Download skew-primitive elements of quantum groups and braided lie algebras

SKEW-PRIMITIVE ELEMENTS OF QUANTUM GROUPS
AND BRAIDED LIE ALGEBRAS
BODO PAREIGIS
Contents
1. Quantum groups, Yetter-Drinfel'd algebras, and G-graded algebras
2. Skew primitive elements
3. Symmetrization of B -modules
4. Lie algebras
5. Properties of Lie algebras
6. Derivations and skew-symmetric endomorphisms
7. Lie structures on C -graded modules
References
n
pn
3
5
8
11
13
14
18
20
In the study of Lie groups, of algebraic groups or of formal groups, the concept of
Lie algebras plays a central role. These Lie algebras consist of the primitive elements.
It is diÆcult to introduce a similar concept for quantum groups. Many important
quantum groups have braided Hopf algebras as building blocks. As we will see most
primitive elements live in these braided Hopf algebras. In [P1] and [P2] we introduced
the concept of braided Lie algebras for this type of Hopf algebras. In this paper we
will give a survey of and a motivation for this concept together with some interesting
examples.
By the work of Yetter [Y] we know that the category of Yetter-Drinfel'd modules
is in a sense the most general category of modules carrying a natural braiding on
the tensor power of each module (instead of a symmetric structure). The study of
algebraic structures in such a category is a generalization of the study of group graded
algebraic structures. We will describe the braid structure in the category of YetterDrinfel'd modules, the concept of a Hopf algebra in this category, and explain the
reason why we want to study such braided Hopf algebras.
Date : April 15, 1997.
1991 Mathematics Subject Classication.
Primary 16A10.
1
2
BODO PAREIGIS
One of the big obstacles in this theory is the fact, that the set of primitive elements P (H ) of a braided Hopf algebra H does not form a Lie algebra in the ordinary
or slightly generalized sense. We will show, however, that there is still an algebraic
structure on P (H ) consisting of partially dened n-ary bracket operations, satisfying certain generalizations of the anti-symmetry and Jacobi relations. We call this
structure a braided Lie algebra. This will generalize ordinary Lie algebras, Lie super
algebras, and Lie color algebras. Furthermore we will show that the universal enveloping algebra of a braided Lie algebra is again a braided Hopf algebra leading us
back to quantum groups.
Primitive elements of an ordinary Hopf algebra L are elements x 2 L satisfying
(x) = x 1 + 1 x. The set of primitive elements P (L) of L forms a Lie algebra
induced by the Lie algebra structure [x; y] := xy yx on L. In fact one veries that
([x; y]) = [x; y] 1 + 1 [x; y] if x; y 2 P (L).
Since primitive elements are cocommutative they can only generate a cocommutative Hopf subalgebra
of L. More general elements, skew-primitive elements with
0
(x) = x g + g x, are needed to generate quantum groups or general (noncommutative noncocommutative) Hopf algebras. But the skew-primitive elements do
not form a Lie algebra anymore.
Many quantum groups are Hopf algebras of the special form L = kG?H = kG H
where H is a braided graded Hopf algebra over a commutative group G [L, Ma, R, S].
In this situation the primitive elements of H are skew-primitive elements of L. So
the structure of a braided Lie algebra on the set of primitive elements in H induces
a similar structure on a subset of the skew-primitive elements of L.
The central idea leading to the structure of braided Lie algebras is the concept of
symmetrization. For any module P in the category of Yetter-Drinfel'd modules the
n-th tensor power P of P has a natural braid structure. We construct submodules
P ( ) P for any nonzero in the base eld k , that carry a (symmetric) S structure. This is essentially an eigenspace construction for a family of operators. The
Lie algebra multiplications will be dened on these S -modules P ( ) for primitive
n-th roots of unity .
In the group graded case, there is a fairly explicit construction of these symmetrizations. In the last section of this paper we describe them in the C -graded case for a
cyclic group of prime power order.
Apart from the explicit examples of braided Lie algebras we gave in [P1] we showed
in [P2] that the set of derivations Der (A) of an algebra A in YD forms a braided Lie
algebra. This is based on the existence of inner hom-objects in YD . In Theorem
6.3 we will show that the category YD is a closed monoidal category. We also
construct another large family of braided Lie algebras consisting of skew-symmetric
endomorphisms of a Yetter-Drinfel'd module with a bilinear form. This generalizes
the construction of Lie algebras of classical groups.
I wish to thank Peter Schauenburg for valuable conversations especially on Theorems 2.2 and 6.3.
n
n
n
n
n
n
pn
K
K
K
K
K
K
SKEW-PRIMITIVE ELEMENTS OF QUANTUM GROUPS AND BRAIDED LIE ALGEBRAS
1.
3
G-graded
Quantum groups, Yetter-Drinfel'd algebras, and
algebras
Quantum groups arise from deformations of universal enveloping algebras of Lie
algebras. They often have the following form.
Let K = kG be the group algebra of a commutative group. Let H be a Hopf
algebra in the category of Yetter-Drinfel'd modules over K . Then the biproduct
K ? H is a Hopf algebra [R, Ma, FM], which in general is neither commutative
nor cocommutative. More generally quantum groups are Hopf algebras of the form
H ? K ? H where H and H are dual to each other [L, S].
We investigate the question what the primitive (Lie) elements of these quantum
groups are and whether they carry a specic structure (of a Lie algebra).
Let us rst introduce the concept of Yetter-Drinfel'd modules over a Hopf algebra
K with bijective antipode (see [M] 10.6.10). A Yetter-Drinfel'd module or crossed
module over a Hopf algebra K is a vector space M which is a right K -module and a
right K -comodule such that
+
+
X(x c)
(1)
[0]
(x c) =
[1]
Xx
[0]
c S (c )x
(2)
(1)
c
[1] (3)
for all x 2 M and all c 2 K . Here we useP the Sweedler notation (c) = P c c
with : K ! K K and Æ(x) = x x with Æ : M ! M K . The
Yetter-Drinfel'd modules form a category YD in the obvious way (morphisms are
the K -module homomorphisms which are also K -comodule homomorphisms).
The most interesting structure on YD is given by its tensor products. It is well
known that the tensor product M N of two vector spaces which are K -modules
is again a K -module (via the comultiplication or diagonal of K ). If M and N are
K -comodules then their tensor product is also a K -comodule (via the multiplication
of K ). If M and N are Yetter-Drinfel'd modules over K , then their tensor product
is a Yetter-Drinfel'd module, too. So with this tensor product YD is a monoidal
category.
A monoidal or tensor product structure on an arbitrary category C allows to dene
the notion of an algebra A with a multiplication r : A A ! A which is associative
and unitary (by u : k ! A). Similarly one can dene coalgebras in C . There
is, however, a problem with dening a bialgebra or Hopf algebra H in C . In the
compatibility condition between multiplication and comultiplication of H
(1)
[0]
K
K
(2)
[1]
K
K
K
K
X(h h0)
X h h0 h h0
one usesPin the formation of the right hand side P h h0 h h0 = (r
r)(1 1)( h h h0 h0 ) = (r r)(1 1)( )(h h0 ) a switch or
(1)
(h h0) =
(2)
(1)
(1)
(1)
(2)
(2)
(1)
(1)
exchange function : H H ! H H in the category C .
