Download Quantum groups: A survey of de nitions, motivations, and results

Quantum groups:
A survey of denitions, motivations, and results
Arun Ram*
Department of Mathematics
Princeton University
Princeton, NJ 08544 USA
Introduction
1.1 Goal of this survey
1.2 References for quantum groups
1.3 Some missing topics and where to nd them
1.4 Further references for the background topics
1.5 On reading these notes
1.6 Disclaimer
1.7 Acknowledgements
I. Hopf algebras and quasitriangular Hopf algebras
1. SRMCwMFFs
1.1 De
nition of an algebra
1.2 De
nition of a module
1.3 Motivation for SRMCwMFFs
1.4 De
nition of SRMCwMFFs
2. Hopf algebras
2.1 De
nition of Hopf algebras
2.2 Sweedler notation for the comultiplication
2.3 Hopf algebras give us SRMCwMFFs!
2.4 Group algebras are Hopf algebras
2.5 Enveloping algebras of Lie algebras are Hopf algebras
2.6 De
nition of the adjoint action of a Hopf algebra on itself
2.7 Motivation for the de
nition of the adjoint action
2.8 De
nition of an ad-invariant form on a Hopf algebra
3. Braided SRMCwMFFs
3.1 Motivation for braided SRMCwMFFs
* Research and writing supported in part by an Australian Research Council fellowship
and a National Science Foundation grant DMS-9622985.
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Arun Ram
3.2 De
nition of braided SRMCwMFFs
3.3 Pictorial representation of braiding isomorphisms
3.4 What \natural isomorphism" means
3.5 The braid relation
4. Quasitriangular Hopf algebras
4.1 Motivation for quasitriangular Hopf algebras
4.2 De
nition of quasitriangular Hopf algebras
4.3 Quasitriangular Hopf algebras give braided SRMCwMFFs
5. The quantum double
5.1 Motivation for the quantum double
5.2 Construction of the Hopf algebra A coop
5.3 Construction of the quantum double
5.4 If A is an in
nite dimensional Hopf algebra
5.5 An ad-invariant pairing on the quantum double
II. Lie algebras and enveloping algebras
1. Semisimple Lie algebras
1.1 De
nition of a Lie algebra
1.2 De
nition of a simple Lie algebra
1.3 De
nition of the radical of a Lie algebra
1.4 De
nition of simple modules for a Lie algebra
1.5 De
nition of the adjoint representation of a Lie algebra
1.6 De
nition of the Killing form
1.7 Characterizations of semisimple Lie algebras
2. Finite dimensional complex simple Lie algebras
2.1 Dynkin diagrams and Cartan matrices
2.2 Classi
cation of nite dimensional complex simple Lie algebras
2.3 Triangular decomposition
2.4 Weights and weight spaces
2.5 Classi
cation of simple modules
2.6 Roots and the root lattice
2.7 The inner product on hR
2.8 The Weyl group corresponding to g
3. Enveloping algebras
3.1 Motivation for the enveloping algebra
3.2 De
nition of the enveloping algebra
3.3 A functorial way of realising the enveloping algebra
Quantum Groups: A Survey
3.4 The enveloping algebra is a Hopf algebra
3.5 Modules for the enveloping algebra and the Lie algebra are the same!
3.6 The Lie algebra can be recovered from its enveloping algebra!
3.7 A basis for the enveloping algebra
4. The enveloping algebra of a complex simple Lie algebra
4.1 A presentation by generators and relations
4.2 Triangular decomposition
4.3 Grading on Un; and Un+
4.4 Poincare-Birkho-Witt bases of Un; , Uh, and Un+
4.5 The Casimir element in Ug
III. Deformations of Hopf algebras
1. h-adic completions
1.1 Motivation for h-adic completions
1.2 The algebra Ah]], an example of an h-adic completion
1.3 De
nition of the h-adic topology
1.4 De
nition of an h-adic completion
2. Deformations
2.1 Motivation for deformations
2.2 Deformation as a Hopf algebra
2.3 De
nition of equivalent deformations
2.4 De
nition of the trivial deformation as a Hopf algebra
2.5 Deformation as an algebra
2.6 The trivial deformation as an algebra
IV. Perverse sheaves
1. The category Dcb (X )
1.1 Complexes of sheaves
1.2 The category K (X ) and derived functors
1.3 Bounded complexes and constructible complexes
1.4 De
nition of the category Dcb (X )
2. Functors
2.1 The direct image with compact support functor f!
2.2 The inverse image functor f
2.3 The functor f
2.4 The shift functor n]
2.5 The Verdier duality functor D
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Arun Ram
3. Perverse sheaves
3.1 De
nition of perverse sheaves
3.2 Intersection cohomology complexes
V. Quantum groups
1. Denition, uniqueness, existence
1.1 Making the Cartan matrix symmetric
1.2 The Poisson homomorphism
1.3 The de
nition of the quantum group
1.4 Uniqueness of the quantum group
1.5 De
nition of q-integers and q-factorials
1.6 Presentation of the quantum group by generators and relations
2. The rational form of the quantum group
2.1 De
nition of the rational form of the quantum group
2.2 Relating the rational form and the original form of the quantum group
3. Integral forms of the quantum group
3.1 De
nition of integral forms
3.2 Motivation for integral forms
3.3 De
nition of the non-restricted integral form of the quantum group
3.4 De
nition of the restricted integral form of the quantum group
VI. Modules for quantum groups
1. Finite dimensional Uh g-modules
1.1 As algebras Uhg = Ugh]]
1.2 De
nition of weight spaces in a Uh g-module
1.3 Classi
cation of modules for Uh g
2. Finite dimensional Uq g-modules
2.1 Construction of the Verma module M () and the simple module L()
2.2 Twisting L() to get L( )
2.3 Classi
cation of the nite dimensional irreducible modules for Uq g
2.4 Weight spaces for Uq g-modules
VII. Properties of quantum groups
1. Triangular decomposition and grading
1.1 Triangular decomposition
1.2 The grading on Uh n+ and Uh n;
Quantum Groups: A Survey
5
2. The inner product h i
2.1 The pairing between Uh b; and Uh b+
2.2 Extending the pairing to an ad-invariant pairing on Uh g
2.3 Duality between matrix coecients for representations and Uq g
3. The universal R-matrix
3.1 Motivation for the R-matrix
3.2 Existence and uniqueness of R
3.3 Properties of the R-matrix
4. An analogue of the Casimir element
4.1 De
nition of the element u
4.2 Properties of the element u
4.3 Why the element u is an analogue of the Casimir element
5. The element Tw0
5.1 The automorphism Sh
5.2 De
nition of the element Tw0
5.3 Properties of the element Tw0
6. The Poincar
e-Birkho-Witt basis of Uh g
6.1 Root vectors in Uh g
6.2 Poincare-Birkho-Witt bases of Uh n; , Uh h, and Uh n+
6.3 The PBW-bases of Uh n; and Uh n+ are dual bases with respect to h i (almost)
7. The quantum group is a quantum double (almost)
7.1 The identi
cation of (Uhb+ ) coop with Uh b;
7.2 Recalling the quantum double
7.3 The relation between D(Uh b+ ) and Uh g
7.4 Using the R-matrix of D(Uh b+ ) to get the R-matrix of Uh g
8. The quantum Serre relations occur naturally
8.1 De
nition of the algebras Uh b+ and Uh b;
8.2 The dierence between the algebras Uh b and the algebras Uh b
8.3 A pairing between Uhb+ and Uh b;
8.4 The radical of h i is generated by the quantum Serre relations
VIII. Hall algebras and perverse sheaf algebras
1. Hall algebras
1.1 Quivers
1.2 Representations of a quiver
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Arun Ram
1.3 De
nition of the Hall algebra
1.4 Connecting Hall algebras to the quantum group
2. An algebra of perverse sheaves
2.1 ;-graded vector spaces and the varieties EV with GV action
2.2 De
nition of the categories QV and QT QW
2.3 The Grothendieck group K associated to the categories QV
2.4 De
nition of the multiplication in K
2.5 De
nition of the pseudo-comultiplication r: K ! K K
2.6 The symmetric form on K
2.7 De
nition of the elements L~ 2 K
2.8 The connection between K and the quantum group
2.9 Dictionary between K and Uq b+
2.10 De
nition of the constant M 0 ( !) which was used in (2.7)
2.11 De
nition of the vector spaces Hj+2dim(Gn) (u!(t s B1 t s B2)) from (2.6)
2.12 Some remarks on Part II of Lusztig's book
3. The connection between representations of quivers and perverse sheaves
3.1 Relating orbits and isomorphism classes of representations of ;
3.2 Realizing the structure constants of the Hall algebra in terms of orbits
3.3 Rewriting the Hall algebra in terms of functions constant on orbits
3.4 The isomorphism between K and K
IX. Link invariants from quantum groups
1.1 Knots, links and isotopy
1.2 Link invariants
1.3 Braids
1.4 Every link is the closure of a braid
1.5 Markov equivalence
1.6 Quantum dimensions and quantum traces
1.7 Quantum traces give us link invariants!
References
Introduction
7
Introduction
(1.1) Goal of this survey
The theory of quantum groups began its development in about 1982-1985. It is now
10 years since the 1986 ICM address of V.G. Drinfel'd ignited a wild frenzy of research
activity in this area and things related to it. During this time quantum groups have
become a \household" term in Lie theory in much the same way that Kac-Moody Lie
algebras did in the 1970's. Given that quantum groups are now a part of every day Lie
theory it seems desirable that there are treatments of the subject which are accessible to
graduate students.
It has been my goal to produce a survey which is accessible to graduate students, and
which contains the necessary background and the main results in the theory. I have chosen
to make this a compendium of motivation, de
nitions and results. A secondary goal has
been to write this in a relatively small space (long works are usually too daunting) and
with this in mind I have chosen not to include any proofs. In many cases, providing a full
proof would require introducing and developing some fairly sophisticated tools.
My main focus in these notes is to give a description of what the Drinfel'd-Jimbo
quantum groups are, how one arrives at them and why they are natural. In the last
chapter I shall explain how the Drinfel'd-Jimbo quantum groups are applied to get link
invariants such as the Jones polynomial.
(1.2) References for quantum groups
Drinfel'd's paper in the proceedings of the ICM 1986 is a dense summary of many of
the amazing results that he had obtained. This paper still remains a basic reference.
Dr] V.G. Drinfeld, Quantum Groups, in Proceedings of the International Congress of
Mathematicians, A.M. Gleason ed., pp. 798-820, American Mathematical Society,
Providence 1987.
Between 1987 and 1995 literally thousands of papers on quantum groups have been
published. The book by V. Chari and A. Pressley which appeared in 1994 has 70 pages
of references in minuscule type! Instead of wading through this mass of literature I have
decided to only refer you to the books on quantum groups which have begun to appear
recently, as follows:
CP] V. Chari and A. Pressley, \A Guide to Quantum Groups", Cambridge University
Press, Cambridge, 1994.
Ja] J. Jantzen, \Lectures on Quantum Groups", Graduate Studies in Mathematics
Vol. 6, American Mathematical Society, 1995.
Jo] A. Joseph, \Quantum groups and their Primitive Ideals", Ergebnisse der Mathematik und ihrer Grenzgebiete 3 Folge, Bd. 29, Springer-Verlag, New YorkBerlin, 1995.
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Arun Ram
Ka] C. Kassel, \Quantum groups", Graduate Texts in Mathematics 155, SpringerVerlag, New York, 1995.
Lu] G. Lusztig, \Introduction to Quantum Groups", Progress in Mathematics 110,
Birkhauser, Boston, 1993.
Ma] S. Majid, \Foundations of quantum group theory", Cambridge University Press,
1995.
SS] S. Shnider and S. Sternberg, \Quantum groups: From Coalgebras to Drinfel'd
Algebras", Graduate Texts in Mathematical Physics Vol. 2, International Press,
Cambridge, MA 1995.
I recommend CP] for obtaining a basic understanding of what quantum groups are,
where they came from, what the main results are, and what was known as of about the
end of 1993. It contains only easy proofs and sketches of more involved proofs, very
often referring the reader to the original papers for the full details of proofs. This book,
however, is very useful for understanding what is going on. The recent book Ja] is written
speci
cally for graduate students. It has an excellent choice of topics, thorough descriptions
of the motivations at each stage and detailed proofs. The book SS] treats the deformation
theory aspect of quantum groups in detail and the book Lu] is the only one that covers
the connection between the quantum group and perverse sheaves.
(1.3) Some missing topics and where to nd them
There are many beautiful things in the theory of quantum groups that we won't even have
time to mention. A few of these are:
(a) Canonical and crystal bases and the Littelmann path model for representations,
see Jo] Chapt. 5-6 and Ja] Chapt. 9-11.
(b) Yangians, see CP] Chapt. 12.
(c) Quasi-Hopf algebras and twisting, see CP] Chapt. 16 and SS] Chapt. 8.
(d) The Knizhnik-Zamalodchikov equation and hypergeometric functions, see CP]
Chapt. 16, Ka] Chapt. 19 and SS] Chapt. 12.
(e) Lie bialgebras, Poisson Lie groups, and symplectic leaves, see CP] Chapt. 1.
(f) Representations at roots of unity and the connection to representations of algebraic groups over a nite eld, see CP] Chapt. 11 and AJS].
(g) The connection between representations of quantum groups at roots of unity and
representations of ane Lie algebras at negative level, see CP] Chapt. 11 and
Chapt. 16 and KL].
(1.4) Further references for the background topics
Chapters I-IV consist of background material needed for the material on quantum
groups. These chapters are:
Introduction
9
I. Hopf algebras and braided tensor categories
II. Lie algebras and enveloping algebras
III. Deformations of Hopf algebras
IV. Perverse Sheaves
The following book contains a very nice up-to-date account of the theory of Hopf algebras,
and it also includes some useful things on quantum groups.
Mo] S. Montgomery, \Hopf Algebras and their Actions on Rings", Regional Conference
Series in Mathematics 82, American Mathematical Society, 1992.
The book by Chari and Pressley CP] contains a nice introduction to monoidal categories
and braided monoidal categories.
The following little book is a beautiful summary of the main results in semisimple Lie
theory.
Se] J.-P. Serre, \Complex Semisimple Lie algebras", Springer-Verlag, New York,
1987.
Comprehensive accounts of the theory of Lie algebras and enveloping algebras can be found
in Bourbaki and in the book by Dixmier.
Bou] N. Bourbaki, \Groupes et Algebres de Lie, Chapitres I-VIII", Masson, Paris,
1972.
Dix] J. Dixmier, \Enveloping algebras", Amer. Math. Soc. (1994) originally published in French by Gauthier-Villars, Paris 1974 and in English by North Holland,
Amsterdam 1977.
The following are standard (and very useful) texts in Lie theory.
Hu] J. Humphreys, \Introduction to Lie algebras and representation theory", Graduate Texts in Mathematics 9, Springer-Verlag, New York-Berlin, (3rd printing)
1980.
K] V. Kac, \In
nite dimensional Lie algebras", Birkhauser, Boston, 1983.
The most comprehensive reference for modern deformation theory, especially in regard to
deformations of Hopf algebras, is the book by Shnider and Sternberg SS] listed above.
The book CP] also contains a very informative chapter on deformation theory.
Unfortunately, to my knowledge, there is no good introductory text on the theory of
perverse sheaves. The classical reference is the following monograph.
BBD] A. Beilinson, J. Bernstein, and P. Deligne, Faisceaux pervers, Asterisque 100
(1982), Soc. Math. France.
On the other hand, much of the background material to perverse sheaves, such as homological algebra and sheaf theory is classical and appears in many books. The rst few
chapters of the following book contain an introduction to these topics.
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Arun Ram
KS] M. Kashiwara and P. Schapira, \Sheaves on Manifolds", Grundlehren der mathematischen Wissenschaften 292, Springer-Verlag, New York-Berlin, 1980.
(1.5) On reading these notes
I advise the reader to begin immediately with Chapter V and nd out what a quantum
group is. One can always peek back at the earlier chapters and nd out the de
nitions later.
This makes it more fun and provides good motivation for learning the earlier background
material. It also avoids getting bogged down before one even gets to the quantum group.
In a number of places I have chosen to make these notes \nonlinear". There have been
some occasions when I have decided to repeat some de
nition or some statement. Also in
a few places, I have used some terms and notations that have not been de
ned yet, with
an appropriate reference to the place later in the text where the de
nitions and notations
can be found. I have done this with the intention of making each section a somewhat
complete set of ideas without disrupting any particular section with a myriad of lengthy
de
nitions. Even though we may wish it so, ideas in mathematics are not really linear and
this has been reected in these notes. The reader should feel free to skip around in the
notes whenever the inclination arises.
I have included a complete table of contents in the hope that it will be helpful to the
reader as a tool for nding de
nitions and for organizing and motivating the structures.
For the same reason I have given every small section a title. This way the reader can follow
the process of the development, as well as the details. Think of the table of contents as a
ow chart for the mathematics.
(1.6) Disclaimer
Even though the theory of quantum groups is less than 15 years old I shall not undertake the complicated task of giving appropriate references and credits concerning the
sources of the theorems and their rst proofs. I refer the reader to the above books on
quantum groups for this information.
Let me stress that none of the theorems stated in this manuscript are due to me with
two possible exceptions. Chapt. I Proposition (5.5) and Chapt. VII Theorem (5.2) are
more general than I know of in the existing literature. Chapt. I Proposition (5.5) is well
known in the context of the quantum group and I am only pointing out here that the well
known proof, see Ta] Prop. 2.2.1, works for any quantum double. Chapt. VII Theorem
(5.2) is a nontrivial, but very natural, extension of well known results which appear, for
example, in Ja] Chapt. 8. The crucial part of the proof is similar to the proof of Ja]
Lemma 8.3.
I have tried to indicate, at the beginning of each chapter, where one can nd proofs of
the theorems stated in that chapter. In many instances I have had to make minor changes
in notations and statements in order to be consistent with the de
nitions that I have given.
Especially since I have not included proofs the reader should be watchful and open to the
possibility that there may be some minor errors.
Introduction
(1.7) Acknowledgements
11
First and foremost I thank Hans Wenzl for introducing me to the world of quantum
groups and encouraging me to pursue research in related topics. He taught me the basics
of quantum group theory and notes from a course he gave at University of California, San
Diego have been tremendously useful over the years. I thank all of the audience members
in my course in quantum groups at University of Wisconsin during Spring of 1994 for their
interest, their suggestions and for coming so early in the morning.
I thank Gus Lehrer for inviting me to Australia, for making my year there a wonderful
one and for suggesting my name for various invitations via which these notes have come
into being. I thank Chuck Miller and John Cossey for the invitation to give a series of
lectures on quantum groups to graduate students at the Workshop on Algebra, Geometry
and Topology at Australian National University in Canberra during January 1996. These
notes are an expanded version of the notes I distributed there. I thank Michael Murray
and Alan Carey for inviting me to speak at the Australian Lie Groups Conference 1996
and for inviting me to contribute to these proceedings. Finally, I thank Dave Benson for
some very helpful proofreading.
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Arun Ram
I. Hopf algebras and quasitriangular Hopf algebras
Let k be a eld. Unless otherwise speci
ed all maps between vector spaces over k are
assumed to be k-linear and, if V is a vector space over k, then idV : V ! V denotes the
identity map from V to V .
The proofs of most of the statements in this chapter can be found in Mo]. The proof
that the antipode is an antihomomorphism (2.1) is given in Sw] 4.0.1. The statement of
Theorem (5.3), giving the construction of the quantum double, is given explicitly in D1]
x13, and the proof can be found in Ma] p. 287-289. A statement similar to Proposition
(5.5) is in Ta] Prop. 2.2.1 and the proof is similar to the proof given there.
1. SRMCwMFFs
(1.1) Denition of an algebra
An algebra over k is a vector space A over k with a multiplication
m: A A ;!
A
a b 7;! a b = ab
and an identity element 1A 2 A such that
(a) m is associative, i.e. (ab)c = a(bc), for all a b c 2 A, and
(b) 1A a = a 1A = a, for all a 2 A.
