On some universal algebras associated to the category of Lie bialgebras

17 Pages·0.21 MB·English

by  B. Enriquez

ON SOME UNIVERSAL ALGEBRAS ASSOCIATED TO
1 THE CATEGORY OF LIE BIALGEBRAS
0
0
2 B. ENRIQUEZ

n
a Abstract. Inourpreviouswork(math/0008128),westudiedthesetQuant(K)
J
ofalluniversalquantizationfunctorsofLie bialgebrasoverafieldK ofcharac-
8
teristic zero,compatible with duals anddoubles. We showedthat Quant(K) is
2
canonicallyisomorphicto aproductG0(K)×∐∐(K),whereG0(K)is auniversal

] group and ∐∐(K) is a quotient set of a set B(K) of families of Lie polynomials
A
bytheactionofagroupG(K). WeproveherethatG0(K)isequaltothemulti-
Q plicativegroup1+~K[[~]]. SoQuant(K)is ‘ascloseasitcanbe’to∐∐(K). We
. also prove that the only universal derivations of Lie bialgebras are multiples
h
t of the composition of the bracket with the cobracket. Finally, we prove that
a the stabilizer of any element of B(K) is reduced to the 1-parameter subgroup
m
generated by the corresponding ‘square of the antipode’.
[

2
v
9
5 1. Main results
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1 1.1. Results on Quant(K). Let K be a field of characteristic zero. In [2], we
0
introduced the group G (K) of all universal automorphisms of the adjoint repre-
1 0
0 sentations of K[[~]]-Lie bialgebras.
h/ Let us recall the definition of G (K) more explicitly. Let ~ be a formal variable
0
at andletLBA~bethecategoryofLiebialgebrasoverK[[~]], which aretopologically
m free K[[~]]-modules. An element of G (K) is a functorial assignment (a,[,],δ ) 7→
0 a
v: ρa, where for each object (a,[,],δa) of LBA~, ρa is an element of EndK[[~]](a), such
Xi that (ρa mod ~) = ida, ρa∗ = (ρa)t, and for any x,y in a, [ρa(x),y] = ρa([x,y]),
and ρ is given by a composition of tensor products of the bracket and cobracket
a
r
a of a, this composition being the same for all Lie bialgebras (we express the latter
condition by saying that a 7→ ρ is universal).
a
View 1 + ~K[[~]] as a multiplicative subgroup of K[[~]]×. There is a unique
map α : 1+~K[[~]] → G (K), such that for any Lie bialgebra a, (α(λ)) = λid .
0 a a
This map makes 1+~K[[~]] a subgroup of G (K).
0
We will show
Theorem 1.1. G (K) is equal to its subgroup 1+~K[[~]].
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In [2], we defined Quant(K) as the set of all isomorphism classes of universal
quantization functors of Lie bialgebras, compatible with duals and doubles. If we
Date: January 2001.
1
2 B. ENRIQUEZ
denote by LBA and QUE the categories of Lie bialgebras and of quantized uni-
versal enveloping algebras over K, and by class : QUE → LBA the semiclassical
limit functor, then a universal quantization functor of Lie bialgebras, compatible
with duals and doubles, is a functor Q : LBA → QUE, such that
1) class◦Q is isomorphic to the identity;
2) (universality) there exists an isomorphism of functors between a 7→ Q(a)
and a 7→ U(a)[[~]] (these are viewed as functors from LBA to the category of
K[[~]]-modules) with the following properties: if we compose this isomorphism
with the symmetrisation map U(a)[[~]] → S(a)[[~]], and if we transport the op-
erations of Q(a) on S(a)[[~]], then the expansion in ~ of these operations yields
maps Si(a) ⊗Sj(a) → Sk(a) and Si(a) → Sj(a) ⊗Sk(a); we require that these
maps be compositions of tensor products of the bracket and cobracket of a, these
compositions being independent of a (see [1])
3) if Q∨ (resp., D(Q)) denotes the QUE-dual (resp., Drinfeld double) of an
object Q of QUE, and D(a) denotes the double Lie bialgebra of an object a of
LBA, then there are canonical isomorphisms Q(a∗) → Q(a)∨ and Q(D(a)) →
D(Q(a)); moreover, the universal R-matrix of D(Q(a)) should be functorial (see
[2]).