(1)
(2)
(2)
(2)
(2)
4
BODO PAREIGIS
There exists such a nontrivial morphism : M N ! N M in the category
YDKK of Yetter-Drinfel'd modules. It is given by
X n m n 2 N M:
(2)
: M N 3 m n 7!
[0]
[1]
This is a natural transformation with the additional property of a braiding which we
will discuss later. So we know now how to dene a Hopf algebra H in YD . Observe
that these Hopf algebras are not ordinary Hopf algebras since the condition for the
compatibility between multiplication and comultiplication involves the new switch
morphism.
Given a Hopf algebra H in YD we can dene the biproduct K ? H [R] between
K and H . The
underlying vector space is the tensor product K H . We denote the
elements by P c h =: P c #h . The (smash product) multiplication is given by
X
(3)
(c#h)(c0#h0) := cc0 #(h c0 )h0
and the (smash coproduct) comultiplication is given by
X
(4)
(c#h) = (c #(h ) ) (c (h ) #h ):
If K is a Hopf algebra and H is a Hopf algebra in YD then K H becomes a Hopf
algebra with this multiplication and comultiplication, called the biproduct K ? H (see
[R, M, Ma]).
We will be mainly interested in the case where K = kG is the group ring of
a commutative group. It is well known that the kG-comodules are precisely the Ggraded vector spaces ([M] Example 1.6.7). We denote this category by M . From the
comodule structure on a G-graded vector space M = 2 M we can construct a kGmodule structure such that M becomes a Yetter-Drinfel'd module. This construction
depends
on a bicharacter : G G ! k given by a group homomorphism : G Z G
! k . Then it is easy to verify that M is in YD with the kG-module structure
m g := (h; g )m
for homogeneous elements m 2 M , h 2 G and g 2 G. So any bicharacter denes
a functor M ! YD . This functor preserves tensor products. In particular any
algebra or coalgebra in M is also an algebra resp. coalgebra in YD . Since M
can be considered as a monoidal subcategory of YD via and thus has a switch
map : M N ! N M we can also dene Hopf algebras in M and they are
also preserved by the functor induced by . In this situation turns out to be simply
(m n ) = (h; g )n m :
Although one may dene Yetter-Drinfel'd categories YD for arbitrary Hopf algebras K with bijective antipode hence in particular for arbitrary group rings kG
(where G is not commutative) the above functor that induces Yetter-Drinfel'd modules from kG-comodules with a bicharacter can only be constructed for commutative
groups G.
K
K
K
K
i
i
i
i
(1)
(1)
(2)
(1) [0]
(2)
(1) [1]
(2)
K
K
kG
h G
h
kG
kG
h
kG
kG
kG
h
h
h
kG
kG
kG
kG
kG
h
g
g
kG
h
K
K
kG
SKEW-PRIMITIVE ELEMENTS OF QUANTUM GROUPS AND BRAIDED LIE ALGEBRAS
5
If A is an algebra in M then kG#A carries the induced algebra structure
(5)
(g#a )(g0#a 0 ) = gg0#(h; g0)a a 0 :
If H is a Hopf algebra in M then the Hopf algebra L = kG ? H has the comultiplication
X
(6)
(g#a ) = (g#b ) (gk#b )
kG
h
h
h h
kG
L
h
k
hk
1
if (a ) = P 2 b b where lower indices stand for the degree of homogeneous
elements.
Example 1.1. Let G = C = hti be the cyclic group with three elements, generator
t, and let (t; t) = 2 k be a primitive 3-rd root of unity. Then H = k [x]=(x ) with
(x) = x 1 + 1 x is a Hopf algebra in YD where x has degree t, the generator
of C .
To see that k[x]=(x ) is a Hopf algebra in YD we check that (x ) = (x) .
We observe that (x 1)(1 x) = x x but (1 x)(x 1) = x x. So we have
(x) = (x 1+1 x) = x 1+(1+ + )(x x )+(1+ + )(x x)+(1 x ) =
0 = (0) = (x ).
This example shows, that products and powers of primitive elements behave totally
dierent in YD from how they behave in Vec, the category of vector spaces. This
new behavior is central to the following observations on Lie algebras.
The biproduct kC ? k[x]=(x ) is isomorphic to the Hopf algebra kht; xi=(t
1; x ; xt tx) if we associate t#1 and t resp. 1#x and x.
2
k G
H
h
k G k
hk
1
3
3
kG
kG
3
kG
kG
3
3
3
3
2
2
3
2
2
3
3
3
kG
kG
3
3
3
3
2.
The last example shows the importance of elements x 2 H (a Hopf algebra in
YD ) with (x) = x 1+1 x. These elements are called primitive elements. They
act like derivations. The primitive elements, homogeneous of degree g 2 G, form a
vector space P (H ). In [P1] (after the proof of 3.2) and [P2] Lemma 5.1 we proved
Lemma 2.1. The set of primitive elements of a Hopf algebra H in YD is a YetterDrinfel'd module P (H ).
L 2 P (H ) is also in M (via the same
If K = kG and H 2 M then P (H ) =
bicharacter ).
In general and especially in Hopf algebras of the form K ? H we have to consider
more general conditions for primitive elements. An element g 6= 0 in a Hopf algebra
L (in Vec) is called a group-like element if
(g) = g g:
This implies "(g) = 1. By (4) a group-like element g 2 K denes a group-like element
g #1 2 L = K ? H .
Skew primitive elements
K
K
g
K
K
kG
g G
g
kG
6
BODO PAREIGIS
Let g; g0 2 L be group-like elements. An element x 2 L is called a (g0; g)-primitive
element or a skew primitive element if
(x) = x g + g0 x:
This implies "(x) = 0. Observe that a primitive element is (1; 1)-primitive, since 1 is
a group-like
element.
0
The (g ; g)-primitive elements form a vector space P 0 (L). For x 2 P 0 (L) we
have
(g0 x) = (g0 g0 )(x g + g0 x) = g0 x g0 g + 1 g0 x;
so we get an isomorphism P 0 (L) 3 x 7! g0 x 2 P 0 (L). Thus it suÆces to
study the spaces P0 (L).
If f : L ! L is a Hopf algebra
homomorphism then f obviously preserves
group-like
elements
g 2 L and (g 0 ; g )-primitive elements x 2 L are mapped into
(f (g0); f (g))-primitive
elements f (x) 2 L0. So we get a homomorphism f : P 0 0 (L)
0
! P 0 (L ) and in particular a homomorphism f : P (L) ! P
(L ).
