Quantization of $r-Z$-quasi-Poisson manifolds and related modified classical dynamical $r$-matrices

0.13 MB·English

by  Gilles Halbout

Quantization of r−Z-quasi-Poisson manifolds and related
8
0
modified classical dynamical r-matrices
0
2

n
GillesHalbout
a
J

7 InstitutdeMathe´matiquesetdeMode´lisationdeMontpellier
1
UMR5149del’Universite´MontpellierIIetduCNRS

] CC5149,PlaceEuge`neBataillon
A
F-34095MontpellierCEDEX5
Q
e-mail:[email protected]
.
h
t February 2,2008
a
m
[

Abstract
1
v ¥
LeX beaC -manifoldandgbea finitedimensionalLiealgebraactingfreelyonX.
9
Let r∈L 2(g) be such thatZ =[r,r]∈L 3(g)g. In this paper we provethat everyquasi-
8
7 Poisson (g,Z)-manifold can be quantized. This is a generalizationof the existence of a
2 twistquantizationofcoboundaryLiebialgebras([EH])inthecaseX =G(whereGisthe
1. simplyconnectedLiegroupcorrespondingtog).Wededuceourresultfromageneralized
0 formalitytheorem.InthecaseZ=0,wegetanewproofoftheexistenceof(equivariant)
8 formality theoremand so (equivariant)quantizationof Poisson manifold(cf. [Ko, Do]).
0
As a consequenceof ourresults, we get quantizationof modifiedclassical dynamicalr-
:
v matricesoverabelianbasesinthereductivecase.
i
X
r 0. Introduction
a
Throughoutthis paper, the groundfield will be R. Letg be a finite dimensionalLie
algebra with a fixed element r ∈L 3(g) such that [r,r]=Z ∈L 3(g)g. In [AK, AKM],
quasi-PoissonmanifoldswereintroducedasageneralizationofPoissong-manifoldswith
PoissonbracketsatisfyingtheJacobiidentityuptoaninvarianttrivectorcorrespondingto
Z. Moreprecisely:
Definition0.1. Aquasi-Poisson(g,Z)-manifoldisag-manifoldXwithaninvariantbivec-
torp suchthattheSchoutenbracket[p ,p ] equalsg ⊗3(Z),whereg : g→Vect(X)isthe
S
actionhomomorphism.
TheSchoutenbracketwillbedescibedlater. ThusthePoissonbracket{−,−}associated
top satisfies
{{f,g},h}+{{g,h},f}+{{h,f},g}=m (g ⊗3(Z)(f⊗g⊗h)),
0
1
wherem istheusualmultiplication. Intheframeworkofdeformationquantization(see
0
[BFFLS1,BFFLS2]),EnriquezandEtingofdefinedthequantizationofquasi-Poissonma-
nifoldsin[EE1]: leth¯ beaformalparameterandF =1+h¯2Z+O(h¯3)∈(U(g)⊗3)g[[h¯]]
6
beanassociatorforg(Drinfeldprovedin[Dr], Proposition3.10,thatsuchanassociator
alwaysexists).
Definition0.2. AquantizationofX associatedtoF isaninvariantstar-product⋆onX,i.e.
¥
aninvariantbidifferentialoperatoronC (X), whichsatisfies f ⋆g= fg+O(h¯)andthe
equation
f⋆g−g⋆f =h¯{f,g}+O(h¯2),
andisassociativeinthetensorcategoryof(U(g)[[h¯]],F )-modules. Thismeans,
m (m ⊗1)=m (1⊗m )g ⊗3(F ),
⋆ ⋆ ⋆ ⋆
onC¥ (X)⊗3,wherem (f⊗g)= f⋆g.
