Koszul duality in deformation quantization and Tamarkins approach to Kontsevich formality

Koszul dualit y in deformati on quan tization and T amarkin’s approach to Kon tsevic h formalit y Boris Shoikhet Abstract Let α be a quadratic Poisson bivector on a vector space V . Then one can also consider α as a quadratic Poisson bivector on the vector space V ∗ [1]. Fixed a univer- sal deformation quant ization (prediction o f s ome complex weigh ts to all K ontsevic h graphs [K97]), we hav e defor mation quan tization of the b oth algebr as S ( V ∗ ) and Λ( V ). These are graded quadratic algebras, and therefor e K oszul algebras. W e prov e that for some universal de fo rmation qua nt ization, indep endent on α , these t wo a lgebras are Ko szul dual. W e characterize s o me deformation q ua ntizations for which this theorem is true in the framework o f the T amarkin’s theo ry [T1 ]. In tro duction This pap er is devo ted to the theorem that there exists a u niv ersal deformation quan tiza- tion compatible with the Koszul du alit y , as it is explained in the Abstract. Let us ﬁrstly form ulate it here in a m ore d etail, and then outline the main id eas of the p ro of. 0.1 Let V b e a ﬁnite-dimensional vec tor space o v er C . W e denote b y T poly ( V ) the graded Lie algebra of p olynomial p olyv ector ﬁelds on V , w ith the Sc houten-Nijenh uis brac k et. F or a Z -graded vec tor space W d enote by W [1] the graded space for w h ic h ( W [1 ]) i = W 1+ i , that is, the space ”shifted to the left” . The f ollo wing simple statemen t is very fundamental for this work: Lemma. Ther e is a c ano nic al isomorph ism of gr ade d Lie algebr as D : T poly ( V ) → T poly ( V ∗ [1]) . The m ap D maps a bi-homog eneous p olyvec tor ﬁeld γ on V , γ = x i 1 . . . x i k ∂ ∂ x j 1 ∧ · · · ∧ ∂ ∂ x j ℓ to the p olyv ector ﬁ eld D ( γ ) = ξ j 1 . . . ξ j ℓ ∂ ∂ ξ i 1 ∧ · · · ∧ ∂ ∂ ξ i k on the space V ∗ [1]. Here { x i } is a basis in V ∗ , an d { ξ i } is the dual basis in V [ − 1]. It is a goo d place to recall t he Hoc hsc hild-Kostant- Rosen b erg theorem b y whic h the cohomological Ho chsc hild complex of the algebra A = S ( V ∗ ) endow ed with the 1 Gerstenhab er b rac k et has cohomology isomorp hic to T poly ( V ) as graded Lie algebra. That is, the Gerstenh ab er brac k et b ecomes the Schouten-Nijenh uis brac k et on the lev el of cohomology . This theorem is rela ted to the lemma abov e (which is certainly clear just strai ght- forw ardly , without th is more sophisticated argument ), as follo ws. Consid er the algebras A = S ( V ∗ ) a nd B = Λ( V ) = F un ( V ∗ [1]). The alge bras A and B are Koszul dual (see, e.g. [PP]). Bernh ard Keller pro ve d in [Kel1] (see also the discuss ion b elo w in Sec- tions 1.4-1.6) that the cohomological Ho c hsc hild complexes Ho ch q ( A ) and Ho ch q ( B ) are quasi-isomorphic w ith all structures when A and B are quad r atic Koszul and Koszul dual algebras. (F or the Hochsc hild cohomology it w as kn o wn b efore, s ee the references in lo c.cit). In our case H H q ( A ) = T poly ( V ) and H H q ( B ) = T poly ( V ∗ [1]). 0.2 The isomorp h ism D from the lemma ab o v e do es not c hange the grading of the p olyv ector ﬁeld, but it maps i -p olyv ector ﬁelds with k -linear co eﬃcien ts to k -p olyvec tor ﬁ elds with i -linear co eﬃcients. In particular, it maps qu adratic bivec tor ﬁelds on V to quadratic biv ector ﬁelds on V ∗ [1]. Moreo v er, D maps a Poisson quadratic b iv ector on V to a P oisson quadr atic biv ector on V ∗ [1], b ecause it is a map of Lie algebras. In [K97], Maxim Kon tsevic h ga ve a form ula for deformation quantiza tion of algebra S ( V ∗ ) by a P oisson bivec tor α on V (the v ector spaced V should b e ﬁnite-dimensional). His formula is organized as a sum o ve r admissib le graphs, and eac h graph is tak en with the K on tsevic h w eigh t W Γ . In particular, this W Γ dep end s only on th e graph Γ and do es not dep end on dimension of the s p ace V . Consider no w all these complex n um b ers W Γ as un deﬁned v aria bles. Then the as- so ciativit y give s an inﬁ nite num b er of quadratic equations on W Γ . K on tsevic h’s pap er [K97] then sho ws that these equ ations hav e at least one complex solution. Actually there is a lot of essen tially diﬀeren t solutions, as is cle ar from [T] (see the discussion in Sec- tion 3 of this p ap er). Any suc h deformation quant ization is called