Une algèbre de Hopf associée à une paire de Lie

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Differential geometry

A Hopf algebra associated with a Lie pair ^✩

Une algèbre de Hopf associée à une paire de Lie

Zhuo Chen

^a

, Mathieu Stiénon

^b

, Ping Xu

^b

aDepartmentofMathematics,TsinghuaUniversity,China bDepartmentofMathematics,PennStateUniversity,UnitedStates

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received6August2013

Acceptedafterrevision16September2014 Availableonline3October2014 PresentedbyCharles-MichelMarle IntributetoAlanWeinsteinontheoccasion ofhisseventiethbirthday

The quotient L/A[−¹] ^{ofapair A}→^{L ofLie algebroids is aLie algebra object inthe} derived category D^b(A) ofthe category A ^ofleftU(A)-modules,theAtiyah class

α

L/A beingitsLiebracket.Inthisnote,wedescribetheuniversalenvelopingalgebraoftheLie algebraobjectL/A[−¹]^{andweprovethatitisaHopfalgebraobjectinDb}(A).

©^{2014Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

r é s um é

Le quotient L/A[−¹] ^{d’unepaire A}→^L d’algébroïdes de Lie est un objet algèbre de Lie danslacatégorie dérivée D^b(A)de lacatégorieA ^{desmodulesàgauchesur}U(A). Danscettenote,nousdécrivonsl’algèbreenveloppanteuniverselledel’objetalgèbredeLie L/A[−¹]^{etnousprouvonsquecelle-ciconstitueunobjetalgèbredeHopfdansDb}(A)^.

©^{2014Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

Versionfrançaiseabrégée

SoitLunealgébroïdedeLiesurunevariétédifférentiableMet Aunesous-algébroïdede L–nousdironsque

(

L

,

A

)

est unepairedeLie.Laclassed’Atiyah

α

E d’un A-moduleE relativeàlapaire

(

L

,

A

)

,quifutintroduitepar[1],estl’obstruction à l’existence d’une L-connexion A-compatible sur E. Elle unifie les classes définies par Atiyah pour les fibrés vectoriels holomorphesetparMolinopourlesvariétésfeuilletées.

SoitA ^lacatégorie(abélienne)des A-modules,c’est-à-direlesmodules surl’algèbreenveloppanteuniverselleU

(

A

)

de l’algébroïdedeLie A.LequotientL

/

A d’unepairedeLie

(

L

,

A

)

–plusprécisément,sonespacedesectionsC^∞ –constitue un A-module[1].Saclassed’Atiyah

α

L/Arelativeàlapaire

(

L

,

A

)

détermineunmorphisme L

/

A

[−

¹

] ⊗

^L

/

A

[−

¹

] →

^L

/

A

[−

¹

]

delacatégoriedérivéeD^b

(A)

^{,faisantde L}

/

A

[−

¹

]

^{unobjetalgèbredeLiedansDb}

(A)

^.

Danscettenote, nousconstruisonsl’algèbreuniverselle enveloppantedel’objet algèbrede LieL

/

A

[−

¹

]

^{dans Db}

(

A

)

et nousprouvonsquecelle-ciestunobjetalgèbredeHopfdansD^b

(A)

.Pourcefaire,nousétablissonsuneapplicationdetype Hochschild–Kostant–Rosenberget,reprenantlesidéesdeRamadoss [6],remplaçonsL

/

A

[−

¹

]

^{parlecomplexeL}

(D

_poly¹

)

,qui est,quantàlui,quasi-isomorphe.

✩ ResearchpartiallysupportedbyNSFgrantDMS1101827,NSAgrantH98230-12-1-0234,andNSFCgrants11001146and11471179.

E-mailaddresses:zchen@math.tsinghua.edu.cn(Z. Chen),stienon@psu.edu(M. Stiénon),ping@math.psu.edu(P. Xu).

http://dx.doi.org/10.1016/j.crma.2014.09.010

1631-073X/©^{2014Académiedessciences.PublishedbyElsevierMassonSAS.}All rights reserved.

Lecas particulierde TX

[−

¹

] ∈

^Db

(

X

)

correspondantà lapairedeLie

(

TX

⊗

C

,

T⁰_X^,1

)

associéeàunevariétécomplexe X joue un rôleimportant dansla théoriedesinvariantsde Rozansky–Witten desvariétés de dimension 3 etenthéoriede l’indice.Desapplicationsdurésultatci-dessusdanscesdomainesserontdéveloppéesailleurs.