⋆
They also conjectured that such quantizations always exist when the action of g on the
quasi-Poisson manifold X is free. Note that when the action is not free, Fronsdal([Fr])
gavein 1978a counter-examplewheresuchquantizationis impossibleeveninthe sym-
plectic case. From now on, we will suppose that the manifold X is a G-bundle over a
manifoldM, whereGisthesimplyconnectedLiegroupcorrespondingtog. Inthecase
G={e},theconjectureisequivalenttotheexistenceofstar-productsandwasprovedby
Kontsevich([Ko]).InthecaseZ=0,theconjecturefollowsfromtheequivariantformality
theoremofDolgushev([Do]).
Inthegeneralcase,g ⊗3(Z)commuteswithalltheleftinvariantpolyvectorfieldsinthe
followingsense:
[g ⊗3(Z),X] =0, forallinvariantpolyvectorfieldsX. (0.1)
S
Moreover,forF anassociator,g ⊗3(F )commuteswithalltheinvariantdifferentialopera-
torsinthefollowingsense:
[g ⊗3(F ),C] =0, forallinvariantdifferentialoperatorC (0.2)
G
(theGerstenhaberbracket[−,−] willbedescribedlaterinthispaper). Fromnowon,if
G
s∈L k(g),wewilldenotesinsteadofg ⊗k(s)whennoconfusionispossible.
Inthispaper,weprovethatthereexists(aleast)oneassociatorforgsuchthatEnriquez-
Etingof’sconjectureistrue:
Theorem 0.3. Let r∈L 3(g) such that [r,r]=Z ∈L 3(g)g. There exists F =1+h¯2Z+
6
O(h¯3)∈(U(g)⊗3)g[[h¯]] and a deformation g of the Lie algebra g such that for every
h¯
invariantbivectorp satisfying[p ,p ] =g ⊗3(Z),thequasi-Poissonmanifold(X,p )admits
S
aquantizationassociatedto(F ,g )i.e.amultiplicationassociativeinthetensorcategory
h¯
of(U(g )[[h¯]],F )-modules.
h¯
Toprovethistheorem,wewillconstructaformalitybetweeninvariantpolyvectorand
polydifferentialoperatorasstated in Theorem7.3. We first provea localversionof this
theorem in the case X =Rn×g. Using Fedosov’s resolutions we will be able to get a
global version. We then get the wanted invariant star-product on the manifold X and
classificationofsuchdeformations.Wewillthendiscusstherelationwithquantizationof
modifiedclassicaldynamicalr-matrices.
2
Remark0.4. Asaparticularcase,ourresultsgiveanewproofofKontsevich(andDolgu-
shevforequivariant)formalitytheorem.OnecanseethisapproachasrelatedtoMerkulov’s
work(see[Me])forquantizationofLiebialgebras.Inourworktheuseofagradedversion
of Etingof-Kazhdantheoremwas acrucialstep to gofromquantizationof Liebialgebra
toquantizationofPoissonmanifolds.
Thepaperisorganizedasfollows:
-InSection1,werecalldefinitionsofL¥ -structuresandformalitymorphisms.
- In Section2, we givea gragedversionof quantizationof Lie bialgebras: in particular,
wegetdifferentialgradedEtingof-Kazhdanquantization/dequantizationfunctors.
- In Sections 3 and 4, we construct two useful functors between Lie and Gerstenhaber
algebras“uptohomotopy”andprovetheexistenceoftworesolutionsforthosealgebras.
-InSection5, weprovetheexistenceofL¥ -morphismsbetweenDG Liebialgebrasand
theGerstenhaberalgebraoftheirEtingof-Kazhdanquantization
- In Section 6, we transpose the algebra structures into the category of (U(g)[[h¯]],F )-
modules. We define the graded Lie bialgebra g˜ =R⊕V[1]⊕V∗⊕g, the direct sum of
theEisenbergLiealgebraE=R⊕V[1]⊕V∗andtheLiebialgebra(g,[r,−])whichcorre-
spondslocallytothealgebraofinvariantpoly-vectors.Weprovetheexistenceofthelocal
wantedL¥ -morphism.