u niversal b ecause the complex num b ers W Γ do not dep end on the v ector sp ace V . The case of a quad r atic Poi sson b ivecto r α is distinguished, by the follo wing lemma: Lemma. L et S ( V ∗ ) α b e a universal deformation quantizat ion of S ( V ∗ ) by a quadr atic Poisson bive ctor α . Then the algebr a S ( V ∗ ) α is gr a de d. This me ans that for f ∈ S i ( V ∗ )[[ ~ ]] and g ∈ S j ( V ∗ )[[ ~ ]] , the pr o duct f ⋆ g ∈ S i + j ( V ∗ )[[ ~ ]] . Pr o of. By th e Kontse vic h form ula, f ⋆ g = f · g + X k ≥ 1 ~ k X Γ ∈ G k, 2 W Γ B Γ ( f , g ) (1) where G k , 2 is the set of admissible graphs with t w o vertic es on the r eal line and k v ertices in the upp er half-plane (see [K97], Section 1 for details). No w eac h graph Γ from G k , 2 2 has k v ertices at the half-plane, and 2 k edges. One can compute the grading degree of B Γ ( f , g ) as follo ws. It is th e sum of degrees o f quanti ties asso ciated with all v ertice s (whic h is deg f + deg g + k deg α = deg f + d eg g + 2 k ) minus the num b er of edges (equal to 2 k by deﬁ n ition of an admissible graph ) b ecause eac h edge diﬀerent iate once, and then decreases the d egree by 1). Th is diﬀerence is equal to deg f + d eg g . In particular, for quadr atic deformation qu an tization the map x i · x j 7→ x i ⋆ x j giv es a C [[ ~ ]]-linear endomorphism of th e sp ace S 2 ( V )[[ ~ ]] whic h is clearly non-d egenerate. W e can ﬁnd an in v erse to it, then w e can presen t the star-algebra as the quotien t of the tensor algebra T ( V ∗ ) b y the set of quadr atic rela tions R ij ∈ V ∗ ⊗ V ∗ , one relation for eac h pair of ind ices 1 ≤ i < j ≤ d im V . W e conclude, that the Kontsevic h deformation quan tization of S ( V ∗ ) by a quadr atic P oisson bivecto r is a graded quadratic algebra. 0.3 W e actually get t wo quadr atic asso ciativ e alge bras for any u niv ersal deformation quan- tizatio n, one is th e deformation quan tizatio n of S ( V ∗ ) b y the quadratic Poisson b iv ec- tor α , and another one is the deformatio n quan tizati on of Λ( V ) = F un ( V ∗ [1]) by the quadratic P oisson biv ector D ( α ). Denote these t wo algebras by S ( V ∗ ) ⊗ C [[ ~ ]] α and Λ( V ) ⊗ C [[ ~ ]] D ( α ) . In th e present pap er we pr o v e th e f ollo wing resu lt: Theorem. Ther e exists a universal d eformation quantization such that the two alge- br a s S ( V ∗ ) ⊗ C [[ ~ ]] α and Λ( V ) ⊗ C [[ ~ ]] D ( α ) ar e Koszul dual as algebr as over C [[ ~ ]] . In p articular, Ext q S ( V ∗ ) ⊗ C [[ ~ ]] α − M od ( C [[ ~ ]] , C [[ ~ ]]) = Λ( V ) ⊗ C [[ ~ ]] D ( α ) (2) and Ext q Λ( V ) ⊗ C [[ ~ ]] D ( α ) − M od ( C [[ ~ ]] , C [[ ~ ]]) = S ( V ∗ ) ⊗ C [[ ~ ]] α (3) The T amarkin ’s deformation quantization deﬁne d fr om any Drinfeld’s asso ciat or (which is cle arly universal) satisﬁes the c ondition of The or em. See S ection 1 of this p ap er for an ov e rview of K oszul dualit y , and of Koszul dualit y o v er a discrete v aluation r ing. R e mark. T o c onsider S ( V ∗ ) and Λ( V ) as Koszul d u al algebras, th e Ext group s ab o v e should b e take n in the catego ry of Z -graded mo dules ov er Z -graded alg ebras. Without considering the Z -graded categ ory , the Koszul d ual to Λ( V ) is S [[ V ]]. It is ev erywhere implicitly assum ed that we w ork in the Z -graded category . 0.4 No w let us outline our strategy ho w to pro ve this theorem. W e ﬁ rstly ”lift the T h eorem” on the lev el of complexes. W e do it as follo ws. 3 Let A a nd B b e t w o associativ e a lgebras, and le t K b e a dg B − A -mod ule (this means that it is a left B -mo dule and left A -mo du le, and the left act ion commutes w ith the right action). Deﬁne then a diﬀeren tial graded category with 2 ob jects, a and b , as follo ws. W e set Mor( a, a ) = A , Mor( b, b ) = B , Mor( b, a ) = K , Mor ( a, b ) = 0. T o mak e this a d g catego ry the only what we need is that A and B are algebras, and K is a B − A -mo dule. D enote this category by cat ( A, B , K ), see Section 5 f or more detail. Consider t he H o chsc hild co homological complex Ho c h q ( cat ( A, B , K )) o f this dg catego ry . There are natural p ro jections p A : Ho ch q ( cat ( A, B , K )) → Ho c h q ( A ) and p B : Ho ch q ( cat ( A, B , K )) → Hoc h q ( B ). The B.Kelle r’s theorem [Kel1] giv es suﬃcient conditions for p A and p B b eing q u asi-isomorphisms. Th ese conditions are that the nat- ural maps B → RHom M od − A ( K, K ) (4) and A opp → RHom B − M od ( K, K ) (5) are quasi-isomorphisms. An easy computation sh o ws that in the case when A is Koszul algebra, B = A ! opp is the opp osite to the Koszul dual algebra, and K is the Koszul complex of A , the Keller’s condition is s atisﬁed (see Section 5). R e mark. According to the Remark at the end of Section 0.3, w e should work with the Z -graded category . Therefore, our Ho c hsc hild complexes should b e also compatible w ith this gradin g. More precisely , the Hoc hschild co c hains should b e ﬁn ite sums of graded co c hains. See S ection 4.2.1 where it is explicitly stated. Consider the case when A = S ( V ∗ ) ⊗ C [[ ~ ]] and B = Λ( V ) ⊗ C [[ ~ ]]. Denote in this case the catego ry cat ( A, B , K ) where K is the Koszul complex of A , j ust b y cat . Consider the follo wing solid arro w d iagram diagram: Ho c h q ( A ) T poly ( V ) U S 8 8 q q q q q q q q q q U Λ & & M M M M M M M M M M F / / Ho c h q ( cat ) p A g g N N N N N N N N N N N p B w w p p p p p p p p p p p Ho c h q ( B ) (6) The r igh t ”horn” w as just deﬁn ed. Th e maps U S and U Λ in th e left ”h orn ” are the f ollo wing. W e consider some universal L ∞ map U : T poly ( V ) → Ho ch q ( S ( V ∗ )). This means that we attribute some complex n umbers to eac h K on tsevic h graph in his formalit y morphism in [K97], bu t wh ich are not n ecessarily the Kontsevic h inte grals (bu t the ﬁr st T a ylor comp onen ts is ﬁxed, it is the Ho chsc hild-Kostan t-Rosen b erg map). The wo rd ”univ ersal” again means that these num b ers are the same f or all space s V . Then w e 4 apply this L ∞ morphism to our sp ace V , it is U S , and the comp osition of D : T poly ( V ) → T poly ( V ∗ [1]) with the L ∞ morphism U : T poly ( V ∗ [1]) → Hoch q (Λ( V )), constructed from the same u niv ersal L ∞ map. The all solid a rrows (6) are quasiisomorph ism s. Therefore, th ey are homotopically in v ertible (see Section 4), and w e can sp eak ab out the h omotopical comm utativit y of this diagram. Theorem. Ther e exists a universal L ∞ morphism U : T poly ( V ) → Hoc h q ( S ( V ∗ )) such that the solid arr ow diagr am (6) is homo topic a l ly c ommutative. The L ∞ morphism c or- r e sp onding by the T amarkin ’s the ory (se e Se ctions 2 and 3) to any choic e of the Drinfeld asso ciator satisﬁes this pr op erty. W e ﬁrstly explain why our th eorem ab out Koszul d u alit y follo ws from this theorem, and, secondly , h o w to prov e this new theorem. 0.5 If we kno w the h omotopical comm utativit y o f the solid arrow diagram (of quasi- isomorphismes) (6), w e can construct the dott ed arro w F whic h is a G ∞ quasi- isomorphism F : T poly ( V ) → Ho c h q ( cat ), wh ic h divides the d iagram into t w o homotopi- cally comm utativ e triangles. Then, if α is a qu ad r atic Poisson b iv ector ﬁeld on V , the L ∞ part of F deﬁnes a solution of the Maurer-Cartan equation F ∗ ( α ) in Ho c h q ( cat ). A solution of the Maurer-Cartan equ ation in Ho ch q ( cat ) deforms the follo wing four things sim ultaneously: the algebra structures on A and B , the d iﬀerential in K , and th e bi- mo dule structure. Using v ery simple argu m en ts we then can pro ve that this deform ed complex K is a free resolution of the deformed A , and the deformed b imo dule isomor- phisms (4)-(5) giv e the Koszul d ualit y theorem. See Section 7 for detail. 0.6 Here we outline the main id eas of Th eorem 0.4. First of all, the t wo maps p A and p B in the righ t ”horn” of the diagram (6) are m aps of B ∞ algebras (see [Kel1]). Here B ∞ is the braces dg op erad, wh ic h acts on the Hochsc hild cohomologica l complex of an y algebra or d g category . F ormally it is deﬁned as follo ws: a B ∞ mo dule structure on X is a dg bialgebra structure on th e cofree coalgebra cogenerated b y X [1] s uc h that th e coalgebra structure co incides with the giv en on e. The action of B ∞ on the Ho c hsc hild complex Ho c h q ( A ) of an y dg algebra (or d g category) A is constructed by Getzler-Jones [GJ] via the braces op erations. No w d eﬁne analogously the dg op erad B Lie . A ve ctor space Y is an algebra ov e r B Lie iﬀ there is a dg L ie bialgebra str ucture o n the free Lie coalgebra cogenerat ed by Y [1] suc h that the Lie coalgebra structure coincides with the giv en on e. The op erads B Lie and B ∞ are quasi-isomorphic b y the Etingof-Kazhdan (de)quantiza tion. The constru ction of 5 quasi-isomorphism of op erads B Lie → B ∞ dep end s on the c hoice of Drinfeld’s asso ciator [D]. The op erad B Lie is quasi-isomorphic to the Gerstenhab er h omotopical op erad G ∞ , as is explained in [H], S ection 6 (see also discussion in S ection 2 of this pap er). Finally , the Gerstenhab er op erad is Koszul, and G ∞ is its K oszul resolution. Thus, any B ∞ algebra can b e considered as a G ∞ algebra. As G ∞ is a resolution of the Gerstenhab er op erad G , all three dg op erads G ∞ , B ∞ , and B Lie , are quasi-isomorphic to th eir cohomology G . (There is no canonical morph ism from B Lie to B ∞ . Any suc h quasi-iso morph ism giv es a G ∞ structure on the Ho c hsc hild cohomologica l complex of an y dg category . An y Drinfeld asso ciator [D] giv es, via the Etingof-Kazhdan (de)quant ization, suc h a morph ism of op erads .) No w consider the en tire diagram (6) as a diagram o f G ∞ algebras and G ∞ maps, where the G ∞ action on the Ho c hsc hild complexes is as ab o v e, it d ep ends on the c hoice of a map B Lie → B ∞ . Then, if our diagram is homotopically not comm utativ e, it deﬁnes some G ∞ automorphism of T poly ( V ). This G ∞ automorphism is clearly Aﬀ ( V )-equiv ariant. First of all, w e pr o v e th at on the lev el of cohomology the diagram (6) is comm utativ e. This is in a sense the only n ew computation wh ic h we mak e in this pap er (see Section 5). Th us, we can take the logarithm of this automorphism, w hic h is G ∞ -deriv at ion. By the T amarkin’s G ∞ -ridigit y of T poly ( V ), an y Aﬀ -equiv a riant deriv ation is homotopically inner. But an y inner deriv ation acts n on-trivially on cohomology! O n th e other hand , a G ∞ -morphism homotopically equiv al ent to identi t y , acts b y the identit y on cohomology . This p ro v es that our diagram is homotopically comm utativ e. The only prop ert y of this diagram whic h w e ha v e used is that it is deﬁned ov er G ∞ and is co mmutati ve on the lev el of cohomology . 0.7 When th e author s tarted to atta c k this problem, he started to pro v e the homotopical comm utativit y of the diagram (6) b y ”physical” methods. Namely , the Kon tsevic h’s formalit y in the original pr o of giv en in [K97] is a p articular case of the AKS Z m o del on op en d isc [AKSZ], also called b y Cattaneo and F elder the ”Poisson sigma-mod el”. As usual in op en theories, w e should imp ose some b ou n dary conditions f or the restrictions of th e ﬁelds to the circle S 1 = ∂ D 2 . Maxim Kontsevi c h considers the b ound ary condition ” p = 0” on all arcs. Th is, certainly together w ith other mathematical insigh ts, led h im in [K97] to th e formalit y theorem. Our idea w as to divide S 1 b y t w o p arts, ﬁxing t w o p oin ts { 0 } and {∞} (in the Kon tsevic h’s case only {∞} is ﬁxed). T h en, we imp ose the b oundary cond ition ” x = 0” on all left arcs, and ” p = 0” on all r igh t arcs. Th is seems to b e v ery reasonable, and the author hop ed to construct in th is w a y an L ∞ quasi-isomorphism F (t he dotted arro w in (6)), making the t w o triangles homotopically commuta tiv e. 6 Someho w, he d id not succeed in that. F rom the p oint of view of this pap er, it seems that the r eason for that is the follo wing. The auth or wo rked with the Kont sevic h’s propagator in [K 97], namely , with ϕ ( z 1 , z 2 ) = 1 2 π i d Log ( z 1 − z 2 )( z 1 − z 2 ) ( z 1 − z 2 )( z 1 − z 2 ) (7) (here z 1 and z 2 are distinct p oin ts of the complex upp er half-plane). In this p ap er we deal with the T amarkin’s quan tization. Conjecturally (see [K99]) when this formalit y morphism is constructed from the Knizhnik-Zamolodchik o v Drin- feld’s asso ciator, it coi ncides (as a unive rsal L ∞ morphism, see ab o v e) with the L ∞ morphism, constructed from the ”another K ontsevic h’s propagator”. T his is ”the h alf” of (7): ϕ 1 ( z 1 , z 2 ) = 1 2 π i d Log z 1 − z 2 z 1 − z 2 (8) Kon tsevic h pro v ed (u npub lished) that this p ropagator also leads to an L ∞ morphism from T poly ( V ) to Hoc h q ( S ( V ∗ )). If this conject ur e (that the T amarkin’s t heory in the Knizhnik-Zamolo dchik o v case giv es this pr opagator) is tru e, w e should try to elab orate the physica l idea describ ed ab ov e (with the t w o b oundary conditions) for this propagator. The r eason is that it is n ot a pr iori clear that th e Kontsevi c h’s ﬁ r st propagator ϕ ( z 1 , z 2 ) comes fr om any Drinfeld’s asso ciator, and therefore from the T amarkin’s theory . W e are going to come bac k to these questions in the sequel. 0.8 W e tried to make the exp osition as self-con tained as p ossible. In particular, we pro ve in Section 2.4 th e main Lemma in the T amarkin’s p ro of [T1] of the Kon tsevic h form ality , b ecause we use it here not only for the ﬁrst cohomology as in [T1] and [H], and also for 0- th cohomology . W e giv e a simp le pr o of of it for all cohomology for completeness. As well, w e repr o duce in S ection 4.2 the p ro of of Keller’s theorem from [Kel1], b ecause in [Kel1] some details are omitted. Nev ertheless, in one p oin t we did not o ve rcome some v agueness. This is the usin g of the h omotopical relation for maps of dg op erads or algebras ov er dg op erads. Some im p lications lik e ”homotopical maps of dg op erads induce homotopical morphisms of algebras” in S ection 3, are stated without p ro ofs. Finally in Section 5 we giv e a construction of the homotopica l catego ry of dg Lie algebras through th e ”right cylinder” in the sense of [Q], whic h is suitable for the proof of the Main Theorem in Section 7. 0.9 The pap er is organized as follo ws: 7 In Sect ion 1 we d ev elop the Koszul dualit y for algebras o v er a discrete v aluati on rings. Our main example is the algebras o v er C [[ ~ ]], and we should ju stify that the main theorems of K oszul dualit y for asso ciativ e algebras hold in this context ; In Section 2 we giv e a brief exp osition of th e T amarkin’s theory [T1]. The Hin ich’s pap er [H] is a v ery goo d surve y , b u t w e ac hiev e some more clarit y in the computation of deformation cohomology of T poly ( V ) o v er the op erad G ∞ of homotop y Gerstenh ab er algebras. As we ll, in the App en dix after Section 2.5 we giv e a deduction of the existence of Kon tsevic h formalit y ov er Q fr om its existence o v er C , which d iﬀers fr om th e Drinfeld’s approac h [D]. This deduction seems to b e n ew; In Section 3 w e touc h some u nsolv ed problems in the T amarkin’s theory an d l ea v e them unsolve d, w ee only n eed to kno w h er e th at an y map of op erads G ∞ → B ∞ deﬁned up to homotop y , deﬁn es a universal G ∞ map T poly ( V ) → Ho ch q ( S ( V ∗ )) where th e G ∞ structure on Ho c h q ( S ( V ∗ )) is deﬁ ned via the map of op erads. T he deform ation quant i- zations f or w hic h ou r Main Theorem is true b elong to the image of the map X deﬁ ned there; In Section 4 for intro d uce diﬀeren tial graded ca tegories, give a co nstru ction of the Keller’s dg categ ory from [Kel1 ] associated with a Kel ler’s triple, and reformulate our Main Theorem in this new setting. W e get a more general statemen t, which is, ho wev e r, more easy to p ro v e; A very short S ection 5 is just a p lace to relax b efore the long computation in S ection 6, here w e recall the explicit construction [Sh3] of the Quillen’s homoto pical category via the r ight cylinder. Th e adv an tage of this construction is that it is immed iately clear from it that tw o homotopical L ∞ maps map a s olution of th e Maurer -C artan equation to gauge equiv alen t solutions (Lemma 5.2); In Section 6 w e construct the Ho chsc hild-Kostan t-Rosen b erg map f or the Keller’s dg category . This computation is done in terms of graph s, closed to the ones f rom [K97]. Originally the author got this computation truing to constru ct the L ∞ morphism F : T poly ( V ) → Hoch q ( cat ( A, B , K )) divid ing the diagram (6) into t wo commutati ve tri- angles, b y ”physical” metho ds. The compu tation here is the only what the author succeed to do in th is direction; The ﬁnal Section 7 is the culmination of our story . Here we dedu ce the Main Theorem on Koszul dualit y in deformation quan tization f rom Theorem 4.4. Th e idea is that from the diagram (6) associates with a quadratic P oisson bivect or α on V a solution of the Maurer-Cartan equation in the Ho chsc hild co mplex of the Keller’s dg category . This Maurer-Cartan ele ments deﬁn es an A ∞ deformation of the Keller’s category , and, in p articular, deforms the K oszul complex. Th is is enough to conclude that the t w o deformed algebras are Koszul du al. Ac kno wledgemen ts I am very grateful to Maxim K on tsevic h who taugh t me his formalit y theorem and many related topics. Discussions w ith Victor Ginzburg, P a v el Etingof and Bernhard Keller were 8 v ery v aluable for m e. It w as Victor Ginzbu rg wh o pu t my atten tion on the assump tions on A 0 in the theory of Koszul du alit y , related w ith ﬂatness of A 0 -mo dules, and explained to me why A 0 is semisimple in [BGS]. And it was Bernhard K eller w h o explained to me in our corresp ondence some foundations ab out dg categories, as well as his constructions from [Kel1]. But more than to the others, I am indebted to Dima T amarkin. Discussions with Dima after my talk at the Noth w estern Univ ersit y on th e sub ject of the pap er, an d thereafter, sh ed new light to many of my pr evious constr u ctions, and ﬁn ally h elp ed me to pr o v e the Main Th eorem of this pap er. I expr ess my gratitude to th e MIT and to the Unive rsity of Chicago whic h I visited in Octob er-No v em b er 2007 and where a part of this wo rk was d one, for a v ery stim ulating atmosphere and for the p ossibilit y of man y v aluable discussions, as w ell as for their hospitalit y and particular ﬁn ancial supp ort. I am grateful to th e r esearc h grant R1F105L1 5 of the Un iv ersit y of Lux emb ourg for partial ﬁ n ancial su pp ort. 1 Koszul dualit y for algebras o v er a discrete v aluation ring Here we give a brief ov erview of the K oszul d ualit y . Our main reference is S ection 2 of [BGS]. In lo c.cit., the zero d egree comp onent A 0 is supp osed to b e a (non-comm utativ e) semisimple algebra o v er th e base ﬁeld k . F or our ap p lications in deform ation quan tiza- tion, w e should consider algebras o v er C [[ ~ ]]. F or this reason, w e sho w that th e theory of Koszul dualit y may b e deﬁned o v er an arb itrary comm utativ e discrete v aluatio n ring. This result seems to b e n ew, although L.P ositselski announced in [P] that the zero d egree comp onent A 0 ma y b e an arbitrary algebra ov er the base ﬁeld. 1.1 The main classical example of Koszul dual alg ebras are th e alge bras A = S ( V ∗ ) and B = Λ( V ), where V is a ﬁnite-dimensional ve ctor sp ace ov er th e b ase ﬁ eld k . In general, supp ose A 0 is a ﬁxed k -algebra. Koszulness is a pr op ert y of a graded algebra A = A 0 ⊕ A 1 ⊕ A 2 ⊕ . . . (9) that is, A i · A j ⊂ A i + j (10) In our example w ith S ( V ∗ ) and Λ( V ) the algebra A 0 = k , it is the simp lest p ossib le case. In general, all A i are A 0 -bimo dules. There is a n atural pro jection p : A → A 0 whic h end o ws A 0 with a (left) A -module structure. Denote by A − Mo d the category of all left A -mo d u les, an d by A − mo d the catego ry of graded left A -mo dules. The A -mo d ule A 0 alw a ys has a f ree resolution in A − mo d · · · → M 2 → M 1 → M 0 → 0 (11) 9 suc h that M i is a grad ed A -mo d ule generated b y elemen ts of d egrees ≥ i . Ind eed, the bar-resolution · · · → A ⊗ k A ⊗ 2 + ⊗ k A 0 → A ⊗ k A + ⊗ k A 0 → A ⊗ k A 0 → 0 (12) ob eys this pr op erty . (Here A + = A 1 ⊕ A 2 ⊕ . . . ). Th is motiv ates the follo wing deﬁnition: Deﬁnition. A graded algebra (9) is called Koszul if the A -mo dule A 0 admits a pro j ectiv e resolution (11) in A − mo d such th at eac h M i is ﬁnitely generated by element s of degree i . F or our example with the symmetric and the exterior algebra, such a resolution is the follo wing Koszul c omplex : · · · → S ( V ∗ ) ⊗ Λ 3 ( V ) ∗ → S ( V ∗ ) ⊗ Λ 2 ( V ) ∗ → S ( V ∗ ) ⊗ V ∗ → S ( V ∗ ) → 0 (13) with the d iﬀerential d = dim V X i =1 x i ⊗ ∂ ∂ ξ i (14) Here { x i } is a basis in the v ector space V ∗ and { ξ i } is the corresp on d ing basis in V ∗ [1]. The diﬀeren tial is a gl ( V )-in v a riant elemen t, it do es n ot dep end on the c hoice of basis { x i } of the vec tor space V ∗ . 1.2 Here we explain s ome consequences of the deﬁnition of Koszul a lgebra, leading to the concept of Koszul dualit y for quadratic algebras. In Sections 1.2.1-1.2.3 A 0 ma y b e arbitrary ﬁnite-dimensional alg ebra ov er the ground ﬁeld k , and in Sections 1.2.4-1 .2.6 w e sup p ose that A 0 is a semisimple ﬁ nite-dimensional algebra ov e r k (see [BGS]). 1.2.1 Let A b e a graded algebra. T hen the space E x t q A − Mo d ( A 0 , A 0 ) is naturally bigr ad e d . W e write Ext n A − Mo d ( A 0 , A 0 ) = ⊕ a + b = n Ext a,b ( A 0 , A 0 ). F rom the bar-resolutio n (12 ) we see that for a general algebra A the only non-zero Ext a,b ( A 0 , A 0 ) app ear for a ≤ − b (h ere a is the cohomologica l grading and b is the inn er grading). I n the Koszul case the only nonzero summ ands are E x t a, − a . Let u s analyze this condition for a = 1 and a = 2. 1.2.2 Lemma. Supp ose A is a g r ade d algebr a. 1. If Ext 1 A − Mo d ( A 0 , A 0 ) = Ext 1 , − 1 ( A 0 , A 0 ) (that is, al l Ext 1 , − b = 0 for b > 1 ), the algebr a A is 1-gener ate d. The latter me ans that the algebr a A i n the form of (9) is gener at e d over A 0 by A 1 ; 10 2. if, furthermor e, Ext 2 A − Mo d ( A 0 , A 0 ) = Ext 2 , − 2 ( A 0 , A 0 ) (that is, Ext 2 , − ℓ ( A 0 , A 0 ) = 0 for ℓ ≥ 3 ), the algebr a A is quadr atic. This me ans that A = T A 0 ( A 1 ) /I wher e I is a gr ad e d ide al gener ate d as a two-side d ide al by I 2 = I ∩ A 2 . See [BGS], Section 2.3. This L emm a imp lies that an y K oszul algebra is quadratic. So, in f act th e Koszulness is a p rop erty of quadratic algebras. 1.2.3 F rom no w on, we use the notation I = I 2 for the in tersection o f the grad ed id eal I in T A 0 ( A 1 ) with A 2 . Any qu adratic algebra is uniquely deﬁ ned by the triple ( A 0 , A 1 , I ⊂ A 1 ⊗ A 0 A 1 ). Using th e b ar-complex (12 ), it is v ery easy to compute the ”diagonal p art” ⊕ ℓ Ext ℓ, − ℓ ( A 0 , A 0 ) ⊂ Ext q A − Mo d ( A 0 , A 0 ) f or an y algebra A . Let u s formulate the answe r. Deﬁne f rom a triple ( A 0 , A 1 , I ) another triple ( A ∨ 0 , A ∨ 1 , I ∨ ), as follo ws. Sup p ose A 1 and I are ﬂat A 0 -bimo dules. W e set A ∨ 0 = A 0 , A ∨ 1 = Hom A 0 ( A 1 , A 0 )[ − 1]. De ﬁn e no w I ∨ . Denote ﬁrstly A ∗ 1 = Hom A 0 ( A 1 , A 0 ). There is a pairing ( A 1 ⊗ A 0 A 1 ) ⊗ ( A ∗ 1 ⊗ A 0 A ∗ 1 ) → A 0 whic h is non-degenerate. Denote by I ∗ the su b space in A ∗ 1 ⊗ A 0 A ∗ 1 dual to I . Denote b y I ∨ = I ∗ [ − 2], it is a sub s pace in A ∨ 1 ⊗ A 0 A ∨ 1 . T he triple ( A 0 , A ∨ 1 , I ∨ ) generates some quadratic algebra, d enote it by A ∨ . Let no w A b e an y 1-generated n ot necessarily qu adratic algebra. Then the quadr atic p art A q is w ell-deﬁned. Let A b e a quotient of T A 0 ( A 1 ) b y graded not necessar- ily quadr atic ideal. W e d eﬁne A q as the quadratic algebra asso ciated with the triple ( A 0 , A 1 , I ∩ A 2 ). There is a c anonical surjection A q → A whic h is an isomorphism in degrees 0, 1, and 2. Lemma. L et A b e a 1-gene r ate d algebr a over A 0 . Then the diagonal c ohomolo gy ⊕ ℓ Ext ℓ, − ℓ ( A 0 , A 0 ) as algebr a is c anonic al ly i somorphic to the algebr a op p ose d to ( A q ) ∨ . Her e by the opp ose d algebr a to an algebr a B we understand the pr o duct b 1 ⋆ opp b 2 = b 2 ⋆ b 1 . It is a direct consequence from the b ar-resolution (12 ). In p articular, l et no w a graded 1-generated algebra A b e Koszul. Then Ext q A − Mo d ( A 0 , A 0 ) = ( A ∨ ) opp . This fol lo ws from the id en tit y A = A q for a qu adratic algebra A (in particular, for Koszul A ), and from the equalit y of the all Exts to it s diagonal part for any Koszul algebra. 1.2.4 The inv erse is also true, u nder an assumption on A 0 . Lemma. Supp ose A 0 is a simple ﬁnite-dimensional algebr a over k and A is a quadr atic algebr a over A 0 . Then if Ext q A − Mo d ( A 0 , A 0 ) i s e qual to its diagonal p art, then A is Koszul. In p ar ticular, if Ext q A − Mo d ( A 0 , A 0 ) = ( A q ∨ ) opp , then A is Koszul. 11 See [BGS], Prop osition 2.1.3. Let us commen t why we need here a condition on A 0 . The pro jectiv e resolution which w e need to prov e that A is Koszul is constru cted inductive ly . W e constru ct a resolution · · · → P 3 → P 2 → P 1 → P 0 → 0 (15) satisfying the p rop erty of Deﬁn ition 1.2 and such that the diﬀerential is injectiv e on P i i . W e set P 0 = A . T o p erform the step of induction, set K = k er( P i → P i − 1 ). W e hav e: Ext i +1 A − Mo d ( A 0 , A 0 ) = Hom A − Mo d ( K, A 0 ). F rom the cond ition of lemma w e conclude that K is generate d b y the part K i +1 of inner degree i + 1 (here for simp licit y we su pp ose that A has trivial cohomolo gical grading). Th en we p u t P i +1 = A ⊗ A 0 K i +1 . But then w e need to c hec k that the ima ge of the map P i +1 → P i is K . F or this w e n ecessarily need to kn o w that K i +1 is a ﬂ at left A 0 -mo dule. F or this it is suﬃcien tly to kn o w that K is. So we need a theorem lik e th e follo wing: the k ernel of an y map of go o d (ﬂat, etc.) A 0 -mo dules is again a ﬂat A 0 -mo dule. It do es not f ollo w from any general things, it is a prop erty of A 0 . It is the case if any mo d ule is ﬂat, as in the case of a ﬁ nite-dimensional simple algebra. F or another p ossible condition, see Section 1.3. 1.2.5 Prop osition. Supp ose A is a quadr atic algebr a deﬁne d f r o m a triple ( A 0 , A 1 , I ) wher e A 1 and I ar e ﬂat A 0 -bimo dules. Supp o se A is Koszul. Then ( A ∨ ) opp is also Koszul. R e mark. It is clear that A is K oszul iﬀ A opp is Koszul. W e give a ske tc h of pro of, which is essential ly giv en by th e construction of the Koszul complex. F or A = S ( V ∗ ) the K oszul complex is constru cted in (13). Let A = ( A 0 , A 1 , I ) b e a quadratic algebra, and let A 1 and I ⊂ A 1 ⊗ A 0 A 1 b e ﬂat A 0 -bimo dules. W e deﬁne the Koszul complex · · · → K 3 → K 2 → K 1 → K 0 → 0 (16) W e set K i = A ⊗ A 0 K i i (17) where K i i = \ ℓ A ⊗ ℓ 1 ⊗ A 0 I ⊗ A 0 A ⊗ i − ℓ − 2 1 (18) In particular, K 0 0 = A 0 , K 1 1 = A 1 , K 2 2 = I . T he diﬀerent ial d : K i → K i − 1 is deﬁned as the restriction of the map ˆ d : A ⊗ A 0 A ⊗ i 1 → A ⊗ A 0 A ⊗ ( i − 1) 1 giv en as a ⊗ v 1 ⊗ · · · ⊗ v i 7→ ( av 1 ) ⊗ v 2 ⊗ · · · ⊗ v i − 1 (19) Clearly d 2 = 0. The complex (16) is called the Koszul c omplex of the quadratic algebra A . 12 Lemma. L e t A = ( A 0 , A 1 , I ) b e a quadr atic algebr a, A 1 and I ﬂat A 0 -bimo dules. Sup- p ose, additional ly, that A 0 is a ﬁnite-dimensional semisimple algebr a over k . Then its Koszul c omplex is acyclic exc ept de gr e e 0 iﬀ A is Koszul. See [BGS], Theorem 2.6.1 for a pro of. In the pr o of it is essential that the mo dules K i i a ﬂat left A 0 -bimo dules . In general the tensor pro du ct of t w o ﬂ at mo dules is ﬂ at, but there is no theorem wh ic h guarantee s the same ab out the intersect ion of t wo ﬂat sub m o dules. In the case wh ic h is considered in [BGS], any mo dule ov er a ﬁnite-dimensional semisimp le algebra is ﬂat. Let u s note that the part ”only if” also follo ws from Lemm a 1.2.4. The Pr op osition follo ws from this Lemma easily . Indeed, it is clear that K i = A ⊗ A 0 [( A ! ) ∗ ] i [ − i ] and that the K oszul complex K of a Koszul algebra satisﬁes the Deﬁnition 1.1. Th en the du al complex K ∗ also satisﬁes th e Deﬁnition 1 .1 and can b e w r itten a s K ∗ = A ∗ ⊗ A 0 A ! . W e immediatel y c hec k that it coincides with th e K oszul complex of the qu adratic algebra A ! b ecause ( A ! ) ! = A for any quadratic algebra A . Then from its acyclicit y follo ws that A ! is Koszul. 1.2.6 W e summ arize the discussion ab o v e in the follo wing theorem. Theorem. L et A = ( A 0 , A 1 , I ) b e a quadr atic algebr a , A 1 and I b e ﬂat A 0 -bimo dules, and A 0 b e semisimple ﬁnite-dimensional algebr a over k . Then A is Koszul if and only if the quadr atic dual A ! is also Koszul, and in this c ase Ext i A − Mo d ( A 0 , A 0 ) = [( A ! ) opp ] i [ − i ] (20) and Ext i A ! − Mo d ( A 0 , A 0 ) = [ A opp ] i [ − i ] (21) for any inte ger i ≥ 0 . 1.3 In the context of deformation quant ization, all our algebras are algebras o v er the form al p ow er series C [[ ~ ]], therefore, A 0 = C [[ ~ ]]. The theory of Koszul alg ebras as it is devel - op ed in [BGS] do es not co v er this case. In this Su bsection w e explain that in the theory of Koszul du alit y A 0 ma y b e an y comm utativ e discrete v aluation ring (see [AM], [M]). L.P ositselski announced in [P] th at A 0 ma y b e any algebra o v er k . Recall the deﬁnition of a d iscrete v aluation ring. Deﬁnition. A comm utativ e ring is called a discrete v aluation ring if it is an integ rally closed domain with only one nonzero pr ime ideal. In particular, a discrete v aluation ring is a lo cal r ing. 13 The tw o main examples are the follo wing: (1) Let C b e an aﬃne algebraic curve, and let p ∈ C b e a non-singular p oin t (not necessarily closed). Then the lo cal ring O p ( C ) is a discrete v aluation ring (recall that in dimension 1 integ rally closed=nonsingular); (2) let C and p b e as ab o v e; w e can consider the completion of th e lo cal ring O p ( C ) b y the p ow ers of th e m aximal ideal. Denote this ring b y b O p ( C ), this is a discrete v aluat ion ring. In particular, k [[ ~ ]] is a discrete v aluation ring. It is known that an y discrete v aluation ring is No etherian and is a p rincipal id eal domain (see [M], T heorem 11.1). T o extend the theory of Section 1.2 to the case w h en A 0 is a discrete v al uation r ing w e need to kno w that the intersectio n of ﬂ at submo dules ov er a discrete v aluation ring (Section 1.2.5), and the k ernel of a map of ﬂat mo d ules o v er a discrete v aluation r in g (Section 1.2.4) are ