
V
F
Finally, if C is a tortile tensor category, EF is a tortile bimonoid in V.
To obtain Ulbrich’s [Ulb89] setting, we take V = Modop
R for a commutative
ring R . For each R-coalgebra C, we have
Mod V (C)op = ComodR (C) .
We use the notation ComodR (C)c to denote the full subcategory consisting
of C-comodules M for which the underlying R-module UC M is Cauchy.
Tannaka duality
Consider a small category C and a functor F : C
values FX are Cauchy R-modules. The coend
X
FX ⊗R (FX)∗
EF =
115
/ Mod whose
R
becomes an R-coalgebra and we have
N
C
F
ComodR (EF )
U
ModR
(since we can apply our previous theory to F regarded as going from C op
to V = Modop
R ). Notice that N actually lands in ComodR (EF )c .
If C is a tensor category and F is a tensor functor then EF becomes
an R-bialgebra and N becomes a tensor functor. If C is left autonomous
then EF becomes a Hopf algebra with invertible antipode. If C is a tortile
tensor category then EF becomes a cotortile R-bialgebra (quantum group!)
and N becomes a balanced tensor functor.
An important case of Tannaka duality is the characterization of those
/ Mod for some
/ Mod equivalent to U : Comod (H)
F :C
c
R
H
R
R
Hopf algebra H. This can be investigated by looking at when the functor
/ Comod (E ) is an equivalence.
N : C
R F c
The question arises here as to whether EF ∼
= C when the equality
/ Mod holds for a coalgebra C . We cannot
F = UC : Comod(C)c
R
use the technique of Example 16.1 since, although C is a C-comodule, it is
generally not Cauchy as an R-module.
Proposition 16.3 If C is a coalgebra over a field R and U denotes the
/ Mod , then there is a coalgebra isoforgetful functor U : Comod(C)c
R
morphism
EU ∼
= C.
Proof. We need to show that C has the universal property of EU ; that
/ (f ⊗ 1) ◦ δ determines a natural bijection between
is, the assignment f / X and families of R-module morphisms
R-module morphisms F : C
/ X⊗ U (M ) natural in M ∈ Comod(C) .
θM : U (M )
c
We need to apply the fundamental theorem on coalgebras (see Sweedler
[Swe69, p.46]): (when R is a field) “the sub-coalgebra generated by an
element of C is Cauchy”.
Given a family θM , we must define f (c) for each c ∈ C. Let M be any subcoalgebra of C which contains c and is finite dimensional. Such M exist
116
Quantum Groups: A Path to Current Algebra
by the above fundamental theorem, and can be regarded as C-comodules.
Put f (c) = (1 ⊗ ε )θM (c). This is independent of the choice of M since θM
/ (f ⊗ 1) ◦ δ is
is natural in M . The proof that this gives the inverse to f now easy.
Exercise 16.1 Give a precise definition of the “obvious pair of arrows”
in the defining equalizer for EF .
Exercise 16.2 Complete the proof of Proposition 16.3.
Bradshaw: Clothes Peg Figure Period, [Wal94, Plate 67].
17
Adjoining an antipode to a bialgebra
Tannaka duality allows the possibility of taking an R-bialgebra A , applying
some categorical construction to ComodR (A)c , and asking whether the
result again has the form ComodR (B)c for some R-bialgebra B .
An example of an appropriate categorical construction is adjoining leftdual objects to a tensor category. To each tensor category C , there is a
/ A (C)
left autonomous tensor category A (C) and a tensor functor C
which induces a natural equivalence between the category of tensor func/ D and the category of tensor functors C
/ D for all left
tors A (C)
autonomous tensor categories D . (See [JS91a], [JSV96] and [KSW02].)
/ Mod is a tensor functor whose values FX
Suppose that F : C
R
are Cauchy R-modules. Then we obtain a corresponding tensor functor
/ Mod .
F̂ : A (C)
R
Proposition 17.1 EF̂ is the reflection of the R-bialgebra EF into the category of Hopf R-algebras.
Proof. Let H be a Hopf R-algebra. Then we have that ComodR (H)c is
a left autonomous tensor category. Thus we have an equivalence between
/ Comod (H) , over Mod and tensor functors
tensor functors A (C)
c
R
R
/
C
ComodR (H)c over ModR . By the left adjoint property of E ,
/ H correspond to bialgebra
it follows that bialgebra morphisms EF̂
/
morphisms EF
H, as required.
This gives a construction for adjoining an antipode to a bialgebra over
a field R ; that is, a construction for a left adjoint to the inclusion of the
category HopfR of Hopf algebras in the category BigR of bialgebras. Given
/ Mod . By Proposition
a bialgebra A , put F = UA : ComodR (A)c
R
.
By
Proposition
17.1,
the
Hopf algebra H = EF̂ is
16.3, we have A ∼
E
= F
the required reflection.
If we require the adjoined antipode to be invertible, we must replace
(C)
in the above by A(C) which is the free autonomous tensor category
A
on the tensor category C . And so on.
117
118
Bradshaw: Sash Bradshaw Group, [Wal94, Plate 54].
18
The quantum general linear group
again
Let V be an n-dimensional vector space over a field k . Given an invertible
q ∈ k , let (y , z) be the tortile Yang–Baxter operator on V defined in
Chapter 14. By Example 13.1 and Example 13.2, there are strict tensor
functors
/ Mod
/ Mod
, G : T
M : B
k
k
taking + , c+,+ , θ+ to be V , y , z (where we are identifying B with the
subcategory of T whose objects are positively signed sets, with arrows
being ribbons which do not bend around).
Applying Tannaka duality ideas (Chapter 16) to M and G, we obtain
a cobalanced bialgebra EM and a cotortile bialgebra EG .
