Comparison of Poisson structures and Poisson-Lie dynamical r-matrices

0.2 MB·English

by  B. Enriquez

COMPARISON OF POISSON STRUCTURES AND POISSON-LIE
5 DYNAMICAL r-MATRICES
0
0
2 BENJAMINENRIQUEZ,PAVELETINGOF,ANDIANMARSHALL

n
a Abstract. We construct a Poisson isomorphism between the formal Poisson manifolds g∗
J and G∗, where gis a finite dimensional quasitriangular Lie bialgebra. Here g∗ is equipped
3 with its Lie-Poisson (or Kostant-Kirillov-Souriau) structure, and G∗ with its Poisson-Lie
structure. WealsoquantizethePoisson-Liedynamicalr-matricesofBalog-Feh´er-Palla.

]
A
Q
.
h Introduction and main results
t
a We constructPoissonisomorphismsbetweenthe formalPoissonmanifoldsg∗ andG∗,where
m g is a finite dimensionalquasitriangularLie bialgebra. Hereg∗ is equipped with its Lie-Poisson
[ (or Kostant-Kirillov-Souriau)structure, and G∗ with its Poisson-Lie structure.

Thisresultmaybeviewedasageneralizationoftheformalversionof[GW](laterreprovedin
2
v [Al],Theorem1,and[Bo]), where GinzburgandWeinsteinconstructaPoissondiffeomorphism
2 between the real Poisson manifolds k∗ and K∗, where K is a compact Lie group and k is its
4 Lie algebra. It can also be viewed as a new result in the subject of linearization of Poisson
3
structures; e.g., in contrast with our result, it has been shown in [Ch] that not all Poisson
2
structures on Poisson-Lie groups are linearizable.
1
4 Our result relies on constructing a map g(λ) : g∗ → G satisfying a differential equation.
0 We give two constructions of g(λ): the first one uses the theory of the classical Yang-Baxter
/ equation and gauge transformations; the second one relies of the theory of quantization of Lie
h
t bialgebras.
a
WethenapplythisresulttothequantizationofPoisson-Liedynamicalr-matricesintroduced
m
in [FM1].
:
v We now describe our results in more detail.
i
X
0.1. Comparison of Poisson structures. Let (g,r) be a finite dimensional quasitriangular
r Lie bialgebra over a field k of characteristic 0. Recall that this means that g is a Lie algebra,
a
r ∈g⊗2, t :=r+r2,1 is invariant, and CYB(r):=[r1,2,r1,3]+[r1,2,r2,3]+[r1,3,r2,3]=0. The
Lie cobracketon g is defined by δ(x):=[x⊗1+1⊗x,r] for any x∈g.
TheLiebracketongdefinesalinearPoissonstructureong∗;wealsodenotebyg∗ theformal
neighborhood of the origin in this vector space, which is then a formal Poisson manifold. On
theotherhand,wedenotebyG∗ theformalPoisson-LiegroupwithLiebialgebrag∗. Ourmain
result is:
Theorem 0.1. The formal Poisson manifolds g∗ and G∗ are isomorphic.
We now explicitly describe an isomorphism g∗ → G∗. Denote by G the formal group with
Lie algebra g and by Map (g∗,G) the space of formal maps g : g∗ → G, such that g(0) = 1;
0
this is the space of maps of the form ex(λ), where x(λ)∈g⊗S(g) . 1
>0
b
1We denote by S(g) the degree completion ⊕k≥0Sk(g) of the symmetric algebra S(g), and set S(g)>i =
⊕k>iSk(g). b b b
b 1
2 BENJAMINENRIQUEZ,PAVELETINGOF,ANDIANMARSHALL
Map (g∗,G) has a group structure, defined by (g ∗g )(λ) := g (Ad∗(g (λ))(λ))g (λ). Its
0 1 2 2 1 1
subspace of all maps g(λ) such that
g−1d (g )(λ)−g−1d (g )(λ)+hid⊗id⊗λ,[g−1d (g )(λ),g−1d (g )(λ)]i=0 (1)
1 2 1 2 1 2 1 3 1 2 3 2
is a subgroup Maph0am(g∗,G). Here g1−1d2(g1)(λ) = Pig−1∂εig(λ)⊗ei is viewed as a formal
function g∗ → g⊗2, (εi), (e ) are dual bases of g∗ and g, g−1d (g ) = (g−1d (g ))i,j and
i i j i 1 2 1
∂ g(λ)=(d/dε) g(λ+εξ). We will also denote by g−1d (g ) the same quantity, viewed as
ξ |ε=0 13 2 13
an element of g⊗2⊗S(g).
We have a group bmorphism θ : Mapham(g∗,G) → Aut (g)op to the group (with opposite
0 1
structure) of Poisson automorphisms of g∗ with differential at 0 equal to the identity, taking
g(λ) to the automorphism λ7→Ad∗(g(λ))(λ).
Proposition 0.2. There exists a formal map g(λ)∈Map (g∗,G), such that
0
(g )−1d (g )−(g )−1d (g )+Ad(g⊗g)−1(r )+hid⊗id⊗λ,[(g )−1d (g ),(g )−1d (g )]i=ρ .
1 2 1 2 1 2 0 1 3 1 2 3 2 AM
(2)
(identity of formal maps g∗ → ∧2(g)). 2 Here r = (r −r2,1)/2, and ρ is the Alekseev-
0 AM
Meinrenken r-matrix ([AM, BDF]) given by
ρ (λ)=(id⊗ϕ(adλ∨))(t),
AM
where λ∨ =(λ⊗id)(t) and ϕ(z):=−1 + 1cotanhz.
z 2 2
The group Mapham(g∗,G) acts simply and transitively on the space of solutions g(λ) of (2),
0
as follows: (α∗g)(λ)=g(Ad∗(α(λ))(λ))α(λ).
Proposition 0.3. Let g(λ) ∈ Map (g∗,G) be as in Proposition 0.2. There exists a unique
0
isomorphism g∗(λ):g∗ →G∗, defined by the identity
g(λ)eλ∨g(λ)−1 =L(g∗(λ))R(g∗(λ))−1.
HereL,R:G∗ →GaretheformalgroupmorphismscorrespondingtotheLiealgebramorphisms
L,R:g∗ →g given by L(λ):=(λ⊗id)(r), R(λ):=−(λ⊗id)(r2,1).
Inotherwords, wehave(non-Poisson) formalmanifold isomorphisms g∗ →a G←b G∗,a(λ)=
g(λ)eλ∨g(λ)−1, b(g∗)=L(g∗)R(g∗)−1, and g∗(λ)=b−1◦a(λ).
The isomorphism (α∗g)∗(λ):g∗ →G∗ corresponding to (α∗g)(λ) is such that (α∗g)∗(λ)=
g∗(θ(α)(λ)).
It follows that the set of all isomorphisms g∗ → G∗ constructed in Propositions 0.2, 0.3
is a principal homogeneous space under the image of θ : Mapham(g∗,G) → Aut (g∗). When
0 1
g is semisimple, any derivation g → Sk(g) is inner, so θ is surjective. So in that case, our
constructionyields allthe Poissonisomorphismsg∗ →G∗ taking 0to 1 andwithdifferentialat
0 equal to the identity.
0.2. Quantization of Poisson-Lie dynamical r-matrices. In [BFP, FM1], there was in-
troduced the Poisson-Lie dynamical r-matrix
id+a(g∗)2ν 1id+a(g∗)
ρ (g∗)= id⊗ ν − (t).
FM (cid:16) (cid:0) id−a(g∗)2ν 2id−a(g∗)(cid:1)(cid:17)
Here a(g∗) : G∗ → GL(g) is defined by a(g∗) = Ad(L(g∗)R(g∗)−1) and ν is a fixed element in
k. Then ρFM is a formal map G∗ →∧2(g), i.e., an element of ∧2(g)⊗OG∗. It is a Poisson-Lie
classical dynamical r-matrix for (G∗,g,Z ), where Z =(ν2− 1)[t1,2,t2,3] (see also [EEM]).
ν ν 4
2Weview∧n(g)asasubspaceofg⊗n.
COMPARISON OF POISSON STRUCTURES AND POISSON-LIE DYNAMICAL r-MATRICES 3
A quantization of this dynamical r-matrix is the data of a quantization U (g) of the Lie
~
bialgebra g, and a pair (J¯,Φ¯), where:
•Φ¯ isanassociatorforU (g),i.e.,Φ¯ ∈U (g)⊗3satisfies34thepentagonequationΦ¯2,3,4Φ¯1,23,4Φ¯1,2,3 =
~ ~
Φ¯1,2,34Φ¯12,3,4, the invariance condition [Φ¯,(∆⊗b id)◦∆(a)] =0 for any a ∈U (g), and the ex-
~
pansion Φ¯ =1+~2φ , where φ ∈U (g)⊗3 satisfies sign(σ)σ((φ ) )=Z ;
2 2 ~ b Pσ∈S3 2 |~=0 ν
•J¯∈U (g)⊗2⊗U (g)′satisfiesthedynamicaltwistequationJ¯2,3,4J¯1,23,4Φ¯1,2,3 =J¯1,2,34J¯12,3,4,
~ ~
theinvariancecbonbdition[J¯,(∆⊗id)◦∆(a)]=0foranya∈U~(g),andtheexpansionJ¯=1+~¯j1,
where ¯j ∈U (g)⊗2⊗U (g)′ satisfies (¯j −¯j2,1,3) =ρ .
1 ~ b ~ 1 1 |~=0 FM
b
Theorem 0.4. Such a quantization exists (see Theorem 2.1).
1. Construction of isomorphisms g∗ →G∗
Theorem 0.1 follows from Propositions 0.2 and 0.3. In this Section, we give two proofs of
these propositions: (a) a Poisson geometric proof (Section 1.1), and (b) a proof based on the
theory of quantization of Lie bialgebras (Section 1.2).