Now let L = K ? H and let h 2 H be primitive and homogeneous of degree g 2 K ,
a group-like element in K , i.e. (h) = h 1+1 h and Æ(h) = P h h = h g
with respect to the K -comodule structure Æ : H ! H K of H . Then by (6)
(1#h) = 1#h g#1 + 1#1 1#h
hence 1#h 2 P (L). So the following is a monomorphism
P (H ) 3 h 7! 1#h 2 P (L):
(g ;g )
1
1
1
(g ;g )
1
1
(g ;g )
(1;g
1
1
1
g)
(1;g )
(g ;g )
(f (g );f (g ))
(1;g )
H
(1;f (g ))
[0]
[1]
L
(1;g )
g
(1;g )
Theorem 2.2. Let K be a Hopf algebra with bijective antipode and H be a Hopf
YDKK . Let L = K ? H . For every group-like element g 2 K we have
P(1;g)(L) = P(1;g)(K )#1 1#Pg (H ):
algebra in
Let : K ? H ! K be dened by (c#h) = c"H (h) and : K ! K ? H by
(c) = c#1. Then one checks easily that and are Hopf algebra homomorphisms
and that = idK . Thus and preserve group-like elements and skew-primitive
elements. In particular we have for any group-like element g 2 K that (g) = g#1 2 L
is group-like. We identify the group-like elements g in K with the group-like elements
(g ) in L. For any (1; g )-primitive element x 2 L the element (x) 2 K is also
(1; g)-primitive. Furthermore and dene a direct sum decomposition K ? H =
Im() Ker().
We have already seen 1#Pg (H ) P(1;g)(L). Furthermore if c 2 P(1;g)(K ) then
(c) = c#1 2 P(1;g)(L) so that P(1;g)(L) P(1;g)(K )#1 + 1#Pg (H ).
Given a (1; g)-primitive element x 2 L = K ? H for g 2 K . We study how x
decomposes with respect to the direct sum decomposition x = (x) + (x (x)).
The element (x) is (1; g)-primitive since x is (1; g)-primitive. So (x) 2 P(1;g)(L)#1.
Proof.
SKEW-PRIMITIVE ELEMENTS OF QUANTUM GROUPS AND BRAIDED LIE ALGEBRAS
7
Furthermore (x) 2 P (L) implies y := x (x) 2 P (L) \ Ker(). We
haveP (y) = y g + 1 y and ( 1) (y) = 1 y, since y 2 Ker(). Let
y = c#h 2 L = K ? H then
1 y = ( 1) (y) = P
( 1)(P(c #(h ) ) (c (h ) #h ))
= c " ((h ) ) c (h ) #h :
We apply 1 " #1 and get
P 1#" (c)h = (1 " #1)(P 1 c#h)
= P c " (c )#" ((h ) )" ((h ) )h = P c#h;
so we know
y = x (x) = 1#h
for some h 2 H .
Since y is skew-primitive we get 0 = " (y) = " (h). Furthermore (y) = y g +
1 y = 1#h g#1 + 1#1 1#h = P(1#(h ) ) ((h ) #h ) implies
X
h g #1 + 1 1 #h = (h ) (h ) #h
so that by applying 1 "#1 we get h 1 + 1 h = P
h h = (h). i.e. h is
primitive. Furthermore with 1 1#" we get h g = P h h = Æ(h) where Æ : H
! H K is the given comodule struktur of H . Thus h is homogeneous of degree g
so that
x (x) 2 1#P (H ):
So we have shown P (L) P (K )#1 1#P (H ).
(1;g )
(1;g )
L
L
L
(1)
(1) H
(1) [0]
(1) [0]
(2)
(2)
(1) [1]
(1) [1]
(2)
(2)
K
K
K
(1) K
(2)
H
(1) [0]
L
K
(2)
H
L
(1) [0]
H
(1) [1]
K
(1) [0]
(1) [1]
(2)
(1) [1]
(2)
(1)
(2)
[0]
H
[1]
g
(1;g )
(1;g )
g
Corollary 2.3. Let G be a commutative group, be a bicharacter of G. Let H be a
Hopf algebra in MkG . Let L = kG ? H . For every g 2 G we have
P(1;g)(L) = k (g 1)#1 1#Pg (H ):
Proof. The only thing to check is P(1;g)(kG) = k (g 1). We have (g 1) =
g g 1 1 = (g 1) g + 1 (g 1). Conversely if x = i gi is in P(1;g ) (kG)
then by comparing coeÆcients one obtains x = 0(g0 1).
P
0
In particular we have P 0 (kG ? H ) = 1#P (H ).
In order to study the (g ; g)-primitive elements in L = kG H it suÆces now to
study P (H ). We are interested in obtaining an algebraic structure on P (H ) similar
to the Lie algebra structure on the primitive elements of an ordinary Hopf algebra
in Vec. The usual Lie multiplication on P (H ) induced by the multiplication of the
Hopf algebra H in YD cannot be used as the following example shows.
Example 2.4. Let x; y 2 P (H ) and H a Hopf algebra in YD . If we dene [x; y ] :=
xy r (x y ) then
[x; y] = [x; y] 1 + 1 [x; y] + x y (x y):
(1;1)
1
kG
kG
H
K
K
2
8
BODO PAREIGIS
So in general the element [x; y] 2 H will not be a primitive element unless (x y) =
x y.
However, in Example 1.1 we found the fact that x may be primitiveif x is primitive.
3.
B
We want to nd a reasonable algebraic structure (of a generalized Lie algebra) on
the set of primitiveelements of a Hopf algebra in the category YD of Yetter-Drinfel'd
modules. We also want to get a generalized Lie algebra from every (noncommutative)
algebra A in YD by suitable denition of Lie multiplications with the help of the
algebra multiplication. We expect that the Lie products are (skew-)commutative and
satisfy some kind of Jacobi identity. The (skew-)commutativity of an ordinary Lie
algebra P results from the action of S (the symmetric group) on P P . In the case
of an algebra A made into a Lie algebra this skew-commutativity results from the
following composition of maps
[:; :] = r Æ SkSymm : A A ! SkSymm(A A) ! A
where SkSymm denotes the set of anti-symmetric tensors in A A and the antisymmetrization process itself. In general the Lie multiplication must only be dened
on SkSymm(P P ) since
[x; y] = [x y] = 21 [x y y x]
where (x y y x) 2 SkSymm(P P ).
This is a special case of the following more general
observation. If a nite group
P
G acts on a module M then the map M 3 m 7! 2 gm 2 G-Inv(M ) sends any
m 2 M into the set of G-invariant elements G-Inv(M ) = fm 2 M j8g 2 G : gm = mg:
This process is only possible for nite groups G. In the above case S acts on P P by
(x y ) = y x. We want to use this process to dene a generalized Lie algebra. We
will not restrict ourselves to binary multiplications, since the Jacobi identity indicates
that higher order multiplications might be of interest, too. Furthermore generalized
Lie multiplications will only be partially dened, on subspaces of P : : : P .