Equivalently, an algebra over k is a vector space A over k with a multiplication m: AA ! A
and a unit : k ! A such that
(a) m is associative, i.e. m (m idA) = m (idA m), and
(b) (unit condition) m ( idA ) = m (idA ) = idA .
The relationship between the identity 1A 2 A and the unit : k ! A is (1) = 1A . If we are
being precise we should denote an algebra over k by a triple (A m ) or (A m 1A) but we
shall usually be lazy and simply write A.
(1.2) Denition of a module
Let A be an algebra over k. An A-module is a vector space M over k with an A-action
A M ;!
M
a m 7;! a m = am
such that
(a) (ab)m = a(bm), for all a b 2 A and m 2 M , and
(b) 1Am = m, for all m 2 M .
I. Hopf algebras and quasitriangular Hopf algebras
13
Let M and N be A-modules. An A-module morphism from M to N is a map ': M ! N
such that
'(am) = a'(m) for all a 2 A and m 2 M .
The set of A-module morphisms from M to N is denoted HomA (M N ). An A-module is
nite dimensional if it is nite dimensional as a vector space over k.
(1.3) Motivation for SRMCwMFFs
Our interest will be in special algebras for which the category of nite dimensional Amodules has a lot of nice structure. We want to be able to take the tensor product of two
A-modules and get a new A-module, we want to be able to take the dual of an A-module
and get a new A-module and we want to have a 1-dimensional \trivial" A-module.
(1.4) Denition of SRMCwMFFs
Let A be an algebra over k. The category of nite dimensional A-modules is a strict rigid
monoidal category such that the forgetful functor is monoidal (a SRMCwMFF for short) if
(a) For every pair M N of nite dimensional A-modules there is a given A-module
structure on M N ,
(b) For every nite dimensional A-module M there is a given A-module structure on
M = Homk (M k),
(c) There is a distinguished one-dimensional A-module 1 with a distinguished basis
element 1 2 1,
and the following conditions are satis
ed:
(1) For all nite dimensional A-modules M , N , and P ,
(M N ) P = M (N P )
as A-modules*.
(2) The maps
M
1 M ;!
1 m 7;! m
and
M
M 1 ;!
m 1 7;! m
are A-module isomorphisms.
(3) For each nite dimensional A-module M , the maps
M M ;!
1
' m 7;! '(m) 1
and
1 ;!
PM M
1 7;! i mi 'i
are A-module morphisms.
* Strictly speaking we can only identify (M N ) P and M (N P ) up to coherent
natural isomorphisms. If we are being precise this is crucial, but conceptually these two
spaces are \equal".
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Arun Ram
In condition (3) the set fmi g is a basis of M and the set f'i g is the dual basis in M , i.e.
'i 2 M is such that 'i (mj ) = ij for all i j .
The distinguished one-dimensional A-module 1 is called the trivial A module.
2. Hopf algebras
(2.1) Denition of Hopf algebras
A Hopf algebra is a vector space A over k with
a multiplication,
a comultiplication,
a unit,
a counit,
an antipode,
m: A A ;! A,
: A ;! A A,
: k ;! A,
: A ;! k, and
S : A ;! A,
such that
(1) m is associative,
(2)
m (idA m) = m (m idA )
is coassociative,
(idA ) = ( idA ) (3) (unit condition),
m (idA ) = m ( idA) = idA (4) (counit condition),
(5)
(idA ) = (
idA ) = idA is an algebra homomorphism,
m = (m m) (idA idA) ( )
(6) is an algebra homomorphism,
m = (7) (antipode condition),
I. Hopf algebras and quasitriangular Hopf algebras
15
m (idA S ) = m (S idA ) = :
In condition (5) the algebra structure on A A is given by
(a b)(c d) = ac bd for all a b c d 2 A,
and the map is given by
: A A ;! A A
a b 7;! b a:
In condition (6) we have identi
ed the vector space k k with k. One can show that the
antipode S : A ! A is always an anti-homomorphism,
S (ab) = S (b)S (a) for all a b 2 A.
(2.2) Sweedler notation for the comultiplication
Let A be a Hopf algebra over k. If a 2 A we write
(a) =
X
a
a(1) a(2)
to express (a) as an element of A A. This unusual notation is called Sweedler notation
and is a standard notation for working with Hopf algebras. Don't let it bother you, we are
simply trying to write (a) so that it looks like an element of A A, without having to
go through the rigmarole of actually choosing a basis in A.
(2.3) Hopf algebras give us SRMCwMFFs!
Let (A m S ) be a Hopf algebra over k.
(a) If M1 and M2 are A-modules de
ne an A-module structure on M1 M2 by
a(m1 m2 ) = (a)(m1 m2 ) =
X
a
a(1)m1 a(2) m2 for each a 2 A, m1 2 M1, and m2 2 M2.
(b) De
ne 1 to be the vector space 1 = k 1 and de
ne an action of A on 1 by
a 1 = (a) 1 for each a 2 A.
(c) If M is a nite dimensional A-module de
ne an A-module structure on M =
Homk (M k) by
(a')(m) = '(S (a)m) for each a 2 A, ' 2 M , and m 2 M .
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Arun Ram
The point is that if A is a Hopf algebra then, with the de
nitions in (a)-(c) above, the
category of nite dimensional A-modules is very nice it is a strict rigid monoidal category
such that the forgetful functor is monoidal.
(2.4) Group algebras are Hopf algebras
Let G be a group. The group algebra of G over k is the vector space kG of nite k-linear
combinations of elements of G,
kG =
(X
g
)
cg g cg 2 k and all but a nite number of cg = 0 with multiplication given by the k-linear extension of the multiplication in G. A G-module
is a kG-module.
(a) If M1 and M2 are G-modules de
ne a G-module structure on M1 M2 by
g(m1 m2 ) = gm1 gm2 for all g 2 G, m1 2 M1, and m2 2 M2.
(b) The trivial G-module is the 1-dimensional vector space 1 with G-action given by
g v = v for all g 2 G, v 2 1.
(c) If M is a nite dimensional G-module de
ne a G-module structure on M =
Homk (M k) by
(g')(m) = '(g;1m) for all g 2 G, m 2 M , and ' 2 M .
With these de
nitions the category of nite dimensional G-modules is a strict monoidal
category such that the forgetful functor is monoidal.
The group algebra kG is a Hopf algebra if we de
ne
(a) a comultiplication,
: kG ! kG kG, by
(g) = g g for all g 2 G,
(b) a counit, : kG ! k, by
(g) = 1 for all g 2 G,
(c) and an antipode, S : kG ! kG, by
S (g) = g;1 for all g 2 G.
I. Hopf algebras and quasitriangular Hopf algebras
17
(2.5) Enveloping algebras of Lie algebras are Hopf algebras
Let g be a Lie algebra over k and let Ug be its enveloping algebra. (See II (1.1) and II
(4.2) for de
nitions of Lie algebras and enveloping algebras.)
(a) If M1 and M2 are g-modules we de
ne a g-module structure on M1 M2 by
x(m1 m2 ) = xm1 m2 + m1 xm2 for all x 2 g, m1 2 M1, and m2 2 M2.
(b) The trivial g-module is the 1-dimensional vector space 1 with g-action given by
xv = 0 for all x 2 g, v 2 1.
(c) If M is a nite dimensional g-module we de
ne a g-module structure on M =
Homk (M k) by
(x')(m) = '(;xm) for all x 2 g, ' 2 M , and m 2 M .
With these de
nitions the category of nite dimensional g-modules is a strict rigid monoidal
category such that the forgetful functor is monoidal.
The enveloping algebra Ug of g is a Hopf algebra if we de
ne
(a) a comultiplication,
: Ug ! Ug Ug, by
(x) = x 1 + 1 x for all x 2 g,
(b) a counit, : Ug ! k, by
(x) = 0 for all x 2 g,
(c) and an antipode, S : Ug ! Ug, by
S (x) = ;x for all x 2 g.
(2.6) Denition of the adjoint action of a Hopf algebra on itself
Let (A m S ) be a Hopf algebra. The vector space A is an A-module where the
action of A on A is given by
X
A A ;! P A
where
(
a
)
=
a(1) a(2):
a b 7;! a a(1) bS (a(2))
a
The linear transformation of A determined by the action of an element a 2 A is denoted
ada . Thus,
X
ada (b) = a(1) bS (a(2)) for all b 2 A.
a
18
Arun Ram
(2.7) Motivation for the denition of the adjoint action
Let M be an A-module and let : A ! End(M ) be the corresponding representation of A,
i.e. the map
: A ;! End(M )
a 7;! (a)
where (a) is the linear transformation of M determined by the action of a. Note that
End(M ) = M M as a vector space. On the other hand M M is an A-module. If we
view A as an A-module under the adjoint action then the composite map
: A ! End(M ) = M M
is a homomorphism of A-modules.
(2.8) Denition of an ad-invariant bilinear form on a Hopf algebra
Let A be a Hopf algebra with antipode S and let M be an A-module. A bilinear form
( ): M M ! k
m n 7! (m n) is invariant if (am1 m2 ) = (m1 S (a)m2)
for all a 2 A, m1 m2 2 M . This is equivalent to the condition that the map ( ) is a
homomorphism of A-modules when we identify k with the trivial A-module 1..
A bilinear form
( ): A A ! k is ad-invariant if (ada (b1) b2) = (b1 adS(a) (b2))
for all a b1 b2 2 A. In other words, the bilinear form is invariant if we view A as an
A-module via the adjoint action.
3. Braided SRMCwMFFs
(3.1) Motivation for braided SRMCwMFFs
Our interest here will be in even more special algebras for which the category of nite
dimensional A-modules is \braided". Speci
cally, we want the two tensor product modules
M N and N M to be isomorphic.
(3.2) Denition of braided SRMCwMFFs
Let A be an algebra over k. The category of nite dimensional A-modules is a braided strict
rigid monoidal category such that the forgetful functor is monoidal (a braided SRMCwMFF
for short) if it is a strict rigid monoidal category such that the forgetful functor is monoidal
and
I. Hopf algebras and quasitriangular Hopf algebras
19
(a) There is a family of braiding isomorphisms
R" MN : M N ;! N M
which are natural isomorphisms (in the sense of the theory of categories).
(b) For all nite dimensional A-modules M N P
R" M NP = (R" MP idN ) (idM R" NP )
R" MN P = (idN R" MP ) (R" MN idP ) and
R"1M = idM = R"M1 where 1 denotes the trivial module and we identify M , 1 M , and M 1.
(3.3) Pictorial representation of braiding isomorphisms
Sometimes it is convenient to denote the isomorphism R" MN : M N ;! N M by the
picture
M N
N M
With this notation the relations de
ning a braided SRMCwMFF can be written in the
form
M N P
(M N ) P
=
P (M N )
P M N
M N P
M (N P )
=
(N P ) M
N P M
20
Arun Ram
1 M
M 1
M
=
M 1
=
M
1 M
(3.4) What \natural isomorphism" means
Let M M 0 N N 0 be A-modules and let : M ! M 0 and : N ! N 0 be A-module iso-
morphisms. Then the naturality condition on the isomorphisms R"MN means that the
following diagrams commute.
idN M 0 N
M N ;!
?
RM N?
y
N M
??
yRM
idN 0
M N
?
idM RM N?
y
N
;! M N 0
??
yRM N
0
idM N 0 M
N M ;!
;! N M 0
Pictorially we have
M N
M N
M N
M0 N = N M
and
N M0
M N
N M0
M N0 = N M
N0 M
N0 M
(3.5) The braid relation
The relations in (3.3) imply the following relation which is usually called the braid relation.
M N P
M N P
M (N P )
M N P
= M (P N ) = (N P ) M =
P N M
(P N ) M
P N M
P N M
I. Hopf algebras and quasitriangular Hopf algebras
21
where the middle equality is a consequence of the naturality property and the fact that
the map R"NP is an isomorphism.
4. Quasitriangular Hopf algebras
(4.1) Motivation for quasitriangular Hopf algebras
In addition to the de
nition of a braided SRMCwMFF the following observations help to
motivate the de
nition of a quasitriangular Hopf algebra.
Let (A m S ) be a Hopf algebra and let be the k-linear map
: A A ;! A A
a b 7;! b a:
Let
op = so that, if a 2 A and
(a) =
X
a
a(1) a(2) then
op (a) = X a
a
(2) a(1) :
Then (A m op S ;1) is a Hopf algebra.
The map : A A ! A A is an algebra automorphism of A A (the algebra
structure on A A is as given in (2.1)) and the following diagram commutes
A
??
;!
AA
??
y
idAy
A
;!
AA
op
Sometimes we are lucky and we can replace by an inner automorphism.
(4.2) Denition of quasitriangular Hopf algebras
A quasitriangular Hopf algebra is a pair (A R) where A is a Hopf algebra and R is an
invertible element of A A such that
op (a) = R
(a)R;1 for all a 2 A, and
( idA )(R) = R13 R23 and (idA )(R) = R13 R12 where, if R =
R12 =
P a b then
i i
X
ai bi 1 R13 =
X
ai 1 bi and R23 =
X
1 ai bi :
22
Arun Ram
(4.3) Quasitriangular Hopf algebras give braided SRMCwMFFs
Let (A R) be a quasitriangular Hopf algebra. For each pair of nite dimensional A-modules
M N de
ne
R"MN : M N ;! P N M
m n 7;!
bi n ai m
P
where R = ai bi 2 A A. Then the category of nite dimensional A-modules is a
braided strict rigid monoidal category such that the forgetful functor is monoidal.
5. The quantum double
(5.1) Motivation for the quantum double
In general it can be very dicult to nd quasitriangular Hopf algebras, especially ones
where the element R is dierent from 1 1. The construction in (5.3) says that, given a
Hopf algebra A, we can sort of paste it and its dual A together to get a quasitriangular
Hopf algebra D(A) and that the R for this new quasitriangular Hopf algebra is both a
natural one and is nontrivial.
(5.2) Construction of the Hopf algebra A coop
Let (A m S ) be a nite dimensional Hopf algebra over k. Let A = Homk (A k)
be the dual of A. There is a natural bilinear pairing h i: A A ;! k between A and A
given by
h ai = (a) for all 2 A and a 2 A.
Extend this notation so that if 1 2 2 A and a1 a2 2 A then
h1 2 a1 a2i = h1 a1ih2 a2i:
We make A into a Hopf algebra, which is denoted A coop , by de
ning a multiplication
and a comultiplication on A via the equations
h12 ai = h1 2 (a)i and h op () a1 a2i = h a1a2 i
for all 1 2 2 A and a a1 a2 2 A. The de
nition of op is in (4.1).
(a) The identity in A coop is the counit : A ! k of A.
(b) The counit of A coop is the map
: A ! k
7! (1)
where 1 is the identity in A.
(c) The antipode of A coop is given by the identity hS () ai = h S ;1(a)i, for all
2 A and all a 2 A.
I. Hopf algebras and quasitriangular Hopf algebras
23
(5.3) Construction of the quantum double
We want to paste the algebras A and A coop together in order to make a quasitriangular
Hopf algebra D(A). There are three main steps.
(1) We paste A and A coop together by letting
D(A) = A A coop :
Write elements of D(A) as a instead of as a .
(2) We want the multiplication in D(A) to reect the multiplication in A and the multiplication in A coop . Similarly for the comultiplication.
(3) We want the R-matrix to be
X
R = bi bi
i
where fbi g is a basis of A and fbi g is the dual basis in A .
The condition in (2) determines the comultiplication in D(A),
(a) = () (a) =
P
X
a(1)(1) a(2)(2) a
P
where (a) = a a(1) a(2) and () = (1) (2) : The condition in (2) doesn't
quite determine the multiplication in D(A). We need to be able to expand products like
(a11 )(a22 ). If we knew
1a2 =
X
j
bj j for some elements j 2 A coop and bj 2 A
then we would have
(a1 1)(a22) =
X
j
(a1bj )(j 2 )
which is a well de
ned element of D(A). Miraculously, the condition in (3) and the equation
R (a)R;1 =
op (a)
for all a 2 A,
force that if 2 A coop and a 2 A then, in D(A),
a =
a =
X
h(1) S ;1(a(1) )ih(3) a(3)ia(2) (2) and
a
X
a
h(1) a(1)ih(3) S ;1(a(3) )i(2)a(2)
24
where, if
Arun Ram
is the comultiplication in D(A),
( id) (a) =
X
a
a(1) a(2) a(3) and ( id) () =
X
(1) (2) (3) :
These relations completely determine the multiplication in D(A). This construction is
summarized in the following theorem.
Theorem. Let A be a nite dimensional Hopf algebra over k and let A coop be the Hopf
algebra A = Homk (A k) except with opposite comultiplication. Then there exists a
unique quasitriangular Hopf algebra (D(A) R) given by
(1) The k-linear map
A A coop ;! D(A)
a
7;! a
is bijective.
(2) D(A) contains A and A coop as Hopf subalgebras.
(3) The element R 2 D(A) D(A) is given by
R=
X
i
bi bi where fbig is a basis of A and fbi g is dual basis in A coop.
In condition (2), A is identi
ed with the image of A 1 under the map in (1) and A coop
is identi
ed with the image of 1 A coop under the map in (1).
(5.4) If A is an innite dimensional Hopf algebra
It is sometimes possible to do an analogous construction when A is in
nite dimensional
if onePis careful about what the dual of A is and how to express the (now in
nite) sum
R = i bi bi . To get an idea of how this is done see VII (7.1) and Lu] Chapt. 4.
(5.5) An ad-invariant pairing on the quantum double
Proposition. Let (A m S ) be a Hopf algebra. The bilinear form on the quantum
double D(A) of A which is de ned by
ha b i = h S (a)ih S ;1(b)i for all a b 2 A and all 2 A coop,
satis es
hadu (x) yi = hx adS(u) (y)i for all u x y 2 D(A).
The proposition says that the bilinear form is ad-invariant, as de
ned in (2.8). This bilinear
form is not necessarily symmetric,
hy xi = hx S 2(y)i for all x y 2 D(A).
II. Lie algebras and enveloping algebras
25
II. Lie algebras and enveloping algebras
All of the statements in x1 are proved in Se] Chapts. I-III. The statements in x3,
except possibly (3.6), are proved in Dix] Chapt. 2. The proof that the Lie algebra can be
recovered from its enveloping algebra (3.6) can be found in Bou] II x1.4. The classi
cation
theorem for semisimple Lie algebras, Theorem (2.2), is proved in Se] VI x4 Theorem 7.
The results in (2.4) and (2.5) on the classi
cation of nite dimensional modules for simple
Lie algebras are proved in Se] VII x1-4. Theorem (2.8) is proved in Bou] Chapt 6 x1.3
and Proposition (2.8) is proved in Bou] Chapt 6 x1.6 Cor. 2 and Cor. 3.
1. Semisimple Lie algebras
(1.1) Denition of a Lie algebra
Let k be a eld. A Lie algebra over k is a vector space g over k with a bracket ] : g g ! g
which satis
es
x x] = 0 for all x 2 g,
x y z]] + z x y]] + y z x]] = 0 for all x y z 2 g.
The rst relation is the skew-symmetric relation and is equivalent to x y] = ;y x], for
all x y 2 g, provided that char k 6= 2. The second relation is the Jacobi identity. A Lie
algebra g over k is nite dimensional if it is nite dimensional as a vector space over k,
and it is complex if k = C .
(1.2) Denition of a simple Lie algebra
An ideal of g is a subspace a g such that
x a] 2 a for all x 2 g, and a 2 a .
A Lie algebra g is abelian if x y] = 0 for all x y 2 g. A nite dimensional Lie algebra g
over a eld k of characteristic 0 is simple if
(1) g is not the one dimensional abelian Lie algebra,
(2) The only ideals of g are 0 and g.
(1.3) Denition of the radical of a Lie algebra
Let g be a nite dimensional Lie algebra over a eld k of characteristic 0. If a g is an
ideal of g de
ne
D1 a = a and Dn a = Dn;1 a Dn;1 a] for n 2.
26
Arun Ram
An ideal a of g is solvable if there exists a positive integer n such that Dn a = 0. The
radical of g is the largest solvable ideal of g. A nite dimensional Lie algebra is semisimple
if its radical is 0.