- In Section 7, we show that this L¥ -morphism can be globalized and prove our main
theorem.
-InSection8,wediscussrelationbetweenourquantizationandquantizationofmodified
classicaldynamicalr-matrices.
Notations
Weusethestandardnotationforthecoproduct-insertionmaps:wesaythatanorderedset
is a pair of a finite set S and a bijection {1,...,|S|}→S. For I ,...,I disjoint ordered
1 m
subsetsof{1,...,n},(U,D )aHopfalgebraanda∈U⊗m,wedefine
aI1,...,Im =s I1,...,Im◦(D (|I1|)⊗···⊗D (|Im|))(a),
withD (1)=id,D (2)=D ,D (n+1)=(id⊗n−1⊗D )◦D (n),ands I1,...,Im :U⊗(cid:229) i|Ii|→U⊗nisthe
morphismcorrespondingtothemap{1,...,(cid:229) |I|}→{1,...,n}taking(1,...,|I |)toI ,
i i 1 1
(|I |+1,...,|I |+|I |)toI ,etc. WhenU iscocommutative,thisdefinitiondependsonly
1 1 2 2
onthesetsunderlyingI ,...,I .
1 m
Untiltheendofthispaper,althoughwewilloftenomittomentionit,wewillalways
dealwithgradedstructures.
Acknowledgements
I would like to thank D. Calaque and B. Enriquez for discussions. Part of this work
was done during the program “Poisson Sigma Models, Lie Algebroids, Deformations,
andHigherAnalogues”attheErwinSchro¨dingerInternationalInstituteforMathematical
Physics.
3
1 L¥ -structures
1.1 Definitions
LetusrecalldefinitionsofL¥ -algebrasandL¥ -morphisms.LetAbeagradedvectorspace.
WedenoteT A=T (A[−1])thefreetensoralgebra(withoutunit)ofAwhich,equipped
+ +
with the coshuffle coproduct, is a bialgebra. We also denoteC(A)=S(A[−1]) the free
gradedcommutativealgebrageneratedbyA[−1],seenasaquotientofT A. Thecoshuffle
+
coproductisstillwelldefinedonC(A)whichbecomesacofreecocommutativecoalgebra
onA[−1]. WealsodenoteL A=S(A[1]),theanalogousgradedcommutativealgebragen-
eratedbyA[1](inparticular,forA ,A ∈A,A L A standsforthecorrespondingquotient
1 2 1 2
ofA [1]⊗A [1]inL A). We willusethenotationsTnA,L nAandCn(A)fortheelements
1 2 +
ofdegreen.
Definition 1.1. A vector space A is endowedwith a L¥ -algebra(Lie algebra“up to ho-
motopy”)structureif there aredegreeonelinearmapsd1,...,1: L kA→A[1]such thatthe
associatedcoderivations(extendedwithrespecttothecofreecocommutativestructureon
L A)d:L A→L A,satisfyd◦d=0wheredisthecoderivation
d=d1+d1,1+···+d1,...,1+···.
In particular,a differentialLie algebra(A,b,[−,−])is a L¥ -algebrawith structuremaps
d1 =b[1], d1,1=[−,−][1]andd1,...,1: L kA→A[1]are0fork≥3. Onecannowdefine
thegeneralizationofLiealgebramorphisms:
Definition1.2. AL¥ -morphismbetweentwoL¥ -algebras(A1,d1=d11+···)and(A2,d2=
d1+···)isamorphismofcodifferentialcofreecoalgebras,ofdegree0,
2
j : (L A ,d )→(L A ,d ).
1 1 2 2
Inparticularj ◦d =d ◦j . Asj isamorphismofcofreecocommutativecoalgebras,j
1 2
isdeterminedbyitsimageonthecogenerators,i.e.,byitscomponents: j 1,...,1:L kA →
1
A [1].