Theorem 18.1 There are k-bialgebra isomorphisms (see Example 9.8):
EM ∼
= Mq (n)
,
EG ∼
= GLq (n) .
Proof. Let A be the bialgebra EM . It comes equipped with a universal
/ A⊗M Z for Z ∈ B . In particular, we have
linear function δZ : M Z
/
/ A⊗V ⊗V is the composite
δ = δ+ : V
A⊗V , and δ+,+ : V ⊗V
V ⊗V
δ ⊗δ
while y : V ⊗V
A⊗V ⊗A⊗V
1⊗σ⊗1
A⊗A⊗V ⊗V
µ⊗1⊗1
A⊗V ⊗V
/ V ⊗V becomes a comodule morphism
V ⊗V
δ+,+
y
V ⊗V
A⊗V ⊗V
1⊗y
δ+,+
A⊗V ⊗V
Putting δ(εi ) = j xij ⊗εj , it is a straightforward, but tedious, matter to
check that commutativity of the above square is equivalent to the elements
119
120
Quantum Groups: A Path to Current Algebra
X = {xij | i, j = 1, . . . , n} satisfying the defining relations for the quantum
matrix monoid Mq (n) (see Example 9.8). We therefore have a bialgebra
/ A which can be seen to be invertible.
morphism Mq (n)
To introduce an antipode to the bialgebra A and thereby obtain the
/ A⊗V ∗ for δ :
Hopf algebra H, we must introduce a left dual κ : V ∗
/ A⊗V . Since T is the free autonomous tensor category on B , we have
V
H = EG (as in Chapter 17). If we put
κ(εi∗ ) =
wij ⊗εj∗
j
/ k and d : k
/ V ∗ ⊗V to
and express what it means for e : V ∗ ⊗V
be H-comodule morphisms, we obtain the conditions
wim xjm = δij
and
xmi wmj = δij ,
m
m
which mean that the matrix wij is the inverse of the transpose of xij .
The Hopf algebra H is therefore obtained from A by adjoining elements wij
subject to the above two conditions. By means of a quantum Cramer’sRule
(checked in Chapter 3 for the n = 2 case), we can take wij = t detq Xij
(see Example 9.8) where t is an adjoined inverse for detq (X) . In this way
we see that H ∼
= GLq (n) .
Corollary 18.2 Mq (n) is a cobalanced bialgebra and GLq (n) is a cotortile
bialgebra.
/ k , and cotwist given
The cobraiding given by γ : GLq (n)⊗GLq (n)
/
k , satisfy the equations
by τ : GLq (n)⊗GLq (n)
y(εi⊗ εj )
=
γ(xim , xjr ) εm⊗ εr
m,r
q n εi
=
τ (xim ) εm .
m
This means:



γ(xim , xjr )
τ (xij )
1
(q − q −1 )
=

q

0
=
q n δij .
for i = j , m = j , r = i
for i < j , m = i , r = j
for i = j = m = r
otherwise
19
Solutions to Exercises
Chapter 4
4.2
For a ∈ Z/(2), b ∈ Z/(5), put x = a ⊗ b ∈ Z/(2)⊗Z Z/(5). So we have:
5x = 5(a ⊗ b) = a ⊗ 5b = a ⊗ 0 = a ⊗ (0 × 0) = 0a ⊗ 0 = 0 ⊗ 0 = 0 ,
2x = 2(a ⊗ b) = 2a ⊗b = 0 ⊗ b = 0 .
Hence x = (5 − 2 × 2)x = 0 − 0 = 0. Elements of the form x generate,
so Z/(2)⊗Z Z/(5) = {0}.
4.3
(a)
Let ⊗ denote ⊗Z . The multiplication and unit are given by
(R⊗S)⊗(R⊗S)
∼ Z⊗Z η⊗η
Z=
(b)
1⊗σ⊗1
(R⊗R)⊗(S⊗S)
µ⊗µ
R⊗S
R⊗S .
This makes it clear that µ is an abelian group morphism, so we
automatically have distributivity. Now in terms of generating
elements, the multiplication is (r ⊗ s)(r ⊗ s ) = (rr ) ⊗ (ss ) and
the unit is 1 = 1 ⊗ 1 . Associativity and unit conditions only
need to be checked on generators where they clearly follow from
these conditions in R and S.
/ R ⊗S .
Yes, ϕ(r) = r ⊗ 1 does define a ring morphism ϕ : R
ϕ =
R∼
= R⊗Z
1R ⊗η
R⊗S
is clearly an abelian group morphism. It remains to check that
multiplication and unit are preserved:
• ϕ(rr ) = (rr ) ⊗ 1 = (r ⊗ 1)(r ⊗ 1) = ϕ(r) ϕ(r ) ;
• ϕ(1) = 1 ⊗ 1 = 1 using the definition in (a).
(c)
ϕ /
ψ
R⊗Z S o
S , with ϕ(r) = r ⊗ 1 and ψ(s) = 1 ⊗ s ;
Let R
these are ring morphisms as in (b). These give our “coprojecf /
g
T o
S with f, g ring morphisms
tions”. Now given R
121
122
Quantum Groups: A Path to Current Algebra
and T commutative, we must show there is a unique ring mor/ T with h ◦ ϕ = f , h ◦ ψ = g . These
phism h : R⊗S
last equations force us to define h(r ⊗ s) = h (r ⊗ 1)(1 ⊗ s) =
/T ,
h(r ⊗ 1)h(1 ⊗ s) = f (r) g(s) . It is easily checked that R×S
/ f (r) g(s) is bilinear. So h does give an abelian group
(r, s) morphism. It remains to show h preserves multiplication and
unit.
h (r ⊗ s)(r ⊗ s ) = h rr ⊗ ss = f (rr ) g(ss )
= f (r) f (r ) g(s) g(s )
=
=
4.4
f (r) g(s) f (r ) g(s )
h(r ⊗ s) h(r ⊗ s ) .