In [2], we also introduced an explicit set ∐∐(K) of equivalence classes of fam-
ilies of Lie polynomials, satisfying associativity relations, and we constructed a
canonical injection of ∐∐(K) in Quant(K). Moreover, we constructed an action of
G (K) on Quant(K) and showed that the map
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G0(K)×∐∐(K) → Quant(K)
given by the composition G0(K) × ∐∐(K) ⊂ G0(K) × Quant(K) → Quant(K) (in
which the second map is the action map of G (K) on Quant(K)) is a bijection.
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Theorem 1.1 therefore implies
Corollary 1.1. If a = (a,[,],δ ) is an object of LBA and λ ∈ (1+~K[[~]]), let
a
aλ be the object of LBA~ isomorphic to (a,[,],λδ). The group 1 + ~K[[~]] acts
freely on Quant(K) by the rule (λ,Q) 7→ Q , where Q is the functor a 7→ Q(a )
λ λ λ
and Q is the natural extension of Q to a functor from LBA~ to QUE. Then the
b
map
b 1+~K[[~]] ×∐∐(K) → Quant(K)
given by the compositio(cid:0)n 1 + ~K[(cid:1)[~]] × ∐∐(K) ⊂ 1 + ~K[[~]] × Quant(K) →
Quant(K) is a bijection. (cid:0) (cid:1) (cid:0) (cid:1)
Therefore Quant(K) is ‘as close as it can be’ to ∐∐(K).
1.2. Universal (co)derivations of Lie bialgebras. Let D (resp., C) be the
space of all universal derivations (resp., coderivations) of Lie bialgebras. More
explicitly, D (resp., C) is the linear space of all functorial assignments a 7→ λ ,
a
where for each object a of LBA, λ belongs to End(a), is universal in the above
a
sense, and is a derivation (resp., coderivation) of the Lie algebra structure of
UNIVERSAL ALGEBRAS ASSOCIATED TO LIE BIALGEBRAS 3
a. It is well-known that if [,] and δ are the bracket and cobracket maps of a,
a a
then [,] ◦δ is a derivation of a; e.g., if a is finite-dimensional, if a ⊗b is
a a i∈I i i
the canonical element of a⊗a∗ and if we set u = i∈I[ai,bi], thePn we have the
identity ([,]a◦δa)(x) = [u,x] in the double Lie algePbra of a; and since [,]a∗ ◦δa∗
is a derivation of a∗, its transpose [,] ◦δ is a coderivation of a. Then
a a
Theorem 1.2. D and C both coincide with the one-dimensional vector space
spanned by the assignment a 7→ [,] ◦δ .
a a
If V is a vector space, we denote by F(V) the free Lie algebra generated by
V. Then the assignment c 7→ F(c) is a functor from the category LCA of Lie
coalgebras to LBA. The proof of Theorem 1.2 implies the following analogous
statement for the subcategory of LBA of free Lie algebras of Lie coalgebras.
Proposition 1.1. Let c 7→ λ be a functorial assignment, where for each object
c
c of LCA, λ is a both a derivation and a coderivation of F(c). Then there exists
c
a scalar λ, such that for any object c of LBA, λ = λ[,] ◦δ .
c F(c) F(c)
1.3. Isotropy of the action of G(K) on B(K). We record here the definition
of ∐∐(K). Let B(K) be the set of families (Bpq)p,q≥0, such that for each p,q, Bpq
belongstoFL [[~]], B (x) = B (x) = x, B = B ifp 6= 1, B (x,y) = [x,y],
p+q 10 01 p0 0p 11
and for any integers p,q,r, the identity
B B (x ,... ,x |y ,... ,y )···
αr p1q1 1 p1 1 q1
Xα>0(pβ)β=1,...,α∈Partα(Xp),(qβ)β=1,...,α∈Partα(q) (cid:0)
···B (x ,... ,x |y ,... ,y )|z ,... ,z
pαqα βα=−11pβ+1 p βα=−11qβ+1 q 1 r
P P (cid:1)
= B x ,... ,x |
pα 1 p
Xα>0(qβ)β=1,...,α∈Partα(qX),(rβ)β=1,...,α∈Partα(r) (cid:0)
B (y ,... ,y |z ,... ,z )···B (y ,... ,y |z ,... ,z )
q1r1 1 q1 1 r1 qαrα βα=−11qβ+1 q βα=−11rβ+1 r
holds; here FL is the multilinear part Pof the free Lie algPebra over K wit(cid:1)h n
n
generators, and Part (n) is the set of α-partitions of n, i.e. the set of families
α
(n ,... ,n ) of positive integers such that n +···+n = n. Define G(K) as the
1 α 1 α
subset of FL [[~]] of families (P ) , such that P (x) = x. Then we are
n≥1 n n n≥1 1
going to dQefine a group structure on G(K), and an action of G(K) on B(K); ∐∐(K)
is the quotient set B(K)/G(K).
Recallfirst thatif(c,δ )isa Liecoalgebra andB ∈ B(K), thenthere isaunique
c
Hopf algebra structure on the completed tensor algebra T(c)[[~]] with coproduct
∆c : T(c)[[~]] → T(c)⊗2[[~]] defined by
B
∆c (x) = x⊗1+1⊗x+ ~p+q−1α (δ(Bpq)(x))
B pq c
X
p,q|p+q≥1
for any x ∈ c, where for any P in FLn[[~]], δc(P) is the map from c to c⊗n dual
to the map from (c∗)⊗n to c∗ defined by P (when c is finite-dimensional), and
4 B. ENRIQUEZ
α is the map from c⊗p+q[[~]] to T(c) ⊗ T(c)[[~]] sending x ⊗ ··· ⊗ x to
pq 1 p+q
(x ⊗···⊗x )⊗(x ⊗···⊗x ).
1 p p+1 p+q
If P = (P ) belongs to G(K), and (c,δ ) is a Lie coalgebra, define ic as the
n n≥1 c P
unique automorphism of T(c)[[~]], such that for any x in c, we have
ic (x) = x+ ~n−1δ(Pn)(x).
P c
Xn≥2
Then the product ∗ : G(K)×G(K) → G(K) and the operation ∗ : G(K)×B(K) →
B(K) are uniquely determined by the conditions that for any Lie coalgebra (c,δ ),
c
and any P,Q in G(K) and any B in B(K), we have
ic = ic ◦ic and ∆c = (ic ⊗ic )◦∆c ◦(ic )−1.
P∗Q P Q P∗B P P B P
Then one checks that for any B in B(K), there exists a unique family (S ) ,
n n≥2
where S ∈ FL [[~]], such that for any Lie coalgebra (c,δ ), the antipode Sc of
n n c B
the bialgebra (T(c)[[~]],m ,∆c ) is such that for any x ∈ c,
0 B
Sc(x) = −x+ ~n−1δ(Sn)(x)
B
X
n|n≥2
(m isthe multiplication mapin T(c)[[~]]). It follows that there is a unique family
0
(S ) , where S ∈ FL [[~]], such that
n n≥2 n n
e e (Sc)2(x) = x+ ~n−1δ(Sn)(x)
B
X e
n|n≥2
for any x ∈ c. Let us set S (x) = x and set S2 = (S ) . Then S2 is an
1 B n n≥1 B
element of G(K). Moreover, G(K) is a pro-unipotent group. It follows that one
e e
can define the logarithm log(S2) of S2, and the corresponding one-parameter
B B
subgroup exp(K[[~]]log(S2)).
B
Since for any Lie coalgebra (c,δ ), we have ((SC)2⊗(Sc)2)◦∆c = ∆c ◦(Sc)2,
c B B B B B
we also have S2 ∗B = B. It follows that for any element g of exp(K[[~]]log(S2)),
B B
we have g ∗B = B, so exp(K[[~]]log(S2)) is contained in the isotropy group of
B
B.
Proposition 1.2. For any B in B(K), the isotropy group of B for the action of
G(K) on B(K) is equal to exp(K[[~]]log(S2)).
B
2. Proof of Theorem 1.1
Let us define E as the set of all universal K[[~]]-module endomorphisms of Lie
bialgebras. More explicitly, an element E is a functorial assignment (a,[,],δ ) 7→
a
ǫa ∈ EndK[[~]](a), where a is an object of LBA~ and the universality requirement
means that ǫ is given by a composition of tensor products of the bracket and
a
cobracket of a, this composition being the same for all Lie bialgebras. Then E
is a K[[~]]-module, and G (K) is a subset of E. We will first give a description
0
UNIVERSAL ALGEBRAS ASSOCIATED TO LIE BIALGEBRAS 5
of E is terms of multilinear parts of free Lie algebras (Proposition 2.1). Then a
computation in free algebras will prove Theorem 1.1.
2.1. Description of E. Define FL as the multilinear part in each generator of
n
thefreeLiealgebraoverKwithngenerators. LetS actdiagonallyonFL ⊗FL
n n n
by simultaneous permutation of the generators x ,... ,x and y ,... ,y of each
1 n 1 n
factor.
We define a linear map p 7→ (a 7→ i(p) ) from ⊕ (FL ⊗FL ) [[~]] to E
a n|n≥1 n n Sn
as follows.
b
If Q is an element of FL , which we write Q = Q x ···x , then
n σ∈Sn σ σ(1) σ(n)
we set P
1
δ(Q)(x) = Q (id⊗n−1⊗δ )◦···◦δ (x) (σ(1)...σ(n))
a σ a a a
n
σX∈Sn (cid:0) (cid:1)
for any x ∈ a.
If p = P ⊗Q is an element of FL ⊗FL , define i(p) as the endomor-
α α α n n a
phism ofPa such that
i(p)a(x) = Pα(δ(Qα)(x)) (1)
Xα
for any x ∈ a. This maps factors through a linear map p 7→ i(p) from (FL ⊗
a n
FLn)Sn to EndK[[~]](a), and induces a linear map p 7→ i(p)a from ⊕n|n≥1(FLn ⊗
FL ) [[~]] to End (a) (here ⊕ is the ~-adically completed direct sum).
n Sn K[[~]] b b
Then if a is finite-dimensional over K[[~]], and if we express the canonical
element of a⊗a∗ as a ⊗b ,bthen
i∈I i i
P
i(p) (x) = hx,Q (b ,... ,b )iP (a ,... ,a ).
a α i1 in α i1 in
Xα i1,.X..,in∈I
Proposition 2.1. The linear map i from ⊕ (FL ⊗FL ) [[~]] to E defined
n|n≥1 n n Sn
by p 7→ (a 7→ i(p) ) is a linear isomorphism.
a
b b
Proof. If nband m are ≥ 1, define En,m as the vector space of all universal
linear homomorphisms from a⊗n to a⊗m (‘universal’ again means that these ho-
momorphisms are compositions of tensor products of the bracket and cobracket
map, this composition being the same for each a). Then E is just E . The
1,1
direct sum ⊕ E may be defined formally as the smallest ~-adically
n,m|n,m≥1 n,m
complete vector subspace of the space of all functorial assignments a 7→ ρ ∈
a
⊕ Hbom (a⊗n,a⊗m),containingtheassignmentsa 7→ id ∈ Hom (a,a),
n,m|n,m≥1 K[[~]] a K[[~]]
the bracket and the cobracket operations, stable under the external tensor prod-
ubctsoperationsHomK[[~]](a⊗n,a⊗m)⊗HomK[[~]](a⊗n′,a⊗m′) → HomK[[~]](a⊗n+n′,a⊗m+m′),
underthenaturalactionsofthesymmetricgroupsSn andSm onHomK[[~]](a⊗n,a⊗m),
and under the composition operation.