1.1. Geometric construction.
1.1.1. Construction of g(λ) (proof of Proposition 0.2). One checks that Mapham(g∗,G) is a
0
prounipotent Lie group with Lie algebra {α ∈ g⊗S(g) |Alt◦d(α) = 0}. This Lie algebra is
≥1
isomorphic to (S(g) ,{−,−}) under d:f 7→d(f).b5
>1
Let us denotebby G the set of all g ∈Map (g∗,G) satisfying (2). This is a subvariety of the
0
proalgebraicvariety Map (g∗,G). One checks that (S(g) ,{−,−})acts by vector fields on G,
0 >1
by b
g−1δ (g)=hid⊗id⊗λ,[d (f ),g−1d (g )]i−d (f )∈g⊗S(g) , (3)
f 3 2 12 3 12 1 2 >0
b
and that the right infinitesimal action of Map (g∗,G) on itself is given by the same formula.
0
It follows that the rightaction ofMap (g∗,G) on itself restricts to an actionof Mapham(g∗,G)
0 0
on G.
We nowprovethat if G is nonempty, then Mapham(g∗,G) acts simply and transitivelyonG.
0
Let us show that the action is simple. If g,g′ ∈ G and α ∈ Mapham(g∗,G) are such that
0
g∗α=g′, then let a:=log(α). Assume that a6=0 and let n be the smallest integer such that
the component a of a in g⊗Sn(g) is 6= 0. Then log(g′)−log(g) = a modulo g⊗S(g) ,
n n >n
which implies that g 6=g′. b
Let us now prove that the action is transitive. Let g,g′ ∈G. Set A:= log(g), A′ :=log(g′).
Then A,A′ ∈ g⊗S(g) . Assume that A 6= A′ and let n be the smallest integer such that
>0
the component of Ab′ −A in g⊗Sn(g) is nonzero; we denote by (A′ −A) this component.
n
Comparingequations(2)forg andg′,wegetAlt◦d((A′−A) )=0. Itfollowsthatthereexists
n
a ∈ Sn+1(g), such that (A′−A) = Alt◦d(a), i.e., (A′−A) ∈ LieMapham(g∗,G). Let exp
n n 0 ∗
be the exponential map of Mapham(g∗,G) and α := exp ((A′−A) ) ∈ Mapham(g∗,G). Then
0 ∗ n 0
log(α∗g) = A(Ad∗(α(λ))(λ)) +(A′ −A) modulo g⊗S(g) ; therefore the difference of
n ≥n+1
logarithms6ofα∗gandg′coincidemodulog⊗S(g) . Wborkingbysuccessiveapproximations,
≥n+1
we constructβ ∈Mapham(g∗,G) suchthat β∗bg =g′. This provesthatthe actionis transitive.
0
3Wedenote by⊗thetopologicaltensorproduct, definedasfollows: ifV,W aretopologicalvector spaces of
theformV =V0[[xb1,...,xn]],W =W0[[y1,...,ym]],thenV⊗W :=V0⊗W0[[x1,...,ym]].
4WesetΦ¯12,3,4=(∆⊗id⊗id)(Φ¯),etc. b
Alt(5ξFo⊗rfρ)=:=α(ξ⊗+fξ∈2,..∧.,nn(,g1)+⊗·S·b·(+g),ξnw,1e,.s.e.,tnd−(1ρ))⊗:=f.Piα⊗ei⊗(d/dε)|ε=0f(λ+εεi) and if ξ ∈∧n−1(g)⊗g,
6Herethelogarithmistheinverseoftheordinaryexponential mapg⊗S(g)→Maph0am(g∗,G).
b
4 BENJAMINENRIQUEZ,PAVELETINGOF,ANDIANMARSHALL
Let us now prove that G is nonempty. Recall that r =(r−r2,1)/2. If g ∈Mapham(g∗,G),
0 0
set (r )g := l.h.s. of (2).
0
Lemma 1.1. Assume that log(g)=−r/2 modulo g⊗S(g) . Set ρ:=(r )g and assume that
>1 0
ρ=ρ +α, where ρ ,α∈∧2(g)⊗S(g), ρ is g-inbvariant and α∈g⊗S(g) . Then
inv inv inv ≥n
b 1 b
CYB(ρ)−Alt(dρ)=Z1,2,3− [r1,234,α2,3,4]
2
modulo g⊗3⊗S(g) .
≥n+1
b
Proof of Lemma. This statement can be proved directly. It can also be viewed as the
classical limit of the following statement. Let U = U(g)[[~]], let J ∈ (U⊗2)×, and let Φ :=
b
(J1,2J12,3)−1J2,3J1,23. If K ∈ (U⊗2)× and we set Φ¯ := (K1,23)−1(K2,3)−1J1,2K12,3, then Φ¯
b
satisfies
Φ¯2,3,4Φ¯1,23,4Φ1,2,3 =(Φ¯2,3,4,(K1,234)−1)Φ¯1,2,34Φ¯12,3,4(K123,4,Φ1,2,3).