So h is a ring morphism. Since the definition of h was forced, it
is unique.
/
A module M : R
S is an abelian group with an abelian group
/ M , written µ(r ⊗ m ⊗ s) = rms , satisfying
morphism µ : R⊗M ⊗S
r (rms)s = (r r)m(ss ) , 1m1 = m . One can see that this agrees
with the definition given in Chapter 4 (given left R- , right S-scalar
multiplications satisfying (rm)s = r(ms) we define rms = (rm)s;
all the distributive laws precisely summarize to trilinearity (over Z);
conversely, given µ, define the two scalar multiplications by rm =
rm1 and ms = 1ms). Now a left R⊗S op -module is an abelian group
/ M , written
M with an abelian group morphism µ : (R⊗S)⊗M
⊗
⊗
⊗
⊗
µ(r s m) = (r
s)m , satisfying (1 1)m = m and (r r ⊗ ss )m =
(r ⊗ s ) (r ⊗ s)m . Clearly to give the abelian group morphisms µ, µ
are the “same thing” via the diagram:
R⊗M ⊗S
µ
∼
= 1⊗σ
R⊗S⊗M
4.5
M .
µ
Moreover the conditions on µ directly translate to those on µ.
/M
/N
/L
Suppose R
S
T
U . As an abelian group we have:
M ⊗S N ⊗T L = B
where B is the subset of the abelian group FZ (M × N × L) consisting
of all elements of the forms:
(m + m , n, l) − (m, n, l) − (m , n, l) ,
(m, n + n , l) − (m, n, l) − (m, n , l) ,
(m, n, l + l ) − (m, n, l) − (m, n, l ) ,
(ms, n, l) − (m, sn, l) , (m, nt, l) − (m, n, tl) .
Solutions to Exercises
123
The equivalence class of (m, n, l) is denoted by m ⊗ n ⊗ l. We now define
/
U . Then
r(m ⊗ n ⊗ l)µ = (rm) ⊗ n ⊗ (lµ) yielding M ⊗S N ⊗T L : R
∼
HomU
R (M ⊗S N ⊗T L, K) = Mult (M, N, L; K) .
Chapter 5
5.2
We use the Fundamental Theorem of Morita Theory. Suppose M
is finitely generated and projective. By Theorem 5.2, we have the
/ R satisfying
/ M ∗⊗ M , e : M ⊗ M ∗
morphisms d : R
R
R
M∗
1⊗e
M ∗ ⊗R M ⊗R M ∗
M ∗ = 1M ∗
and
(e ⊗ 1) ◦ (1 ⊗ d) = 1M .
d⊗1
d /
σ /
σ /
M ∗ ⊗R M
M ⊗R M ∗ , e = M ∗ ⊗R M
Put d = R
e
/ R . We can now apply Theorem 5.2(iii) (replacing M ,
M ⊗R M ∗
N , e, d with M ∗ , M , e , d respectively) and by (iv) M ∗ is finitely
generated and projective.
5.3
∗
/ Hom (M, L) is “natural” in M (and
Notice that ρM
L : M ⊗R L
R
in L too for that matter), meaning that for any module morphism f :
/ N : R / S , the following “f -square” commutes:
M
N ∗ ⊗R L
ρN
L
HomR (N, L)
f ∗ ⊗1
◦f
M ∗ ⊗R L
ρM
L
HomR (M, L) .
Suppose now that M is a retract of a Cauchy module N ; so we have
/ M , r ◦ i = 1 and ρN invertible. We can
/N, r : N
i: M
L
M
show that the composite
Hom (M, L)
◦r
Hom (n, L)
−1
(ρN
L)
N ∗ ⊗L
i∗ ⊗1
M ∗ ⊗L
is an inverse for ρM
L . This is seen as follows:
∗
N −1
N −1
ρM
◦ ( ◦ r) = ( ◦ i) ◦ ρN
◦ ( ◦ r)
L ◦ (i ⊗ 1) ◦ (ρL )
L ◦ (ρL )
(by the i-square)
= ( ◦ i) ◦ 1Hom(N,L) ◦ ( ◦ r) = ( ◦ i) ◦ ( ◦ r)
=
◦ (ri) =
◦ 1M = 1Hom(M,L)
124
Quantum Groups: A Path to Current Algebra
and
−1
∗
N −1
∗
(i∗ ⊗ 1) ◦ (ρN
◦ ( ◦ r) ◦ ρM
◦ ρN
L)
L = (i ⊗ 1) ◦ (ρL )
L ◦ (r ⊗ 1)
(by the r-square)
= (i∗ ⊗ 1) ◦ (r∗ ⊗ 1) = (ri)∗ ⊗ 1 = 1M ∗ ⊗ 1L = 1M ∗ ⊗L
establish the inverses, as required.
Chapter 6
6.1
(a)
(b)
(c)
(d)
/ End (M ⊗N ) , whereby g / m ⊗ n /
We have that
G
R
(gm)⊗ (gn) , is a monoid morphism since 1 / 1M⊗N and gh
/ m ⊗n / (hm) ⊗ (hn) / (ghm) ⊗ (ghn) . It extends to a
/ End (M ⊗ N ) . So M ⊗ N
unique R-algebra morphism R(G)
R
R
R
is an R(G)-module.