The reason for the fact that the Lie bracket [x; y] of two primitive elements x; y 2
P (H ) is not primitive in Example 2.4 results from the following observation. The
operation of the switch map : P P ! P P induces only an operation of the
group Z rather than Z=(2) on P P . Observe that Z= B is the 2-nd braid group,
whereas Z=(2) = S is the 2-nd symmetric group.
The switch morphism : P P ! P P satises the (quantum-) Yang-Baxter
equation
( 1)(1 )( 1) = (1 )( 1)(1 )
hence it induces the action of the n-th braid group B on P . The braid group B is
generated by elements ; : : : ; ( acting on P by switching the i-th and (i +1)-st
2
3
Symmetrization of
n -modules
K
K
K
K
2
1
2
g G
2
2
2
1
n 1
i
n
n
n
n
SKEW-PRIMITIVE ELEMENTS OF QUANTUM GROUPS AND BRAIDED LIE ALGEBRAS
9
component) and has relations
i j = j i for ji j j 2;
i i+1 i = i+1 i i+1 :
So the n-th symmetric group
Sn with generators 1 ; : : : ; n 1 is a canonical quotient
2
of Bn (by i 7! i) by i = id. This observation is the reason that is called a
braiding for
the category YD .
Any S -module is a B -module by the residue homomorphism B ! S . Conversely, is there a way to construct an S -module from a given B -module in a canonical way? Why do we want to consider S -modules rather than B -modules? The
main reason is, as we saw above, that B is an innite group and S is a nite group.
The rst question leads to the process of symmetrization of a B -module or braid
module as follows. Any element 2 k induces an algebra automorphism : kB
! kB by ( ) := , due to the fact that the relations for the braid group
are homogeneous. The algebra homomorphism : kB ! kB ! kS induces a
forgetful functor M ! M which has the right
adjoint Hom (kS ; ) : M
! M. So any braid module M and any 2 k induces a module Hom (kS ; M )
over the symmetric group. Since the algebra homomorphism is surjective we get
a submodule
M ( ) := Hom (kS ; M ) Hom (kB ; M ) = M:
In [P2] following Denition 2.3 we proved
M ( ) = fm 2 M j (m) = m 8 2 B ; i = 1; : : : ; n 1g
and computed the action of S on M ( ) as
(7)
(m) = (m):
So we have kS -submodules M ( ) M for every 2 k. Since they are constructed similar to eigenspaces for the eigenvalues they form direct sums in M .
If P 2 YD then P = P : : :
P is in YD and B acts on P . The symmetrization with respect to 2 k gives a module P ( ) 2 YD ([P2] Theorem
2.5).
Now we consider the special case K = kG and a G-graded vector space M 2 M
for a commutative group G with a bicharacter : G Z G ! k . As we saw in
the rst section M can be considered as a Yetter-Drinfel'd module in YD . The
space M decomposes into homogeneous components M = 2 M . The components
themselves are again Yetter-Drinfel'd modules, since M is G-graded.
Assume that M is in YD and that B operates on M by morphisms in the category YD . Then B operates also on the homogeneous components M . Since
M ( ) is a Yetter-Drinfel'd module it decomposes into homogeneous components
M ( ) = M ( ).
K
K
n
n
n
n
n
n
n
n
n
n
n
n
n
i
i
n
kSn
n
kBn
n
kBn
kSn
n
kBn
kBn
kBn
1
n
2
kBn
2
i
n
n
n
n
1
i
n
K
K
i
2
n
K
K
n
n
n
K
K
kG
kG
kG
g G
kG
kG
g
g
kG
kG
n
g
g
n
g
10
and P x
BODO PAREIGIS
Let P 2 M
: : : x 2 P be an element with the x homogeneous
of degree g 2 G. Then the action of 2 B looks particularly simple
X
X
( x : : : x ) = (g ; g )(g ; g )( x : : : x ):
One shows thatPthe element P x : : : x is in P ( ) i (g ; g )(g ; g )(P x : : : x ) = ( x : : : x ) for all i 6= j . This implies (g ; g )(g ; g ) = for
all i 6= j , if the given element is not zero. In [P2] Proposition 7.2 we proved
Proposition 3.1. Let 2 k be given. Then
M P :::
P :
P ( ) =
kG
g1
n
gn
i
2
i
2
g1
i
i+1
i
g1
2
gn
gn
g1
gi
n
i
i+1
g1
n
gn
gn
n
g1
f(g1 ;:::;gn ) -familyg
is called a -family if (gi; gj )(gj ; gi) = 2
gn
j
i
i
j
j
i
i
j
g1
2
gn
Here (g ; : : : ; g )
for all i 6= j .
Example 3.2. We close this section with some important examples.
1. If G = f0g then (0; 0) = 1 and YD = Vec. The braiding in Vec is the usual
switch map (x y) = y x hence = id. If M is a B -module then the equation
(m) = m has a non-trivial solution only if = 1. Hence P ( ) = 0 for all
6= 1. So the only nontrivial "symmetrization" is P (1) = P ( 1) = P . Since
= id the braid group B acts on P by ordinary permutations generated by the
action of the canonical switch map. If 2 B and its canonical image in S then
the action (7) given by the symmetrization on z 2 P is (z) = sgn()(z), where is acting as ordinary switch permutation.
2. Let G = Z=2Z= f0; 1g the cyclic group of order two with the (only) nontrivial
bicharacter (i; j ) = ( 1) . Then M is the category of (2-graded) super vector
spaces with the braiding (x y ) = ( 1) y x . Again = id so that P ( ) = 0
for all 6= 1. The only symmetrization is P (1) = P ( 1) = P .
3. Let G be an arbitrary nite abelian group with bicharacter such that (h; g) =
(g; h) . Then M is the category of (G-graded) color vector spaces with the
braiding (x y ) = (h; g)y x . Again = id so that P ( ) = 0 for all 6= 1.
The only symmetrization is P (1) = P ( 1) = P .
4. The rst interesting example is G = Z=3Z = f0; 1; 2g with the bicharacter
(i; j ) = where is a primitive 3-rd root of unity. Then M is the category of
3-graded vector spaces with
the braiding (x y ) = y x . The homogeneous
L
elements m 2 P ( ) = f
g P : : : P for P 2 M have to
satisfy (m) = m = m. The possibilities for are 1; 1; ; ; ; . By
computing all possible -families we get for example
(P P )( 1) = (P P ) + (P P );
(P P P )() = (P P P ) (P P P );
P ( ) = P P ;
(P P P )( ) = 0;
P ( ) = 0:
n
1
kG
kG
2
1
2
n
2
i
n
n
2
n
n
n
n
n
ij
n
n
kG
i
ij
j
j
n
i
n
2
n
n
kG
1
h
g
g
n
n
2
h
n
n
ij
kG
i
n
1
2
2ij
j
g1
(g1 ;:::;gn ) -family
2
1
1
kG
0
1
2
6
2
2
2
i
2
6
1
6
j
gn
0
6
ij
2
2
2
SKEW-PRIMITIVE ELEMENTS OF QUANTUM GROUPS AND BRAIDED LIE ALGEBRAS
11
The particular choice of the number of tensor factors in this example will become clear
in the next section. The action of the symmetric group on these symmetrizations is
(x y ) = y x ;
(x y z ) = y x z ;
i.e. the ordinary permutation { and this holds for all elements in S and also for
elements in P P P ,
(x y z u v w ) = y x z u v w :
4.