Here (a,b) = aba−1b−1. The statement of the lemma is recovered when J,K ∈ (U⊗2)× have
the form 1−~r/2+o(~), and K is admissible with classical limit g(λ): the contrbibution of
(Φ¯2,3,4,(K1,234)−1) is −1[r1,234,α2,3,4], while the commutator (K123,4,Φ1,2,3) does not con-
2
tribute (as the classical limit of Φ is proportional to Z and hence invariant). (cid:3)
Let us now prove that G is nonempty. We will construct a sequence g ∈Map (g∗,G), such
n 0
that ρ :=(r )gn satisfies ρ =ρ modulo ∧3(g)⊗S(g) .
n 0 n AM ≥n+1
If n=0, we set g :=exp(−r/2), then ρ =0=ρ b modulo ∧3(g)⊗S(g) . Assume that
0 0 AM ≥1
wedeterminedg andletusconstructg . Setα:=ρ −ρ ,andletα b bethecomponent
n n+1 n AM n+1
of α in ∧2(g)⊗Sn+1(g). Then Lemma 1.1 implies that
1
CYB(ρ +α )−Alt(d(ρ +α ))=Z1,2,3− [r1,234,α2,3,4] mod ∧3(g)⊗Sn+1(g).
AM n+1 AM n+1 2 n+1
Since ρ satisfies the modified classical dynamical Yang-Baxter equation, the component
AM
in ∧3(g) ⊗ Sn(g) of this identity yields Alt(dα ) = 0, so that we have α = Alt(dβ)
n+1 n+1
for some β ∈ g ⊗ Sn+2(g). Then we set gn+1 := exp∗(−β) ∗ gn, ρn+1 := (r0)gn+1. Then
ρ −ρ = ρ − ρ − Alt(dβ) = α −Alt(dβ) = 0 modulo ∧3(g)⊗S(g) . By
n+1 AM n AM n+1 ≥n+1
successive approximations,we then construct g such that (r )g =ρ . So G is nbonempty.
0 AM
1.1.2. Poisson isomorphism g∗ →G∗ (proof of Proposition 0.3). In this section, we show that
the fact that g(λ) satisfies (2) implies that λ 7→ g∗(λ) is a Poisson isomorphism. We will
freely use the formalism of differential geometry, even though we work in the formalsetup; the
computations below make sense (and prove the desired result) when working over an Artinian
k-ring.
According to [STS], the image of the Poisson bracket {−,−}G∗ on G∗ under the formal
isomorphism b:G∗ →G is the Poissonbracket on G
{F,H}G(g)=h(dR−dL)F(g)⊗dLH(g),ri+h(dR−dL)F(g)⊗dRH(g),r2,1i,
where g ∈G, F,H are functions on G, dLF(g),dRF(g)∈g∗ are the left and right differentials
defined by hdLF(g),ai=(d/dε)|ε=0F(eεag), hdRF(g),ai=(d/dε)|ε=0F(geεa) for any a∈g.
For ξ ∈g∗, define F ∈O by F (g)=hξ,log(g)i.
ξ G ξ
Lemma 1.2. (dLFξ)(g) = f(12ad∗(logg))(ξ) and (dRFξ)(g) = f(−21ad∗(logg))(ξ), where
f(z)=zez/(sinhz), and ad∗ denotes the coadjoint action of g on g∗.
Proof. Set x := log(g), let a ∈ g, and set a˜ := (d/dε) log(eεaex). The coefficient
|ε=0
in ε of eεaex = ex+εa˜+O(ε2) yields aex = (n!)−1 n−1xka˜xn−1−k. Applying ad(x) to
Pn≥1 Pk=0
this relation, we get a˜ = f(−21adx)(a). Now hdLFξ(g),ai = (d/dε)|ε=0Fξ(eεag) = hξ,a˜i =
COMPARISON OF POISSON STRUCTURES AND POISSON-LIE DYNAMICAL r-MATRICES 5
hξ,f(−1adx)(a)i = hf(1ad∗x)(ξ),ai, which yields the first formula. On the other hand, we
2 2
havefor anyfunction F onG, dRF(g)=Ad∗(g−1)(dLF(g)), where Ad∗ is the coadjointaction
of G on g∗, hence dRFξ(g)=Ad∗(e−x)f(12ad∗x)(ξ)=f(−21ad∗x)(ξ). (cid:3)
Then we get
{F ,F } (g)=had∗(x)(ξ)⊗ad∗(x)(η),(id⊗ϕ(ad(x))(t)+r i−had∗(x)(ξ)⊗η,ti, (4)
ξ η G 0
where x=log(g) and ϕ and r are as in Proposition 0.2.
0
Ontheotherhand,let{−,−}′G bethe imageofthePoissonbracket{−,−}g∗ bya:g∗ →G.
Setfξ(λ):=Fξ◦a(λ),thenfξ(λ)=hξ⊗λ,(Ad(g(λ))⊗id)(t)i. Iff ∈Og∗,λ∈g∗,definedf(λ)∈
gby hα,df(λ)i=(d/dε)|ε=0f(λ+εα)foranyα∈g∗. Wehave{f,g}g∗(λ)=hλ,[df(λ),dg(λ)]i.
Then df (λ)=hξ⊗id,A(λ)i, where
ξ
A(λ)=(Ad(g(λ))⊗id)(t)+[(d g )g−1,Ad(g(λ))(λ∨)⊗1].