/ Hom (M, L) be the R-module morLet µ̂(g) : HomR (M, L)
R
phisms given by µ̂(g)(µ)(m) = gµ(g −1 m). Then we have that
µ̂(1)(µ)(m) = µ(m),
= ghµ(h−1 g −1 m) =
and µ̂(gh)(µ)(m)
−1
g µ̂(h)(µ)(g m) = µ̂(g) ◦ µ̂(h) (µ)(m).
/ End Hom (M, L) is a monoid morphism. Hence
So µ̂ : G
R
R
HomR (M, L) becomes an R(G)-module.
evM g ◦ (m ⊗ µ) = evM (gm ⊗ gµ) = (gµ)(gm) = g µ(g −1 gm) =
g µ(m) = g evM (m ⊗ µ), so evM preserves the R(G)-action.
/ Hom (M, M ⊗ N ) , n We also need d : N
R
R
to be an R(G)-module morphism. We have
/ (m / m ⊗ n)
d(gn)(m) = m ⊗ gn = g (g −1 m) ⊗ n
= gd(n)(g −1 m) = gd(n) (m) ,
so d(gn) = gd(n). The required isomorphism is the restriction
of
HomR N, HomR (M, L)
f
(g ◦ ) ◦ d
o
∼
=
/
HomR (M ⊗R N, L)
e ◦ (1M ⊗ f )
g
to R(G)-module morphisms f and g; since e and d are such, so
then are e ◦ (1M ⊗ f ) and (g ◦ ) ◦ d when f and g are. (This
will be generalized from R(G) to an arbitrary Hopf algebra in
Chapter 9.)
Solutions to Exercises
125
Chapter 7
7.1
There is a little abuse of notation here since the four parts to part
(a), for example, are elements of C⊗R C, R⊗R C⊗R C, C⊗R C⊗R R,
C⊗R R⊗R C respectively. But these modules are canonically isomorphic, and so “=” really means “corresponds under the canonical isomorphism to”.
(a)
To prove the first part:
σ
C⊗C
ε⊗1
C⊗C
R⊗C
1⊗δ
δ
∼
=
C
σ
1⊗ε
C⊗R
δ ⊗1
C⊗C⊗R
∼
=
R⊗C⊗C
∼
=
δ
C⊗C
(b)
Similarly for the second:
C⊗C⊗C
δ
C
δ
δ ⊗1
C⊗C
1⊗δ
C⊗C⊗C
ε⊗1⊗1
∼
=
ε⊗1⊗1
R⊗C⊗C
1⊗σ
R⊗C⊗C
∼
=
σ
C⊗C
(c)
For the third:
C⊗C⊗C
δ
δ ⊗1
C
δ
C⊗C
∼
=
1⊗σ
ε⊗1⊗1
R⊗C⊗C
C⊗C⊗C
ε⊗1⊗1
1⊗σ
R⊗C⊗C
1⊗1⊗ε
1⊗ε
∼
=
C⊗R
∼
=
R⊗C⊗R
ε⊗ε⊗1
1⊗ε⊗1
1⊗σ
R⊗R⊗C
126
Quantum Groups: A Path to Current Algebra
Chapter 9
9.1
(a)
Suppose I is a coideal of C, that is, a submodule satisfying
δ(I) ⊆ I⊗C + C⊗I and ε(I) = 0. The composite morphism
C
δ
C⊗C
ρ⊗ρ
C/I ⊗ C/I
maps I to 0 since (ρ ⊗ ρ)δ(I) ⊆ (ρ ⊗ ρ)(I⊗C + C⊗I) = ρ(I) ⊗ ρ(C)
+ ρ(C) ⊗ ρ(I) = 0 ⊗ C/I + C/I ⊗ 0 = 0. So there exists a unique
/ C/I ⊗ C/I such that δ ◦ ρ =
module morphism δ : C/I
(ρ ⊗ ρ) ◦ δ. Similarly, ε(I) = 0 implies that there exists a unique
/ R with ε ◦ ρ = ε. These properties of δ, ε will mean
ε : C/I
/
C/I is a coalgebra morphism once we know C/I is a
ρ: C
/ C/I ⊗ C/I ,
coalgebra. To prove coassociativity of δ : C/I
take the coassociativity diagram for C/I and precompose with
/ C/I :
ρ: C
C
ρ
C/I
δ
C/I ⊗ C/I
ρ⊗ρ
δ ⊗1
1⊗δ
C/I ⊗ C/I ⊗ C/I
ρ⊗ρ⊗ρ
δ
C⊗C
δ ⊗1
1⊗δ
C⊗C⊗C
The result commutes by coassociativity of C. But ρ is surjective;
so the coassociativity diagram for C/I commutes. Similarly we
/ R is a counit. If C is a bialgebra and
can prove ε : C/I
I is also an ideal, certainly C/I becomes an algebra. All that
remains to check are the extra bialgebra conditions (see Proposition 7.5). The main one, showing that δ preserves multiplication, is obtained by precomposing the diagram with ρ ⊗ ρ and
using the corresponding condition for C. This gives the result
since ρ ⊗ ρ is surjective.
Since ρ is a bialgebra morphism, the only possible way C/I can
become a Hopf algebra is to have ν ◦ ρ = ρ ◦ ν. This forces us to
ask whether ν(I) ⊆ I for the antipode of C.
(b)
Put B = R
x, y, z. The given equations define algebra mor/ B ⊗B, ε : B
/ R since B is free as an algephisms δ : B
bra. By Proposition 7.5, it remains to see that these morphisms
make B a coalgebra.
First look at the coassociativity:
(δ ⊗ 1)δ(x) = (δ ⊗ 1)(x ⊗ x) = x ⊗ x ⊗ x
= (1 ⊗ δ)(x ⊗ x) = (1 ⊗ δ)δ(x) .