2
Let E be a graded vector space. Let us denote cT(E) the cofree tensor coalgebra
of E with coproductD ′. Equipped with the shuffle product• (defined on the cogenera-
torscT(E)⊗cT(E)→E aspr⊗e +e ⊗pr, wherepr: cT(E)→E isthe projectionand
e is the counit), it is a bialgebra. Let cT (E) be the augmentation ideal. We denote
+
cT(E)=cT (E)/(cT (E)•cT (E))thequotientbytheshuffles. Ithasagradedcofree
+ + +
Liecoalgebrastructure(withcoproductd =D ′−D ′op). ThenS(cT(E)[1])hasastructure
ofcofreecoGerstenhaberalgebra(i.e. equippedwithcofreecoLieandcofreecocommu-
tativecoproductssatisfyingcompatibilitycondition). WeusethenotationcTn(E)forthe
elementsofdegreen.
Remark1.3. OnecouldalsodefineG¥ -structures.MostoftheL¥ -morphismconstructed
in this paper are also G¥ -morphismsbetween correspondingG¥ -structures. Definitions
andextensionstoG¥ -structurescanbefoundin[Ha].
4
2 Etingof-Kazhdan functors
2.1 QUEand QFSHalgebras
We recall some facts from [Dr] (proofs and definitions can be found in [Gav]). Let us
denote by QUE the category of quantized universal enveloping (QUE) algebras and by
QFSH the categoryof quantizedformalseries Hopf(QFSH) algebras. Let us recallthe
definitionofFSHandQFSHalgebras:
Definition2.1. AFSHalgebraisaHopfalgebraofpowerseriesisomorphicasanalgebra
toK[[{u|i∈J}]](forsomesetJ).
i
There is an equivalence of categories between the category of FSH algebra and the
categoryofLiecoalgebra(LCalgebra),sendingO toh=O /O 2 whereO isthe
h h+ h+ h+
maximalidealofO .
h
Definition2.2. AQFSHalgebraisaHopfalgebraH,whichisatopologicallyfreeK[[h¯]]-
module,suchthatH :=H/h¯H isisomorphictoaFSHalgebra
0
LetusgiveanexampleofaFSHalgebra,veryimportantinthispaper:letV beavector
space andcTV (definedin the previoussection) thecofreecoalgebra,equippedwith the
shuffleproduct. LetusnowcompletecTV. ThealgebracTV isagradedalgebrawithV
beingthesetofelementsofdegree1. LetusdenoteMcTV thesetofelementsofdegree
≥1. Finally,wedenotecTV thecommutativecofreebialgebra,McTV-adiccompletionof
cTV.
c
Proposition2.3. [Ha]cTV is the FSHalgebraOcTV associatedwith the Lie coalgebra
cTV =cT V/(cT V)2,whichisthecofreeLiecoalgebraoverV.
+ +
c
WehavecovariantfunctorsQUE→QFSH,U 7→U′ andQFSH→QUE,O 7→O∨.
Thesefunctorsarealsoinversetoeachother.
U′ isasubalgebraofU definedasfollows: foranyorderedsubset S ={i ,...,i }⊆
1 k
{1,...,n} with i1<···<ik, definethemorphism jS :U⊗k−→U⊗n by jS (a1⊗···⊗
ak):=b1⊗···⊗bn with bi:=1 if i∈/S and bim :=am for 1≤m≤k; thenset D S :=
jS ◦D (k), D 0/ :=D (0),and
d S := (cid:229) (−1)n−|S ′|D S ′ , d 0/ :=e .
S ′⊂S
Weshallalsousethenotation d (n):=d , d (0):=d ,andtheusefulformula
{1,2,...,n} 0/
d (n)=(id −e )⊗n◦D (n).
U
Finally,wedefine
U′:= a∈U d (n)(a)∈hnU⊗n (⊆U)
(cid:8) (cid:12) (cid:9)
andendowitwiththeinducedtopolog(cid:12)y.