(1.4) Denition of simple modules for a Lie algebra
Let g be a Lie algebra over a eld k. A g-module is a vector space V over k with a g-action
gV
;!
V
x v 7;! x v = xv
such that
x y] v = x(yv) ; y(xv) for all x y 2 g, and v 2 V .
A representation of g on a vector space V is a map
: g ;! End(V ) such that (x y]) = (x)(y) ; (y)(x)
x 7;! (x)
for all x y 2 g. Every g-module V determines a representation of g on V (and vice versa)
by the formula
(x)v = xv for all x 2 g, and v 2 V .
A submodule of a g-module V is subspace W V such that xw 2 W for all x 2 g and
w 2 W . A simple or irreducible g-module is a g-module V such that the only submodules
of V are 0 and V . A g-module V is completely decomposable if V is a direct sum of simple
submodules.
(1.5) Denition of the adjoint representation of a Lie algebra
Let g be a nite dimensional Lie algebra over a eld k. The vector space g is a g-module
where the action of g on g is given by
gg
;!
g
x y 7;! x y]:
The linear transformation of g determined by the action of an element x 2 g is denoted
adx . Thus,
adx (y) = x y] for all y 2 g.
The representation
ad: g ;! End(g)
x 7;! adx
is the adjoint representation of g.
(1.6) Denition of the Killing form
II. Lie algebras and enveloping algebras
27
Let g be a nite dimensional Lie algebra over a eld k. The Killing form on g is the
symmetric bilinear form h i : g g ! k given by
hx yi = Tr(adx ady ) for all x y 2 g.
The Killing form h i is invariant, i.e.
hx y] zi + hy x z]i = 0 for all x y z 2 g.
(1.7) Characterizations of semisimple Lie algebras
Theorem. A nite dimensional Lie algebra g over a eld k of characteristic 0 is semisimple
if any of the following equivalent conditions holds:
(1) g is a direct sum of simple Lie subalgebras.
(2) The radical of g is 0.
(3) Every nite dimensional g module is completely decomposable and g = g g].
(4) The Killing form on g is non-degenerate.
2. Finite dimensional complex simple Lie algebras
(2.1) Dynkin diagrams and Cartan matrices
A Dynkin diagram is one of the graphs in Table 1. A Cartan matrix is one of the matrices
in Table 2. The (i j ) entry of a Cartan matrix is denoted j (Hi). Notice that every
Cartan matrix satis
es the conditions,
(1) i(Hi ) = 2, for all 1 i r,
(2) j (Hi ) is a non positive integer, for all i 6= j ,
(3) i(Hj ) = 0 if and only if j (Hi ) = 0.
If C is a Cartan matrix the vertices of the corresponding Dynkin diagram are labeled by
i 1 i r, such that i (Hj )j (Hi ) is the number of lines connecting vertex i to vertex
j . If j (Hi ) > i (Hj ) then there is a > sign on the edge connecting vertex j to vertex
i , with the point towards i . With these conventions it is clear that the Cartan matrix
contains exactly the same information as the Dynkin diagram each can be constructed
from the other.
(2.2) Classication of nite dimensional complex simple Lie algebras
Fix a Cartan matrix C = (j (Hi))1ijr . Let gC be the Lie algebra over
generators
X1; X2; : : : Xr; H1 H2 : : : Hr X1+ X2+ : : : Xr+
C
given by
28
Arun Ram
and relations
for all 1 i j r,
Hi Hj ] = 0,
Hi Xj+] = j (Hi )Xj+,
for all 1 i j r,
Hi X ;] = ;j (Hi)X ; ,
j
j
Xi+ Xj;] = ij Hi,
for 1 i j r,
|Xi+ Xi+{z Xi+} Xj+]] ] = 0,
;j (Hi )+1 brackets
X ; X ; X ; X ; ]] ] = 0,
|
i
i{z
i} j
;j (Hi )+1 brackets
for i 6= j .
Theorem. Let C be a Cartan matrix and let gC be the Lie algebra de ned above.
(1) The Lie algebra gC is a nite dimensional complex simple Lie algebra.
(2) Every nite dimensional complex simple Lie algebra is isomorphic to gC for some
Cartan matrix C .
(3) If C C 0 are Cartan matrices then
gC
'
gC
0
if and only if C = C 0 :
(2.3) Triangular decomposition
Fix a Cartan matrix C = (i(Hj ))1ijr and let g = gC . De
ne
n; = Lie subalgebra of g generated by X1; X2; : : : Xr; .
h = C -span fH1 H2 : : : Hr g,
n+ = Lie subalgebra of g generated by X1+ X2+ : : : Xr+ ,
The elements X1; X2; : : : Xr; H1 : : : Hr X1+ X2+ : : : Xr+ are linearly independent in g
and
g = n; h n+ II. Lie algebras and enveloping algebras
29
The Lie subalgebra h g is a Cartan subalgebra of g and the Lie subalgebra b = h n+ is
a Borel subalgebra of g. The rank of g is r = dim h.
(2.4) Weights and weight spaces
Fix a Cartan matrix C = (j (Hi ))1ijr and let g = gC . Let h = HomC (h C ) and de
ne
the fundamental weights !1 : : : !r 2 h by
!i (Hj ) = ij for 1 i j r.
P
Let V be a g-module and let = ri=1 i !i 2 h . The subspace
V = fv 2 V j hv = (h)v for h 2 hg
= fv 2 V j Hi v = i v for 1 i rg
is the -weight space of V . Vectors v 2 V are weight vectors of V of weight , wt(v) = .
The weights of the g-module V are the elements 2 h such that V 6= 0. If is a weight
of V , the multiplicity of in V is dim(V ). A highest weight vector in a g-module V is a
weight vector v 2 V such that n+ v = 0 or, equivalently, a weight vector v 2 V such that
Xi+v = 0, for 1 i r.
The set of dominant integral weights P + and the weight lattice P are the subsets of
h given by
r
r
X
X
P + = N !i and P = Z!i respectively,
where N = Z0.
i=1
i=1
(2.5) Classication of simple g-modules
Theorem. Let g be a nite dimensional complex simple Lie algebra. Every nite dimensional g-module V is a direct sum of its weight spaces and all weights of V are elements of
P,
M
V =
V :
2P
Theorem. Let g be a nite dimensional complex simple Lie algebra.
(1) Every nite dimensional irreducible g-module V contains a unique, up to constant
multiples, highest weight vector v+ 2 V and wt(v+) 2 P + .
(2) Conversely, if 2 P + , then there is a unique (up to isomorphism) nite dimensional
irreducible g-module, V , with highest weight vector of weight .
(2.6) Roots and the root lattice
30
Arun Ram
Fix a Cartan matrix C = (j (Hi))1ijr and let g = gC . The adjoint action of g on g
(see (1.5)) makes g into a nite dimensional g-module. An element 2 P , 6= 0 is a root
if the weight space g 6= 0. A root is positive, > 0, if g n+ and negative, < 0, if
g n; . We have
dim g = 1 for all roots ,
n; =
M
<0
g h = g0 n+ =
M
>0
g and g = n; h n+ :
The roots i , 1 i r, given by gi = C Xi+ are the simple roots. The Cartan matrix is
the transition matrix between the simple roots and the fundamental weights,
i =
r
X
j =1
i (Hj )!j for 1 i r.
The root lattice is the lattice Q P h given by Q =
(2.7) The inner product on hR
r
X
i=1
Zi.
Let g be a nite dimensional complex simple Lie algebra and let C = (j (Hi))1ijr
be the corresponding Cartan matrix. There exist unique positive integers d1 d2 dr
such that gcd(d1 dr ) = 1 and the matrix (dij (Hi ))1ijr is symmetric. The integers
d1 d2 dr are given explicitly by
Ar Dr ,
di = 1 for all 1 i r,
E6 E7 E8 :
Br :
di = 1 for 1 i r ; 1, and dr = 2,
Cr :
di = 2, for 1 i r ; 1, and dr = 1,
F4 :
d1 = d2 = 1, and d3 = d4 = 2,
G2 :
d1 = 3, and d2 = 1.
Let 1 : : : r be the simple roots for g. De
ne
hR =
r
X
i=1
R i so that hR is a real vector space of dimension r. De
ne an symmetric inner product on hR
by
(i j ) = di i (Hj )
for i j r,
where the values j (Hi ) are the entries of the Cartan matrix corresponding to g.
(2.8) The Weyl group corresponding to g
II. Lie algebras and enveloping algebras
31
Let g be a nite dimensional complex simple Lie algebra and let R be the set of roots of g
and let 1 : : : r be the simple roots. For each root 2 R de
ne a linear transformation
of hR by
s() = ; ( _) where _ = (2) :
The Weyl group corresponding to g is the group of linear transformations of hR generated
by the reections s , 2 R,
W = < s j 2 R > :
The simple reections in W are the elements si = si , 1 i r.
Theorem. Let g be a nite dimensional complex simple Lie algebra and let W be the
Weyl group corresponding to g.
(a) The Weyl group W is a nite group.
(b) The Weyl group W can be presented by generators s1 : : : sr and relations
s2i = 1
s|i sj s{zi sj } = s|j si s{zj si }
mij factors
where
8 2
>
<
mij = > 34
: 6
mij factors
1 i r
for i 6= j ,
if i (Hj )j (Hi ) = 0,
if i (Hj )j (Hi ) = 1,
if i (Hj )j (Hi ) = 2,
if i (Hj )j (Hi ) = 3.
Let w 2 W . A reduced decomposition for w is an expression
w = si1 si2 si`(w)
of w as a product of generators which is as short as possible. The length `(w) of this
expression is the length of w.
Proposition. Let g be a nite dimensional complex simple Lie algebra and let W be
the Weyl group corresponding to g.
(a) There is a unique longest element w0 in W .
(b) Let w0 = si1 siN be a reduced decomposition for the longest element of W . Then
the elements
1 = i1 2 = si1 (i2 ) : : : N = si1 si2 siN 1 (iN )
;
are the positive roots of g.
32
Arun Ram
3. Enveloping algebras
(3.1) Motivation for the enveloping algebra
A Lie algebra g is not an algebra, at least as de
ned in I (1.1), because the bracket is not
associative. We would like to nd an algebra, or even better a Hopf algebra, Ug, for which
the category of modules for Ug is the same as the category of modules for g. In other
words we want Ug to carry all the information that g does and to be a Hopf algebra.
(3.2) Denition of the enveloping algebra
L
Let g be a Lie algebra over k. Let T (g) = k0 gk be the tensor algebra of g and let J
be the ideal of T (g) generated by the tensors
x y ; y x ; x y] where x y 2 g:
The enveloping algebra of g, Ug, is the associative algebra
T (g) :
Ug =
J
There is a canonical map
0 : g ;! Ug
x 7;! x + J :
The algebra Ug can be given by the following universal property:
Let : g ! A be a mapping of g into an associative algebra A over k
such that
(x y]) = (x)(y) ; (y)(x)
for all x y 2 g, and let 1 and 1A denote the identities in Ug and A
respectively. Then there exists a unique algebra homomorphism :
Ug ! A such that (1) = 1A and = 0 , i.e. the following diagram
commutes.
0
;! Ug
??
&
y
g
A
(3.3) A functorial way of realising the enveloping algebra
If A is an algebra over k, as de
ned in I (1.1), then de
ne a bracket on A by
x y] = xy ; yx for all x y 2 A.
This de
nes a Lie algebra structure on A and we denote the resulting Lie algebra by L(A)
to distinguish it from A. L is a functor from the category of algebras to the category of
II. Lie algebras and enveloping algebras
33
Lie algebras. U is a functor from the category of Lie algebras to the category of algebras.
In fact U is the left adjoint of the functor L since
Homalg (Ug A) = HomLie (g L(A))
for all Lie algebras g and all algebras A.
(3.4) The enveloping algebra is a Hopf algebra
The enveloping algebra Ug of g is a Hopf algebra if we de
ne
(a) a comultiplication,
: Ug ! Ug Ug, by
(x) = x 1 + 1 x for all x 2 g,
(b) a counit, : Ug ! k, by
(x) = 0 for all x 2 g,
(c) and an antipode, S : Ug ! Ug, by
S (x) = ;x for all x 2 g.
(3.5) Modules for the enveloping algebra and the Lie algebra are the same!
Every g-module M is a Ug-module and vice versa, since there is a unique extension of the
action of g on M to a Ug-action on M .
(3.6) The Lie algebra can be recovered from its enveloping algebra!
An element x of a Hopf algebra A is primitive if
(x) = 1 x + x 1:
It can be shown that if char k = 0 then the subspace g of Ug is the set of primitive elements
of Ug. Thus, if char k = 0, we can \determine" the Lie algebra g from the algebra Ug and
the Hopf algebra structure on it.
(3.7) A basis for the enveloping algebra
The following statement is the Poincare-Birkho-Witt theorem.
Suppose that g has a totally ordered basis (xi )i2. Then the elements
xi1 xi2 xin
in the enveloping algebra Ug, where i1 i2 in is an arbitrary
increasing nite sequence of elements of #, form a basis a Ug.
34
Arun Ram
5. The enveloping algebra of a complex simple Lie algebra
(5.1) A presentation by generators and relations
Let g be a nite dimensional complex simple Lie algebra and let C = (j (Hi ))1ijr be
the corresponding Cartan matrix. Then the enveloping algebra Ug of g can be presented
as the algebra over C generated by
X1; X2; : : : Xr; H1 H2 : : : Hr X1+ X2+ : : : Xr+
with relations
for all 1 i j r,
Hi Hj ] = 0,
Hi Xj+] = j (Hi )Xj+,
for all 1 i j r,
Hi Xj;] = ;j (Hi)Xj; ,
Xi+ Xj;] = ij Hi,
X
s+t=1;j (Hi )
for 1 i j r,
(;1)s 1 ; sj (Hi ) (Xi)sXj (Xi)t = 0
for i 6= j ,
where, if a b 2 Ug, we use the notation a b] = ab ; ba. Note that since
|a a{z a} b]] ] =
` brackets
X
s+t=`
(;1)s
`
s t
s a ba for any two elements a b 2 Ug and any positive integer `, the relations for Ug are exactly
the same as the relations for g given in (2.2).
(5.2) Triangular decomposition
Let g be a nite dimensional complex simple Lie algebra as presented in (2.2). Recall from
(2.3) that g has a decomposition
g = n; h n+ where
n; = Lie subalgebra of g generated by X1; X2; : : : Xr; .
h = C -span fH1 H2 : : : Hr g,
n+ = Lie subalgebra of g generated by X1+ X2+ : : : Xr+
II. Lie algebras and enveloping algebras
35
It follows from this and the Poincare-Birkho-Witt theorem that
Ug = Un; Uh Un+ as vector spaces.
(5.3) Grading on Un+ and Un;
Let g be a nite dimensional complex simple Lie algebra as presented in (2.2). Let
1 : : : r be the simple roots for g and let
Q+ =
X
i
N i where N = Z0.
P
For each element = ri=1 i i 2 Q+ de
ne
(Un+ ) = span-fXi+1 Xi+p j Xi+1 Xi+p has j -factors of type Xj+ g
(Un; ) = span-fXi;1 Xi;p j Xi;1 Xi;p has j -factors of type Xj; g:
Then
M ;
M +
Un; =
(Un ) and Un+ =
(Un ) 2Q+
as vector spaces.
2Q+
(5.4) Poincar
e-Birkho-Witt bases of Un; , Uh, and Un+
Let g be a nite dimensional complex simple Lie algebra as presented in (2.2), let n+ ,
n; and h be as in (2.3) and recall the root spaces g from (2.6). Let W be the Weyl
group corresponding to g. Fix a reduced decomposition of the longest element w0 2 W ,
w0 = si1 siN and de
ne
1 = i1 2 = si1 (i2 ) : : : N = si1 si2 siN 1 (iN ):
The elements 1 : : : N are the positive roots g and the elements ;1 : : : ;N are the
negative roots of g.
For each root , x an element X 2 g.
Since g is 1-dimensional X is uniquely de
ned, up to multiplication by a constant. Since
;
n; =
M
<0
g n+ =
it follows that
M
>0
g h = span-fH1 H2 : : : Hr g and g = n; h n+ fX1 : : : XN g
is a basis of n+ ,
fX;1 : : : X;N g
fH1 H2 : : : Hr g
is a basis of n; , and
is a basis of h.
36
Arun Ram
Then, by the Poincare-Birkho-Witt theorem,
fXp11 Xp22 XpNN j p1 : : : pN 2 Z0g
fX;nN1 X;n22 X;n11 j n1 : : : nN 2 Z0g
fH1s1 H2s2 H1sr j s1 : : : sN 2 Z0g
(5.5) The Casimir element in Ug
is a basis of Un+ ,
is a basis of Un; , and
is a basis of Uh:
Let g be a nite dimensional simple complex Lie algebra and let h i be the Killing form
on g (see (1.6)). Let fbi g be a basis of g and let fbi g be the dual basis of g with respect
to the Killing form. Let c be the element of the enveloping algebra Ug of g given by
c=
X
i
bi bi :
Then
c is in the center of Ug.
Any central element of Ug must act on each nite dimensional simple module by a constant.
For each dominant integral weight let V be the nite dimensional simple Ug-module
indexed by (see (2.5)). Let be the element of hR given by
= 21
X
>0
where the sum is over all positive roots for g. Then the element
c acts on V by the constant ( + + ) ; ( ),
where inner product on hR is as given in (2.7).
II. Lie algebras and enveloping algebras
Ar;1 :
Br :
Cr :
37
1
2
3
r;2
r;1
1
2
3
r;1
r
1
2
3
r;1
r
r;1
Dr :
1
2
3
1
3
4
r;2
5
6
5
6
7
5
6
7
r
E6 :
2
1
3
4
E7 :
2
1
3
4
8
E8 :
2
F4 :
1
2
3
4
G2 :
1
2
Table 1. Dynkin diagrams corresponding to nite dimensional complex simple Lie algebras
38
Ar;1 :
0 2 ;1 0 B
;1 2 ;1 B
B
0 ;1 2 B
..
...
B
.
B
@
0
0
0 2 ;1 0 BB ;1 2 ;1 B 0 ;1 2 Cr : B
...
BB ...
@
0
0
02
BB 0
E6 : B
BB ;01
@0
0
2
0
;1
0
0 0
01
0C
C
0C
.. C
. C
C
;1 2 ;1 A
0 ;2 2
;1 0
0 ;1
2 ;1
;1 2
0 ;1
0
0
0
0
;1
2
0 ;1
02
BB 0
BB ;1
BB 0
BB 0
B@ 0
0
01
0C
0C
CC
0C
;1 A
2
0
2
0
;1
0
0
0
0 0
E8:
F4:
01
0C
C
0C
.. C
. C
CA
;1 2 ;1
0 ;1 2
0 2 ;1 0 BB ;1 2 ;1 B 0 ;1 2 Br : B
...
BB ...
@
0 2 ;1 0 BB ;1 2 ;1 BB
BB
@
02
BB 0
BB ;1
E7: B
BB 00
@0
..
.
0
0
0
...
;1
0
0
0
2
0
;1
0
0
0 0
;1 0
0 ;1
2 ;1
;1 2
0 ;1
0
0
0
0 2 ;1 0 0 1
B@ ;1 2 ;2 0 CA
0 ;1 2 ; 1
0
0 1
0 C
C
0 C
.. C
. C
C
;1 2 ;2 A
0 ;1 2
0
0
Dr :
0 ;1 2
0
0
0
;1
2
0 ;1
0 0
0 0
Arun Ram
0
0
0
0
;1
2
;1
0
G2 :
0
0
0
0
0
;1
2
;1
01
0C
.. C
. C
C
2 ;1 ;1 C
C
;1 2 0 A
;1 0 2
;1 0
0 ;1
2 ;1
;1 2
0 ;1
0
0
0
0
0
;1
2
0 ;1
0 0
0 1
0 C
0 C
C
0 C
C
0 C
CC
0 C
;1 A
2
0
0
0
0
;1
2
;1
01
0C
0C
C
0C
C
0C
C
;1 A
2
2 ;1 ;3 2
Table 2. Cartan matrices corresponding to nite dimensional complex simple Lie algebras
III. Deformations of Hopf algebras
39
III. Deformations of Hopf algebras
The basic material on completions given in x1 can be found in many books, in particular, AM] Chapt 10. The book SS] has a comprehensive treatment of deformation theory.