Let P be a Yetter-Drinfel'd module in YD . Then P ( ) is an S -module. We will
have to consider morphisms [ ] : P ( ) ! P in YD . If we suppress the summation
index and the summation sign then we may write the bracket operation on elements
z = x : : : x 2 P ( ) as [x ; : : : ; x ] := [z ]. Furthermore we dene
(8)
x : : : x := (z )
Observe that the components x ; : : : ; x in these expressions are interchanged and
changed according to the action of the braid group resp. the symmetric group on
P ( ), so x : : : x is only a symbolic expression, not the usual permutation
of the tensor factors given by the permutation of the indices.
We need another submodule of P whose special properties will not be investigated.
Dene
P ( 1; ) := P P ( ) \ fz 2 P j8 2 S : (1 ) (1 )(z ) = z g:
Since this is a kernel (limit) construction in YD , P ( 1; ) is again an object in
YD .
For z = x y : : : y 2 P ( 1; ) we write y : : : y x y : : : y :=
: : : (z ). If the morphisms [ ] : P ! P and [ ] : P ! P are suitably dened
then we write
(9)
[y ; : : : ; [x; y ]; : : : ; y ] := [:; [:; :] ; :] : : : (z):
Now we have the tools to give the denition of a braided Lie algebra.
Denition 4.1. A Yetter-Drinfel'd module P together with operations in YD
[:; :] : (P : : : P )( ) = P ( ) ! P
for all n 2 N and all primitive n-th roots of unity 6= 1 is called a braided Lie algebra
or a Lie algebra in YD if the following identities hold:
(1) ("anti"-symmetry ) for all n 2 N, for all primitive n-th roots of unity 6= 1,
for all 2 S , and for all z 2 P ( )
[z] = [(z)];
i
1
1
j
1
1
j
i
1
1
1
3
2
1
2
1
2
1
1
1
1
1
1
1
1
1
1
1
Lie algebras
K
K
n
n
n
1
1
(n)
n
1
(1)
n
n
1
(1)
n
n
K
K
(n)
n
n+1
n
1
n
K
K
K
K
n
1
i 1
n+1
n+1
n
1
i
1
n
n+1
1
n
2
1
2
n i 1
2
2
i 1
i
n
1
K
K
n
K
K
n
n
12
BODO PAREIGIS
(2) (1. Jacobi identity ) for all n 2 N, for all primitive n-th roots of unity 6= 1,
and for all z = x : : : x 2 P ( )
n+1
n+1
1
X[x ; [x ; : : : ; x^ ; : : : ; x
n+1
i
i
1
X
]] = [:; [:; :]](1 : : :i)(z) = 0;
n+1
n+1
i=1
i=1
where we use notation (8) (and where (1 : : : i) is a cycle in S ),
(3) (2. Jacobi identity ) for all n 2 N, for all primitive n-th roots of unity 6= 1,
and for all z = x y : : : y 2 P ( 1; ) we have
n
n+1
n
1
X
[x; [y ; : : : ; y ]] = [y ; : : : ; [x; y ]; : : :; y ]
n
n
1
i
1
n
i=1
where we use notation (9).
Observe that the bracket operations are only partially dened and should not be
considered as multilinear operations, since P ( ) P is just a submodule in YD
and does not necessarily decompose P
into an n-fold tensor product. The elements in
P ( ) are, however, of the form z =
x :::
x .
Clearly the braided Lie algebras in YD form a category LYD . Before we investigate its properties we discuss some examples.
Example 4.2. 1. If G = f0g as in Example 3.2.1. then the only required morphism
for a braided Lie algebra P is [ ] : P ( 1) ! P since 1 is a primitive 2-nd root of
unity. This is the usual bracket operation of Lie algebras. The action of B on P P
is given by the canonical switch map (x y) = y x. The induced action of S with
respect to = 1 is then
(x y ) = y x
by (7). Thus axiom 1. gives [x; y] = [(x y)] = [y; x]; the usual anti-symmetry
relation. With this action of S on P ( 1) = P P one gets the usual Jacobi identity
from both braided Jacobi identities.
2. Let G = Z=2Z = f0; 1g with the nontrivial bicharacter (i; j ) = ( 1) . In
Example 3.2.2. we saw that the only non-trivial symmetrization occurs with respect
to = 1, a primitive 2-nd root of unity. So the only bracket is dened on (P P )( 1) = P P . The operation of B on P P is the braid action and (x y ) =
( 1)( 1) y x . So we get [x ; y ] = [(x y )] = [y ; x ] if at least one of the
degrees i or j is zero and we get [x ; y ] = [(x y )] = [y ; x ]. In this case we get
the notion of Lie super algebras since the braided Jacobi identities translate to the
Jacobi identity for Lie super algebras.
3. Let G be an arbitrary nite abelian group with bicharacter such that (h; g) =
(g; h) . Again we get only one bracket operation [ ] : P P ! P and anti-symmetry
and Jacobi identities translate to those for Lie color algebras.
n
n
k
n
k;1
K
K
K
K
k;n
K
K
2
2
2
2
2
ij
ij
i
i
j
1
1
i
2
j
i
1
j
1
j
1
i
1
1
j
SKEW-PRIMITIVE ELEMENTS OF QUANTUM GROUPS AND BRAIDED LIE ALGEBRAS
13
4. The example G = Z=3Z= f0; 1; 2g with bicharacter (i; j ) = where is a
primitive 3-rd root of unity has three bracket operations
[ ] : P ( 1) = (P P ) + (P P ) ! P;
[ ] : P () = (P P P ) (P P P ) ! P;
[ ] : P ( ) = P P ! P:
Here the 1. Jacobi identity means for example
[x ; [x ; x ; x ]] + [x ; [x ; x ; x ]] + [x ; [x ; x ; x ]] + [x ; [x ; x ; x ]] = 0;
and the 2. Jacobi identity
[x; [y ; y ; y ]] = [[x; y ]; y ; y ] + [y ; [x; y ]; y ] + [y ; y ; [x; y ]]:
Further explicit examples of braided Lie algebras can be found in [P1].
5.
The denition of braided Lie algebras, although it generalizes the notion of the
known Lie algebras, Lie super algebras, and Lie color algebras, gains its interest from
the properties that these Lie algebras have. We cite some of these properties in brief.
Theorem 5.1. ([P2] Corollary 4.2) Let A be an algebra in YD . Then A carries the
ij
2
0
3
1
2
3
1
4
1
2
2
1
1
6
3
3
1
4
1
2
0
3
3
2
6
2
6
1
1
1
2
2
2
4
2
4
3
1
1
2
2
3
3
Properties of Lie algebras
K
K
X r (z):
structure of a Lie algebra AL with the symmetric multiplications
[{] : A ( ) ! A
n
[z] :=
dened by
n
2
Sn
=6 1
ThisKdenes a Kfunctor { : AYD ! LYDKK from the category AYDKK of algebras
in YDK to LYDK .
for all n 2 N and all roots of unity L
in k .