2 1 1
So
{f (λ),f (λ)}=hξ⊗η⊗λ,[A1,3(λ),A2,3(λ)]i.
ξ η
This decomposes as the sum of four terms (we set x¯=Ad(g(λ))(λ∨)):
(a) hξ⊗η⊗λ,(Ad(g(λ))⊗2⊗id)([t13,t23])i=−hξ⊗η,(ad(x¯)⊗id)(t)i;
(b)hξ⊗η⊗λ,[[(d g )g−1,Ad(g (λ))(λ∨)],Ad(g (λ))(t23)]i=had∗(x¯)⊗2(ξ⊗η),−Ad(g)⊗2(g−1d (g ))i;
3 1 1 1 2 1 2 1
(c)hξ⊗η⊗λ,[Ad(g (λ))(t13),[(d g )g−1,Ad(g (λ))(λ∨)]]i=had∗(x¯)⊗2(ξ⊗η),Ad(g)⊗2(g−1d (g ))i;
1 3 2 2 2 2 2 1 2
(d)
hξ⊗η⊗λ, [(d g )g−1,Ad(g (λ))(λ∨)],[(d g )g−1,Ad(g (λ))(λ∨)]
(cid:2) 3 1 1 1 1 3 2 2 2 2 (cid:3)
=had∗(x¯)(ξ)⊗ad∗(x¯)(η)⊗λ,[(d g )g−1,(d g )g−1]i.
3 1 1 3 2 2
We have {Fξ,Fη}′G(g)={fξ,fη}g∗(a−1(g)); then x¯=x, so {Fξ,Fη}′G ={Fξ,Fη}G iff
had∗(x)(ξ)⊗ad∗(x)(η),Ad(g)⊗2(g−1d (g )−g−1d (g ))
2 1 2 1 2 1
+hid⊗id⊗λ,[d (g )g−1,d (g )g−1]ii− id⊗ϕ(ad(x)))(t)−r i=0,
3 1 1 3 2 2 (cid:0) 0
for which a sufficient condition is that g(λ) satisfies (2).
Remark 1.3. Formula (4) implies that the image of {−,−} under the map log : G → g is
G
given by
1 1
{f,g}(x)=hdf(x)⊗dg(x), ad(x)⊗ ad(x)coth( ad(x)) (t)+ad(x)⊗2(r )i;
0
(cid:0) (cid:0)2 2 (cid:1)(cid:1)
this is a result of [FM2].
1.2. Construction based on quantization of Lie bialgebras. We now give a proof of
Theorem 0.1 based on the theory of quantization of Lie bialgebras.
1.2.1. Construction of g(λ) (proof of Proposition 0.2). Set U := U(g)[[~]], U′ := {x∈ U|∀n ≥
0,δ (x) ∈ ~nU⊗n} = U(~g[[~]]) ⊂ U (here δ = (id−η ◦ ε)⊗n ◦ ∆(n), η is the unit map
n n
k[[~]]→U). Asba k[[~]]-algebra, U′ is a flat deformation of S(g)=k[[g∗]].
Let Φ∈U⊗3 be an admissible associator. This means thabt
b
~2
Φ∈1+ [t1,2,t2,3]+~2U⊗3, ~log(Φ)∈(U′)⊗3,
24 b b
(U,m,∆ ,R = 1,Φ) is a quasitriangular quasi-Hopf algebra. We also require ε(i)(Φ) = 1,
0 0
i= 1,2,3 (here ε(1) =ε⊗id⊗id, etc., ε is the counit, m and ∆ are the undeformed product
0
andcoproductofU). Accordingto[EH],anyuniversalLieassociatorgivesrisetoanadmissible
associator.
6 BENJAMINENRIQUEZ,PAVELETINGOF,ANDIANMARSHALL
Accordingto[EK],thereexistsatwistkillingΦ,andaccordingto[EH],thistwistcanthenbe
madeadmissiblebyasuitablegaugetransformation. TheresultingtwistJ satisfiesthefollowing
conditions: J ∈U⊗2, J =1−~r/2+o(~), ~log(J)∈(U′)⊗2, (ε⊗id)(J)=(id⊗ε)(J)=1,
b b
Φ=(J2,3J1,23)−1J1,2J12,3. (5)
Then UJ := (U,m,∆J,R) is a quasitriangular Hopf algebra quantizing (g,r). Here ∆J(x) =
J∆ (x)J−1 and R=J2,1e~t/2J−1.
0
We have Ker(ε)∩U′ ⊂ ~U, therefore log(J) ∈ U⊗U′. Then its reduction mod ~, denoted
g(λ) = g1,2 = log(J) , belongs to U(g)⊗S(g) = Ub(g)[[g∗]] (formal series on g∗ with coeffi-
|~=0
cientsinU(g)). Thereductionmod~of(5)bisbg12,3 =g1,3g2,3. Sincewealsohave(ε⊗id)(g)=1,
we get g =exp(A), with A∈g⊗S(g) .
>0
b
Lemma 1.4. g(λ) satisfies (2).