Solutions to Exercises
127
Similarly for y:
(δ ⊗ 1) δ(z) = (δ ⊗ 1)(1 ⊗ z + z ⊗ x)
= 1 ⊗ 1 ⊗ z + (1 ⊗ z + z ⊗ x) ⊗ x
= 1 ⊗ (1 ⊗ z + z ⊗ x) + z ⊗ x ⊗ x
= (1 ⊗ δ)(1 ⊗ z + z ⊗ x) = (1 ⊗ δ)δ(z) .
Then check the counit conditions:
(ε ⊗ 1) δ(x) = (ε ⊗ 1)(x ⊗ x) = ε(x)x = x
= (1 ⊗ ε)(x ⊗ x) = (1 ⊗ ε)δ(x) ,
and similarly for y:
(ε ⊗ 1)δ(z) = (ε ⊗ 1)(1 ⊗ z + z ⊗ x) = z + 0x = z
= 0 + z = (1 ⊗ ε)(1 ⊗ z + z ⊗ x) = (1 ⊗ ε) δ(z) .
(c)
We have:
δ(xy − 1) (x ⊗ x)(y ⊗ y) − 1 ⊗ 1 = xy ⊗ xy − 1 ⊗ 1
= (xy − 1) ⊗ xy + 1 ⊗ xy − 1 ⊗ 1
= (xy − 1) ⊗ xy + 1 ⊗ (xy − 1) ⊆ I⊗B + B⊗I .
(d)
We must check:
H
δ
ν ⊗1
H⊗H
1⊗ν
ε
H⊗H
µ
H
η
R
µ(ν ⊗ 1) δ(x) = µ(ν ⊗ 1)(x ⊗ x) = µ(y ⊗ x) = yx ≡ 1 = νε(x)
≡ xy = µ(x ⊗ y) = µ(1 ⊗ ν)(x ⊗ x) = µ(1 ⊗ ν) δ(x) .
Similarly for y:
µ(ν ⊗ 1) δ(z) = µ(ν ⊗ 1)(1 ⊗ z + z ⊗ x) = µ(1 ⊗ z + (−zy) ⊗ x)
= z − zyx ≡ 0 = ηε(z) = 0
= −zy + zy = µ(1 ⊗ (−zy) + z ⊗ y)
= µ(1 ⊗ ν)(1 ⊗ z + z ⊗ x) = µ(1 ⊗ ν) δ(z) .
So H is a Hopf algebra. By Proposition 9.1, ν reverses both
multiplication and comultiplication. The formulas for ν r (z) are
trivial for r = 0 or 1. Also ν 2n (z) = xn zy n implies:
ν 2n+1 (z) = ν(y)n ν(z)ν(x)n = xn (−zy)y n = −xn zy n+1
128
Quantum Groups: A Path to Current Algebra
which gives:
ν 2n+2 (z) = −ν(y)n+1 ν(z)ν(x)n = −xn+1 (−zy)y n
= xn+1 zy n+1 .
So the formulas follow by induction.
If ν had finite order we would have either xn z = zxn , or xn z =
−zxn+1 , which are both false in H.
(e)(i)
Evaluate each of ν, δ and ε at (xn zy n − z) :
ν(xn zy n − z) = ν(y)n ν(z)ν(x)n − ν(z) = xn (−zy)y n + zy
= −xn zy n+1 + zy = (xn zy n − z)(−y) ∈ In ,
δ(x zy − z) = (xn ⊗ xn )(1 ⊗ z + z ⊗ x)(y n ⊗ y n ) − 1 ⊗ z − z ⊗ x
n
n
= (xn ⊗ xn z + xn z ⊗ xn+1 )(y n ⊗ y n ) − 1 ⊗ z − z ⊗ x
= xn y n ⊗ xn zy n + xn zy n ⊗ xn+1 y n − 1 ⊗ z − z ⊗ x
= 1 ⊗ xn zy n + xn zy n ⊗ x − 1 ⊗ z − z ⊗ x
= 1 ⊗ (xn zy n − z) + (xn zy n − z) ⊗ x
∈ In ⊗H + H⊗In ,
ε(xn zy n − z) = 0 .
So In is a Hopf ideal in H. Also compute:
ν(xn − 1) = y n − 1 ≡ −y n (xn − 1) ∈ Jn ,
δ(xn − 1) = xn ⊗ xn − 1 ⊗1 = (xn − 1) ⊗ xn + 1 ⊗ (xn − 1)
∈ Jn ⊗H + H⊗Jn ,
ε(x − 1) = 1n − 1 = 0 .
n
So Jn is a Hopf ideal in H.
(e)(ii)
ν 2n (x) = x and ν 2n (y) = y since ν just switches x and y.
ν 2n (z) = xn zy n ≡ z
ν
2n
(mod In )
(z) = x zy ≡ |z| = z
n
n
(mod Jn ) .
Chapter 10
10.1
Let M be a Cauchy R-module. The diagrams (coassociativity and
counit) showing E to be a coalgebra are:
MM∗
1⊗d⊗1
M M ∗M M ∗
1⊗1⊗1⊗d⊗1
1⊗d⊗1⊗1⊗1
M M ∗M M ∗M M ∗
Solutions to Exercises
MM∗
1⊗d⊗1
129
e⊗1⊗1
M M ∗M M ∗
MM∗ .
1⊗1⊗e
These follow from functionality of ⊗R and Theorem 5.2.
/ δ̂ characterizes a bijection between R-linear functions
Certainly δ / C . The
/ C⊗ M and R-linear functions ω : M ⊗ M ∗
δ: M
R
R
/ ∨ ω is given by:
inverse assignment ω 1⊗d
ω ⊗1
∨
ω = M
M M ∗M
CM .