Theorem (2.6) is stated and proved in SS] Prop. 11.3.1.
1. h-adic completions
(1.1) Motivation for h-adic completions
We will be working with algebras over C h]], the ring of formal power series in a variable
h with coecients in C . A typical element of C h]] which is not in C h] is the element
2
3
eh = 1 + h + h2! + h3! + :
The ring C h]] is just C h] extended a little bit so that some nice elements that we want
to write down, like eh , are in C h]].
An algebra over C h]] is a vector space over C h]], i.e. a free C h]]-module, which has
a multiplication and an identity which satisfy the conditions in I (1.1). If A is an algebra
over C then we can extend coecients and get a new algebra A C C h]] which is over
C h]]. But sometimes this new algebra is not quite big enough so we need to extend it a
little bit and work with the h-adic completion Ah]] which contains all the nice elements
that we want to write down.
Continuing in this vein we will want to consider the tensor product Ah]] Ah]].
Again, this algebra is not quite big enough and we extend it to get a slightly bigger object
Ah]]^ Ah]] so that all the elements we want are available.
(1.2) The algebra Ah]], an example of an h-adic completion
If A is an algebra over k then the set
Ah]] = fa0 + a1h + a2 h2 + j ai 2 Ag
of formal power series with coecients in A is the completion of the kh]]-module kh]]k A
in the h-adic topology. The kh]]-linear extension of the multiplication in A gives Ah]]
the structure of a P
kh]]-algebra. The ring Ah]] is, in general, larger than kh]] k A. For
each element a = j0 aj hj 2 Ah]] the element
h2 h3 X (ha) `
2
3
ha
= 1 + a0h + (a0 + 2a1 ) 2 + (a0 + 3(a0a1 + a1 a0) + 6a2) 3! + e =
`0 `!
is a well de
ned element of Ah]].
40
Arun Ram
(1.3) Denition of the h-adic topology
Let k be a eld and let h be an indeterminate. The ring kh]] is a local ring with unique
maximal ideal (h). Let M be a kh]]-module. The sets
m + hn M m 2 M , n 2 N ,
form a basis for a topology on M called the h-adic topology. De
ne a map d: M M ! R
by
d(x y) = e;v(x;y) for all x y 2 M ,
where e is a real number e > 1 and v(x) is the largest nonnegative integer n such that
x 2 hn M . Then d is a metric on M which generates the h-adic topology.
(1.4) Denition of an h-adic completion
Let M be a kh]]-module. The completion of the metric space M is a metric space M^
which contains M in a natural way and which has a natural kh]]-module structure. The
completion M^ of M is de
ned in the usual way, as a set of equivalence classes of Cauchy
sequences of elements of M . Let us review this construction.
A sequence of elements fpn g in M is a Cauchy sequence in the h-adic topology if for
every positive integer ` > 0 there exists a positive integer N such that
pn ; pm 2 h` M for all m n > N ,
i.e. pn ; pm is \divisible" by h` for all n m > N . Two Cauchy sequences P = fpn g and
Q = fqn g are equivalent if the sequence fpn ; qn g converges to 0, i.e.
Q if for every ` there exists an N such that pn ; qn 2 h` M for all n > N .
The set of all equivalence classes of Cauchy sequences in M is the completion M^ of M .
The completion M^ is a kh]]-module where the operations are determined by
P
P + Q = fpn + qn g and aP = fapn g
where P = fpn g and Q = fqn g are Cauchy sequences with elements in M and a 2 kh]].
De
ne a map
: M ;!
M^
m 7;! (m m m : : :)]
i.e. (m) is the equivalence class of the sequence fpn g such that pn = m for all n. This
map is injective and thus we can view M as a submodule of M^ .
2. Deformations
III. Deformations of Hopf algebras
41
(2.1) Motivation for deformations
We are going to make the quantum group by deforming the enveloping algebra Ug of a
complex simple Lie algebra g as a Hopf algebra. This last condition is important because
the enveloping algebra Ug does not have any deformations as an algebra.
(2.2) Deformation as a Hopf algebra
Assume that (A m S ) is a Hopf algebra over k. Let Ah]]^ Ah]] denote the completion of Ah]] kh]] Ah]] in the h-adic topology. A deformation of A as a Hopf algebra
is a tuple (Ah]] mh h h h Sh) where
^ Ah]]
mh : Ah]]^ Ah]] ;! Ah]]
h : Ah]] ;! Ah]]
h : kh]] ;! Ah]] h : Ah]] ;! kh]] and Sh : Ah]] ;! Ah]]
are kh]]-linear maps which are continuous in the h-adic topology, satisfy axioms (1) - (7)
in the de
nition of a Hopf algebra, and can be written in the form
mh = m + m1 h + m2 h2 + h = + 1 h + 2 h2 + h = + 1 h + 2 h2 + h = + 1 h + 2 h2 + Sh = S + S1 h + S2 h2 + where, for each positive integer i,
mi : A A ;! A
i : A ;! A A
i : k ;! A i : A ;! k and Si : A ;! A
are k-linear maps which are extended rst kh]]-linearly and then to the h-adic completion.
We shall abuse language (only slightly) and call (Ah]] mh h h h Sh) a Hopf algebra
over kh]].
(2.3) Denition of equivalent deformations
Two Hopf algebra deformations (Ah]] mh h h h Sh) and (Ah]] m0h 0h 0h Sh0 ) of a
Hopf algebra (A m S ) are equivalent if there is an isomorphism
fh : (Ah]] mh h h h Sh) ;! (Ah]] m0h 0h 0h 0h Sh0 )
of h-adically complete Hopf algebras over kh]] which can be written in the form
fh = idA + f1h + f2 h2 + such that, for each positive integer i, fi : A ! A is a k-linear map which is extended
kh]]-linearly to kh]] k A and then to the h-adic completion Ah]].
42
Arun Ram
(2.4) Denition of the trivial deformation as a Hopf algebra
Let (A m S ) be a Hopf algebra. The trivial deformation of A as a Hopf algebra is
the Hopf algebra (Ah]] mh h h h Sh) over kh]] such that mh = m, h = , h = ,
h = and Sh = S (extended to Ah]]).
(2.5) Deformation as an algebra
Assume that (A m ) is an algebra over k. Let Ah]]^ Ah]] denote the completion of
Ah]] kh]] Ah]] in the h-adic topology. A deformation of A as an algebra is a tuple
(Ah]] mh h) where
mh : Ah]]^ Ah]] ;! Ah]]
h : kh]] ;! Ah]]
are kh]]-linear maps which are continuous in the h-adic topology, satisfy the axioms the
de
nition of an algebra (see I (1.1)) and can be written in the form
mh = m + m1 h + m2 h2 + h = + 1 h + 2 h2 + where, for each positive integer i,
mi : A A ;! A
i : k ;! A
are k-linear maps which are extended rst kh]]-linearly and then to the h-adic completion.
We shall abuse language (only slightly) and call (Ah]] mh h) an algebra over kh]].
This de
nition is exactly like the de
nition of a deformation as a Hopf algebra in (2.2)
above except that we only need to start with an algebra and we only require the result to
be an algebra. We can de
ne equivalence of deformations as algebras in exactly the same
way that we de
ned them for deformations as Hopf algebras except that we only require
the isomorphism fh to be an algebra isomorphism instead of a Hopf algebra isomorphism.
(2.6) The trivial deformation as an algebra
Let (A m ) be an algebra. The trivial deformation of A as an algebra is the algebra
(Ah]] mh h) over kh]] such that mh = m and h = (extended to Ah]]). The deformation of the quantum group given in V (1.3) is even more incredible if one keeps the
following theorem in mind.
Theorem. Let g be a nite dimensional complex simple Lie algebra and let Ug be the
enveloping algebra of g. Then Ug has no deformations as an algebra (up to equivalence of
deformations).
In other words, all deformations of Ug as an algebra are equivalent to the trivial deformation
of Ug.
IV. Perverse sheaves
43
IV. Perverse sheaves
To any reader that has not met sheaves before: I suggest that you don't read this
section, only refer to it a few times while you are reading Chapter VIII of these notes. The
most important thing, from the point of view of these notes, is to understand the basic
structures given in Chapter VIII anyone who is going to study these topics in more depth
can always come back and learn these de
nitions later.
A large part of the material in this section is basic material about derived categories.
This material can usually be found in texts which treat homological algebra. Everything
in this section, except the de
nition and properties of perverse sheaves given in x3 can be
found in KS] Chapt. I-III. The de
nition of a perverse sheaf is in BBD] 4.0 and the proof
of Theorem (3.1) is in BBD] Theorem 1.3.6. The Theorems in (3.2) are proved in BBD]
2.1.9-2.1.11 and Theorem 4.3.1, respectively. We shall not review the de
nition of sheaves,
it can be found in many textbooks, see KS] Chapt. II.
1. The category Dcb(X )
(1.1) Complexes of sheaves
Let X be an algebraic variety. A complex of sheaves on X is a sequence of sheaves Ai on
X and morphisms of sheaves di : Ai ! Ai+1,
; d 2 A;1 ;!
d 1 0 d0 1 d1
A = ;!
A ;! A ;! ;
such that
;
di+1di = 0:
The morphisms di : Ai ! Ai+1 are called the dierentials of the complex A. Let A and B
be complexes of sheaves. A morphism f : A ! B is a set of maps fn : An ! B n such that
the diagram
d 2
d 1
d0 A1 ;!
d1
;!
A;1 ;!
A0 ;!
;
;
??
yf
??
yf
;1
0
??
yf
1
d 2
d 1
d0 B 1 ;!
d1 ;!
B ;1 ;!
B 0 ;!
;
;
commutes.
The ith cohomology sheaf of a complex A is the sheaf
Ai ! Ai+1 )
Hi (A) = ker(
im(Ai;1 ! A )
i
We have a well de
ned complex of sheaves H(A) given by
d 2 ;1
d 1 0
d0 H1 (A) ;!
d1 ;!
H (A) ;!
H (A) ;!
;
;
44
Arun Ram
B is a morphism f : A ! B such that the induced morA quasi-isomorphism f : A;!
phism H(f ): H(A) ! H(B ) is an isomorphism. Note that every isomorphism is a quasiisomorphism but not the other way around (even though the notation may be confusing).
(1.2) The category K (X ) and derived functors
Let X be an algebraic variety. Let A and B be complexes of sheaves on X . Two morphisms
f : A ! B and g: A ! B are homotopic if there is a collection of morphisms ki : Ai ! B i;1
such that
fn ; gn = kn+1 dn + dn;1 kn :
The motivation for this de
nition is that if f and g are homotopic then H(f ) = H(g).
De
ne K (X ) to be the category given by
Objects: Complexes of sheaves on X .
Morphisms: A K (X )-morphism from a complex A to a complex B is an homotopy
equivalence class of morphisms from A to B .
This just means that, in the category K (X ), we identify homotopic morphisms.
Let A be acomplex of sheaves on X . An injective resolution of A is a quasiisomorphism A ;!
J such that J i is injective (an injective object in the category of sheaves
on X ) for all i. Let Sh(X ) denote the category of sheaves on X and let F : Sh(X ) ! Sh(X )
be a functor. The right derived functor of F is the functor RF : K (X ) ! K (X ) given by
; (d 2) F (J ;1) F;!
(d 1 )
(d0 )
(d1 )
RF (A) = F (J ) = F;!
F (J 0) F;!
F (J 1) F;!
;
;
where J is an injective resolution of A. The ith derived functor of F is the functor
Ri F : K (X ) ! Sh(X ) given by
RiF (A) = Hi (F (J ))
where J is an injective resolution of A. In other words Ri F (A) is the ith cohomology sheaf
of the complex RF (A).
(1.3) Bounded complexes and constructible complexes
A complex of sheaves A is bounded if there exists a positive integer n such that Am = 0
and A;m = 0 for all m > n.
F
An algebraic stratication of an algebraic variety X is a nite partition X = X
of X into strata such that
(a) For each , the stratum X is a smooth locally closed algebraic subvariety in X,
(b) The closure of each stratum is a union of strata, and
(c) The Whitney condition holds (see Verdier Ver]).
Let l be a prime number and let Q l be the algebraic closure of the eld Q l of l-adic
F numbers.
A sheaf F on X is Q l -constructible if there is an algebraic strati
cation X = X such
that, for each , the restriction of F to X is a locally constant sheaf of nite dimensional
IV. Perverse sheaves
45
vector spaces over Q l . A complex A 2 K (X ) is Q l -constructible if Hi (A) is Q l -constructible
for all i.
(1.4) Denition of the category Dcb (X )
Let X be a variety. Let A and B be complexes of sheaves on X . De
ne an equivalence
relation on diagrams
C ;! B
A ;
C ;! B
in K (X ) which have A and B asend points by saying that the diagram A ;
is equivalent to the diagram A ; C 0 ;! B if there exists a commutative diagram
C
x?
. ? &
D ;! B
A ;!
??
- y %
C0
A denotes that the map is a quasi-isomorphism. The bounded derived
The notation C ;
category of Q l -constructible sheaves on X is the category Dcb (X ) given by
Objects: Bounded, Q l -constructible complexes of sheaves on X .
Morphisms:
A morphism from A to B is an equivalence class of diagrams
C ;! B:
A ;
This de
nition of morphisms is a formal mechanism that inverts all quasi-isomorphisms.
It ensures (in a coherent way) that \inverses" of quasi-isomorphisms are morphisms, i.e.
that A ; B is a morphism from
A to B .
E ;! C in Db (X ) one can show
Given two morphisms A ; D ;! B and B ;
c
that there always exists a commutative diagram
A
.
D
.
&
F
B
&
E
.
&
C
D ;! B and B ;
E ;!
and one de
nes the composition of the two morphisms
A
;
F ;! C .
C to be the morphism de
ned by the diagram A ;
2. Functors
46
Arun Ram
(2.1) The direct image with compact support functor f!
A map g: X ! Y between locally compact algebraic varieties is compact if the inverse
image of every compact subset of Y is a compact subset of X .
Let f : X ! Y be a morphism of locally compact algebraic varieties. Let F be a sheaf
on X . The support, supp s, of a section s of F on an open set V is the complement in V
of the union of open sets U V such that sjU = 0.
The direct image with compact support sheaf f!F , is the sheaf on Y de
ned by setting
;(U f!F ) = fs 2 ;(f ;1(U ) F ) j f : supp s ! U is compactg
for every open set U in Y . (For a sheaf F on X and an open set U in X , ;(U F ) = F (U ).)
This de
nes a functor f! : Sh(X ) ! Sh(Y ), where Sh(X ) denotes the category of sheaves
on X .
Let f : X ! Y be a morphism of locally compact algebraic varieties. The direct image
with compact support functor f!: Dcb(X ) ! Dcb (Y ) is given by
f! = Rf!
so that f! is the right derived functor of the functor f!: Sh(X ) ! Sh(Y ).
(2.2) The inverse image functor f
Let f : X ! Y be a morphism of algebraic varieties. Let F be a sheaf on Y . The inverse
image sheaf f F is the sheaf on X associated to the presheaf
V 7;! U lim
F (U )
f (V )
for all V open in X
where the limit is over all open sets U in Y which contain f (V ). This de
nes a functor
f : Sh(Y ) ! Sh(X ), where Sh(X ) denotes the category of sheaves on X . It is very
common to denote this functor by f ;1 but we shall follow BBD] and Lu] and use the
notation f .
The inverse image functor f : Dcb (Y ) ! Dcb (X ) is given by
f = Rf so that f is the right derived functor of the functor f : Sh(Y ) ! Sh(X ).
(2.3) The functor f
Let f : X ! Y be a morphism of algebraic varieties. Let A 2 Dcb (X ). Then f A is the
unique (up to isomorphism) complex on Y such that
A = f (f A):
Actually, I have cheated here: We can only be sure that the complex f A is well de
ned if
f is a locally trivial principal G-bundle, A is a semisimple G-equivariant complex on X and
IV. Perverse sheaves
47
we require f A to be a semisimple complex on Y , see Lu] 8.1.7 and 8.1.8 for de
nitions
and details.
(2.4) The shift functor n]
Let A be a complex of sheaves on X . For each integer n de
ne a new complex An] , with
dierentials dn]i, by
(An])i = An+i The shift functor is the functor
and
(dn])i = (;1)n dn+i :
n]
Dcb (X ) ;!
Dcb (X )
A ;! An]:
(2.5) The Verdier duality functor D
This de
nition is too involved for us to take the energy to repeat it here, we shall refer the
reader to KS] x3.1. The main thing that we will need to know is that this functor exists.
3. Perverse sheaves
(3.1) Denition of perverse sheaves
Let X be an algebraic variety. The support, supp
F , of a sheaf F on X is the complement
of the union of open sets U X such that F U = 0.
A complex A 2 Dcb (X ) is a perverse sheaf if
(a) dim supp Hi (A) = 0 for i 0 and dim supp Hi (A) ;i for i < 0, and
(b) dim supp Hi (D(A)) = 0 for i 0 and dim supp Hi (D(A)) ;i for i < 0,
where D(A) is the Verdier dual of A.
An abelian category is a category which has a direct sum operation and for which
every morphism has a kernel and a cokernel. See KS] I x1.2 for a precise de
nition.
Theorem. The full subcategory of Dcb (X ) whose objects are perverse sheaves on X is an
abelian category.
(3.2) Intersection cohomology complexes
Theorem. Let Y X be a smooth locally closed subvariety of complex dimension d > 0
and let L be a locally constant sheaf on Y . There is a unique complex IC (Y L) in Dcb (X )
such that
(1) Hi (IC (Y L)) = 0, if i < ;d,
48
Arun Ram
H;d (IC (Y L)) jY = L,
dim supp Hi (IC (Y L)) ;i, if i > ;d,
dim supp Hi (D(IC (Y L))) ;i, if i > ;d,
The complexes IC (Y L) are the intersection cohomology complexes and an explicit con(2)
(3)
(4)
struction of these complexes is given in BBD] Prop. 2.1.11.
Theorem. The simple objects of the category of perverse sheaves are the intersection
complexes IC (Y L) as L runs through the irreducible locally constant sheaves on various
smooth locally closed subvarieties Y X .
V. Quantum Groups
49
V. Quantum groups
The de
nition of the quantum group and the uniqueness theorem, Theorem (1.4), are
stated in D1] x6 Example 6.2. Theorem (1.4) appears with proof in SS] Theorem 11.4.1.
The statements in (3.3) and (3.4) can be found in CP] 9.2.1 and 9.3.1 and the treatment
there gives references for where to nd the proofs.
1. Denition, uniqueness, and existence
(1.1) Making the Cartan matrix symmetric
Let g be a nite dimensional complex simple Lie algebra and let C = (j (Hi))1ijr
be the corresponding Cartan matrix. There exist unique positive integers d1 d2 dr
such that gcd(d1 dr ) = 1 and the matrix (dij (Hi ))1ijr is symmetric. The integers
d1 d2 dr are given explicitly by
Ar Dr ,
di = 1 for all 1 i r,
E6 E7 E8 :
Br :
di = 1 for 1 i r ; 1, and dr = 2,
Cr :
di = 2, for 1 i r ; 1, and dr = 1,
F4 :
d1 = d2 = 1, and d3 = d4 = 2,
G2 :
d1 = 3, and d2 = 1.
(1.2) The Poisson homomorphism
Let : g ! g g be the C -linear map given by
(Hi) = 0 (Xi) = di (Xi Hi ; Hi Xi ) 1 i r:
There is a unique extension of the map : g ! g g to a C -linear map : Ug ! Ug Ug
such that
(xy) = (x) (y) + (x) (y) for all x y 2 Ug.
(1.3) The denition of the quantum group
A Drinfel'd-Jimbo quantum group Uh g corresponding to g is a deformation of Ug as a Hopf
algebra over C such that
(1) Poisson condition :
op
h (a) ; h (a) (mod h) = (a mod h) for all a 2 U g.
h
P
h
P
(If h (a) = a a(1) a(2) then op
h (a) = a a(2) a(1) .)
(2) Cartan subalgebra condition:
50
Arun Ram
There is a subalgebra Uh h Uh g such that
op
(a) Uh h is cocommutative, i.e.
h (a) = h (a), for all a 2 Uh h,
(b) The mapping Uh h=hUh h ! Ug is injective with image Uh.
(3) Cartan involution condition:
There is a mapping : Uh g ! Uh g such that
(a) 2 = idUh g ,
(b) (Uhh) = Uh h,
(c) is an algebra homomorphism and a coalgebra antihomomorphism, i.e.
(ab) = (a)(b) for all a b 2 Uh g, and
op
h ((a)) = ( ) h (a) for all a 2 Uh g,
(d) mod h is the Cartan involution.
(1.4) Uniqueness of the quantum group
Theorem. Let g be a nite dimensional complex simple Lie algebra. The Drinfel'd-Jimbo
quantum group Uh g corresponding to g is unique (up to equivalence of deformations).
(1.5) Denition of q-integers and q-factorials
For any symbol q de
ne
n ;n
n]q = qq ;;qq;1 n]q ! = n]q n ; 1]q 2]q 1]q and
m
m]q ! for all positive integers m n,
=
n q n]q !m ; n]q !
(1.6) Presentation of the quantum group by generators and relations
Note the similarities (and the dierences) between the following presentation of the quantum group by generators and relations and the presentation of the enveloping algebra of g
given in II (2.2).
Theorem. Let g be a nite dimensional complex simple Lie algebra and let C =
(j (Hi ))1ijr be the corresponding Cartan matrix. The Drinfel'd-Jimbo quantum group
Uh g corresponding to g can be presented as the algebra over C h]] generated (as a complete
C h]]-algebra in the h-adic topology) by
X1; X2; : : : Xr; H1 H2 : : : Hr X1+ X2+ : : : Xr+
V. Quantum Groups
51
with relations
for all 1 i j r,
Hi Hj ] = 0,
Hi Xj+] = j (Hi )Xj+,
for all 1 i j r,
Hi X ;] = ;j (Hi)X ; ,
j
j
di hHi ; e;di hHi
Xi+ Xj;] = ij e edi h ;
e;di h ,
X
(;1)s 1 ; j (Hi )
s
s+t=1;j (Hi )
for 1 i j r,
edi h
(Xi)sXj (Xi)t = 0
for i 6= j
and with Hopf algebra structure given by
h (Hi ) = Hi 1 + 1 Hi h (Xi+ ) = Xi+ edi hHi + 1 Xi+ h (Xi; ) = Xi; 1 + e;di hHi Xi; Sh (Hi ) = ;Hi Sh (Xi+) = ;Xi+ e;di hHi Sh (Xi;) = ;edi hHi Xi; (H ) = (X +) = (X ;) = 0
h
i
h
i
h
i
Cartan subalgebra Uhh]] Uh g, and Cartan involution : Uhg ;! Uh g determined by
(Xi+) = ;Xi; (Xi;) = ;Xi+ (Hi ) = ;Hi :
2. The rational form of the quantum group
The rational form of the quantum group is an algebra which is similar to the algebra
Uh g except that it is over an arbitrary eld k. There are two reasons for introducing this
algebra.
(1) In the case when k = C (q) is the eld this new algebra Uq g has \integral forms"
which can be used to specialize q to special values.
(2) In the case when k = C and q is a power of a prime then part of this algebra
appears naturally as a Hall algebra of representations of quivers or, equivalently,
as a Grothendieck ring of G-equivariant perverse sheaves on certain varieties EV .
(2.1) Denition of the rational form of the quantum group
52
Arun Ram
Many authors use the following form Uq g of the quantum group as the denition of the
quantum group.
Let g be a nite dimensional complex simple Lie algebra and let C = (j (Hi))1ijr
be the corresponding Cartan matrix. Let k be a eld and let q 2 k be an nonzero element
of k. The rational form of the Drinfel'd-Jimbo quantum group Uq g corresponding to g is
the algebra Uq g over k generated by
F1 F2 : : : Fr K1 K2 : : : Kr K1;1 K2;1 : : : Kr;1 E1 E2 : : : Er with relations
KiKj = Kj Ki ,
for all 1 i j r,
KiKi;1 = Ki;1 Ki = 1,
for all 1 i r,
KiEj Ki;1 = qdi j (Hi ) Ej ,
KiFj Ki;1 = q;di j (Hi ) Fj ,
i ; Ki;1 ,
EiFj ; Fj Ei = ij K
qdi ; q;di
X
s+t=1;j (Hi )
(;1)s 1 ; j (Hi )
X
s+t=1;j (Hi )
s
(;1)s 1 ; j (Hi )
s
for all 1 i j r,
for 1 i j r,
qdi
EisEj Eit = 0
s
t = 0
Fi Fj Fi
qdi
for i 6= j
for i 6= j
and with Hopf algebra structure given by
(Ki ) = Ki Ki (Ei) = Ei Ki + 1 Ei (Fi ) = Fi 1 + Ki;1 Fi S (Ki) = Ki;1 S (Ei) = ;Ei Ki;1 S (Fi ) = ;Ki Fi (Ki) = 1 (Ei ) = 0 (Fi ) = 0:
It is very common to take q to be an indeterminate and to let k = C (q ) be the eld of
rational functions in q.
(2.2) Relating the rational form and the original form of the quantum group
The relations in the rational form of the quantum group are obtained from the relations
in the presentation of Uh g by making the following replacements:
eh ;! q ehdi Hi ;! Ki Xi; ;! Fi Xi+ ;! Ei :
V. Quantum Groups
53
The ring Uq g is an algebra over k and q 2 k while the ring Uh g is an algebra over C h]]
where h is an indeterminate. They have many similar properties. Most of the theorems
about the structure of the algebra Uh g have analogues for the case of the algebra Uq g. The
category of modules for Uq g is very similar to the category of module for the enveloping
algebra Ug. One should note, however, in contrast to Chapt. VI Theorem (1.1) which says
that Uh g = Ugh]], it is not true that Uq g is isomorphic to Ug, even if k = C and q 2 k.
This fact complicates many of the proofs when one is trying to generalize results from the
classical case of Ug to the quantum case Uq g.
3. Integral forms of the quantum group
There are two dierent commonly used integral forms of a C (q )-algebra Uq g, the \nonrestricted integral form" UA g and the \restricted integral form" UAres g. Let us begin by
de
ning integral forms precisely.
(3.1) Denition of integral forms
Let q be an indeterminate and let Uq be an algebra over C (q), the eld of rational functions
in q. An integral form of Uq is a A = Zq q;1] subalgebra UA of Uq such that the map
UA A C (q) ;! Uq
is an isomorphism of C (q) algebras. In other words, upon extending scalars from Zq q;1]
to C (q) the algebra UA turns into Uq .
(3.2) Motivation for integral forms
The purpose of de
ning integral forms of algebras is that we can use them to specialize
the variable q to certain elements of Q , or R , or C , etc. Let UA be an integral form of an
algebra Uq over C (q) and let 2 C , 6= 0. The specialization at q = (over C ) of UA is
the algebra over C given by
U = UA A C where the equation qc = c
describes how C is an A = Zq q;1]-module. Similarly, we can de
ne specializations of UA
over any eld. With this last de
nition in mind we see that one could regard an integral
form of Uq as an A = Zq q;1] subalgebra UA such that Uq is the specialization of UA over
C (q ) at q = q .
(3.3) Denition of the non-restricted integral form of the quantum group
Let q be an indeterminate and let k = C (q) be the eld of rational functions in q. Let
Uq g be the corresponding rational form of the quantum group. For each 1 i r, de
ne
elements
i ; Ki;1 :
Ki 0]qdi = K
qdi ; q;di
54
Arun Ram
The non-restricted integral form of Uq g is the A = Zq q;1] subalgebra UA g of Uq g generated by the elements
F1 F2 : : : Fr K11 K21 : : : Kr1 K1 0] K2 0] : : : Kr 0] E1 E2 : : : Er :
The Hopf algebra structure on Uq g restricts to a well de
ned Hopf algebra structure on
UA g.
(3.4) Denition of the restricted integral form of the quantum group
Let q be an indeterminate and let k = C (q) be the eld of rational functions in q. Let
Uq g be the corresponding rational form of the quantum group. The restricted integral
form of Uq g is the A = Zq q;1] subalgebra UAres g of Uq g generated by the elements
K11 K21 : : : Kr1 and the elements
`
`
Fi(`) = `F] id ! and Ei(`) = `E] id ! for all 1 i r and all ` 1.
qi
qi
(The notation for the q-factorials is as in (1.5).) The Hopf algebra structure on Uq g restricts
to a well de
ned Hopf algebra structure on UAres g. It is nontrivial to prove that UAres g is an
integral form of Uq g.
VI. Modules for quantum groups
55
VI. Modules for quantum groups
The isomorphism theorem in (1.1) is found (with proof) in D2] p. 330-331. The proof
of this theorem uses several cohomological facts,
H 2 (g Ug) = 0 H 1(h Ug=(Ug)h ) = 0 and H 1 (g Ug Ug) = 0:
The correspondence theorem in (1.3) is also found in D2] p.331. All of the results in
section 2 can be found, with detailed proofs, in Ja] Chapt. 5.
1. Finite dimensional Uhg-modules
(1.1) As algebras, Uh g = Ugh]]
The algebra Ugh]] is just the enveloping algebra of the Lie algebra g except over the ring
C h]] (and then h-adically completed) instead of over the eld C . It acts exactly like the
algebra Ug, the only dierence is that we have extended coecients.
The following theorem says that the algebra Uh g and the algebra Ugh]] are exactly
the same! In fact we have already seen that this must be so, since Ug has no deformations
as an algebra (Chapt. III Theorem (2.6)). One might ask: If Uh g and Ugh]] are the
same then what is big deal about quantum groups? The answer is: They are the same as
algebras but they are not the same when you look at them as Hopf algebras.
Theorem. Let g be a nite dimensional complex simple Lie algebra and let Uh g be the
Drinfel'd-Jimbo quantum group corresponding to g. Then there is an isomorphism of
algebras
' : Uh g ;! Ugh]] such that
(a) ' = idUg (mod h) and
(b) 'h = idh where, in the second condition, h = C -span fH1 : : : Hr g Uh g.
(1.2) Denition of weight spaces in a Uh g module
A nite dimensional Uh g-module is a Uh g-module that is a nitely generated free module
as a C h]]-module. If M is a nite dimensional Uh g-module and 2 h de
ne the -weight
space of M to be the subspace
M = fm 2 M j am = (a)m for all a 2 hg:
The dimension of the weight space M is the number of elements in a basis for it, as a
C h]]-module.
56
Arun Ram
(1.3) Classication of modules for Uh g
Theorem (1.1) says that Uh g and Ugh]] are the same as algebras. Since the category of
nite dimensional modules for an algebra depends only on its algebra structure it follows
immediately that the category of nite dimensional modules for Uh g is the same as the
category of modules for Ugh]].
Theorem. There is a one to one correspondence between the isomorphism classes of nite
dimensional Uh g-modules and the isomorphism classes of nite dimensional Ug-modules
given by
1;1
Uh g-modules !
Ug-modules
M
! M=hM
V h]]
!
V
where the Uh g module structure on V h]] is de ned by the composition
Ugh]] ;! End(V h]]):
Uh g;!
It follows from condition (b) of Theorem (1.1) that, under the correspondence in the
Theorem above, weight spaces of Uh g-modules are taken to weight spaces of Ug-modules
and their dimension remains the same. Furthermore, irreducible nite dimensional Ugmodules correspond taken to indecomposable Uh g-modules and vice versa. (Note that
hV h]] is always a Uh g-submodule of the Uh g-module V h]].)
The previous theorem combined with Chapt. II Theorem (2.5) gives the following
corollary.
Corollary. Let P + = Pri=1 N !i be the set of dominant integral weights for g, as in (2.4).
For every 2 P + there is a unique (up to isomorphism) nite dimensional indecomposable
Uh g-module L() corresponding to .
2. Finite dimensional Uq g-modules
The category of nite dimensional modules for the rational form Uq g of the quantum
group is slightly dierent from the category of nite dimensional modules for Uh g. The
construction of the nite dimensional irreducible modules for Uq g is similar to the construction of these modules in the case of the Lie algebra g. Let us describe how this is
done.
(2.1) Construction of the Verma module M () and the simple module L()
Let g be a nite dimensional complex simple Lie algebra and let Uq g be the corresponding
rational form of the quantum group over a eld k and with q 2 k. We shall assume that
VI. Modules for quantum groups
57
char k 6= 2 3 and q is not a root of unity in k.
Let 2 P be an element of the weight lattice for g. The Verma module M () is be the
Uq g-module generated by a single vector v where the action of Uq g satis
es the relations
Ei v = 0 and Ki v = q(i) v for all 1 i r.
The map
Uq n; ;! M ()
y 7;! yv
is a vector space isomorphism.
The module M () has a unique maximal proper submodule. For each 2 P de
ne
L() = MN()
where N is the maximal proper submodule of the Verma module M ().
Theorem. Let g be a nite dimensional complex simple Lie algebra and let Uq g be the
corresponding rational form of the quantum group over a eld k with q 2 k. Assume that
char k =
6 2 3 and that q is not a root of unity in k. Let 2 P be an element of the weight
lattice of g and let L() be the Uq g-module de ned above.
(a) The module L() is a simple Uq g-module.
(b) The module L() is nite dimensional if and only if is a dominant integral weight.
(2.2) Twisting L() to get L( )
Let Q be the root lattice corresponding to g as given in II (2.6) and let : Q ! f1g be
a group homomorphism. The homomorphism induces an automorphism : Uq g ! Uq g
of Uq g de
ned by
: Uq g ;!
Uq g
7;! (i)Ei
7;!
Fi
Ki1 7;! (i)Ki1 where 1 : : : r are the simple roots for g. Let 2 P be an element of the weight lattice
Ei
Fi
and let L() be the irreducible Uq g-module de
ned in (2.1). De
ne a Uq g-module L( )
by de
ning
(a) L( ) = L() as vector spaces,
(b) Uq g acts on L( ) by the formulas
u ? m = (u)m for all u 2 Uq g, m 2 L(),
58
Arun Ram
where is the automorphism of Uq g de
ned above.
(2.3) Classication of nite dimensional irreducible modules for Uq g
Theorem. Let g be a nite dimensional complex simple Lie algebra and let Uq g be the
rational form of the quantum group over a eld k. Assume that char k =
6 2 3 and q 2 k is
not a root of unity in k. Let P + be the set of dominant integral weights for g and let Q
be the root lattice for g (see II (2.6)).
(a) Let 2 P + and let : Q ! f1g be a group homomorphism. The modules
L( ) de ned in (2.2) are all nite dimensional irreducible Uq g-modules.
(b) Every nite dimensional Uq g-module is isomorphic to L( ) for some 2 P +
and some group homomorphism : Q ! f1g.
(2.4) Weight spaces for Uq g-modules
Retain the notations and assumptions from (2.3) and let (,) be the inner product on hR
de
ned in II (2.7). Let M be a nite dimensional Uq g-module. Let : Q ! f1g be a
group homomorphism and let 2 P . The ( )-weight space of M is the vector space
M() = fm 2 M j Ki m = (i)q(i) m for all 1 i r.g
The following proposition is analogous to Chapt. II Proposition (2.4).
Proposition. Every nite dimensional Uq g-module is a direct sum of its weight spaces.
The following theorem says that the dimensions of the weight spaces of irreducible
Uq g-modules coincide with the dimensions of the weight space of corresponding irreducible
modules for the Lie algebra g.
Theorem. Let 2 P + be a dominant integral weight and let be a group homomorphism
: Q ! f1g. Let V be the simple g-module indexed by the and let L( ) be the
irreducible Ug -module indexed by the pair ( ). Then, for all 2 P and all group
homomorphisms : Q ! f1g,
;
;
dimk L( ) = dimC (V )
and
;
dimk L( ) = 0 if 6= .
VII. Properties of quantum groups
59
VII. Properties of quantum groups
Let g be a nite dimensional complex simple Lie algebra and let Uh g be the Drinfel'dJimbo quantum group corresponding to g that was de
ned in V (1.3). We shall often
use the presentation of Uh g given in V (1.6). In this chapter we shall describe some of
the structure which quantum groups have. In many cases this structure is similar to the
structure of the enveloping algebra Ug.
The proofs of the triangular decomposition and the grading on the quantum group
given in x1 can be found in Ja] 4.7 and 4.21. The proof of the statements in (2.1) and (2.3),
concerning the pairing h i, can be found in Ja] 6.12, 6.18, 8.28, and 6.22. The statement in
(2.2) follows from Chapt I, Prop. (5.5). The theorem giving the existence and uniqueness
of the R-matrix is stated in D2] p.329 and the uniqueness is proved there. The existence
follows from (7.4) see also Lu] Theorem 4.1.2. The properties of the R-matrix stated in
(3.3) are proved in D2] Prop. 3.1 and Prop. 4.2. Proofs of the statements in the section
on the Casimir element can be found in D2] Prop 2.1, Prop 3.2 and Prop. 5.1.
Theorem (5.2a) is proved in Ja] 8.15-8.17 and Lu] 39.2.2. Theorem (5.2b) is a nontrivial, but very natural, extension of well known results which appear, for example, in Ja]
Chapt. 8. The proof is a combination of the methods used in CP] 8.2B and Ja] 8.4 and a
calculation similar to that in the proof of Ja] Lemma 8.3. The properties of the element
Tw0 given in (5.3) are proved in the following places: The formula for (Tw0 )Tw0 is proved
in CP] 8.2.4 The formula for Tw;01 is proved by a method similar to Ja] 8.4 The formula
for h(Tw0 ) is proved in CP] 8.3.11 and the remainder of the formulas are proved in CP]
8.2.3.
The construction of the Poincare-Birkho-Witt basis of Uh g given in section 6 appears
in detail in Ja] 8.18-8.30. The statement that Uh g is almost a quantum double, Theorem
(7.3), appears in D1] x13, and an outline of the proof can be found in CP] 8.3. The proof
of Theorem (8.4) can be gleaned from a combination of Ja] 6.11 and 6.18. Both of the
books Lu] and Jo] also contain this fact.
1. Triangular decomposition and grading
(1.1) Triangular decomposition of Uh g
The triangular decomposition of the quantum group Uh g is analogous to the triangular
decomposition of the Lie algebra g and the triangular decomposition of the enveloping
algebra Ug given in II (2.3) and II (4.2).
Proposition. Let g be a nite dimensional complex simple Lie algebra and let Uh g be
60
Arun Ram
the corresponding Drinfel'd-Jimbo quantum group as presented in V (1.6). De ne
Uh n; = subalgebra of Uh g generated by X1; X2; : : : Xr; ,
Uh h = subalgebra of Uh g generated by H1 H2 : : : Hr ,
Uh n+ = subalgebra of Uh g generated by X1+ X2+ : : : Xr+.
The map
Uh n; Uh h Uh n+
u; u0 u+
is an isomorphism of vector spaces.
;! Uh g
7;! u; u0 u+
(1.2) The grading on Uh n+ and Uh n;
The gradings on the positive part Uhn+ and on the negative part Uh n; of the quantum
group Uh g are exactly analogous to the gradings on the postive part Un+ and the negative
part Un; of the enveloping algebra Ug which are given in II (4.3).
Proposition. Let g be a nite dimensional complex simple Lie algebra and let Uh g be
the corresponding Drinfel'd-Jimbo quantum group as presented in V (1.6). Let 1 : : : r
be the simple roots for g and let
Q+ =
X
i
N i where N = Z0.
P
For each element = ri=1 i i 2 Q+ de ne
(Uh n+ ) = span-fXi+1 Xi+p j Xi+1 Xi+p has j -factors of type Xj+g
(Uhn; ) = span-fXi;1 Xi;p j Xi;1 Xi;p has j -factors of type Xj; g:
Then
as vector spaces.
Uh n; =
M
2Q+
(Uhn; ) and Uh n+ =
2. The inner product h i
M
2Q+
(Uhn+ ) VII. Properties of quantum groups
61
In some sense the nonnegative part Uh b+ of the quantum group is the dual of the
nonpositive part Uh b; of the quantum group. This is reected in the fact that there is a
nondegenerate bilinear pairing between the two. Later we shall see that this pairing can
be extended to a pairing on all of Uh g. The extended pairing is an analogue of the Killing
form on g in two ways:
(1) it is an ad-invariant form on Uh g, and
(2) upon restriction to g it coincides (mod h) with the Killing form.
(2.1) The pairing between Uh b; and Uh b+
Let g be a nite dimensional complex simple Lie algebra and let Uh g be the corresponding
Drinfel'd-Jimbo quantum group as presented in V (1.6). De
ne
Uh b; = subalgebra of Uh g generated by X1; X2; : : : Xr; and H1 : : : Hr ,
Uh b+ = subalgebra of Uh g generated by X1+ X2+ : : : Xr+ and H1 : : : Hr .
Theorem.
(1) There is a unique C h]]-bilinear pairing
h i : Uh b; Uh b+ ;! C h]] which satis es
(a) h1 1i = 1
(b) hHi Hj i = j d(Hi) j
(c) hXi; Xj+i = ij edi h ;1 e;di h (d) hab ci = ha b h(c)i for all a b 2 Uh b; and c 2 Uh b+ ,
;
+
(e) ha bci = h op
h (a) b ci for all a 2 Uh b and b c 2 Uh b .
(2) The pairing h i is nondegenerate.
(3) The pairing h i respects the gradings on Uh n+ and Uh n; in the following sense:
(a) Let 2 Q+ .
If 6= then (Uh b; ) (Uhb+ ) = 0.
(b) Let 2 Q+ . The restriction of the pairing h i to (Uh n; ) (Uhn+ ) is a nondegenerate pairing
h i: (Uhn; ) (Uh n+ ) ! C h]]:
If is the Cartan involution of Uh g as given in V (1.6) and Sh is the antipode of Uh g then
h(u+) (u;)i = hu; u+i and hSh(u; ) Sh(u+ )i = hu; u+ i
62
Arun Ram
for all u; 2 Uh b; and u+ 2 Uh b+ .
(2.2) Extending the pairing to an ad-invariant pairing on Uh g
The triangular decomposition (1.1) of Uh g says that Uh g = Uh n; Uh h Uh n+ and that
every element u 2 Uh g can be written in the form u; u0 u+ ,
where u; 2 Uh n; , u0 2 Uh h, and u+ 2 Uh n+ . We can use this to extend the pairing
de
ned in (2.1) to a pairing
h i: Uhg Uh g ;! C h]] de
ned by the formula
u;u0u+ u;u0u+ = u; S (u0u+ )u; S ;1(u0u+)
1 1 1 2 2 2
1 h 2 2
2 h 1 1
for all u;1 u;2 2 Uh n; , u01 u02 2 Uh h, and u+1 u+2 2 Uh n+ , where Sh is the antipode of Uh g.
Then
hadu (v1 ) v2i = hv1 adSh (u) (v2 )i for all u v1 v2 2 Uh g,
This formula says that the extended pairing h i is an ad-invariant pairing as de
ned in I
(5.5). The pairing h i: Uhg Uh g ! C h]] is not symmetric, see I (5.5).
(2.3) Duality between matrix coecients for representations and Uq g.
Let g be a nite dimensional complex simple Lie algebra and let Uq g be the rational form
of the quantum group over a eld k, where char k 6= 2 3 and q 2 k is not a root of unity.
Let Q be the root lattice for g.
Theorem. Let M be a nite dimensional Uq g module such that all weights of M satisfy
2 2 Q. Then, for each pair n 2 M and m 2 M there is a unique element u 2 Uq g such
that
n (vm) = hv ui for all v 2 Uq g,
where h i is the bilinear form on Uq g given by (2.1) after making the substitutions in V
(2.2).
The function
cmn : Uq g ! C (q) de
ned by cmn (v) = hn vmi
is the (m n )-matrix coecient of v acting on M . The above theorem gives a duality
between matrix coecient functions and Uq g. It also says that every element of Uq g is
determined by how it acts on nite dimensional Uq g-modules.
3. The universal R-matrix
(3.1) Motivation for the R-matrix
VII. Properties of quantum groups
63
The following theorem states that there is an element R such that the pair (Uhg R)
is a quasitriangular Hopf algebra. In particular, this implies that the category of nite
dimensional modules for the quantum group Uhg is a braided SRMCwMFF.
(3.2) Existence and uniqueness of R
Let g be a nite dimensional complex simple Lie algebra and Uh g be the corresponding
quantum group as presented in V (1.6). Recall the Killing form on g from II (1.6).
Let fH~ i g be an orthonormal basis of h with respect to the Killing form and de
ne
t0 =
r
X
~
P
i=1
Hi H~ i :
If 2 Q+ (see (1.2)) and = ri=1 i i where 1 : : : r are the simple roots, de
ne
n to be the smallest number of positive roots > 0 whose sum is equal to .
The element R is not quite an element of Uh g Uh g so we have to make the tensor
product just a tiny bit bigger. To do this we let Uh g^ Uh g denote the h-adic completion
of the tensor product Uh g Uhg, see III x1.
Theorem. There exists a unique invertible element R 2 Uh g^ Uh g such that
R h (a)R;1 =
R has the form R =
X
2Q
+
op (a)
h
;
for all a 2 Uh g, and
exp h t0 + 21 (H 1 ; 1 H ) P where
P 2 (Uh n; ) (Uh n+ ) ,
P
P
H = ri=1 i Hi , if = i i i ,
P is a polynomial in Xi+ 1 and 1 Xi;, 1 i r, with coecients in C h]],
such that
the smallest power of h in P with nonzero coecient is hn .
(3.3) Properties of the R-matrix
Recall V (1.6) that Uh g is a Hopf algebra with comultiplication h, counit h , and antipode
Sh and that Uh g comes with a Cartan involution . The following formulas describe
P the
relationship between the R-matrix and the Hopf algebra structure of Uhg. If R = ai bi
then let
R12 =
X
ai bi 1 R13 =
and let
X
ai 1 bi and R23 =
R21 =
X
bi ai :
X
1 ai bi 64
Arun Ram
Let : Uhg ! Uh g be the C -algebra automorphism of Uh g given by (h) = ;h, (Xi) =
Xi , and (Hi) = Hi . With these notations we have
( h id)(R) = R13 R23 and (id h)(R) = R13 R12 (
h id)(R) =1 = (id h )(R)
(Sh id)(R) = (id Sh;1)(R) = R;1 and (Sh Sh )(R) = R
( )(R) = R21 and ( )(R) = R;1 :
4. An analogue of the Casimir element
(4.1) Denition of the element u
Let g be a nite dimensional complex simple Lie algebra and let Uh g be the corresponding
Drinfel'd-Jimbo quantum group as presented in V (1.6). The antipode Sh: Uh g ! Uh g is
an antiautomorphism of Uh g, see I (2.1). This means that the map Sh2 : Uhg ! Uh g is an
automorphism of Ug . The following theorem says that this automorphism is inner!
Theorem. Let R P2 Uh g^ Uh g be the universal
R-matrix of Uh g as de ned in (3.2).
P
Suppose that R = ai bi and de ne u = S (bi)ai : Then u is invertible and
uau;1 = Sh2 (a) for all a 2 Uh g.
(4.2) Properties of the element u.
The relationship of the element u to the Hopf algebra structure of Uh g is given by the
formulas
h (u) = (R21 R12 );1 (u u) Sh (u) = u and h (u) = 1
P
where
= R = ai bi is the universal R-matrix of Uh g given in (3.2), and R21 =
P b Ra12. The
inverse of the element u is given by
i
i
u;1 =
X
Sh;1 (dj )cj where R;1 =
X
cj dj :
(4.3) Why the element u is an analogue of the Casimir element
VII. Properties of quantum groups
65
Let ~ be the element of h such that i (~) = 1 for all simple roots i of g. An easy check
on the generators of Uh g shows that
eh~ae;h~ = Sh2 (a) for all a 2 Uh g.
It follows that
the element e;h~u = ue;h~ is a central element in Uh g.
Any central element of Uh g must act on each nite dimensional simple Uh g-module by a
constant. For each dominant integral weight let L() be the nite dimensional simple
Uh g-module indexed by (see VI (1.3)). As in II (4.5), let be the element of hR given by
= 21
X
>0
where the sum is over all positive roots for g. Then the element
e;h~u acts on L() by the constant q;(++)+() ,
where q = eh and the inner product in the exponent of q is the inner product on hR given
in II (2.7). Note the analogy with II (4.5). It is also interesting to note that
(e;h~u)2 = uSh (u):
5. The element Tw
0
(5.1) The automorphism Sh
Let W be the Weyl group corresponding to g and let w0 be the longest element of W (see
II (2.8)). Let s1 : : : sr be the simple reections in W . For each 1 i r there is a unique
1 j r such that w0 si w0;1 = sj . The map given by
(Xi ) = Xj and (Hi ) = Hj where w0 si w0;1 = sj for 1 i r,
extends to an automorphism of Uh g. Let ~ be the anti-automorphism of Uh g de
ned by
~(Xi) = Xi and ~(Hi ) = Hi . This is an analogue of the Cartan involution. Let Sh be
the antipode of Uh g as given in V (1.6). These are both anti-automorphisms of Uh g. The
composition
(Sh ~ ): Uhg ! Uh g
is an automorphism of Uh g. The following result says that this automorphism is inner.
(5.2) Denition of the element Tw0
66
Arun Ram
Let g be a nite dimensional complex simple Lie algebra and let Uh g be the corresponding quantum group as presented in V (1.6). Let q = eh and for each 1 i r
let
+r
+r
i ) F (r) = (Xi ) and Ki = ehdi Hi Ei(r) = (X
r] d ! i
r] d !
qi
qi
where the notation for q-factorials is as in V (1.5). For each 1 i r, de
ne
Ti =
X
(;1)b qb;ac+(c+a;b)(c;a) Ei(a) Fi(b) Ei(c)Kic;a abc0
where the sum is over all nonnegative integers a,b, and c.
Theorem.
(a) The elements Ti satisfy the relations
T| i Tj T{zi Tj } = T| j Ti T{zj Ti }
mij factors
mij factors
for i 6= j ,
where the mij are as given in II (2.8).
(b) Let w0 = si1 siN be a reduced word for the longest element of the Weyl group W ,
see II (2.8). De ne
Tw0 = Ti1 TiN :
Then Tw0 is invertible and
Tw0 aTw;01 = (Sh ~ )(a) for all a 2 Uh g.
(5.3) Properties of the element Tw0
Let u 2 Uhg be the analogue of the Casimir element for Uh g as given in x4 and let be
the C -algebra automorphism of Uh g given in (3.3). Let ~ be the C -linear automorphism
of Uh g given by ~ (h) = ;h, ~ (Xi) = Xi, and ~ (Hi ) = ;Hi . Then
(Tw0 )Tw0 = u and Tw;01 = ~ (Tw0 ):
The relationship between the element Tw0 and the Hopf algebra structure of Uh g is given
by the formulas
1
;1
h (Tw0 ) = R;
12 (Tw0 Tw0 ) = (Tw0 Tw0 )R21 Sh(Tw0 ) = Tw0 eh~ and h (Tw0 ) = 1
P
where
P b Ra12. = R = ai bi is the universal R-matrix of Uhg given in (3.2), and R21 =
i
i
VII. Properties of quantum groups
67
6. The Poincar
e-Birkho-Witt basis of Uhg
(6.1) Root vectors in Uh g
Let g be a nite dimensional complex simple Lie algebra and let Uh g be the corresponding
quantum group as presented in V (1.6). Let Ti be the elements of Uh g given in (5.2).
De
ne an automorphism i : Uhg ! Uhg by
i(u) = Ti uTi;1 for all u 2 Uh g,
Let W be the Weyl group corresponding to g. Fix a reduced decomposition w0 =
si1 siN of the longest element w0 2 W , see II (2.8). De
ne
1 = i1 2 = si1 (i2 ) : : : N = si1 si2 siN 1 (iN ):
;
The elements 1 : : : N are the positive roots g. De
ne elements of Uh g by
X1 = Xi1 X2 = i1 (Xi2 ) : : : XN = i1 i2 iN 1 (XiN ):
These elements depend on the choice of the reduced decomposition. They are analogues of
the elements X and X; in Ug which are given in II (4.4).
;
(6.2) Poincar
e-Birkho-Witt bases of Uhn; , Uh h, and Uh n+
Let g be a nite dimensional complex simple Lie algebra and let Uh g be the corresponding quantum group as presented in V (1.6). Let Uh n; , Uh h, and Uh n+ be the
subalgebras of Uh g de
ned in (1.1). The following bases of Uh n; , Uh h, Uh n+ , and Uh g are
analogues of the Poincare-Birkho-Witt bases of Un; , Uh, and Un+ which are given in II
(4.4).
Theorem. Let X1 : : : XN be the elements of Uh g de ned in (6.1). Then
f(X+1 )p1 (X+2 )p2 (X+N )pN j p1 : : : pN 2 Z0g is a basis of Uh n+ ,
f(X;1 )n1 (X;2 )n2 (X;1 )nN j n1 : : : nN 2 Z0g is a basis of Uh n; ,
fH1s1 H2s2 H1sr j s1 : : : sN 2 Z0g
is a basis of Uh h.
(6.3) The PBW-bases of Uh n; and Uh n+ are dual bases with respect to h i
(almost)
Recall the pairing between Uh b; and Uh b+ given in (2.1).
68
Arun Ram
Theorem. Let w0 = si1 siN be a reduced decomposition of the longest element of
the Weyl group and let j and Xj , 1 j N , be the elements de ned in (6.1). Let
p1 : : : pN n1 : : : nN 2 Z0. Then
N
(X ; )n (X ; )n (X ; )nN (X + )p (X + )p (X + )pN = Y
N
N
1
1
2
2
1
1
2
nj pj (Xi;j )nj (Xi+j )nj
2
j =1
where nj pj is the Kronecker delta.
Furthermore, we have that, for each 1 i r,
(X ;)n (X +)n = (;1)nq;din(n;1)=2 n]qdi ! i
i
(qdi ; q;di )n
where q = eh .
7. The quantum group is a quantum double (almost)
(7.1) The identication of (Uh b+ ) coop with Uhb;
Let g be a nite dimensional complex simple Lie algebra and let Uh g be the corresponding
quantum group as presented in V (1.6). De
ne
Uh b; = subalgebra of Uh g generated by X1; X2; : : : Xr; and H1 : : : Hr ,
Uh b+ = subalgebra of Uh g generated by X1+ X2+ : : : Xr+ and H1 : : : Hr ,
except let us distinguish the elements Hi which are in Uh b+ from the elements Hi which
are in Uh b; by writing Hi+ and Hi; respectively, instead of just Hi in both cases.
The nondegeneracy of the pairing h i between Uh b+ and Uh b; (see (2.1)) shows that
;
Uh b is essentially the dual of Uh b+ . Furthermore, it follows from the conditions
hx1x2 yi = hx1 x2 h(y)i and hx y1y2i = h
op (x) y
1 y2 i
that the multiplication in Uh b; is the adjoint of the comultiplication in Uh b+ and the
opposite of the comultiplication in Uh b; is the adjoint of the multiplication in Uh b+ . Thus
(here we are fudging a bit since Uh b+ is in
nite dimensional),
Uh b;
' (Uhb+ ) coop
as Hopf algebras
where (Uh b+ ) coop is the Hopf algebra de
ned in I (5.2).
(7.2) Recalling the quantum double
VII. Properties of quantum groups
69
Recall, from I (5.3), that the quantum double D(A) of a nite dimensional Hopf algebra
A is the new Hopf algebra
D(A) = f a j a 2 A 2 A coopg = A A coop
with multiplication determined by the formulas
a =
a =
where, if
X
h(1) S ;1(a(1) )ih(3) a(3)ia(2) (2) and
a
X
a
h(1) a(1)ih(3) S ;1(a(3) )i(2)a(2)
is the comultiplication in A and A coop ,
( id) (a) =
X
a
a(1) a(2) a(3) and ( id) () =
X
(1) (2) (3) :
The comultiplication D(A) is determined by the formula
(a) =
P
X
a
a(1) (1) a(2) (2) P
where (a) = a a(1) a(2) and () = (1) (2) .
(7.3) The relation between D(Uh b+ ) and Uh g
With the de
nition of the quantum double in mind it is natural that we should de
ne the
quantum double of Uh b+ to be the algebra
D(Uh b+ ) = (Uh b+ ) coop Uh b+ = Uh b; Uh b+
with multiplication and comultiplication given by the formulas in (7.2). The following
theorem says that the quantum group Uh g is almost the quantum double of Uh b+ , in other
words, Uh g is almost completely determined by pasting two copies of Uh b+ together.
Theorem. Let (Bij ) = C ;1 be the inverse of the Cartan matrix corresponding to g and,
for each 1 i r, de ne
r
X
Hi = Bij Hj 2 Uh g:
j =1
(a) There is a surjective homomorphism : D(Uh b+ ) ;! Uh g determined by
: D(Uh b+ ) ;! Uh g
;
7
!
Xi+
Xi+
D(Uh b+ ) U g:
Hi+
;
7
!
Hi
and thus
= h
ker
;
;
Xi
7;! Xi
Hi;
7;! Hi
70
Arun Ram
(Recall (7.1) that we distinguish the elements Hi which are in Uh b+ from the elements Hi
which are in Uh b; by writing Hi+ and Hi; respectively, instead of just Hi in both cases.)
(b) The ideal ker is the ideal generated by the relations
0r
1
X
Hi; ; @ Bij Hj+ A j =1
where 1 i r.
(7.4) Using the R-matrix of D(Uh b+ ) to get the R-matrix of Uhg
Recall (7.2) that the double D(Uh b+ ) comes with a natural universal R-matrix given by
R~ =
X
i
bi bi where the sum is over a basis fbi g of Uh b+ and fbig is the dual basis in Uh b; with respect
to the form h i given in (2.1). We have used the notation R~ here to distinguish it from
the element R in Theorem (3.2). The element R~ is not exactly in the tensor product
D(Uh b+ ) D(Uh b+ ) but if we make the tensor product just a tiny bit bigger by taking
the h-adic completion D(Uh b+ )^ D(Uh b+ ) of D(Uh b+ ) D(Uhb+ ) then we do have
R~ 2 D(Uh b+ )^ D(Uh b+ ):
The image of R~ under the homomorphism
: D(Uh b+ )^ D(Uh b+ ) ;! Uh g^ Uh g
R~
7;!
R
coincides with the element R given in Theorem (3.2). This means that we actually get the
element R in Theorem (3.2) for free by realising the quantum group as a quantum double
(almost).
8. The quantum Serre relations occur naturally
In this section we will see that the most complicated of the de
ning relations in
the quantum group can be obtained in quite a natural way. More speci
cally, the ideal
generated by them is the radical of a certain bilinear form.
(8.1) Denition of the algebras Uh b+ and Uhb;
Let g be a nite dimensional complex simple Lie algebra and let C = (j (Hi ))1ijr be
the corresponding Cartan matrix.
VII. Properties of quantum groups
71
Let Uh b+ be the associative algebra over C h]] generated (as a complete C h]]-algebra
in the h-adic topology) by
H1 H2 Hr X1+ X2+ Xr+
with relations
Hi Hj ] = 0 and Hi Xj+] = j (Hi )Xj+ for all 1 i j r,
and de
ne an algebra homomorphism h: Uhb+ ! Uhb+ ^ Uh b+ by
h (Hi ) = Hi 1 + 1 Hi and
h (Xi+ ) = Xi+ edi hHi + 1 Xi+ where Uh b+ ^ Uhb+ denotes the h-adic completion of the tensor product Uhb+ C h]]
Uh b+ .
Let Uh b; be the associative algebra over C h]] generated (as a complete C h]]-algebra
in the h-adic topology) by
X1; X2; Xr; H1 H2 Hr with relations
Hi Hj ] = 0 and Hi Xj;] = ;j (Hi )Xj; for all 1 i j r,
and de
ne an algebra homomorphism h: Uhb; ! Uhb; ^ Uh b; by
h (Hi ) = Hi 1 + 1 Hi and
h (Xi; ) = Xi; 1 + e;di hHi Xi; where Uhb; ^ Uh b; denotes the h-adic completion of the tensor product Uhb; C h]]
Uh b; .
(8.2) The dierence between the algebras Uh b and the algebras Uh b
The algebras Uhb+ are much larger than the algebras Uh b+ used in (2.1) since they have
fewer relations between the Xi generators.
(8.3) A pairing between Uh b+ and Uh b;
In exactly the same way that we had a pairing between Uh b+ and Uh b; in (2.1), there is
a unique C h]]-bilinear pairing
h i : Uh b; Uh b+ ;! C h]] which satis
es
(a) h1 1i = 1
(b) hHi Hj i = j d(Hi ) j
(c) hXi; Xj+i = ij edi h ;he;di h (d) hab ci = ha b h(c)i for all a b 2 Uh b; and c 2 Uh b+ ,
;
+
(e) ha bci = h op
h (a) b ci for all a 2 Uh b and b c 2 Uh b .
72
Arun Ram
(8.4) The radical of h i is generated by the quantum Serre relations
Let r; and r+ be the left and right radicals, respectively, of the form h i de
ned in (8.3),
i.e.
r; = fa 2 Uh b; j ha bi = 0 for all b 2 Uh b+ g and
r+ = fb 2 Uh b+ j ha bi = 0 for all a 2 Uh b; g:
Theorem. The sets r; and r+ are the ideals of Uhb; and Uhb+ generated by the elements
1 ; (H ) X
j i
s
(;1)
(X ;)s X ;(X ; )t for i 6= j
s
s+t=1;j (Hi )
X
i
j
i
and
(;1)s 1 ; j (Hi )
s+t=1;j (Hi )
edi h
s
edi h
(Xi+)sXj+ (Xi+)t for i 6= j
respectively.
It follows from this theorem that the quantum group Uh g is determined by the algebras
Uh b+ , Uhb; and the form h i. A construction of the quantum group along these lines
would be very similar to the standard construction of Kac-Moody Lie algebras (see K]
x1.3).
VIII. Hall algebras
73
VIII. Hall algebras
The results in x1 are outlined in CP] x9.3D. The proof of Theorem (1.2) appears
in BGP] Theorem 3.1 and the proof of Theorem (1.4) appears in Lu1] Prop. 5.7. The
material in x2 is a combination of Lu2] and Lu] Part II. In particular, Theorem (2.7)(1) is
proved in Lu] 13.1.2, 12.3.2, and 9.2.7, Theorem (2.7)(2) is proved in Lu] 13.1.5, 13.1.12e,
12.3.3, and 9.2.11, Theorem (2.7)(3) is proved in Lu] 13.1.12d and 12.3.6. The statement
about the symmetric form given in (2.6) is proved in Lu] 12.2.2, 9.2.9 and the references
given there. The proof of the isomorphism theorem in (2.8) is given in Lu] 13.2.11 and
in Lu2] Th. 10.17. The material in x3 appears in Lu1] x9. The isomorphism theorem in
(3.4) is stated in Lu1] 9.6.
1. Hall algebras
The Hall algebra is an algebra which has a basis labeled by representations of quivers
and for which the structure constants with respect to this basis reect the structure of
these representations. The Hall algebra encodes a large amount of information about
the representations of the quiver. Amazingly, this algebra is almost isomorphic to the
nonnegative part of the quantum group.
(1.1) Quivers
A quiver is an oriented graph ;, i.e. a set of vertices and directed edges. The following is
an example of a quiver.
Every Dynkin diagram if type A, D or E can be made into a quiver by orienting the edges.
Note that there are many possible ways of orienting the edges of a Dynkin diagram in
order to make a quiver. For example the quivers
are both obtained by orienting the edges of the Dynkin diagram of type E6.
(1.2) Representations of a quiver
A representation R of a quiver ; over a eld k is a labeling of the graph ; such that
(1) Each vertex i 2 ; is labeled by a vector space Ri over k,
74
Arun Ram
(2) Each edge i ! j in ; is labeled by a (vector space) homomorphism ij : Ri ! Rj .
De
ne morphisms of representations of quivers in the natural way and make the category
of representations of the quiver ;. The dimension of a representation R is the vector
dim(R) = (di ) where, for each vertex i 2 ;, di = dim(Ri ). An irreducible representation
of ; is a representation R of ; such that the only subrepresentations of R are 0 and R.
A representation R of a quiver ; is indecomposable if it cannot be written as R = S T
where S and T are nonzero representations of ;.
Theorem. Let ; be a quiver.
(a) There are a nite number of indecomposable representations of ; if and only if ;
is an oriented Dynkin diagram of type A, D or E .
(b) If ; is an oriented Dynkin diagram of type A, D or E then the indecomposable
representations of ; are in 1-1 correspondence with the positive roots for the Lie
algebra g corresponding to the Dynkin diagram.
(1.3) Denition of the Hall algebra
Let ; be a quiver and let F q be a nite eld with q elements. The Hall algebra or
Grothendieck ring R; of representations of ; is the algebra over C with
(1) basis labeled by the isomorphism classes R] of representations of ; over F q , and
(2) multiplication of two isomorphism classes R] and S ] given by
R] S ] =
X
T ]
T T ]
CRS
where
;
T = Card fP T j P = R T=P = S g :
CRS
(1.4) Connecting Hall algebras to the quantum group
Let ; be a quiver which is obtained by orienting the edges of a Dynkin diagram of type A,
D, or E , and let F q be a nite eld with q elements. Let us describe explicitly two types
of indecomposable representations of ;.
(1) Let i be a vertex of ;. The representation
F if j = i
ei given by Vj = 0q if j 6= i
is an irreducible representation of ;.
(2) Let i ! j be an edge of ;. The representation
F if ` = i or ` = j eij given by V` = 0q otherwise
and
is an indecomposable (but not irreducible) representation of ;.
ij = idFq VIII. Hall algebras
75
The following relations hold in the Hall algebra R;,
eij = ei ej ; ej ei ei eij = qeij ei eij ei = qej eij for each edge i ! j in ;.
It is easier to prove the rst relation by writing it in the form ei ej = eij + ej ei . Combining
the rst two of these relations and the rst and last of these relations respectively, gives
the identities
e2i ej ; (q + 1)ei ej ei + qej e2i = 0 and ei e2j ; (q + 1)ej ei ej + qe2j ei = 0 respectively.
We shall make the Hall algebra a bit bigger by adding the Ki1s that are in the quantum group Uq g. Let g be the nite dimensional complex simple Lie algebra corresponding
to the Dynkin diagram given by ; and let Uq g be the rational version of the quantum group
with k = C and q 2 C the number of elements in the eld F q . Let Uq h be the subalgebra
of Uq g generated by K11 : : : Kr1. Let 1 : : : r be the simple roots corresponding to
the Lie algebra g (see II (2.6)). De
ne
Rf; = algebra generated by R; and K11 : : : Kr1 with the additional relations
Ki R]Ki;1 = q(id(R)) R] for all 1 i r and representations R of ;,
P
where d(R) = rj=1 dim(Rj )j , and the inner product in the exponent of q is the inner
product on hR given in II (2.7).
Theorem. Let ; be a quiver which is obtained by orienting the edges of a Dynkin diagram
of type A, D or E . Let R; be the Hall algebra of representations of ; over the nite eld
f; be the extended Hall algebra de ned above. Let Uq g be the
F q with q elements and let R
rational form of the quantum group with k = C which corresponds to the Dynkin diagram
; and let
Uq b+ = subalgebra of Uq g generated by K11 : : : Kr1 and E1 : : :Er .
Choose elements z1 zr 2 Z such that zi ; zj = 1 if i ! j is an edge in ;. Then the
homomorphism of algebras determined by
Uq b+ ;!
Rf;
7 ! Kzi1
Ki1 ;
Ei ;
7 ! Ki i ei
is an isomorphism.
76
Arun Ram
2. An algebra of perverse sheaves
In this section we shall construct an algebra K from a Dynkin diagram ;. There is a
strong relationship between this algebra and the quantum group Uq g where g is the simple
complex Lie algebra corresponding to the Dynkin diagram ;.
The algebra K is graded,
M
K=
K 2 Q+
in the same way that the quantum group Uq n+ is graded, see VII (1.2). The vector space K
comes with natural shift maps n] which correspond to multiplication by qn in the quantum
group Uq b+ . The algebra K has a natural multiplication which comes from an induction
functor and a natural \pseudo-comultiplication" which comes from a restriction functor.
The multiplication and the pseudo-comultiplication turn out to be almost the same as the
multiplication and the comultiplication on the quantum group Uq b+ . Lastly, the algebra
K has a natural inner product f g that is related to the inner product h i pairing Uq b;
and Uq b+ , (see VII (2.1)).
In Theorem (2.8) we shall see that if we extend the algebra K a little bit, by adding
the Ki1's that are in the quantum group Uq g then we get an algebra Ke such that
Ke ' Uq b+ :
This last fact is very similar to the case of the Hall algebra (1.4) where after extending
the Hall algebra R; by adding the Ki1 's that are in the quantum group Uq g, we got an
algebra Rf; which was also isomorphic to Uq b+ . We shall see in section 3 that this is not a
coincidence, there is a concrete connection between R; and the algebra K. The advantage
of working with the algebra K instead of the Hall algebra R; is that K has more natural
structure than R;, it has:
(a) a natural pseudo-comultiplication r: K ! K K,
(b) a natural inner product f g: K K ! Z((q)),
(c) a natural involution D: K ! K,
(d) a natural basis coming from simple perverse sheaves.
The natural basis coming from simple perverse sheaves is called the canonical basis.
(2.1) ;-graded vector spaces and the varieties EV with GV action
Let ; be a quiver obtained by orienting the edges of a Dynkin diagram of type A, D or
E . For convenience we label the vertices by 1 2 : : : r. Let g be the nite dimensional
complex simple Lie algebra corresponding to the Dynkin diagram given by ;.
Let p be a positive prime integer and let F p be the algebraic closure of the nite eld
F p with p elements. A ;-graded vector space V over F p is a labeling of the graph ; such
that each vertex i is labeled by a vector space Vi over F p . The dimension of a ;-graded
VIII. Hall algebras
77
vector space V is the r-tuple of nonnegative integers dim(V ) = (dim(Vi )). We shall identify
dimensions of ;-graded vector spaces with elements of
X
Q+ = N i
i
so that
dim(V ) =
r
X
i=1
dim(Vi )i where 1 : : : r are the simple roots for g and N = Z0.
Fix an element 2 Q+ and a ;-graded vector space V over F p such that dim(V ) = .
De
ne
Y
M
GV = GL(Vi ) and EV = Hom(Vi Vj )
i
i!j
where the sum in the de
nition of EV is over all edges of ;. There is a natural action of
GV on EV given by
g (ij ) = (gj ij gi;1) if (ij ) 2 EV and g = (g1 gr ) 2 GV .
Let x 2 EV and let W be a ;-graded subspace of V , i.e. Wi Vi for all vertices i in ;.
The subspace W is x-stable if xWi Wj for all edges i ! j in ;. We shall simply write
W V if W is a ;-graded subspace of V and xW W if W is x-stable.
(2.2) Denition of the categories QV and QT QW
The reader may skip this de nition if it looks like too much to swallow. The only important
thing at this stage is that QV is a category of objects and it is contained in a category
called Dcb (EV ).
Let V be a ;-graded vector space over F p and let EV be the variety over F p de
ned in
(2.1). Let Dcb (EV ) be the bounded derived category of Q l -(constructible) sheaves on EV ,
see IV (1.4). Recall that Dcb (EV ) comes endowed with shift functors IV (2.4),
n]: Dcb (EV ) ;! Dcb (EV )
A
7;! An]:
De
ne
QV = the full subcategory of Dcb (EV ) consisting of nite direct sums of simple
perverse sheaves L such that some shift of L is a direct summand of L~
for some partition ~ of = dim(V ).
The complexes L~ are de
ned in (2.7). Let T and W be ;-graded vector spaces over F p .
De
ne
s
QT QW = the complexes L 2 Dcb (ET EW ) such that L = M Ai Bi i=1
for some Ai 2 QT , Bi 2 QW , and some positive integer s.
78
Arun Ram
This is a subcategory of Dcb (ET EW ).
(2.3) The Grothendieck group K associated to the categories QV
Let 2 Q+ and let V be a ;-graded vector space of dimension . Let QV be as in (2.2).
The important thing about QV at the moment is that it is a category related to EV .
The Grothendieck group K(QV ) of the category QV is the C (q)-module generated by
the isomorphism classes of objects in QV with the addition operation given by the relations
B1 B2] = B1] + B2] if B1 B2 2 QV ,
and multiplication by q given by the relations
B n]] = qn B ]
for B 2 QV and n 2 Z,
where the map B ! B n] is the shift functor on Dcb (EV ), see IV (2.4). The structure of
K(QV ) depends only on the element and so we shall often write K in place of K(QV ).
De
ne
M
K=
K :
2Q+
The group K is graded in the same way that Uq n+ is graded, see VII (1.2).
(2.4) Denition of the multiplication in K
Let V be a ;-graded vector space. Let T and W be ;-graded vector spaces such that
W V
and
V=W = T:
If x 2 EV such that xW W then let xW be the linear transformation of W induced by
the action of x on W and let xT be the linear transformation of T = V=W induced by the
action of x on V=W . De
ne
S = fx 2 EV j xW W g
P = fg 2 GV j gW W g
U = fg 2 P j gW = idW gT = idT g:
The groups P and U are subgroups of GV . The group P is the stabilizer of W in GV , it
is a parabolic subgroup of GV . The group U is the unipotent radical of P .
Let QT QW be the subcategory of Dcb (ET EW ) which is de
ned in (2.2). The
diagram
p1
p2
p3
ET EW ;
GV U S ;!
GV P S ;!
EV
(xT xW ) ;7 (g x) 7;! (g x) 7;! gx
induces the diagram
VIII. Hall algebras
79
p1
p2 )
p3 )!
QT QW ;! Dcb (ET EW ) ;!
Dcb (G U S ) (;!
Dcb (G P S ) (;!
Dcb (EV )
where the rst map is the inclusion map.
Theorem. Let V be a ;-graded vector space and let EV be the variety with the GV action
which is de ned in (2.1). Let W and T be ;-graded vector spaces such that W V and
V=W = T . Let QT QW and QV be the categories of complexes of sheaves on ET EW
and EV , respectively, which are de ned in (2.2). There is a well de ned functor
IndVTW : QT QW ;!
;
QV
A
7;! (p3)! (p2) p1 A dim(p1 ) ; dim(p2 )]
where p1 p2, and p3 are as de ned in the diagram above, dim(p1 ) is the dimension of the
bers of the map p1 , and dim(p2 ) is the dimension of the bers of the map p2 .
The multiplication in K is de
ned by the formula
A] B ] = IndVTW (A B ) for A 2 QT and B 2 QW .
With this multiplication K becomes an algebra. The strange shift by dim(p1 ) ; dim(p2 )]
in the de
nition of IndVTW is there to make the multiplication in K match up with the
multiplication in the nonnegative part of the quantum group Uq b+ , see Theorem (2.8)
below.
(2.5) Denition of the pseudo-comultiplication r: K ! K K
Let V be a ;-graded vector space. Let T and W be ;-graded vector spaces such that
W V
and
V=W = T:
If x 2 EV such that xW W then let xW be the linear transformation of W induced by
the action of x on W and let xT be the linear transformation of T = V=W induced by the
action of x on V=W .
De
ne
S = fx 2 EV j xW W g
and let QV be the subcategory of Dcb (EV ) which is de
ned in (2.2). The diagram
S ;!
E E
EV ;
T
W
x ;7 x 7;! (xT xW )
induces the diagram
! Db (E E )
QV ;! Dcb (EV ) ;!
Dcb (S ) ;!
W
c T
80
Arun Ram
where the rst map is the inclusion map.
Theorem. Let V be a ;-graded vector space and let EV be the variety with the GV action
which is de ned in (2.1). Let W and T be ;-graded vector spaces such that W V and
V=W = T . Let QT QW and QV be the categories of complexes of sheaves on ET EW
and EV , respectively, which are de ned in (2.2). There is a well de ned functor
ResVTW : QV ;!
QT QW
B ;
7 ! ;! B dim(p1) ; dim(p2) ; 2dim(GV =P )]
where p1 p2 , and are as de ned above, dim(p1) is the dimension of the bers of the map
p1 , dim(p2 ) is the dimension of the bers of the map p2, and P is the parabolic subgroup
of GV de ned in (2.4).
The pseudo-comultiplication on K is the map r: K ! K K de
ned by
r(A]) = ResVTW (A) if A 2 QV .
The strange shift by dim(p1 ) ; dim(p2) ; 2dim(GV =P )] in the de
nition of ResVTW is
there to make the pseudo-comultiplication in K match up with the comultiplication in the
nonnegative part of the quantum group Uq b+ , see Theorem (2.8) below.
(2.6) The symmetric form on K
Recall that we write K in place of K(QV ) since the structure of K(QV ) depends only on
. For each 2 Q+ , de
ne a bilinear form
f g : K K ! C (q)
by de
ning
B ] B ] = X q;j dim;Hj+2dim(Gn)(u (t s B t s B ))
1
2 !
1
2
j
for B1 B2 2 QV . The vector spaces Hj+2dim(Gn) (u! (t s B1 t s B2)) are de
ned in
(2.10) below. At this stage the important thing is that they depend only on B1, B2 and j .
Use the forms f g , 2 Q+ , to de
ne a bilinear form
f g: K K ! Z((q))
K K = 0
x y = x y on K =
M
2Q
+
K
by setting
if 2 Q+ such that 6= , and
if x y 2 K .
VIII. Hall algebras
81
Theorem. Let V be a ;-graded vector space and let T and W be ;-graded subspaces
such that W V and T = V=W . Let A 2 QT QW and let B 2 QV . Then
A ResV (B) = IndV (A) B TW
TW
The result in this theorem is an analogue of the property of the bilinear form h i on the
quantum group which is given in VII (2.1)(d).
(2.7) Denition of the elements L~ 2 K
Let 2 Q+ and let V be a ;-graded subspace of dimension . A partition of is a sequence
~ = ( 1 : : : m ) of elements of the root lattice Q such that
(1) each j , 1 j m, is a nonnegative integer multiple of a simple root, and
(2) 1 + + m = .
For example we might have ~ = (31 23 0 1 21) if = 61 + 23 . A ag of type ~ in
V is a sequence
f = (V = V (0) V (1) V (m) = 0)
of ;-graded subspaces of V such that dim(V (`;1) =V (`) ) = ` , for all 1 ` m.
Let x 2 EV . A ag f is x-stable if xV (`) V (`) for all 1 ` m. De
ne
F~ = f(x f ) j x 2 EV f is an x-stable ag of type ~ in V g:
The map
~ E
;!
V
~ )!
induces a map Dcb (F~ ) (;!
Dcb (EV ):
(x f ) 7;! x
Let f (~ ) = dim(F~ ) and de
ne
;
L~ = (~ )! 1 dim(F~ )] i.e.
F~
~ )!
Dcb (F~ ) (;!
Dcb (EV )
dim(F~ )]
;!
7;!
Dcb (EV )
1
7;!
L~
where 1 is the constant sheaf on F~ and dim(F~ )] is a shift, see IV (2.4).
Theorem. Let V be a ;-graded vector space of dimension and let T and W be ;-graded
vector spaces such that W V and T = V=W .
(1) Let ~ and ~! be partitions of dim(T ) and dim(W ), respectively. Then
IndVTW (L~ L!~ ) = L~!~ 82
Arun Ram
where, if ~ = ( 1 2 : : : s) and !~ = (!1 : : : !t), then ~ !~ = ( 1 : : : s !1 : : : !t).
(2) Let ~ be a partition of dim(V ). Then
ResVTW L~ =
M
~ ~!
(L~ L!~ )M 0(~ ~!)]
where the sum is over all ~ ~! such that ~ is a partition of dim(T ), ~! is a partition of
dim(W ) and ~ + ~! = ~ . The positive integer M 0(~ ~!) is de ned in (2.9) below.
(3) Let = i be a simple root for g and let V be a ;-graded subspace such that dim(V ) =
i . De ne Li 2 K(QV ) by Li = L~ where ~ = (i ). Then
L ] L ] = 1 :
i
i
1 ; q2
(2.8) The connection between K and the quantum group
We shall make the algebra
K=
M
2Q
+
K
a bit bigger by adding the Ki1's that are in the quantum group Uq g. Let g be the nite
dimensional complex simple Lie algebra corresponding to the Dynkin diagram given by
; and let Uq g be the rational version of the quantum group with k = C (q) where q is
an indeterminate. Let Uq h be the subalgebra of Uq g generated by K11 : : : Kr1. Let
1 : : : r be the simple roots corresponding to the Lie algebra g. De
ne
Ke = algebra generated by K and K11 : : : Kr1 with the additional relations
Ki xKi;1 = q(i )x for all 1 i r and all x 2 K ,
where the inner product in the exponent of q is the inner product on hR given in II (2.7).
De
ne a map j + : K K ! Ke Ke by
P
j + (x y) = xK11 Krr y if x 2 K and y 2 K , where = i i i .
Use the map j + and the pseudo-comultiplication r: K ! K K de
ned in (2.5) to de
ne
a coproduct on Ke by
:
Ke
Ki1
x
;! Ke Ke
7;! Ki1 Ki1
7;! j + r(x)
for 1 i r
for x 2 K,
VIII. Hall algebras
83
where r: K ! K K is the pseudo-comultiplication de
ned in (2.5). Then Ke is a Hopf
algebra!
Theorem. Let Li be as de ned in Theorem (2.7b). The algebra homomorphism determined by
I:
Ke
Li
Ki1
;! Uq b+
7;! Ei 1
7;! Ki
is an isomorphism of Hopf algebras.
(2.9) Dictionary between K and Uq b+
Let us make a small dictionary between the algebra K and the quantum group Uq b+ . Our
intent is to describe, conceptually, the correspondence between the structures inherent
in the algebra K and the structures in the quantum group Uq b+ . The map I is the
isomorphism given in Theorem (2.8).
84
Arun Ram
Ke
Ke is the algebra
generated by
K and the Ki1 s.
K is graded,
K = L 2Q+ K :
is isomorphic to
Uq b+ :
Similarly,
Uq b+ is the algebra
generated by
Uq n+ and the Ki1 s.
Similarly,
The shift functor n] which corresponds to
gives rise to
multiplication by qn in K
Un+
Uq nL+ is graded,
= 2Q+ (Uq n+ ) :
multiplication by qn
in Uq b+ .
The functor IndVTW
corresponds to
the multiplication in Uq n+ .
The functor ResVTW
corresponds to
the comultiplication in Uq b+ .
The inner product f g
corresponds to
the bilinear form h i
pairing Uq b; and Uq b+ .
A partition
~ = (1i1 : : : l il )
indexes L~
which maps,
under I , to
Ei(11) : : : Ei(ll)
where Ei(n) = Ein=n]!.
The Verdier duality
functor D
corresponds to
the C -algebra involution
': Uq n+ ! Uq n+
which sends q 7! q;1 and Ei 7! Ei .
The simple
perverse sheaves
in the various QV
map, under I , to
a canonical basis in Uq n+ .
(2.10) Denition of the constant M 0( !) which was used in (2.7)
Let V be a ;-graded vector space and let T and W be ;-graded subspaces such that
W V and T = V=W . If x 2 EV such that xW W then let xW be the linear transformation of W induced by the action of x on W and let xT be the linear transformation of
T = V=W induced by the action of x on V=W . Let ~ be a partition of dim(V ). If
;
f = V = V (0) V (1) V (m) = 0
is a ag of type ~ in V then de
ne
;
fW = (V \ W ) = (V 0 \ W ) (V (1) \ W ) (V (m) \ W ) = 0
and
;
fT = p(V ) = p(V (0) ) p(V (1) ) p(V (m) ) = 0
where p: V ! V=W
VIII. Hall algebras
85
is the canonical projection.
Let ~ be a partition of dim(T ) and let ~! be a partition of dim(W ), such that ~ + ~! = ~ .
De
ne
)
f
is
an
x
-stable
ag
of
type
~
in
V
,
F~ (~ ~!) = (x f ) xW W
:
and fW is a ag of type ~! in W
(
De
ne a map
and de
ne
: F~ (~ ~!) ;!
F~ F!~
(x f ) 7;! ((xT fT ) (xW fW ))
M 0( !) = dim(p1 ) ; dim(p2) ; 2dim(GV =P ) + dim(F~ ) ; dim(F~ ) ; dim(F!~ ) ; 2dim():
where p1 and p2 are the maps given in (2.4), P is the parabolic subgroup of GV de
ned in
(2.4), and dim(p1), dim(p2 ) and dim() are the dimensions of the bers of the maps p1,
p2 , and , respectively.
(2.11) Denition of the vector spaces Hj+2dim(Gn) (u!(t s B1 t s B2)) from (2.6)
Let ( be a smooth irreducible algebraic variety with a free action of GV such that the
Q l -cohomology of ( is zero in degrees 1 2 : : : m where m is a large integer. Consider the
diagram
s ( E ;!
t Gn(( E )
EV ;
u fpointg:
V
V and the diagram G n(( E ) ;!
V
V
x ;7 (! x) 7;! GV (! x)
These diagrams induce diagrams
t b
s Db (( E ) ;!
Dcb (EV ) ;!
Dc (Gn(( EV ))
V
c
u
!
b
b
Dc (GV n(( EV )) ;! Dc (fpointg):
and
With these notations one has that Hj+2dim(Gn) (u! (t s B1 t s B2)) is a sheaf on the
space fpointg, i.e. a Q l -vector space.
(2.12) Some remarks on Part II of Lusztig's book
The construction of the algebra K and the relationship between it and the quantum group
is detailed in Lusztig's book Lu]. Lusztig works in much more generality there.
(1) Lusztig allows ; to be an arbitrary quiver, rather than just a quiver gotten by orienting
a Dynkin diagram of type A, D or E . It does not require any more theory than what
we have already outlined in order to de
ne the algebra K in this more general setting.
(2) Lusztig wants to construct algebras K which will be isomorphic to the nonnegative
parts of the quantum groups corresponding to general Dynkin diagrams. In order to
do this he must rst consider only diagrams with single bonds and then `fold' the
86
Arun Ram
diagram by analyzing the action of an automorphism of the diagram. The addition
of the folding automorphism into the theory is a nontrivial extension of what we have
developed in these notes.
(3) We have ignored the eect of the orientation of the quiver. If one wants to compare
the algebras K that are obtained by orienting the same quiver in dierent ways one
must analyze a Fourier-Deligne transform between these two dierent algebras. The
amazing thing is that, after one extends the algebras by adding the Ki1s that are in
the quantum group, the two dierent algebras (from the dierent orientations) become
isomorphic!
3. The connection between representations of quivers and perverse sheaves
(3.1) Correspondence between orbits and isomorphism classes of representations of ;
Let ; be a quiver obtained by orienting the edges of a Dynkin diagram of type A, D or
E . For convenience we label the vertices by 1 2 : : : r. Let g be the nite dimensional
complex simple Lie algebra corresponding to the Dynkin diagram given by ;.
Let p be a positive prime integer and let F p be the algebraic closure of the nite eld
F p with p elements. Fix an element 2 Q+ (see VII (1.2)) and a ;-graded vector space V
over F p such that dim(V ) = . De
ne
Y
M
GV = GL(Vi ) and EV = Hom(Vi Vj )
i!j
i
where the sum in the de
nition of EV is over all edges of ;. The natural action of GV on
EV is given by
g (ij ) = (gj ij gi;1) if (ij ) 2 EV and g = (g1 gr ) 2 GV .
The group GV is an algebraic group over F p and EV is a variety over F p with a GV action.
Each element (ij ) 2 EV determines a representation of ; of dimension dim(V ). Each
GV -orbit in EV determines an isomorphism class of representations of ;. Let us make this
correspondence precise.
An orbit index for V is a sequence of positive integers labeled by the positive roots
X
~c = (c )2R+ such that
c = dim(V )
2R+
where R+ is the set of positive roots for g. For each orbit index ~c for V de
ne a representation of ; by
M c
R~c =
e
and let O~c = the GV -orbit in EV corresponding to R~c,
2R+
VIII. Hall algebras
87
where e is the indecomposable representation of ; indexed by the positive root , see
Theorem (1.2b). Then we have a one-to-one correspondence
1;1
GV orbits in EV !
isomorphism classes of representations
of ; of dimension O~c
!
R~c]
(3.2) Realizing the structure constants of the Hall algebra in terms of orbits
Let q be a power of the prime p. Since EV is a variety over F p there is an action of
the the qth power Frobenius map F on EV , see Ca] p. 503. If X is a subset of EV then
let X F denote the set of points of X which are xed under the action of the Frobenius
map F .
Let T and W be ;-graded vector spaces such that W V and T = V=W . Recall the
diagram
p1
p2
p3
ET EW ;
GV U S ;!
GV P S ;!
EV
(xT xW ) ;7 (g x) 7;! (g x) 7;! gx
given in (2.4). Let ~a, ~b, and ~c be orbit indices for T , W and V , respectively. Then we have
p1
ET EW ;
p2
p3
GV U S ;!
GV P S
;!
EV
O~a O~b $ p;1 1 (O~a O~b ) 7;! p2 (p;1 1 (O~a O~b ))
p;3 1 (O~c)
! O~c
Let M = R~a , N = R~b and P = R~c be the representations of ; given in (3.1). By a direct
count, we have
P = Card
CMN
;
p2 (p;1 1 (O~a O~b ))
\
p;3 1 (O~c) F :
P are the structure coecients of the Hall algebra R; given in (1.3).
where CMN
(3.3) Rewriting the Hall algebra in terms of functions constant on orbits
Let q be a power of the prime p. On any variety Y over F p there is an action of the the qth
power Frobenius map F on EV , see Ca] p. 503. If X is a subset of Y then X F denotes
the set of points of X which are xed under the action of the Frobenius map F .
Let l be a positive prime number, invertible in F p . Let Q l be the algebraic closure of
the eld of l-adic numbers. De
ne
K = the vector space of Q l -valued functions on (EV )F which are constant on the
orbits (O~c)F for all orbit indexes ~c for V .
De
ne
K=
M
2Q+
K 88
Arun Ram
where Q+ is as in VII (1.2).
De
ne a multiplication on K as follows. Let T and W be ;-graded vector spaces such
that W V and T = V=W . Recall the diagram
p1
p2
p3
ET EW ;
GV U S ;!
GV P S ;!
EV
(xT xW ) ;7 (g x) 7;! (g x) 7;! gx
given in (2.4). Let = dim(T ) and ! = dim(W ). Given f1 2 K and f2 2 K! de
ne a
function f1 f2 as follows:
If x 2 (EV )F then
X V
(f1 f2)(x) =
CTW f1 (xT )f2 (xW )
xT xW
where the sum is over all xT 2 (ET )F and xW 2 (EW )F , and
F
V = Card( f(y f ) 2 (GV P S ) j p1 (y f ) = (xT xW ) p3 (p2 (y f )) = xg ) :
CTW
Card((GT )F )Card((GW )F )
Let ~c be an orbit index and let ~c be the characteristic function of the orbit O~c, i.e.
x 2 (O~c)F ,
~c(x) = 10 ifotherwise.
Then it follows from the observation in (3.2) that the map
K ;! R;
~c 7;! R~c]
is an isomorphism of algebras, where R; is the Hall algebra de
ned in (1.3).
(3.4) The isomorphism between K and K
Let ~a be an orbit index and let O~a be the corresponding GV -orbit in EV as de
ned in
(3.1). Let F~c be the constant sheaf Q l on the orbit O~c extended by 0 on the complement.
This sheaf can be viewed as the complex of sheaves A, for which A0 = F~c and Ai = 0,
for all i 6= 0. In this way F~c can be viewed as an element of QV , see IV (1.4), and the
isomorphism class F~c] of F~c is an element of K.
for x 2 (EV )F ,
Theorem. Let K be the algebra de ned in x2 and let K be the algebra de ned in (3.3).
For each orbit index ~c let O~c be the corresponding GV orbit in EV , as given in (3.1), and
let ~c be the characteristic function of the orbit O~c. The map
K ;! K
F~c] 7;! ~c
is an isomorphism of algebras.
VIII. Hall algebras
89
This theorem is a consequence of an analogue of the Grothendieck trace formula. The
Grothendieck trace formula, Ca] p. 504, is the formula
jX F j =
2dim(
XX)
i=0
(;1)i Tr(F Hci(X Q l ))
which describes the number of points of X which are xed under a Frobenius map F in
terms of the trace of the action of the Frobenius map on the l-adic cohomology Hci (X Q l )
of the variety X .
Theorems (3.4) and (2.8) together show that there is a natural connection between
the algebra K and the Hall algebra R; which was introduced in (1.3).
90
Arun Ram
IX. Link invariants from quantum groups
The theorems of Alexander and Markov given in (1.4) and (1.5) are considered classical, they can be found in Bi] Theorem 2.1 and Theorem 2.3, respectively. A sketch, with
further references, of the proof of Theorem (1.7) can be found in CP] 15.2. See J] Prop.
6.2 for the proof of Theorem (1.2) and Stb] Lemma 2.5 for the proof of Proposition (1.6).
(1.1) Knots, links and isotopy
A knot is an imbedded circle in R 3 . By circle we mean an S 1 and imbedded is in the sense
of dierential geometry. A link is a disjoint union of imbedded circles in R3 . A link is
oriented if each connected component is oriented. We shall identify a link with its \picture
in the plane".
knot (unknot)
knot (trefoil)
link (Borromean rings)
The conceptual idea of when two links are the same is called ambient isotopy. More
precisely, two oriented links L1 and L2 are equivalent under ambient isotopy if there is an
orientation preserving dieomorphism of R 3 which takes L1 to L2 . In terms of pictures
in the plane L1 and L2 are equivalent under ambient isotopy if the picture for L1 can be
transformed into the picture for L2 by a sequence of Reidemeister moves:
(R1)
!
!
(R2)
(R3)
!
!
!
These moves are applied locally to a region in the picture and all possible orientations of
the strings are allowed. The equivalence relation on pictures in the plane gotten by only
allowing moves (R2) and (R3) is called regular isotopy.
(1.2) Link invariants
Let S be a set. An oriented link invariant with values in S is a map
P : L ;! S
IX. Link invariants from quantum groups
91
from the set L of equivalence classes of oriented links under ambient isotopy to S .
Theorem. There exists a unique oriented link invariant P : L ;! Zx x;1 y y;1] such
that
P
0
= 1 and xP B
@
1
0
CA ; x;1P B@
1
0
CA = yP B@
1
CA :
The unusual notation in the second relation indicates changes to the link in a local region.
The link invariant de
ned in the above Theorem is the HOMFLY polynomial. Other
famous link invariants can be obtained in a similar fashion by specializing x and y, as
follows:
Jones polynomial
x = t;1 and y = t1=2 ; t;1=2 ,
Conway polynomial
x = 1 and y = y,
Alexander polynomial
x = 1 and y = t1=2 ; t;1=2 .
(1.3) Braids
A braid on m-strands consists of two rows of m dots each, one above the other, and m
strands in R 3 such that
(1) each strand connects a dot in the top row to a dot in the bottom row,
(2) the strands do not intersect,
(3) every dot is incident to exactly one strand.
Composition of two braids b1 b2 on m-strands is given by identifying the bottom points of
b1 with the top points of b2 . The following are braids on 6 strands,
b1 =
b2 =
and the product b1 b2 is the braid
b1 b2 =
:
92
Arun Ram
One should note that it is important to be careful in de
ning the word \strand" since the
diagram
is not a legal braid.
The braid group Bm is the group of braids on m strands and it is a famous theorem
of E. Artin that Bm has a presentation by generators
1 2
i;1
i i+1 i+2
m;1 m
gi =
for 1 i m ; 1, and relations
gi gj = gj gi gi gi+1gi = gi+1gi gi+1 if ji ; j j > 1,
for 1 i m ; 2.
(1.4) Every link is the closure of a braid
It will be convenient to \orient" the strands of a braid so that they \travel" from top to
bottom.
The closure (^ m) of a braid 2 Bm on m-strands is the oriented link obtained by joining
together (identifying) each dot in the top row to the corresponding dot in the bottom row.
If
=
^ 3) =
then (
and if
=
^ 3) =
then (
:
IX. Link invariants from quantum groups
93
Theorem. (Alexander) Every oriented link is the closure (^ m) of a braid 2 Bm for
some m.
(1.5) Markov equivalence
The braid group Bm can be embedded into the braid group Bm+1 by adding a strand.
B6
;!
B7
7;!
Two braids 1 2 Bm and 2 2 Bn are Markov equivalent if they are equivalent under the
equivalence relation on tm Bm (disjoint union of Bm ) which is de
ned by the relations
(M1) 0 0 ;1 , for all 0 2 Bk , and
(M2) gk gk;1,
if 2 Bk where in the relation (M2) the products gk and gk;1 are obtained by viewing as an
element of Bk+1 under the imbedding Bk Bk+1 .
Theorem. (Markov) Two braids 1 2 Bm and 2 2 Bn have equivalent closures (^1 m)
and (^2 n) (under ambient isotopy) if and only if 1 and 2 are Markov equivalent.
(1.6) Quantum dimensions and quantum traces
Let g be a nite dimensional complex simple Lie algebra and let Uh g be the corresponding
Drinfel'd-Jimbo quantum group. Let ~ be the element of h such that i (~) = 1 for all
simple roots i , see II (2.6).
Let V be a nite dimensional Uh g module. The quantum dimension of V is
dimq (V ) = TrV (eh~):
If z 2 EndUh g (V ) then the quantum trace of z is
trq (z) = TrV (eh~z) :
Proposition. Let L() be the irreducible Uh g-module of highest weight as given in
VI (1.3) and VI (2.3). Then
dimq (L()) =
Y 1 ; q(+)
()
>0 1 ; q
where q = eh 94
Arun Ram
P
= 21 >0 is the half sum of the positive roots, and the inner product ( ) on hR is as
given in II (2.7).
(1.7) Quantum traces give us link invariants!
Recall that Uh g is a quasitriangular Hopf algebra and that therefore the category of
nite dimensional Uh g-modules is a braided SRMCwMFF. Let
R"V V : V V ;! V V
be the braiding isomorphism from V V to V V . It follows from the identity I (3.5)
that the map
*: Bm ;!
EndUh g (V m )
gi 7;! R"i = id(i;1) R"V V idm;(i+1)
is well de
ned and that *(1 2 ) = *(1 )*(2 ) for all braids 1 2 2 Bm .
Theorem. Let g be a nite dimensional complex simple Lie algebra and let Uh g be the
correponding Drinfel'd-Jimbo quantum group. Let L() be an irreducible Uh g-module of
highest weight (see VI (1.3) and VI (2.3)). Let be the half sum of the positive roots
and let ( ) be the inner product on hR as given in II (2.7). For each braid on m-strands
de ne
m
1
^ m) = <+2>
P (
q
dim (V ) trq (*( ))
q
where q = eh . Then P is a well de ned link invariant.
Remark. The above theorem gives the Jones polynomial when g = sl2 , the simple Lie
algebra corresponding to the Dynkin diagram A1 , and L() is chosen to be the irreducible
representation of Uh g with highest weight = !1 .
References
95
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