Ontheotherway,O∨istheh¯-adiccompletionof(cid:229) h¯−kMk⊂O[1/h¯](hereM ⊂O
k≥0
isthemaximalideal).
5
2.2 The functor DQ
In[GH],ageneralizationofEtingof-Kazhdantheorem([EK])wasprovedinanappendix
byEnriquezandEtingof:
Theorem2.4. Wehaveanequivalenceofcategories
DQF : DGQUE→DGLBAh
from the category of differential graded quantized universal enveloping super-algebras
to that of differential graded Lie super-bialgebras such that if U ∈ Ob(DGQUE) and
a =DQ(U), then U/hU =U(a/ha), where U is the universal algebra functor, taking
adifferentialgradedLiesuper-algebratoadifferentialgradedsuper-Hopfalgebra.
HereF isaDrinfeldassociator. WewilluseanyofthesefunctorsanddenoteitDQ.
3 Two functors
3.1 Functor L-G
Let(h,d ,d)beadifferentialLiebialgebra. LetC(h)=S(h[−1])bethefreegradedcom-
mutative algebra generated by h. Recall from the previous subsection thatC(h) is also
a cofree coalgebra and that coderivationsC(h)→C(h) are defined by their images in
h. Thus, oneeasily checksthatthe coderivation[−,−]: C(h)→C(h) extendingtheLie
bracket(withdegreeshiftedby one)definesa Lie (evenGerstenhaber)algebrastructure
onC(h). Moreover, one can extend maps d: h→h and d : h→S2(h[−1]) on the free
commutative algebraC(h) so that (C(h),[−,−],∧,d+d ) is a differential Gerstenhaber
algebra. The differential d is actually the Chevalley Eilenberg differential: the space
C(h)=S∗(h[−1])isisomorphictothestandardcomplex(L ∗(h))[−∗]andd issimplythe
differentialgivenbytheunderlyingLiecoalgebrastructureofh.
Proposition3.1. [Ha] AnyDGLA morphism f :h →h canbeextendedinto aDGLA
1 2
(andevendifferentialgradedGerstenhaber)morphismC(f):C(h )→C(h )offreecom-
1 2
mutative algebras. This definesan exactfunctorL-G from differentialLie bialgebrasto
differentialGerstenhaberalgebraswhichsendshtoC(h). Quasi-isomorphisms(h ,d )→
1 1
(h ,d )induceaquasi-isomorphisms(C(h ),d ,d )→(C(h ),d ,d ).
2 2 1 1 1 2 2 2
3.2 Functor L-G¥
ConsidernowthecategoryCFDLBofdifferentialLiebialgebraswhicharecofreeasaLie
coalgebra. In other words we are interested in cofree Lie coalgebra cT(E) on a graded
vector space E together with a differential ℓ and a cobracket L on cT(E) that makes it
a differential Lie bialgebra. As cT(E) is cofree, the differentialis uniquely determined
byitsrestrictionto cogeneratorslp: cTp(E)→E. Similarly,theLie bracketisuniquely
determinedbymapsLp1,p2: cTp1(E)L cTp2(E)→E.
Proposition3.2. [Ha]RestrictionmapcTp(E)→E definesanexactfunctorL-G¥ from
CFDLBtothecategoryofG¥ (andsoLie)-algebras.
Until the end of the paper, we will use the notationsTE for T(E[−1]) and cTE for
T(E[1]).
6
4 Two resolutions
4.1 bialgebra structure oncTT U
+
Here,wewilldefineabialgebrastructureoncTT U.Onecanconstructabialgebrastruc-
+
tureonthespaceofHochschildcochainsofanalgebrausingthebraceoperations. Inour
case,wewillfirstlygeneralizethedefinitionofbraceoperationsforageneralHopfalge-
bra. Moreprecisely,let(H,D ,×)beaHopfalgebra(inourcaseH willbetheEtingof-
h¯
KazhdanquantizationU (a)oftheLiebialgebraa). We willdefineabracestructureon
h¯
thecofreetensorcoalgebracTT H ofthefreetensoralgebraT(H[−1])withoutunit. To
+
distinguish the two tensor products, we denote ⊗ the tensor producton T H and ⊠ the
+
tensorproductoncTT H.