K
K
In [P2] Theorem 5.3 we proved
Theorem 5.2. For any algebra A the morphism p : A 3 a 7! a 1 + 1 a 2 A A
is a Lie algebra homomorphism in
YDKK .
An easy consequence of this theorem is
Theorem 5.3. ([P2] Corollary 5.4) Let H be a Hopf algebra in YD . Then the set
of primitive elements P (H ) forms a Lie algebra in YD .
This denes a functor P : HYD ! LYD from the category HYD of Hopf
algebras in YD to LYD . This is the most interesting result which solves the
question for the algebraic structure of the primitive elements of a Hopf algebra in
YD . In particular the braided Lie brackets live also on the set of skew primitive
elements K ? H as partially dened operations.
Theorem 5.4. The functor { : AYD ! LYD has a left adjoint U : LYD
! AYD , called the universal enveloping algebra.
Theorem 5.5. ([P2] Theorem 6.1) The universal enveloping algebra U (P ) of a
braided Lie algebra P is a Hopf algebra in YD .
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
K
L
K
K
K
K
K
K
K
K
K
K
14
BODO PAREIGIS
This denes a left adjoint functor U : LYD ! HYD to P : HYD ! LYD .
Example 5.6. The one dimensional vector space kx considered as a Z=3Z-graded
space with x of degree 1 2 G = Z=3Z= f0; 1; 2g is a braided Lie algebra in YD
with [x x x] = 0 and [x x x x x x] = 0. The universal enveloping algebra
of kx is k[x]=(x ), the Hopf algebra discussed in Example 1.1.
6.
We shall give two examples which show how to construct large families of Lie
algebras from Yetter-Drinfel'd algebras and from Yetter-Drinfel'd modules with a
bilinear form in a similar way as one does for classical Lie algebras.
For this purpose we need inner hom-objects in YD . Let V; W be Yetter-Drinfel'd
modules in YD . Then Hom(V; W ) is a right K -module by
(10)
(fh)(v) = f (vS (h ))h :
This is equivalent to
(11)
(fh )(vh ) = f (vh S (h ))h = f (v)h;
i.e. the evaluation Hom(V; W ) V ! W is a K -module homomorphism.
We dene a map Æ : Hom(V; W ) ! Hom(V; W K ) by
(12)
Æ (f )(v ) := f (v ) f (v ) S (v )
that \dualizes" the right module structure on Hom(V; W ).
Let hom(V; W ) be the pullback (in Vec) in the diagram
hom(V; W ) Æ - Hom(V; W ) K
K
K
K
K
K
K
K
K
kG
kG
3
Derivations and skew-symmetric endomorphisms
K
K
K
K
1
(1)
(2)
1
(3)
(2)
(1)
(2)
(1)
0
0
[0] [0]
[0] [1]
[1]
1
j
?
?
Hom(V; W ) Æ - Hom(V; W K )
hom(V; W ) thus can be written as
P f f 2 Hom(V; W ) K 8v 2 V :
hom
(
V;
W
)
=
f
f
2
Hom(
V;
W
)
j9
P
P
(13)
f (v ) f = f (v ) f (v ) S (v ) = Æ (f )(v )g
Lemma 6.1. hom(V; W ) is a K -comodule.
Proof. Since K is faithfully at we get that
hom(V; W ) K Æ K - Hom(V; W ) K K
j
K
K
?
?
Hom(V; W ) K Æ K - Hom(V; W K ) K
0
0
0
1
1
0
1
[0] [0]
[0] [1]
[1]
0
SKEW-PRIMITIVE ELEMENTS OF QUANTUM GROUPS AND BRAIDED LIE ALGEBRAS
15
again is a pullback. So we get a uniquely determined homomorphism Æ : hom(V; W )
! hom(V; W ) K such that
hom(V; W ) X
BB @@ XXXXXXX
XX(1X
X)
Æ
BB
@Æ@
X
X
XXXz
R
@
Æ
K
B
Hom(V; W ) K K
Æ B hom(V; W ) K
BB
j
K
BB K
?
?
N
Hom(V; W ) K Æ K - Hom(V; W K ) K
commutes. To show that the outer diagram commutes we compute
((Æ 1)Æ (f ))(v) = (Æ 1)(f f )(v)
= (Æ (f ))(v) f
= f (v ) f (v ) S (v ) f
= f (v ) f (v ) S (v ) f (v ) S (v )
= f (v ) (f (v ) S (v ))
= ((1 )Æ (f ))(v)
hence (Æ 1)Æ (f ) = (1 )Æ (f ). Since Æ = ( K )Æ and (Æ K )Æ = (1 )Æ
we get for the induced map Æ the equality ( K K )(Æ K )Æ = (1 )( K )Æ =
( K K )(1 )Æ and thus (Æ K )Æ = (1 )Æ. Consequently hom(V; W ) is a
K -comodule.
Lemma 6.2. hom(V; W ) is a Yetter-Drinfel'd module.
Proof. We rst show that hom(V; W ) is a K -module. Let f 2 hom(V; W ) and h 2 K .
Then
(14)
Æ (fh) = j (f h S (h )f h )
since
Æ (fh)(v ) = (fh)(v ) (fh)(v ) S (v )
= (f (v S (h ))h ) (f (v S (h ))h ) S (v )
= f (v S (h )) h S (h )f (v S (h )) h S (v )
= f (v S (h )) h S (h )f (v S (h )) h S (v )S (h )h
= f (v S (h )) h S (h )f (v S (h )) S (S (S (h ))v S (h ))h
= f ((vS (h )) ) h S (h )f ((vS (h )) ) S ((vS (h )) )h
= f (vS (h ))h S (h )f h
= (f h )(v) S (h )f h
Thus fh 2 hom(V; W ) by denition of hom(V; W ). So hom(V; W ) is a K -submodule
of Hom(V; W ). Furthermore (14) shows also that hom(V; W ) is a Yetter-Drinfel'd
module.
1
1
1
0
0
1
0
0
0
0
1
0
1
0
[0] [0]
[0] [0]
[0] [1]
[0] [1]
[0] [0]
1
[1]
[2]
[0] [1]
[0] [2]
[1]
[1]
1
0
1
1
1
0
0
1
[0]
1
[0]
1
[0]
[0]
1
1
1
0
0
(2)
0
[0] [0]
(2)
[0] [1]
1
(1) [0]
[0]
(4)
[0]
(2)
(1)
[0]
(4)
[0]
(2)
(1)
[0]
(4)
[0]
(2)
(1)
[0]
(3)
[0] [0]
(3)
(2)
(1)
(2)
(1)
(1)
1
(3)
1
(4)
1
(2)
(1)
1
1
(3)
[1]
(2)
1
1
1
1
(1) (1)
[1]
(4)
[1]
(3)
[1]
(4)
[1]
(3)
(4)
[1]
[1]
1
(3)
[0] [1]
(5)
(5)
1
(6)
[1]
(3)
[1]
1
(3)
(4)
(6)
16
BODO PAREIGIS
Theorem 6.3. The category of Yetter-Drinfel'd modules YDKK is a closed monoidal
category.