(That these assignments are mutually inverse follows from the properties of e and d in Theorem 5.2.)
It remains to see that coaction axioms on δ translate precisely to coalgebra morphisms on ω. We shall do the translation for the coaction
axiom:
δ ⊗1
δ
M
CM
CCM
1⊗δ
(where δ is the comultiplication of C). This is equivalent to:
δ ⊗1
MM∗
CM M ∗
δ ⊗1⊗1
CCM M ∗
1⊗δ ⊗1
1⊗1⊗e
Using:
δ ⊗1
MM∗
δ ⊗1⊗1
CM M ∗
CCM M ∗
1⊗e
1⊗1⊗e
δ̂
C
CC
δ
and
1⊗d⊗1
MM∗
M M ∗M M ∗
δ ⊗1⊗1
CM M ∗ M M ∗
δ ⊗1
δ̂ ⊗1
1⊗1⊗δ ⊗1
1⊗e⊗1⊗1
CM M
∗
CM M ∗
1
we see that the axiom becomes:
δ̂
MM∗
C
δ
1⊗d⊗1
M M ∗M M ∗
δ̂ ⊗1
CM M ∗
which is a coalgebra morphism axiom on δ̂.
1⊗δ̂
CC
CC .
130
Quantum Groups: A Path to Current Algebra
Chapter 11
11.1 (a)
cA,I = cA,I ◦ cA,I from the triangle below. Since cA,I is invertible,
cA,I = 1A . Similarly cI,A = 1A .
A
cA,I⊗I
cA,I ⊗1I
A
1I ⊗cA,I
A
The hexagon can be subdivided into commutative regions thus:
ACB
c ⊗1
CAB
1⊗c
1⊗c
cAB,C
ABC
CBA .
cBA,C
c ⊗1
c ⊗1
BAC
(b)
Put γ =
1⊗c
BCA
µi ⊗ vi ∈ A⊗A so that cM,N (m ⊗ n) =
i
(µi n) ⊗ (vi n):
i
m ⊗ µi l ⊗ vi n
c⊗1
µj µi l ⊗ vj m ⊗ vi n
i,j
i
1⊗c
1⊗c
X.
m ⊗ n ⊗l
c⊗1
c ⊗1
µi n ⊗ vi m ⊗ l
1⊗c
i
µi n ⊗ µj l ⊗ vj vi m
i,j
The hexagon gives us the condition:
X=
µj µi l ⊗ µk vi n ⊗ vk vj m =
µk µj l ⊗ vk µi n ⊗ vj vi m
i,j,k
i,j,k
in A⊗A⊗A . Diagrammatically this becomes:
R
γ ⊗γ ⊗γ
A⊗6
σ315264
σ536142
A⊗6
µ⊗µ⊗µ
A⊗3 .
Solutions to Exercises
131
(c)
A
B
C
A
B
C
C
B
A
=
C
B
A
Bradshaw: Elegant Action Figure Group, [Wal94, Plate 79].
132
Bradshaw: “Broad Clothes Peg Figure Group”, Clothes Peg Figure Period, [Wal94,
Plate 71].
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Index
2-category, 100, 102
2-cell, 97
2-category, 100
2-functor, 101, 102
identity, 98
pasting, 97
2-functor, 101, 102
A-module, 101
action, 101
arrow, 101
left, 101
A-point, 55
action, 59, 101, 111
of an A-module, 101
adjoining
an antipode, 117
an inverse for the determinant, 56
left-dual objects, 117
adjoint
left, 111
left 2-adjoint, 111
Lie algebra, 33
algebra
exterior, 31
Hopf, 51
morphism, 27
of endomorphisms, 28
of formal power series, 42
opposite, 30
over R, 27
polynomial algebra, 45
polynomial R-algebra, 46
R-algebra, 27
symmetric, 45
tensor, 28
universal, 87
universal enveloping, 40
algebraic geometry, 5
algebraic varieties, 5
antipode, 7, 51, 105, 117
appropriate
representation, 109
arrow
between monoid arrows, 97
composition, 78
monoid, 97
of category, 6
associativity, 2, 27, 67
autonomous, 82
left, 81
left/right, 82
balanced, 106
YB-operator, 88
bialgebra, 76
Banach spaces, 79
example of closed symmetric tensor
category, 79
bialgebra
balanced, 76
braided, 72
commutative over k, 7
matrix bialgebra, 46
morphism, 44
quasitriangular, 72
R-bialgebra, 42
symmetric, 72
triangular, 72
biclosed, 78
bijection
canonical, 1
bilinear
R-bilinear, 15
universal bilinear function, 15
bimodule
left R-/right S-bimodule, 16
bimonoid, 105
balanced, 106
cobalanced, 108
cobraided, 108
cotortile, 108
135
136
in ModR , 105
in V , 105
strict quasi-bimonoid, 105
tortile, 108, 114
with a braiding, 106
with a twist, 106
Bradshaws, xvii
Age of String, xviii
Bradshaw Foundation, xvii
Edition Limitée, xvii
Grahame Walsh, xvii
silhouette, viii, 8, 20, 26, 36, 46, 58, 66,
92, 116, 118, 131, 132, 141
braid
category, 69
composition, 69
group, of Artin, 69
tensor product, 69
braided, 114
bialgebra, 72
tensor category, 68
braiding, 68, 83, 105
element, 72
for a bimonoid, 105
for a tensor category, 68
Brown, Ronald, xviii
Byrne, Simon, xvi
C ∗ -algebra
commutative, 5
Cartan, Eli, 33
cartesian product, x, 1
of no sets, 1
category, xi
dual, xii
functor category, xiv
functors, xiii
monoidal, 67
morphisms, xi
objects, xi
of braids, 69
of monoids in V , 100
opposite, xii
strict tensor, 104
tensor, 67
Cauchy
H-comodule, 65
module, 21
closed
functor, 86
left-closed, 77
tensor category, 78
co-element
co-T -elements, x
coaction, 62
Index
coalgebra
morphism, 38
over a ring, 37
primitive element, 41
set-like element, 40
cobraiding, 108
coCauchy, 47
cocommutative
coalgebra, 38
codomain
target, xii
coend, 115
coherence theorem
of Mac Lane, 68, 109
commutative, 3, 30
Lie algebra, 35
monoid, 3
R-algebra, 30
rig, 4
commutativity, 3
commutator
defines a Lie bracket, 32
commute with diagonals, 45
comodule, 59, 62