Proof. According to [EE], Φ ∈ U⊗2⊗U′ has the expansion 1+~φ +o(~2), where φ ∈
1 1
b
U⊗2⊗U′ is such that (φ −φ2,1,3) =b−ρ .
b 1 1 |~=0 AM
Ifbx∈U(g)⊗S(g), we denote by x¯ a lift of x in U⊗U′.
Let us expabndblog(J) as A¯+~A1+o(~), with A1b∈U⊗U′.
Then J1,23 =exp(A¯1,3+~(A1,3+d A1,3)+o(~))=J1b,3(1+~g−1d (g )+o(~)).
1 2 1 2 1
Wehave[J1,3,J2,3]=~{g1,3,g2,3}+o(~),so(J12,3)−1[J1,3,J2,3]=~(g1,3g2,3)−1{g1,3,g2,3}+
o(~)=~hid⊗id⊗λ,[g−1d (g ),g−1d (g )]i+o(~). So we get
1 3 1 2 3 2
J12,3 =J2,3J1,3(1+~ψ +o(~)),
1
where ψ ∈U⊗2⊗U′ is such that (ψ −ψ2,1,3) =hid⊗id⊗λ,[g−1d (g ),g−1d (g )]i.
1 b 1 |~=0 1 3 1 2 3 2
Then (5) givesb
1+~φ +o(~)=(1−~g−1d (g )+o(~))(J1,3)−1(J2,3)−1(1−~r/2+o(~))J2,3J1,3(1+~ψ +o(~)).
1 1 2 1 1
The reduction modulo ~ of (J1,3)−1(J2,3)−1rJ2,3J1,3 is Ad(g⊗g)−1(r). Then substracting 1,
dividing by ~, reducing modulo ~ and antisymmetrizing the two first tensor factors, we getthe
lemma. (cid:3)
More generally, assume that J′ ∈ U⊗2 satisfies J′ = 1−~r/2+o(~), ~log(J′) ∈ (U′)⊗2,
(ε⊗id)(J′)=1 and b b
Φ=(J′2,3J′1,23)−1J1,2J′12,3. (6)
Then J′ ∈ U⊗U′, and its reduction g′ modulo ~ satisfies: g′ ∈ exp(g⊗S(g) ), and equation
>0
(2) with g repblaced by g′. Moreover,there exists u∈U× such that J′ =bu2J(u12)−1.
Remark 1.5. Equation(6)canbeinterpretedassayingthatJ′ isavertex-IRFtransformation
relating J1,2 and Φ, and equation (2) for g′ is the classical limit of this statement (see [EN]).
Vertex-IRF transformations are a special kind of non-invariant dynamical gauge transforma-
tions, which map a constant, but non-invariant twist to an invariant, but non-constant (i.e.,
dynamical) one. (cid:3)
Let U′ := Ker(ε)∩U′. Then ~−1U′ ⊂U is a Lie subalgebra for the commutator. This Lie
0 0
algebra acts on the set of solutions of (6) by δ (J)=u2J −Ju12 (δ (J)∈U⊗U′ because it is
u u
equal to [u2,J]−J(u12 −u2)); this means that if ε is a formal parameter wbith ε2 = 0, then
(id+εδ )(J′) is a solution of (6) if J′ is.
u
The reduction modulo ~ of this action may be described as follows. The Lie algebra
(S(g) ,{−,−})acts onthe setofsolutionsof(2)by δ (g)={1⊗f,g}−g·df, i.e.,action(3).
>0 f
b
COMPARISON OF POISSON STRUCTURES AND POISSON-LIE DYNAMICAL r-MATRICES 7
WhenrestrictedtotheLiesubalgebraS(g) ,thisactionistheinfinitesimaloftherightaction
>1
of Mapham(g∗,G) on the set of solutionbs of (2), given by (g∗α)(λ)=g(Ad∗(α(λ))(λ))α(λ).
0
1.2.2. Isomorphism g∗ → G∗ (proof of Proposition 0.3). It follows from [EH] that the subal-
gebras (UJ)′ and U′ of U are equal. According to [Dr1, Gav], (UJ)′ is a flat deformation of
OG∗ := U(g∗)∗. On the other hand, U′ is a flat deformation of Og∗ := S(g). So the equality
i~ : (UJ)′ → U′ induces an isomorphism of Poisson algebras i : OG∗ →∼ bOg∗, and therefore a
Poisson isomorphism g∗ →G∗.
To express this isomorphism explicitly, we will construct maps a:OG →Og∗ and b:OG →
OG∗, such that i=a◦b−1.
We first construct quantized versions of a and b. We have ~log(J) ∈ (U′)⊗2, therefore
J,J−1 ∈ U⊗U′ and Je~tJ−1 ∈ U⊗U′. We then define a : U∗ → U′ by a (f) := (f ⊗
~ ~
id)(Je~tJ−1)b. Now we also have J2,b1 ∈ U⊗U′, hence R,R2,1 ∈ U⊗U′ = U⊗(UJ)′. We then
define thelinearmapb~ :U∗ →(UJ)′ byb~b(f):=(f⊗id)(R2,1R). bThena~◦bb−~1 :U′ →(UJ)′
coincides with i . We define a,b as the reductions modulo ~ of a ,b .