It suÆces to show for a homomorphism g : X V ! W in YD that the
induced map g~ : X ! Hom(V; W ) factors through a Yetter-Drinfel'd homomorphism
g : X ! hom(V; W ). We have
Æ (~g (x))(v ) = g~(x)(v ) g~(x)(v ) S (v )
= g(x v ) g(x v ) S (v )
= g(x v ) x v S (v )
= g~(x )(v) x
= j (~g(x ) x )(v)
or Æ (~g(x)) = j (~g(x ) x ) so that g~(x) 2 hom(V; W ) which denes a homomorphism g : X ! hom(V; W ). Furthermore this proves Æ(g(x)) = g(x ) x =
(g 1)Æ(x) which shows that g : X ! hom(V; W ) is a comodule homomorphism.
Finally we have
g(xh)(v ) = g (xh v )
= g(xh vS (h )h )
= g((x vS (h ))h )
= g(x vS (h ))h
= g(x)(vS (h ))h
= (g(x)h)(v)
so that g(xh) = g(x)h, i.e. g is a Yetter-Drinfel'd homomorphism.
We denote the evaluation map corresponding to id 2 Hom(hom(V; W ); hom(V; W ))
by # : hom(V; W ) V ! W .
Now we consider derivations on algebras A in YD . A derivation from A to A is
a linear map (d : A ! A) 2 hom(A; A) such that
d(ab) = d(a)b + r(1 #)( 1)(d a b)
for all a; b 2 A. Observe that in the symmetric situation this means d(ab) = d(a)b +
ad(b).
Lemma 6.4. Let A be an algebra in YD . Then the set
Der (A) := fd 2 hom(A; A)jd(ab) = d(a)b + r(1 #)( 1)(d a b)8a; b 2 Ag
is a Yetter-Drinfel'd module in YD .
In fact Der (A) is the kernel in YD of the morphism in Hom(end(A); hom(A A; A)) = Hom(end(A) A A; A) given by #(1 r) r(# 1) r(1 #)( 1)). It is
easily checked that Der (A) together with its operation on A is the universal derivation
module on A, i.e. module M in YD together with an operation # : M A ! A
such that (1 r)# = r(# 1) + r(1 #)( 1).
Theorem 6.5. ([P2] Corollary 5.6) The Yetter-Drinfel'd module of derivations
Der (A) of an algebra A is a Lie algebra.
Proof.
K
K
0
[0] [0]
[0] [1]
[0] [0]
[0]
[0] [1]
[0]
[1] [1]
[0]
[0]
[1]
[2]
[1]
[0]
0
[1]
[1]
[1]
[0]
1
(1)
1
1
1
(3)
(2)
(2)
(2)
(1)
(1)
K
K
K
K
K
K
K
K
K
K
(2)
(1)
[1]
SKEW-PRIMITIVE ELEMENTS OF QUANTUM GROUPS AND BRAIDED LIE ALGEBRAS
17
Now let V be in YD with inner endomorphism object A := end(V ). Given a bilinear form h:; :i : V V ! k in YD . We collect the set g of skew-symmetric endomorphisms f 2 end(V ) for which in principle the following hold: hf (v); wi = hv; f (w)i
for all v; w 2 V . Since there is a switch between f and v in these expressions the
correct condition is
Xhv ; f (w)i
hf (v); wi =
K
K
K
K
i
where (f v) = P v f resp.
i
i
i
(# 1)(f v w) = (1 #)( 1)(f v w:
Thus g is the dierence kernel of two morphisms in YD . Hence g is an (universal)
object satisfying the diagrammatic condition
K
K
g
VV
r =
gV
V
r Theorem 6.6. For a Yetter-Drinfel'd module V 2 YDKK with bilinear form = h:; :i :
V V ! k the Yetter-Drinfel'd module g of skew-symmetric endomorphisms is a Lie
K
subalgebra of end (V )L in YDK
.
We give a diagrammatic proof where we use the multiplication r : g g
! A = end(V ) and the fact that the evaluation # : g V ! V of g on V is
associative with respect to this multiplication. For the n-fold multiplication we write
r : g ! A. Let : g ! g be given by := g
g , = ( 1)g
g . Then
Proof.
n
n
n
n
n
n+1
2
(# 1)(rn 1 1) = (
n
n
1) (1 #)( 1)(r 1 1):
n
n
n
We prove this by induction. For n = 1 this is the dening condition for g. The
induction step is
g gn V
r
n
A
r
V
g gn V
V
g gn V
V
g gn V
V
= A =( 1) A =( 1)A = ( 1)
r
r r
g gn V
n+1
A
V
=
r 18
BODO PAREIGIS
g gn V
= ( 1)
g gn V
V
= ( 1)
A r
n+1
A rn
+1
= ( 1)
n+1 n
n+1 n
gn+1V
V
n+1
A
r
r V
In the rst and the last term we indicated the multiplication r : g ! A resp.
r : g ! A. Furthermore we indicated the use of where appropriate.
Now let be a primitive n-th root of unity and z 2 g ( ). Then = ( 1)
and (z) = (z) by (7) and the denition of , where is the image of under the canonical map B ! S . So we get
(# 1)([zP
] v w) =
= (# 1)(r (z) v w)
= P (#P
1)(r 1 1)((z) v w)
= ( 1) P (1 #)( 1)(r 1 1)((z) v w)
= ( 1) (1 #)( 1)(r 1 1)( (z) v w)
= ( 1)(1 #)( 1)([z] v w);
hence [z] 2 g. Thus g is a Lie subalgebra of end(V ).
n
n+1
n+1
n
n(n
n
n(n
2
n
n
2
1)
n 1
1)
n
n
n
n
n
n
n
n
n
n
n
n
n
7.
C
In this section we assume that G = C = Z=(p ) is the cyclic group with p
elements where p 6= 2 is prime and that the eld k has characteristic 6= 2 and contains
a p -th primitive root of unity . We want to get information on the nontrivial
symmetrizations of G-comodules. A bicharacter : G ZG ! k is uniquely dened by the value of (1; 1) as
(i; j ) = (1; 1) 2 k . Since Z=(p ) ZZ=(p ) = Z=(p ) this amounts to a homomor
phism ' : Z=(p ) 3 g 7! 2 k for some element 2k . Such a homomorphism
has a unique image factorization Z=(p ) ! Z=(p ) ! k with g 7! g 7! with the
second homomorphism injective. Then has order p so it is a primitive p -th root of
unity. Without loss of generality we may assume s = t and a primitive p -th root
of unity.
We wish to compute the -symmetrization (P : : : P )( ) = P ( ) where P is a
G = Z=(p )-graded vector space and is an n-th primitive root of unity (with n > 1)
so that we can determineL
the domain of the possible Lie multiplications.
To determine P ( ) = f
g P : : : P (Proposition 3.1) we have
to nd the -families (g ; : : : ; g ) in G i.e. families with (g ; g ) = for all i 6= j .
Since (g ; g ) has order p for some r, there is only a restricted choice for the primitive
root of unity and for the arity of the possible Lie multiplication on P .
Lie structures on
pn -graded modules
t
pt
t
t
ij
t
t
t
t
g
t
s
g
s
s
t
n
t
n
(g1 ;:::;gn ) -family
1
i
j
r
n
g1
gn
i
j
2
2
SKEW-PRIMITIVE ELEMENTS OF QUANTUM GROUPS AND BRAIDED LIE ALGEBRAS
19
We have n > 1 and 2 k a primitive n-th root of unity. We wish to determine
all -families (g ; : : : ; g ) of elements in G = Z=(p ) satisfying (g ; g ) = or
= for all i 6= j . This amounts to = " with 0" = 1. In the case " = +1
we get = 2 Im('). Hence the order of is n = p and = for b 2 Z=(p ).
In the case " = 1 we get0 = 2 Im(') (and 2= Im(') since p is odd). Hence
the order of is n =0 2p and = for b 2 Z=(p ). Together this says = " has
order n = ( ")p for " = 1.
If t0 = 0 then = 1. If = 1 then we get the trivial case P (1) = P . If = 1
the primitive 2-nd root of unity, then (g ; g ) is a -family i it satises (g ; g ) = 1
(the value 1 is not possible)
i g g 0(p ).
0
Assume now that t > 0 hence b 6= 0. Then there are n > 2 components g in a
-family (g ; : : : ; g ). Choose representatives g 2 N with 0 < g < p . Observe that
g 6= 0 since b 6= 0 and b g g (p ). So we can write g = n p with (n ; p) = 1 and
0 r < t and b = qp with (q; p) = 1 and 0 s < t.
We have g g b(p ) for all i 6= j hence n n p p qp (p ). Since g g 6= 0 in
Z=(p ) we get r + r < t hence r + r = s for all i 6= j . Thus r = r for all i = 1; : : : ; n
and s = 2r < t. Since b = qp has order p in Z=(p ) we get t0 = t 2r and
n=(
")p .
Finally we have g (g g ) = n (n n )p 0(p ) hence n n 0(p ) or
n n (p ); 8i 6= j
and b n n p (p ). It is easy to check that any family (g ; : : : ; g ) satisfying these
equations is a -family. This proves the following
Theorem 7.1. Let p 6= 2 be a prime and t 1. Then the ( 1)-families (g ; g ) are
those with g g 0(p ).
2gi gj
t
n
1
2
i
gi gj
gi gj
t
j
2
2
b
t
gi gj
t
t
1
1
2
n
i
i
i
3
2
i
t
i j
t
i
t
i j
s
1
t
1 2
1
b
t
1
2
3
2
b
i
j
i
j
s
j
ri rj
s
t s
i
t
i
ri
i
2
i
t
i j
i
t
t 2r
1
2
i
j
k
i
i
i i
2r
1 2
j
2r
k
t
j
k
t 2r
t 2r
j
t
n
1
n
For any choice of
r such that 0 2r < t,
" 2 f+1; 1g,
m 2 f1; : : : ; pn 1g with (m; p) = 1,
there are -families (g1 ; : : : ; gn ) with = " b , b m2 p2r (pn ), and n = ( 23
The gi can be chosen as gi mpr + aipt r (pn ) with ai 2 f0; : : : ; pr 1g.
These are all families on which a Lie multiplication can be dened.
1
2
1
2
")pt
2r
.
1. Let p = 3 and t = 1. Then there is the ( 1)-symmetrization
P ( 1) = P P + P P ;
since g g 0(3) i one of the factors g is zero. To get all other symmetrizations
observe that r = 0 hence b g g 1(3). So there are 2 cases = and n = 3 or
= and n = 6. The corresponding possible -families are (1; 1; 1) and (2; 2; 2)
resp. (1; 1; 1; 1; 1; 1) and (2; 2; 2; 2; 2; 2) hence
P ( ) = P P P P P P
Example 7.2.
2
0
0
i
1 2
1 1
3
1
1
1
2
2
2
20
BODO PAREIGIS
and
P 6 ( ) = P1 P1 P1 P1 P1 P1
P2 P2 P2 P2 P2 P2:
2. Let p = 3 and t = 2. Then P 2( 1) = P0 P + P P0 + (P3 + P6) (P3 + P6).
For larger -families the only choice for r is r = 0. Since m 2 f1; 2; 4; 5; 7; 8g we get
b 2 f1; 4; 7g. We get families with 9 or 18 elements gi and these elements must all be
equal, gi 2 f1; 2; 4; 5; 7; 8g.
3. Let p = 3 and t = 3. We consider the families of length > 2. There are two
choices for r 2 f0; 1g. So we get9 families with 3, 6, 27, and 54 elements. For the
choice r = 1, m = 4, " = +1 the -families (g19; g2; g3) are composed of gi = 4 3+ ai 9
with ai 2 f0; 1; 2kGg. There is for example the -family (3; 12; 21). So any braided Lie
algebra P in M with G = Z=(27) has a Lie operation [ ] : P3 P12 P21 ! P9.
References
A Schur Double Centralizer Theorem for Cotriangular Hopf Algebras and Generalized Lie Algebras , J. Algebra 168 (1994), 594-614.
[L] Lusztig, George: On Quantum Groups. J. Alg. 131 (1990), 466-475
[Ma] Shahn Majid, Crossed Products by Braided Groups and Bosonization , J. Algebra 163 (1994),
[FM] Davida Fischman and Susan Montgomery,
165-190.
Hopf Algebras and Their Actions on Rings. CBMS 82, AMS-NSF, 1993.
On Lie Algebras in Braided Categories. Banach Center Publications Vol. 40
[M] Susan Montgomery:
[P1] Bodo Pareigis:
(1997). www.mathematik.uni-muenchen.de /personen/professoren.html.
[P2] Bodo Pareigis:
On Lie Algebras in the Category of Yetter-Drinfel'd-Modules. To appear in Ap-
plied Categorical Structures { www.mathematik.uni-muenchen.de /personen/professoren.html.
[R] David E. Radford:
[S]
The Structure of Hopf Algebras with a Projection. J. Alg. 92 (1985), 322-347.
Deformed Enveloping Algebras. New York Journal of Mathematics 2
Yorck Sommerh
auser:
(1996), 1-23.
[Y] Yetter, David N.:
Quantum Groups and Representations of Monoidal Categories.
Camb. Phil. Soc. 108 No 2 (1990), 261-290.
 t Mu
 nchen, Germany
Mathematisches Institut der Universita
E-mail address : [email protected]
Math. Proc.