left, 62
comodule morphism, 62
composable morphisms, xii
composite
pasted, 100
composite morphism, xii
composition
arrow, 78
of braids, 69
vertical, 98
comultiplication, 7, 8, 37
constraint
associativity, 67
left unit, 67
right unit, 67
convergent, 24
convolution
product, 39
structure, 38
coordinate algebra, 9
quantum matrices, 9
coordinate k-algebra, 8
of general linear group, 8
coproduct, xiii
coprojection, 8
coset, 14
H-coset, 14
coterminal
object, xiii
cotwist, 108
counit, 7, 37, 81
Index
Cramer Rule
quantum, 120
derivation, 32
diagonal, 1
formal, 37
in category X, 7
ternary, 2
direct sum
of Lie algebras, 34
of modules, 24
disjoint union
X +Y, x
distributive, 4
domain
source, xii
Drinfeld, Vladimir G., 105
International Congress of
Mathematicians, 1986, ix
dual, 21
category, xii
left dual, 21, 81
opposite, xii
right dual, 81
duality principle, x
dualizing object, 80
left, 79
right, 80
end, 109, 110
is a limit, 113
endomorphism algebra, 28
enrich, 18
abelian group with module structure,
18
equalizer, 109
essentially, 104
evaluation, 77
functor, 95
morphism, 19
exponential series, 30
exterior algebra, 31
Faddeev, Ludwig D., 10
field, 4
finitely generated, 13
forgetful functor, 31, 101, 102, 115
formal
diagonal, 37
power series, 42
Formal Tannaka Duality, 111
free, 14
constructions, 31
module, 14
free module
137
from R to S, generated by X, 16
functor, 86
Freyd, Peter, 82
function
complex-valued, 5
continuous, 5
diagonal, 1
identity, 1
functor, xiii
essentially weak tensor, 103
evaluation, 95
forgetful, 101
natural transformation, xiv
representable, xiv
represented by T , xiii
self-adjoint, 54
tensor, 85
weak tensor functor, 101
functor category, xiv
fundamental theorem
on coalgebras, 115
funny superscripts, 10
Gelfand duality, 5
general linear group, 8
commutative Hopf algebra, 8
coordinate k-algebra, 8
generalised co-element, x
generate, 13
generic point, 10
geometric series, 28
Grassmannian
algebra, 10
group, 2, 7
affine over k, 7
diagrammatic definition, 7
Lie group, 7
R-algebra, 29
topological group, 7
homomorphism
generalised co-element, x
homothety, 96
Hopf algebra, 7, 51
commutative, 7, 53
Hopf monoid, 113
quasi, 105
Hurwitz polynomials, 45
ideal, 29
in an algebra, 29
identity, 1, 2, 27, 84
2-cell, 98
identity morphism, xii
incidence
138
projective plane, xi
indeterminate, 42
initial
object, xiii
injective, 24
morphism, 24
internal hom, 77, 110
left, 77
right, 78
inverse
left- and right-inverses agree, xiii
invertibility, 2
invertible, 2
isomorphism, xiii
morphism, xiii
isomorphic, xiii
isomorphism, xiii, 4
invertible, xiii
isomorphic, xiii
natural, xiv
Jacobi identity, 32
Joyal, André, xvi, 91, 117
k-algebra, 4
coordinate algebra, 8
morphism, 4
Katis, Piergiulio, 117
Kobyzev, Yu., 10
Kronecker delta, 57
left adjoint, 111
left dual, 81
as a functor, 81
of a module, 21
of a signed set, 83
left inverse, xiii
left R-linear, 14
left 2-adjoint, 111
left-closed, 77
Leibniz rule, 32
Lie, Sophus, 33
Lie algebra, 32, 33
adjoint, 33
commutative, 35
direct sum, 34
Lie bracket, 32
morphism, 33
universal enveloping, 33
Lie bracket, 32, 33
Lie group, 33
linearly independent, 14
Mac Lane, Saunders, 109, 113
Mac Lane
Index
coherence theorem, 68, 109
Manin, Yuri I., 10, 64, 65
McCrudden, Paddy, xvi
module
Cauchy, 21
finitely generated, 13
from R to S, 16
left R-module, 13
morphism, 14
projective, 21
right R-module, 13
monoid, 2, 7, 97
affine over k, 7
arrow, 97
category, 100
commutative, 3
diagrammatic definition, 7
homomorphism, 3
morphism, 3
morphisms preserve invertibility, 4
quasi-Hopf, 105
R-algebra, 29
monoidal category
tensor category, 67
monoidal functor
tensor functor, 85
Moore, Ross
LATEX2HTML, xvii
XY-pic, xvi
Morita Theory, 62
fundamental theorem, 22
morphism
algebra, 5
comodule, 62
composable, xii
composite, xii
evaluation, 19
identity, xii
invertible, xiii
left inverse, xiii
map of varieties, 6
module morphism, 14
of bialgebras, 44
of C ∗ -algebras, 5
of coalgebras, 38
of k-algebras, 4, 6
of Lie algebras, 33
of monoids, 3
of R-algebras, 27
of rigs, 4
retraction, 21
right inverse, xiii
morphisms, xi
multilinear
function, 17
Index
multiplication
opposite, 13
scalar, 13
multiplicative matrices, 64
natural family, 68
natural isomorphism, xiv
natural numbers, 4