~ ~ ~
Let us now compute the classical limit of a . Define maps j ,ι ,j′ : U∗ → U′ by j (f) =
~ ~ ~ ~ ~
(f ⊗id)(J), j′(f) = (f ⊗id)(J−1). Recall that the reductions modulo ~ of all three elements
~
J,J−1,e~t in U(g)⊗S(g) are of the form K =exp(k), where k ∈g⊗S(g) .
>0
bb b
Lemma 1.6. If K ∈ U(g)⊗S(g) is of the form exp(k), where k ∈ g ⊗ S(g) , then the
>0
morphism OG = U(g)∗ → Sb(gb) = Og∗ given by f 7→ (f ⊗id)(K) is dual bto the morphism
g∗ →G, λ7→ek(λ). b
Proof. We compose this morphism with the transpose of the inverse of the symmetrization
maptSym−1 :S(g)∗ →U(g)∗. The morphismtSym−1 correspondstothe logarithmmapG→
g. NowthecomposedmorphismS(g)∗ →S(g)isgivenbyf 7→(f⊗id)((Sym−1⊗id)(K)). Now
(Sym−1⊗id)(K) is exp(k), where the expbonential is now taken in S(g)⊗S(g). The morphism
S(g)∗ → S(g), f 7→ (f ⊗id)(exp(k)), is an algebra morphism, taking thbebfunction X 7→ α(X)
on g (α ∈bg∗) to the function λ 7→ α(k(λ)) on g∗, and therefore corresponds to the morphism
g∗ →g, λ7→k(λ). Composing with the exponential, we get the announced morphism. (cid:3)
Itfollowsthatthereductionsmodulo~ofthemorphismsj~,ι~,j~′ aremorphismsOG →Og∗,
correspondingtomorphismsj,ι,j′ :g∗ →Gsuchthatj(λ)=g(λ),ι(λ)=eλ∨,j′(λ)=g(eλ)−1.
Then a~ =m(2)◦(j~⊗ι~⊗j~′)◦∆(2), so a:OG∗ →OG corresponds to the composed map
G∗ dia→g(2) (G∗)3 (j,→ι,j′)G3 prod→uct(2) G. Therefore we get a(λ)=g(λ)eλ∨g(λ)−1.
We now compute the classicallimit of b . Define maps L ,R′ :(UJ)∗ →(UJ)′ by L (f):=
~ ~ ~ ~
(f⊗id)(R2,1), R′(f):=(f⊗id)(R). Then L is anantimorphismofalgebrasanda morphism
~ ~
of coalgebras, while R′ is a morphism of algebras and antimorphism of coalgebras. Their
~
reductions L,R′ modulo ~ are morphisms OG∗ →OG (anti-Poissoncoalgebramorphismin the
caseofL,Poissonanti-coalgebrainthecaseofR′). Using[EGH],appendix,oneshowsthatthese
morphismscorrespondtothemorphismsofformalgroupsL,R′ :G∗ →G(antimorphisminthe
case of R′), corresponding to the morphisms L,R′ : g∗ → g, given by L(λ) := (λ⊗id)(r) and
R′(λ):=(λ⊗id)(r2,1). Here L is a Lie algebra,anti-LiecoalgebramorphismandR′ is ananti-
Liealgebra,Liecoalgebramorphism. Nowb~ =m◦(L~⊗R~)◦∆,sob:OG∗ →OG corresponds
tothecomposedmapG∗ d→iag(G∗)2 (L→,R′)G2 pro→ductG,i.e.,g∗ 7→L(g∗)R′(g∗)=L(g∗)R(g∗)−1.
Finally, the isomorphism i:g∗ →G∗ is equal to b−1◦a, so it takes λ∈g∗ to g∗ ∈G∗ such
that
L(g∗)R(g∗)−1 =g(λ)eλ∨g(λ)−1.
So it coincides with the isomorphism obtained in Proposition 0.3.
8 BENJAMINENRIQUEZ,PAVELETINGOF,ANDIANMARSHALL
1.3. OnthegroupMapham(g∗,G). Inthissection,wegivea”quantum”proofofthefollowing
0
statement, which was used in Section 1.1 (and can also be proved in the setup of this section).
Proposition 1.7. Mapham(g∗,G) is a subgroup of Map (g∗,G).
0 0
Proof. Set U′ = Ker(ε)∩U′. The map of U′ → S(g) of reduction by ~ restricts to U′ →
0 0
S(g) , so we get a map U′ →g. b
>0 0
b Consider the set of allJ′, suchthat ~log(J′)∈(U′)⊗2, the image of ~log(J′) in g⊗2 is zero,
0
and J′2,3J′1,23 =J′1,23. b
This is the set of elements of the form J′(u) = u2(u12)−1, where u ∈ exp(U′/~) is such
0
that the image of log(u) under S(g) → g is zero. Indeed, one recovers u from J′ by u =
>0
(id⊗ε)(J′)−1. b
We denote by A(λ) the image of ~log(J′) by S(g) ⊗S(g) →g⊗S(g) , and set g(λ):=
>0 ≥2 ≥2
exp(A(λ)). Then g(λ)∈Map (g∗,G). b bb b
0
As in Lemma 1.4, one proves that g(λ) satisfies equation (1). Hence we get a map of sets
{J′ as above}→Mapham(g∗,G), which is surjective.
0
Onthe other hand, the setof allJ′(u) is equipped with a product, suchthat J′(u)∗J′(v)=
J′(uv). This product expresses as follows: J′(u)∗J′(v) = u2J′(v)(u2)−1 J′(u); the classical
(cid:0) (cid:1)
limitofthisexpressionistheproductformulaforMap (g∗,G). SoMapham(g∗,G)isasubgroup
0 0
of Map (g∗,G). (cid:3)
0
2. Quantization of ρ
FM
In this section, we prove Theorem 0.4.
Let Φ be a universal Lie associator defined over k with parameter µ = 1. So Φ =
univ univ
1+ 1 [t ,t ]+ terms of degree > 2, where t is the universal version of ti,j (see [Dr3]). Set
24 12 23 ij
Φ :=Φ (2~νt1,2,2~νt2,3) and Φ:=Φ . Then
ν univ 1/2
(U(g)[[~]],m,∆ ,Φ )
0 ν
is a quasi-Hopf algebra; its classical limit is the quasi-Lie bialgebra (g,δ =0,ν2[t1,2,t2,3]). Let
J be an admissible twist killing Φ, and let us twist this quasi-Hopf algebra by J. We obtain
the quasi-Hopf algebra
(U(g)[[~]],m,∆J,ΦJ),
0 ν
where ∆J(x) = J∆ (x)J−1 and ΦJ = J2,3J1,23Φ (J−1)12,3(J−1)1,2. Its classical limit is the
0 0 ν ν
quasi-Lie bialgebra (g,δ(x)=[x1+x2,r],Z ).
ν
Now (U(g)[[~]],m,∆J) is a Hopf algebra quantizing (g,δ), which we denote by U (g), and
0 ~
ΦJ is an associator for this quantized universal enveloping algebra, with classical limit Z .
ν ν
ΦJ clearly satisfies the invariance and pentagon equations. We have J = 1+~j , where
ν 1
j ∈U⊗U′,soJ2,3,J1,23,(J−1)1,23 and(J−1)1,2 arealloftheform1+~k,k ∈U⊗2⊗U′. Hence
1
b
ΦJ = 1b+~ψ , where ψ ∈ U⊗2⊗U′. Set U := U (g), then U′ = U′, so ψ ∈ U⊗b2⊗U′. We
ν ν ν b ~ ~ ~ ν ~b ~
now compute (ψν −ψν2,1,3)|~=0; tbhis is an element of U(g)⊗2⊗OG∗ ≃U(g)⊗2⊗Og∗. b
We have ΦJ = ΦJ +(Φ −Φ)J = 1+~(Φ −Φ)J. Letbus define φ,φ bby Φ = 1+~φ,
ν ν ν ν
Φ =1+~φ , then (ψ ) = (φ −φ)J , therefore
ν ν ν |~=0 (cid:0) ν (cid:1)~=0
(ψ −ψ2,1,3) =((φ −φ2,1,3)J) −((φ−φ2,1,3)J)
ν ν |~=0 ν ν |~=0 |~=0
=Ad(g(λ))⊗2(ρ (λ)−ρν (λ))
AM AM
COMPARISON OF POISSON STRUCTURES AND POISSON-LIE DYNAMICAL r-MATRICES 9
as a formal function g∗ → ∧2(g). Here ρν (λ) = 2νρ (2νλ). Since ρ and ρν are
AM AM AM AM
G-equivariant, this is
1ead(λ¯∨)+id e2νad(λ¯∨)+id
(ρ −ρν )(Ad∗(g(λ))(λ))= id⊗ −ν (t)=ρ (g∗).
AM AM (cid:16) (cid:0)2ead(λ¯∨)−id e2νad(λ¯∨)−id(cid:1)(cid:17) FM
Here we set λ¯ :=Ad∗(g(λ))(λ) and we use the relation L(g∗)G(g∗)−1 =eλ¯∨.
We have proved:
Theorem 2.1. The quantized universal enveloping algebra U (g) = U(g)[[~]]J, together with
~
the pair (J¯,Φ¯) defined by J¯=Φ¯ =ΦJ, is a quantization of the Poisson-Lie dynamical r-matrix
ν
ρ (g∗) over (G∗,g,Z ).
FM ν
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10 BENJAMINENRIQUEZ,PAVELETINGOF,ANDIANMARSHALL
IRMA(CNRS),7 rueRen´eDescartes, F-67084Strasbourg,France
E-mail address: [email protected]
DepartmentofMathematics,MassachusettsInstituteofTechnology,Cambridge,MA02139,USA
E-mail address: [email protected]
D´epartementde Math´ematiques,EPFL, CH-1015 Lausanne,Switzerland
E-mail address: [email protected]