L’algèbreenveloppanteuniverselleU

(

L

)

d’unealgébroïdedeLieréelleLestunbimodulesur R

:=

^C∞

(

M

)

,quis’identifie à l’algèbredes opérateurs différentielssur le groupoïde de Lie local L intégrant L tangents aux s-fibres etinvariants à gauche.Deplus, U(^L

)

admetunecomultiplicationcocommutativeetcoassociative

Δ :

U(^L

) →

U(^L

) ˜⊗

U(^L

)

.Ici,

˜⊗

^désigne leproduittensorieldemodulesàgauchesurR.Enfin,U

(

L

)

estunL-module,puisquetoutélémentl de

Γ (

L

)

agitsurU

(

L

)

parmultiplicationàgauche :

∇

lu

=

^l

·

^u,pourtoutu

∈

U

(

L

)

.

ÉtantdonnéunepairedeLie

(

L

,

A

)

,considéronslequotientD¹_poly

=

_U(^U_{L)Γ (}^(L)_A) ^deU(L

)

parsonidéalà gaucheengendré par

Γ (

A

)

.On vérifie aisément que la comultiplicationde U(^L

)

induit une comultiplication

Δ :

D_poly¹

→

D_poly¹

˜⊗D

¹poly sur D_poly^{1 etque l’actionde L sur} U(^L

)

détermineuneactionde A sur D¹_poly^.Enfait,D¹_poly ^{estunecoalgèbre}cocommutative coassociativesur R,dontlacomultiplicationestcompatibleaveclaA-action.

Soit Ch^b

(A)

la catégorie des complexes de A bornés, et D^b

(A)

la catégorie dérivée de A. Notons Dⁿ_poly la n-ième puissance tensorielleD¹_poly

˜⊗··· ˜⊗D

¹poly de D_poly^{1 et,pourn}

=

^0,posonsD⁰_poly

=

^R.L’opérateurd

:

D^•_poly

→

D_poly^{•+1 définipar} l’équation(1)faitdeD_poly^•

=

_∞

n=⁰Dⁿ_poly ^unobjetdeChb

(A)

^.L’applicationdeHochschild–Kostant–Rosenberg HKR

: Γ

S^•

L

/

A

[−

1

]

→

D_poly^•

estobtenueàpartirdel’inclusionnaturelle

Γ (

L

/

A

) ⊂

D¹_poly^parsymétrisation(voirl’équation(2)).

Proposition0.1. L’applicationHKR

: (

S^•

(

L

/

A

[−

¹

] ),

0

) → (

D^•_poly

,

d

)

est,dansCh^b

(

A

)

,unquasi-isomorphisme.

Théorème0.2.

(i) L’objetD^•_poly^{estunealgèbredeHopfdansDb}

(

A

)

.

(ii) L’algèbreassociative

(

D^•_poly

, ˜⊗ )

estl’algèbreenveloppanteuniverselledeL

/

A

[−

¹

]

^dansDb

(

A

)

. 1. Introduction

Let A be a Lie algebroid over a manifold M. Its space of smooth sections

Γ (

A

)

is a Lie–Rinehart algebra over the commutativering R

=

^C∞

(

M

)

.Byan A-module,wemeanamoduleovertheLie–RinehartalgebracorrespondingtotheLie algebroid A,i.e.amoduleovertheassociativealgebraU

(

A

)

.

Recall that the universal envelopingalgebra U(A

)

of a Lie algebroid A over M is simultaneously an associativealge- bra andan R-bimodule. IncasetheLie algebroid A isreal, U(^A

)

is canonicallyidentified to thealgebra ofleft-invariant s-fiberwisedifferentialoperatorsonthelocalLiegroupoidA integratingA.Letusrecallitsconstruction.

Thevectorspaceg

=

^R

⊕ Γ (

A

)

admitsanaturalLiealgebrastructuregivenbytheLiebracket:

[

^f

+

^X

,

g

+

^Y

] = ρ (

X

)

g

− ρ (

Y

)

f

+ [

^X

,

Y

],

where f

,

g

∈

^{Rand X}

,

Y

∈ Γ (

A

)

.Here

ρ

denotestheanchormap.Leti denotethenaturalinclusionofgintoitsuniversal enveloping algebra U(g) ^andlet V(g) ^{denotethesubalgebra of}U(g)^{generated by i}

(g)

. Theuniversal enveloping algebra U(^A

)

oftheLiealgebroid A isthequotientofV(g)^{bythetwo-sidedidealgeneratedbytheelements oftheformi}

(

f

) ⊗

i

(

g

+

^Y

) −

ⁱ

(

f g

+

^{f Y}

)

with f, g

∈

^{R andY}

∈ Γ (

A

)

.