example of a rig, 4
natural transformation, xiv, 102
naturality, xiv
non-commutating
indeterminates, 9
object
coterminal, xiii
initial, xiii
of category, 6
representing, xiv
terminal, xiii, 7
unit, 67
objects, xi
opposite
algebra, 30
category, xii
dual, xii
multiplication, 13
pasted composite, 100
pasting, 99
2-cells, 97
Planck constant, 9
Poincaré–Birkhoff–Witt, 35
point, 5, 6
as algebra morphism, 5
B-point, 6
B-point of a k-algebra, 6
of a k-algebra, 6
primitive element
in a coalgebra, 41
product
in category X, 6
of modules, 24
of objects, xii
tensor, 67
tensor product, 7
projection, xii, 24
universal property, xii
projective, 21
projective plane, xi
axioms, xi
incidence, xi
reverse, P rev , xi
QIST, 10
139
quadratic algebra, 80
category QA, 80
morphism, 80
quantization, 9
deforming commutative algebras to
non-commutative ones, 9
quantum
Cramer Rule, 120
deformation, 57
determinant, 57
general linear group, 12, 57
group, 108, 115
group over R, 108
inverse scattering transform, 10
matrices, 9
plane, 10, 80
spaces, 9
special linear group, 12
superplane, 10, 81
quantum group
cotortile bimonoid in ModR , 108
quantum spaces
correspond to k-algebras, 9
quasi-bimonoid, 104
in V , 104
quasitriangular
bialgebra, 72
R-algebra, 27
commutative, 30
group, 29
monoid, 29
skew commutative, 31
symmetric, 30
R-coalgebra, 37
representable
functor, xiv
representation, 29
appropriate, 109
of group on monoid, 29
representing object, xiv
restriction of scalars, 59, 101
retract, 21
retraction
morphism, 21
reverse-arrow universal property, 63
ribbons
tangles, 84
YB-operator, 88
rig, 4
commutative, 4
morphism, 4
natural numbers, 4
right inverse, xiii
ring, 4
140
with opposite multiplication, 13
Rivano, see Saavedra Rivano
R-Lie algebra, 32
R-module
derivation, 32
two-sided, 29
Saavedra Rivano, Neantro, 109
Sabadini, Nicoletta, 117
scalar
multiplication, 4
self-adjoint, 54
set-like element
in a coalgebra, 40
Shum Mei Chee, 84, 91
signed sets, 83
skew commutative, 31
small sets, 86
source
of a tangle, 83
space
seen from the other side of your brain,
5
span, 13
strict, 104
strong monoidal functor
tensor functor, 85
submodule, 14
generated by a subset, 15
supergeometry, 10
Sweedler, Moss E., 115
switch, 3, 70
symmetric
R-algebra, 30
tensor category, 69, 79
symmetry
for a tensor category, 68
taking off your belt, 91
Talent, Ross, xvi
tangle, 83
autonomous braided category, 83
geometric, 82
source, 83
tangles on ribbons, 84
tangles on strings, 82
target, 82
Tannaka
duality, 115, 117
duality theorem, 109
target
codomain, xii
of a tangle, 82, 83
tensor
algebra, 28
Index
functor, 85
object, 104
product, 67
tensor category, 67, 104
autonomous, 84
balanced, 73
braided, 68
closed, 78
free autonomous, 117
left autonomous, 117
opposite, 68
strict, 68, 70, 104
symmetric, 69
tortile, 82, 84
tensor functor
balanced, 86, 114, 115
braided, 85, 114
closed, 86
left-closed, 86
monoidal functor, 85
preserves dualizability, 90
preserves duals, 86
preserves products, 87
right-closed, 86
strict, 85
strong monoidal functor, 85
symmetric, 86
takes YB-operator into YB-operator,
89
weak, 85
tensor product, 67
as composition of modules, 16
multiple, 16
of R-modules, 29
of braids, 69
represent bilinear function as module
morphism, 18
tensor–hom, 61
terminal object, xiii, 7
twist, 73, 84, 106
element, 75
two-sided
R-module, 29
Ulbrich, Karl-Heinz, 114
unit, 81
left, 67
object, 67
right, 67
universal
algebra, 87
property, 115
universal enveloping algebra, 33, 40, 46,
87
is a cocommutative bialgebra, 45
141
universal property, x
end, 110
for internal hom, 78
internal hom, 110
reverse-arrow, 63
up to coherent isomorphism, 87
Vaughan, Elaine, xvi
vector space
over R, 13
Verity, Dominic R., 117
Walsh, Grahame, see Badshaws, xvii
Walters, Robert F. C., 117
weak tensor functor
essentially, 103
takes monoids to monoids, 101
whisker, 98–100
Williams, Sam, xvi
XY-pic
Ross Moore, xvi
Kristoffer Rose, xvi
Yang–Baxter, 87
hexagon, 87
YB-operator, 87
YB-hexagon, 87, 93
YB-operator
balanced, 88, 89
compatability with duals, 89
dualizable, 89
given by braiding, 89
in braided tensor category, 89
left-dualizable, 89
tortile, 90, 91, 96
under tensor functors, 89
Yetter, David, 82
Yoneda Lemma, xiv
Bradshaw: Schematized Bradshaw
Group, [Wal94, Plate 45].