+
Definition4.1. We define braceoperationsoncTT H by extendingthe followingmaps
+
givenonthecogeneratorsofthecofreecoalgebracTT H:
+
1. B0=0,
2. B1=bcH (thecoHochschildcoboundaryonT+H),
3. B2 : a ⊠b 7→a ⊗b ,
4. Bn=0forn>2,
5. B0,1=B1,0=id,
6. B0,n=Bn,0=0forn≥1,
7. B1,n :(a ,b ⊠···⊠b )7→
1 n
(cid:229) (−1)ea 1,...,i1+1···i1+k1,...,im+1···im+km,...,n×
0≤i1,...,im+km≤n
il+kl≤il+1
1⊗i1⊗b ⊗1⊗i2−(i1+k1)⊗b ⊗···⊗b ⊗1⊗n−(im+km),
1 2 n
wherek =|b |,n=|a |+(cid:229) k −mande =(cid:229) (k −1)i ,
s s s s s s s
8. Bk,l =0fork>1.
Operations(2),(3)and(4)defineadifferentialdand(5),(6),(7)and(8)defineaproduct
⋆deformingtheshuffleproduct.
Notethat,whenH=U(a),theenvelopingalgebraofaLiealgebraa,T(H[−1])canbe
seenasthespaceofinvariantpolydifferentialoperatorsovertheLiegroupcorresponding
toaandinthatcase,ourdefinitioncoincideswithusualbracesoperations.
Wehave:
Theorem4.2. [Ha]ThebraceoperationsofDefinition4.1defineadifferentialbialgebra
structure on the cofree tensor coalgebra cTT+H, with product ⋆ extending (cid:229) Bp1,p2 and
differentialdextending(cid:229) Bp.
Let us now complete cTT H as in section 2 with V =T H. We get a commuta-
+ +
tive cofree bialgebra cTT+H, the McTT+H-adic completion of cTT+H (where McTT+H
is the maximal ideal of cTT H). Let us consider the free K[[n ]]-module cTT H[[n ]].
c + +
One can now replace the operationsBp,q of Definition 4.1 with K[[n ]]-linear operations
c
n p+q−1Bp,q. Those operations are well defined on the completion cTT H[[n ]] as this
+
spaceiscompleteforthegradinginducedbythedegreeincTT H=cTV plustheh¯-adic
+
c
7
valuation and because the operationswe just defined are homogeneousfor this grading.
Thuswegetamorphismofdifferentialbialgebra
In : (cTT+H,⋆,D ,d)→(cTT+H[[n ]][n −1],⋆n ,D n ,dn )
x7→n −|x|x, (4.3)
where |x| is the degree in cT. The morphism In extends to In : (cTT+H[[n ]],⋆,D ,d)→
(cTT+H[[n ]][n −1],⋆n ,D n ,dn )whichrestrictsto
In′ : (⊕n ncTnT+H[[n ]],⋆,D ,d)→(cTT+H[[n ]][n −1],⋆n ,D n ,dn ) (4.4)
Wehave: b c
Proposition4.3. [Ha]Thealgebra(cTT+H[[n ]],⋆n ,D n ,dn )isaQFSHA.Theunderlying
differentialLiebialgebrastructureoncTT H isgivenbytheGerstenhaberbracket
+
c
[a ,b ] =B1,1(a ,b )−(−1)(|a |−1)(|b |−1)B1,1(b ,a )
G
andcoHochschilddifferential
bcH(a )=[1⊗1,a ]G,
fora ,b ∈TH andthennaturallyextendedoncTT H usingthecofreeLiecobracket.
+
Remark4.4. LetnowHbetheQUEalgebraU=U (a).WehaveprovedthatT U canbe
h¯ +
equippedwithaG¥ -structure.SincethecofreeLiecoalgebrasarerigid,thedifferentialLie
bialgebracorrespondingto cTT U[[n ]]throughEtingof-Kazhdandequantizationfunctor
+
DQisisomorphictocTT U[[n ]]asaK[[n ]]-Liecoalgebra,andisthereforefree.
+
c
4.2 Abialgebra quasi-isomorphismj : U →(cTT U)∨
alg +
c
Wehave:
Proposition4.5. LetU beaQUEalgebra.Onecandefineabialgebraquasi-isomorphism
f : U →cTT U from the bialgebra (U,D ,×) to the bialgebra (cTT U,D ,⋆) whose
alg + h¯ +
structurewasdescribedintheprevioussection.
c
LetU′⊂U (seesection2).
Proposition4.6. [Ha]Wehaveabialgebraquasi-isomorphismj :(U′,×)→(cTT U,⋆ )
alg + h¯
ofQFSHalgebra,where(cTT+U,⋆h¯)is(cTT+U[[n ]],⋆n )/(n =h¯)(cTT+U[[n ]]iscthefree
K[[h¯]]-moduledefinedintheprevioussection:wedtheoperationsBp,qinton p+q−1Bp,q).
c c c
Finally,applyingtoj thederivedDrinfeldfunctor(−)∨,wegetabialgebraquasi-
alg
isomorphismj :U →(cTT U)∨.
alg +
c
8
4.3 ALie bialgebra quasi-isomorphismj : cTA→cTC(cTA)
Lie
LetA be a vectorspace. SupposenowthatthecofreeLie coalgebracTA hasa structure
(cTA,d ,[−,−],d)ofa differentialLie bialgebra. Using thefunctorL-G (see section3),
onegetsadifferentialGerstenhaberalgebra(C(cTA),[−,−],∧,d+d ).Onecanextendthe
structure maps on the cofree Lie coalgebracTC(cTA) and one gets a differentialcofree
Liebialgebra(cTC(cTA),d ′,[−,−],d+d +∧)(wewillsetd1=d+d andd2=∧).
Proposition4.7. [Ha]Letj bethecompositionmapj =cTi◦d¯ ofamap
Lie Lie
d¯ :cTA →cT(cTA),
x 7→x+(cid:229) d¯ (x),
k≥2 k
whered¯ isbuiltusingiteratesofd ,withcTi:cT(cTA)→cTC(cTA)whichiscT ofthein-
k
clusioni:cTA[−1]→C(cTA).Thenj isadifferentialLiebialgebraquasi-isomorphism
Lie
j : cTA→cTC(cTA).
Lie
5 L¥ -morphism for Lie bialgebras
5.1 ALie bialgebra quasi-isomorphismj ′ : a→cTT U
Lie +
Let(a,d )beagradedLiebialgebra. Wewrited =h¯d +h¯2d +···. Let(U (a),D )be
h¯ h¯ 1 2 h¯ h¯
the Etingof-Kazhdancanonicalquantization of (a,d ). We denoteU =U (a) for short.
h¯ h¯
Insection4, we provedtheexistenceofa bialgebrastructureoncTT U andabialgebra
+
quasi-isomorphism j : U →(cTT U)∨. Thanks to Etingof-Kazhdan dequantization
alg +
functor(see section2), andthe factthat(cTT U)∨ is aQUE algebraquantizingcTT U
+ +
(seesection4),wegetaLiebialgebraquasi-isomorphismj ′ : a→cTT U.
Lie +
5.2 Inversionofformalitymorphisms
LetusrecallTheorem4.4ofKontsevich([Ko]):
Theorem5.1. Letg1andg2betwoL¥ -algebrasandF beaL¥ -morphismfromg1tog2.
AssumethatF isaquasi-isomorphism. ThenthereexistsanL¥ -morphismfromg2 tog1
inducingtheinverseisomorphismbetweenassociatedcohomologyofcomplexes.
Remark5.2. WeknowtheexistenceofasimilarG¥ -versionofthistheorem. Thisresult
wouldimplytheexistenceofcorrespondingG¥ -morphisms.
5.3 L¥ -morphism forLie bialgebras
Let us summarize functorsand quasi-isomorphismsconstructedin the previoussections
inthefollowingdiagram:
9
cTC(cTT U) C(cTT U)[1] = C(cTT U)[1] cTT U (cTT U)∨
+ + + + +
j L-G¥ j j L-G j ′ DQ cj
↑ Lie −→ ↑ G¥ ↑ Ger¥ ←− ↑ Lie ←− ↑ alg
cTT U T U[1] C(a)[1] a U =U (a).
+ + h¯
Thus,thankstosection5.2,thecompositionj :C(a)→T U ofj withtheinverseof
+ Ger¥
j givesthewantedquasi-isomorphism.
G¥
Theorem 5.3. [Ha] the map j C(a)→T+U is a L¥ -quasi-isomorphismthat maps v∈
C(a)toAlt(v)∈T Umodh¯.
+
5.4 L¥ -morphism forX =Rn×g
We will now consider X =Rn×g and r∈g∧g such that [r,r]=Z. So (g,[r,−]) is a
Liebialgebra. LetussetV =R. FromnowonwewillconsiderthegradedLiebialgebra
g˜=R⊕V[1]⊕V∗⊕g,thedirectsumoftheEisenbergLiealgebraE=R⊕V[1]⊕V∗and
theLiebialgebra(g,[r,−]). Wewillnowdeduceourmainresultfrom:
Proposition 5.4. There exists a L¥ -quasi-isomorphism j h¯ between (C(g˜),[−,−],[r,−])
and(T U˜,[−,−] ,[1⊗1,−] ). HereU˜ istheEtingof-Kazhdanquantizationofg˜ andso
+ h¯ h¯
U˜ =U(E)⊗U (g) whereU (g) is the Etingof-Kazhdan quantization of (g,[r,−]). The
h¯ h¯
bracket[−,−] denotestheGerstenhaberbracketconstructedinSection4corresponding
h¯
tothecoproductofU˜.
6 Deformed structures and local L¥ -morphism
6.1 Deformedstructures
SupposewearegivenF ∈(U((g)⊗3)g[[h¯]]anassociator. Inparticular,g ⊗3(F )commutes
withalltheinvariantdifferentialoperator.Wehaveinfact:
[C,g ⊗3(F )] =0forallC∈U(g˜)[[h¯]]).
G
Fromnowon,wewillconsiderthetensorcategoryof(U(g)[[h¯]],F )-modules(inwhichwe
wanttoconstructanassociativestar-product).Letusdefinethe“deformed”Gerstenhaber
bracketastheBracketdefinedinSection4butinthenewtensorcategory.WegetanewLie
algebrastructureonU(g˜)[[h¯]]givenbythebracket[−,−]F defined,forD,E ∈U(g˜)[[h¯]],
by
[D,E]F ={D|E}F −(−1)|E||D|{E|D}F ,
whereforD∈U(g˜)⊗d andE∈U(g˜)⊗e,
{D|E}=(cid:229) (−1)(e−1)·iF˜D1,...,i,i+1···i+e,i+e+1,···Ei+1,...,i+e.
i≥0
F˜ correspondstotheobviouschangeofparenthesisinthetensorcategoryof(U(g)[[h¯]],F )-
modules.Forexample,ifAandBaretwo2-cochainsinU(g˜)[[h¯]],onehas
{A,B}F =A12,3B1,2−F −1A1,23B2,3.
10