When A is a Lie algebra, U

(

A

)

is indeed the usual universal envelopingalgebra. On the other hand,when A is the tangentbundle T M,U(^A

)

isthealgebraofdifferentialoperatorsonM.

We usethesymbolA ^{todenotetheAbeliancategoryof A-modules.Abusing}terminology,wesaythatavectorbundle E over Misan A-moduleif

Γ (

E

) ∈

A^.

GivenaLiepair

(

L

,

A

)

ofalgebroids,i.e.aLiealgebroid LwithaLiesubalgebroidA,theAtiyahclass

α

E ofan A-module E relative tothepair

(

L

,

A

)

is definedasthe obstructionto theexistence ofan A-compatible L-connectiononthe vector bundle E.An L-connection

∇

^{onan A-module E issaid tobe} A-compatibleifit extendsthegiven flat A-connection on E andsatisfies

∇

a

∇

l

− ∇

l

∇

a

= ∇

[a,l] foralla

∈ Γ (

A

)

andl

∈ Γ (

L

)

.Thisfairlyrecentlydefinedclass (see[1]) hasasdouble origin,theAtiyahclassofholomorphicvectorbundlesandtheMolinoclassoffoliations.

The quotient L

/

A ofanyLiepair

(

L

,

A

)

is an A-module[1].Its Atiyah class

α

L/A can bedescribed asfollows.Choose an L-connection

∇

^{on L}

/

A extendingthe A-action.Its curvatureisthevectorbundlemap R^∇

: ∧

^2L

→

^End

(

E

)

definedby R^∇

(

l1

,

l2

) = ∇

l1

∇

l2

− ∇

l2

∇

l1

− ∇

[l1,l2],forall l1

,

l2

∈ Γ (

L

)

. Since L

/

A isan A-module, R^∇ vanishes on

∧

^{2A and, therefore,} determines asection R^∇_L/A of A^∗

⊗ (

L

/

A

)

^∗

⊗

^End

(

L

/

A

)

.Itwas proved in[1]that R^∇_L/A isa 1-cocyclefortheLiealgebroid A with values in the A-module

(

L

/

A

)

^∗

⊗

^End

(

L

/

A

)

and that its cohomology class

α

L/A

∈

^H1

(

A

; (

L

/

A

)

^∗

⊗

^End

(

L

/

A

))

is independentofthechoiceoftheconnection.

LetCh^b

(A)

denotethecategoryofboundedcomplexes inAandlet D^b

(A)

denotethecorresponding derivedcategory.

WewriteL

/

A

[−

¹

]

^{todenotethequotientL}

/

A regardedasacomplexinAconcentratedindegree1.

Thefollowingwasprovedin[1].

Proposition1.1.(See[1].)Let

(

L

,

A

)

beaLiealgebroidpair.TheAtiyahclass

α

L/AofthequotientL

/

A relativetothepair

(

L

,

A

)

deter- minesamorphismL

/

A

[−

¹

] ⊗

^L

/

A

[−

¹

] →

^L

/

A

[−

¹

]

^{inthederivedcategoryDb}

(A)

makingL

/

A

[−

¹

]

^{aLiealgebraobjectin Db}

(A)

.

ItiswellknownthateveryordinaryLiealgebragadmitsauniversalenvelopingalgebraU

(g)

,whichisaHopfalgebra.

Wearethus ledtothefollowingnaturalquestions:doesthereexistauniversalenvelopingalgebraforL

/

A

[−

¹

]

^inDb

(A)

and,ifso,isitaHopfalgebraobject?

InthisNote,wegiveapositiveanswertothequestionsabove.Foracomplexmanifold X,theAtiyahclassoftheLiepair

(

TX

⊗

C

,

T⁰_X^,1

)

issimply theusualAtiayhclassoftheholomorphictangentbundle TX recentlyexploitedby Kapranov[2].

It wasproved that theuniversal envelopingalgebraof theLiealgebra object T_X

[−

¹

]

^{in Db}

(

X

)

isthe Hochschildcochain complex

(D

^•_poly

(

X

),

d

)

[5–7]. This result played an important role in the study of several aspects of complex geometry includingtheRiemann–Rochtheorem[5],theCherncharacter[6]andtheRozansky–Witteninvariants[7,8].Applicationsof ourresultwillbedevelopedelsewhere.

2. Hochschild–Kostant–Rosenbergmap

It isknown [9]that theuniversal envelopingalgebra U(^L

)

of aLie algebroid L admitsacocommutative coassociative coproduct

Δ :

U

(

L

) →

U

(

L

) ˜⊗

U

(

L

)

,which is defined on generators as follows:

Δ(

f

) =

^f

˜⊗

¹

=

¹

˜⊗

^f,

∀

^f

∈

^{R and}

Δ(

l

) =

l

˜⊗

¹

+

¹

˜⊗

^l,

∀

^l

∈ Γ (

L

)

.Here, andin the sequel,

˜⊗

^{stands for thetensor product ofleft R-modules. Moreover,} U

(

L

)

is an L-modulesinceeachsectionl ofLactsonU(L

)

byleftmultiplication:

∇

lu

=

^l

·

^u,

∀

^u

∈

U(L

)

.

Now,givenaLiepair

(

L

,

A

)

,considerthequotientD_poly^{1 of}U(^L

)

bytheleftidealgeneratedby

Γ (

A

)

.Itisstraightforward toseethatthecomultiplicationonU

(

L

)

inducesacomultiplication

Δ :

D¹_poly

→

D_poly¹

˜⊗D

_poly^{1 on} D_poly^{1 andtheactionofL} onU

(

L

)

determinesanactionof AonD¹_poly^.

Lemma2.1.ThequotientD¹_poly

=

_U(^U_{L)Γ (}^(L)_A)^issimultaneouslyacocommutativecoassociativeR-coalgebraandanA-module.Moreover, itscomultiplicationiscompatiblewithitsA-action:

∇

X

(Δ

p

) = Δ(∇

Xp

), ∀

X

∈ Γ (

A

),

p

∈

D¹_poly

.

LetDⁿ_poly^{denotethen-thtensorialpower}D¹_poly

˜⊗··· ˜⊗D

¹polyofD¹_poly^and,forn

=

^0,setD⁰_poly

=

^{R.Wedefineacobound-} aryoperatord

:

D_poly^•

→

D_poly^{•+1 on}D_poly^•

=

_∞

n=⁰D_poly^{n by}

d

(

p1

˜⊗··· ˜⊗

^pn

) =

¹

˜⊗

^p1

˜⊗··· ˜⊗

^pn

− (Δ

p1

) ˜⊗··· ˜⊗

^pn

+

^p1

˜⊗ (Δ

p2

) ˜⊗··· ˜⊗

^pn

− · · ·

+ (−

¹

)

ⁿp₁

˜⊗··· ˜⊗

^pn−1

˜⊗(Δ

^pn

) + (−

¹

)

ⁿ⁺¹p₁

˜⊗··· ˜⊗

^pn

˜⊗

¹

,

(1) foranyp1

,

p2

, . . . ,

pn

∈

D_poly^{1 .Sincethe}comultiplication

Δ

iscompatiblewiththeactionofA,theoperatordisamorphism of A-modules.Moreover,

Δ

beingcoassociative,dsatisfiesd²

=

^0.Thus

(

D_poly^•

,

d

)

isanobjectinCh^b

(

A

)

.

Whenendowedwiththetrivialcoboundaryoperator,thespaceofsectionsof

S^•

L

/

A

[−

1

]

=

^∞

k=0

S^k

L

/

A

[−

1

]

=

^∞

k=0

∧

^kL

/

A

[−

k

]

isacomplexof A-modules:

0

→

^R

− →

⁰

Γ (

L

/

A

) − →

⁰

Γ

∧

²

(

L

/

A

)

₀

−

→ Γ

∧

³

(

L

/

A

)

₀

−

→ · · ·

Thenaturalinclusion

Γ (

L

/

A

) →

D_poly^{1 extendsnaturallytothe}Hochschild–Kostant–Rosenbergmap HKR

: Γ

S^•

L

/

A

[−

¹

]

→

D^•_poly byskew-symmetrization:

HKR

(

b₁

∧ · · · ∧

^bn

) =

¹ n

!

σ∈^Sn

sgn

( σ )

b_σ(1)

˜⊗

^bσ(2)

˜⊗··· ˜⊗

^bσ(n)

, ∀

^b1

, . . . ,

b_n

∈ Γ (

L

/

A

).

(2)

Proposition2.2.InCh^b

(A)

,HKRisaquasi-isomorphismfrom

(Γ (

S^•

(

L

/

A

[−

¹

] )),

0

)

to

(D

^•_poly

,

d

)

.

Sketchofproof Assuming L and A are real Lie algebroids,let L ^and A ^{be localLie groupoids} integrating L and A re- spectively. Thesourcemap s

:

L

→

^{M inducesasurjective submersion J}

:

L

/A →

^{M.The rightquotient}L

/A

^isaleft L-homogeneousspacewithmomentummap J [4].Therefore,itadmitsaninfinitesimal L-action,andhenceaninfinitesi- mal A-action.ThecoalgebraD_poly¹ mayberegardedasthespaceofdistributionsonthe J-fibersofL

/A

supportedonM.

Its A-modulestructurethenstemsfromtheinfinitesimal A-actiononL

/

A^{.Then-thtensorialpower}Dⁿ_poly^maybeviewed asthespaceofn-differentialoperatorsonthe J-fibersofL

/A

^{evaluatedalongMandthe}differentialdastheHochschild coboundary.TheconclusionfollowsfromtheclassicalHochschild–Kostant–Rosenbergtheorem.Toprovethepropositionfor complexLiealgebroids,itsufficestoconsiderformalgroupoidsinsteadoflocalLiegroupoids[3].

3. UniversalenvelopingalgebraofL/A[−¹]^inDb(A)

FollowingMarkarian[5],Ramadoss[6],andRoberts–Willerton[7],weintroducethefollowing:

Definition3.1.Ifitexists,theuniversalenvelopingalgebraofaLiealgebraobjectG^inDb

(

A

)

isanassociativealgebraobjectHⁱⁿ D^b

(A)

togetherwithamorphismofLiealgebrasi

:

G

→

HinD^b

(A)

satisfyingthefollowinguniversalproperty:givenanyassociative algebraobjectK^{andanymorphismofLiealgebras f}

:

G

→

K^{in Db}

(A)

^{,thereexistsauniquemorphismof}associativealgebras

f

:

H

→

K^inDb

(

A

)

suchthatf

=

^f

◦

^i.

Inview ofthesimilaritybetween

(

D_poly^•

,

d

)

andtheHochschildcochain complex,wedefine acupproduct

∪

^onD^•_poly bysetting P

∪

^Q

=

^P

˜⊗

^{Q,forall P}

,

Q

∈

D^•_poly^{.It issimpletocheckthat}

d

(

P

∪

^Q

) =

^{d P}

∪

^Q

+ ( −

¹

)

^|P|P

∪

^{d Q}

,

forallhomogeneous P

,

Q

∈

D_poly^{• .}

Proposition3.2.ForanyLiepair

(

L

,

A

)

ofalgebroids,

(D

^•_poly

,

d

, ∪ )

isanassociativealgebraobjectinD^b

(A)

,whichisinfactthe universalenvelopingalgebraoftheLiealgebraL

/

A

[−

¹

]

^inDb

(A)

.

Consider the inclusion

η _:

R

→

Dⁿ_poly, the projection

ε _:

Dⁿ_poly^{R, and the mapst}

:

D^•_poly

→

D^•_poly and

Δ ˜ :

D^•_poly

→

D_poly^•

⊗

RD^•_poly ^defined,respectively,by

t

(

p₁

˜⊗

^p2

˜⊗··· ˜⊗

^pn

) = (−

¹

)

^n(n2−1)p_n

˜⊗

^pn−1

˜⊗··· ˜⊗

^p1 and

Δ( ˜

p₁

˜⊗

^p2

˜⊗··· ˜⊗

^pn

) =

i+^j=ⁿ

σ∈S_i^j

sgn

( σ )(

p_σ(1)

˜⊗··· ˜⊗

^pσ(i)

) ⊗ (

p_σ(i+1)

˜⊗··· ˜⊗

^pσ(n)

),

whereS_i^jdenotesthesetof

(

i

,

j

)

-shuffles.¹

Theorem3.3.ForanyLiepair

(

L

,

A

)

ofalgebroids,

(D

_poly^•

,

d

)

withthemultiplication

∪

^,thecomultiplication

Δ ˜

,theunit

η

,thecounit

ε

, andtheantipode t,isaHopfalgebraobjectin D^b

(A)

^.