Define Efinite and Efinite as the analogues of E and E, where the category
n,m n,m
of Lie bialgebras is replaced by the category of finite-dimensional (over K[[~]])
6 B. ENRIQUEZ
K[[~]]-Lie bialgebras. Then restriction to this subcategory ofLBA~ induces linear
maps E → Efinite and ⊕ E → ⊕ Efinite.
n,m n,m n,m|n,m≥1 n,m n,m|n,m≥1 n,m
Let us define F(n,m) as follows
b b
F(n,m) =
n m
(pij)∈N{1,M...,n}×{1,...,p}(cid:0)Oi=1 FLPmj=1pij ⊗Oj=1 FLPni=1pij(cid:1)Q(i,j)∈{1,...,n}×{1,...,m}Spij;
(ij)
the generators of the ith factor of the first tensor product are x , j = 1,... ,m,
α
α = 1,... ,p , and the generators of the jth factor of the second tensor product
ij
are y(ij), i = 1,... ,n, α = 1,... ,p ; the group S acts by simultaneously
α ij pij
(ij) (ij)
permuting the generators x and y , α ∈ {1,... ,p }.
α α ij
Then there is a unique linear map i : F(n,m) → Efinite, such that if
n,m m,n
p =
n m
Pλ(x(ij);j = 1,... ,m;α = 1,... ,p )⊗ Qλ(y(ij);i = 1,... ,n;α = 1,... ,p ),
i α ij j α ij
Xλ Oi=1 Oj=1
and x ,... ,x belong to a Lie bialgebra a, then
1 m
(i (p)) (x ⊗···⊗x ) = (2)
n,m a 1 m
m
hx ,Qλ(b(i(ij));i = 1,... ,n : α = 1,... ,p )i
j j α ij
Xλ i(111)∈IX,...,ip(nnmm)∈IYj=1
n
Pλ(a(i(ij));j = 1,... ,m;α = 1,... ,p ).
i α ij
Oi=1
Here we write the canonical element of a ⊗ a∗ in the form a(i) ⊗ b(i).
i∈I
Formula (2) has an obvious generalization when a is an arbitPrary Lie bialge-
bra. This means that i factors through a map (also denoted i ) from
n,m n,m
F(n,m) to E ; this map induces linear maps i : F(n,m)[[~]] → E and
m,n n,m n,m
⊕ i : ⊕ F(n,m)[[~]] → ⊕ E .
n,m|n,m≥1 n,m n,m|n,m≥1 n,m|n,m≥b1 n,m
Letusshowthat⊕ i issurjective. Forthis,letusstudy⊕ Im(i ).
b b b n,m|n,m≥1 n,m b n,m≥1 n,m
This is a subspace of the space of all functorial assignments
b b b b
a 7→ ρa ∈ ⊕n,m|n,m≥1HomK[[~]](a⊗n,a⊗m).
Let us show that it shares all thbe properties of ⊕n,m|n,m≥0En,m. The identity is the
image of the element x(11)⊗y(11) of F(1,1), the bracket is the image of the element
1 1
[x(11),x(12)]⊗y(11)⊗y(12) of F(1,2), and the cobracket is the image of the element
1 1 1 1
x(11) ⊗x(21) ⊗[y(11),y(21)] of F(2,1). The fact that ⊕ Im(i ) is stable under
1 1 1 1 n,m≥0 n,m
the composition follows from the following Lemma.
b b
UNIVERSAL ALGEBRAS ASSOCIATED TO LIE BIALGEBRAS 7
Lemma 2.1. Let P and Q be Lie polynomials in FL and FL respectively.
n m
p,λ p,λ
Then there exist an element p = (⊗n P ) ⊗ (⊗m Q )
p∈N{1,...,n}×{1,...,m} λ i=1 i j=1 j
of F(n,m), such that if a is any Pfinite-dimensionalPLie bialgebra over K, and
a(i)⊗b(i) is the canonical element of a⊗a∗, then
i∈I
P
hQ(a(j ),... ,a(j )),P(b(i ),... ,b(i ))i(⊗n a(i ))⊗(⊗m b(j ))
1 m 1 n α=1 α β=1 β
i1,.X..,jm∈I
n
= ( Pp,λ(a(i(ij));j = 1,... ,m;α = 1,... ,p ))
i α ij
p∈N{1,...X,n}×{1,...,m}Xλ i(111)∈IX,...,ip(nnmm)∈I Oi=1
m
⊗( Qp,λ(b(i(ij));i = 1,... ,n;α = 1,... ,p )).
j α ij
Oj=1
Proof of Lemma. We may assume that P and Q have the formP(x ,... ,x ) =
1 n
[x ,[x ,... ,x ]] and Q(y ,... ,y ) = [y ,[y ,... ,y ]]. Then the invariance of
1 2 n 1 m 1 2 m
the canonical bilinear form in D(a) and the fact that a(i) ⊗ b(i) satisfies
i∈I
the classical Yang-Baxter identity in D(a) imply the folloPwing formula. If α is an
integer and k = (k ,... ,k ) is a sequence of integers such that 1 ≤ k < ... <
1 α 1
k < m, let k′ = (k′,... ,k′ ) be the sequence such that k′ = k for i = 1,... ,α,
α 1 m i i
(k′) is decreasing and {k′,... ,k′ } = {1,... ,m}. Then
i i>α 1 m
h[a(j ),[a(j ),... ,a(j )]],[b(i ),[b(i ),... ,b(i )]]i
1 2 m 1 2 n
i1,.X..,jm∈I
a(i )⊗···⊗a(i )⊗b(j )⊗···⊗b(j )
1 n 1 m
m−1 m−α
= κ(s)
Xα=1 k1,...,kα|1≤Xk1<···<kα<m Xs=0 i1,.X..,jm∈I
h[a(j ),... ,[a(j ),a(j )]],[b(i ),[b(i ),... ,b(i )]]i
k′ k′ k′ 2 3 n
m α+s+2 α+s+1
[a(j ),···[a(j ),a(i )]]⊗a(i )⊗···⊗a(i )
k′ k′ 1 2 m
1 α+s
⊗b(j )⊗···⊗[b(j ),b(i )]⊗···⊗b(j ),
1 k′ +1 1 m
α+s
where κ(0) = −1 and κ(s) = (−1)s if s 6= 0. The Lemma then follows by
induction on n and m.
End of proof of Proposition. The other properties of ⊕ E are obvi-
n,m|n,m≥0 n,m
ouslysharedby⊕ Im(i ). So⊕ Im(i )iscontainedin⊕ E
n,m≥0 n,m n,m≥0 n,m b n,m|n,m≥0 n,m
and shares all its properties; since ⊕ E is the smallest vector subspace
b b bn,m|n,m≥0 nb,m b
of the space of functorial assignments a 7→ ρa ∈ ⊕n,m|n,m≥1HomK[[~]](a⊗n,a⊗m)
b
with these properties, we obtain ⊕ Im(i ) = ⊕ E . This
n,m|n,m≥1 nb,m n,m|n,m≥1 n,m
proves that ⊕ i is surjective. Since i = i , this implies that i is
n,m|n,m≥1 n,m b b 1,1 b
surjective.
b b b b b
8 B. ENRIQUEZ
Let us now show that the map i is injective. Let (p ) be a family such that
n n≥1
p ∈ (FL ⊗ FL ) [[~]] and i( p ) = 0. Then if a is any Lie bialgebra,
n n n Sn bn|n≥1 n
then i(p ) = 0. P
n|n≥1 n a b
If VPis any vector space, let us denote by F(V) the free Lie algebra generated
b
by V. If (c,δ ) is a Lie coalgebra, then the map c → ∧2F(c) defined as the compo-
c
sition of c → ∧2c → ∧2F(c) of the cobracket map ofc with the canonical inclusion
extends to a unique cocycle map δ : F(c) → ∧2F(c). Then (F(c),[,],δ ) is
F(c) F(c)
a Lie bialgebra. The assignment c 7→ F(c) is a functor from the category LCA of
Lie coalgebras to LBA. Then if (c,δ ) is any Lie coalgebra, we have
c
i(p ) = 0. (3)
n F(c)
X
n|n≥1
b
DefineFA asthemultilinearpartofthefreealgebrawithgeneratorsx ,... ,x .
n 1 n
ThenS actsonFA ⊗FL bysimultaneouslypermutingthegeneratorsx ,... ,x
n n n 1 n
of FA and y ,... ,y of FL . The injection FL ⊂ FA induces a linear map
n 1 n n n n
(FL ⊗ FL ) → (FA ⊗ FL ) ; since S is finite, this linear map is an
n n Sn n n Sn n
injection. Moreover, the map FL → (FA ⊗FL ) , sending P to the class of
n n n Sn
x ···x ⊗P(y ,... ,y ), is a linear isomorphism.
1 n 1 n
For each n, define p¯ as the element of FL such that the equality
n n
p = x ···x ⊗p¯ (y ,... ,y )
n 1 n n 1 n
holds in (FA ⊗FL ) [[~]].
n n Sn
The restriction of i(p ) to c ⊂ F(c) is a linear map i(p ) from c to
n F(c) n F(c)|c
F(c)[[~]]. The image of this map is actually contained in the degree n part
F(c) [[~]] of F(c)[[~]]b. The space F(c) is a vector subspacbe of c⊗n. Moreover,
n n
formula (1) shows that the composition of i(p ) with the canonical inclusion
n F(c)|c
F(c)n[[~]] ⊂ c⊗n[[~]]coincides with δ(p¯n). Sobif (c,δc) is any Liecoalgebra, the map
δ(p¯n) : c → ⊕ c⊗n[[~]] is zero. Now the linear map FL → {functorial
n|n≥0 n|n≥0 n
Passignments c 7→ τc ∈ Hom(c,c⊗n), where c is an object of LCA} defined by
P 7→ δ(P), is injective. This implies that each p¯ is zero. So i is injective.
n
This ends the proof of Proposition 2.1.
b
Remark 1. More generally, one may show that E is isomorphic to F(m,n).
n,m
Remark 2. Itwould beinteresting tounderstand 1)thealgebrastructure ofF(1,1)
provided by the isomorphism of Proposition 2.1 and 2) the algebra structure of
⊕ F(n,m) provided by Remark 1. As we noted before, the latter alge-
n,m|n,m≥1
bra is also equipped with natural operations of the symmetric groups S and
n
S on each component F(n,m), and external product maps F(n,m) ⊗ F(n′,m′) →
m
F(n+n′,m+m′).
UNIVERSAL ALGEBRAS ASSOCIATED TO LIE BIALGEBRAS 9
2.2. Proof of Theorem 1.1. Let (a 7→ ρ ) belong to G (K). Recall that this
a 0
means that (a 7→ ρ ) belongs to E, in particular, ρ belongs to End (a) for
a a K[[~]]
any object a of LBA~. The condition that (a 7→ ρa) belongs to G0(K) implies
that ρ satisfies the identity ρ ([x,y]) = [ρ (x),y] for any x,y in a. Let p be the
a a a
preimage of ρ by the map i. Then there is a unique sequence (p ) , where
a n n≥1
p belongs to (FL ⊗ FL ) [[~]], such that p = p . Then if we set
n n nbSn n|n≥1 n
p = P(n) ⊗ Q(n), and if a is finite-dimensional,Pand a(i) ⊗ b(i) is the
n α α α i∈I
canonPical element of a⊗a∗, then we have P
ρ (x) = hx,Q(n)(b(i ),... ,b(i ))iP(n)(a(i ),... ,a(i )).
a α 1 n α 1 n
nX|n≥1i1,.X..,in∈I
Then
ρ ([x,y]) = h[x,y],Q(n)(b(i ),... ,b(i ))iP(n)(a(i ),... ,a(i )).
a α 1 n α 1 n
nX|n≥1i1,.X..,in∈I
Lemma 2.2. If ξ belongs to a∗, [ a(i)⊗b(i),ξ⊗1+1⊗ξ] belongs to a∗⊗a ,
i∈I ∗
and we have P
h[x,y],ξi = hx⊗y,[ a(i)⊗b(i),ξ ⊗1+1⊗ξ]i.
Xi∈I
Proof of Lemma. The first statement follows from the fact that a(i)⊗b(i)+
i
b(i)⊗a(i) is D(a)-invariant. P
Let us prove the second statement. The invariance of the bilinear form of
D(a) implies that h[x,y],ξi is equal to hx,[y,ξ]i. This is equal to h [a(i),ξ]⊗
i∈I
b(i),x⊗yi. Since a is an isotropic subspace of D(a), this is the saPme as
h [a(i),ξ]⊗b(i)+a(i)⊗[b(i),ξ],x⊗yi.
Xi∈I
So we get
ρ ([x,y]) = (4)
a
nX|n≥1Xα i1,.X..,in∈I
hx⊗y, [a(i),Q(n)(b(i ),... ,b(i ))]⊗b(i)+a(i)⊗[b(i),Q(n)(b(i ),... ,b(i ))]i
α 1 n α 1 n
Xi
P(n)(a(i ),... ,a(i )).
α 1 n
On the other hand, we have
[ρ (x),y] = hx,Q(n)(b(i ),... ,b(i ))i[P(n)(a(i ),... ,a(i )),y]
a α 1 n α 1 n
nX|n≥0i1,.X..,in∈I
10 B. ENRIQUEZ
so
[ρ (x),y] = (5)
a
hx⊗y,Q(n)(b(i ),... ,b(i ))⊗b(i)i[P(n)(a(i ),... ,a(i )),a(i)].
α 1 n α 1 n
nX|n≥0Xα i1,.X..,in∈IXi∈I
Comparing (4) and (5), and using the first part of Lemma 2.2, we get
[P(n)(a(i ),... ,a(i )),a(i)]⊗Q(n)(b(i ),... ,b(i ))⊗b(i)
α 1 n α 1 n
nX|n≥1Xα i1,..X.,in,i∈I (6)
= P(n)(a(i ),... ,a(i ))⊗a(i)⊗[b(i),Q(n)(b(i ),... ,b(i ))]
α 1 n α 1 n
nX|n≥1Xα i1,..X.,in,i∈I
+P(n)(a(i ),... ,a(i ))⊗[a(i),Q(n)(b(i ),... ,b(i ))]⊗b(i).
α 1 n α 1 n
In fact, it is easy to see that the identity
[P(n)(x ,... ,x ),x]⊗Q(n)(y ,... ,y )⊗y (7)
α 1 n α 1 n
Xn≥1Xα
= P(n)(x ,... ,x )⊗x⊗[y,Q(n)(y ,... ,y )]
α 1 n α 1 n
nX|n≥1Xα
+P(n)(x ,... ,x )⊗[x,Q(n)(y ,... ,y )]⊗y
α 1 n α 1 n
holdsinF(abb) (inthe notationof [2]), which isthe ‘Liepart’ofa universal algebra
for solutions of the classical Yang-Baxter equation (CYBE). Equation (6) is then
a consequence of (7).
Applying the Lie bracket to the two last tensor factors of (6), we obtain
[P(n)(a(i ),... ,a(i )),a(i)]⊗[Q(n)(b(i ),... ,b(i )),b(i)]
α 1 n α 1 n
nX|n≥1Xα i1,..X.,in,i∈I (8)
= P(n)(a(i ),... ,a(i ))⊗[[a(i),b(i)],Q(n)(b(i ),... ,b(i ))].
α 1 n α 1 n
nX|n≥1Xα i1,..X.,in,i∈I
Since a(i)⊗b(i) satisfies CYBE, we have
i∈I
P
a(j)⊗[ [a(i),b(i)],b(j)] = [a(i),a(j)]⊗[b(i),b(j)].
Xj∈I Xi∈I iX,j∈I
So identity (8) is rewritten as
[P(n)(a(i ),... ,a(i )),a(i)]⊗[Q(n)(b(i ),... ,b(i )),b(i)]
α 1 n α 1 n
nX|n≥1Xα i1,..X.,in,i∈I (9)
n
= P(n)(a(i ),... ,[a(i),a(i )],... ,a(i ))
α 1 k n
nX|n≥1Xα Xk=1i1,..X.,in,i∈I
⊗Q(n)(b(i ),... ,[b(i),b(i )]... ,b(i )).
α 1 k n