4. Ramadoss’sapproach:L(D¹_poly)

ToproveProposition 3.2andTheorem 3.3,weessentially followRamadoss’sapproach[6].Let L

(D

¹_poly

)

bethe(graded) freeLiealgebrageneratedover RbyD¹_polyconcentratedindegree1.Inotherwords,L

(D

¹_poly

)

isthesmallestLiesubalgebra ofD^•_poly^containing D_poly^{1 .TheLiebracketoftwovectorsu}

∈

Dⁱ_poly^andv

∈

D_poly^{j isthevector}

[

^u

,

v

] =

^u

˜⊗

^v

− (−

¹

)

^{i j}v

˜⊗

^u

∈

D_poly^{i+j. Actually, L}

(

D¹_poly

)

is made of all linear combinations of elements of the form

[

^p1

, [

^p2

, [· · · , [

^pn−¹

,

pn

] · · ·]]]

^with p1

, . . . ,

pn

∈

D¹_poly^{.Onechecksthat L}

(

D¹_poly

)

isad-stable A-submoduleofD^•_poly^{andthatitsLiebracketisachainmapwith} respecttothecoboundaryoperatord.Therefore

(

L

(

D¹_poly

),

d

)

isaLiealgebraobjectinCh^b

(

A

)

.

1 An(i,j)-shuffleisapermutationσoftheset{¹,2,. . . ,i+^j}^suchthatσ(1)≤σ(2)≤ · · · ≤σ(i)andσ(i+¹)≤σ(i+²)≤ · · · ≤σ(i+^j).

LetS^•

(

L

(D

¹_poly

))

bethesymmetricalgebraofL

(D

¹_poly

)

andlet I

:

^S•

(

L

(D

¹_poly

)) →

D_poly^• bethesymmetrizationmap:

I

(

z₁

· · ·

^zn

) =

¹ n

!

σ∈Sn

sgn

( σ _;

z₁

, . . . ,

z_n

)

z_σ(1)

˜⊗

^zσ(2)

˜⊗··· ˜⊗

^zσ(n)

.

TheKoszul signsgn

( σ ;

^z1

, . . . ,

z_n

)

ofa permutation

σ

ofthe(homogeneous)vectors z₁

,

z₂

, . . . ,

z_n

∈

^S•

(

L

(D

¹_poly

))

isdeter- minedbytherelationzσ(1)

^zσ(2)

· · ·

^zσ(n)

=

^sgn

( σ _;

z1

, . . . ,

zn

)

z1

^z2

· · ·

^zn.

Lemma4.1.ThesymmetrizationI

:

^S•

(

L

(

D¹_poly

)) →

D^•_poly^isanisomorphisminCh^b

(

A

)

.

Using Lemma 4.1andtheHKR quasi-isomorphism,onecan prove thatthe composition

β : Γ (

L

/

A

[−

¹

] ) →

^L

(D

_poly¹

)

of the inclusions

Γ (

L

/

A

[−

¹

] ) ⊂

D¹_poly

⊂

^L

(D

¹_poly

)

is a quasi-isomorphismin Ch^b

(A)

,which intertwines the Lie bracketson

Γ (

L

/

A

[−

¹

])

^andL

(D

_poly¹

)

.

Proposition4.2.

(i) Theinclusion

β : Γ (

L

/

A

[−

¹

]) →

^L

(D

¹_poly

)

isaquasi-isomorphisminCh^b

(A)

^.

(ii) Theinclusion

β : Γ (

L

/

A

[−

¹

]) →

^L

(D

¹_poly

)

isanisomorphismofLiealgebraobjectsinD^b

(A)

^asthediagram

Γ (

L

/

A

[−

¹

]) ˜⊗Γ (

^L

/

A

[−

¹

])

αL/A

β⊗β

L

(

D_poly¹

) ˜⊗

^L

(

D_poly¹

)

[,]

Γ (

L

/

A

[−

1

])

_β L

(

D_poly¹

)

commutesinD^b

(A)

^.

Proposition 3.2andTheorem 3.3nowfollowimmediately.

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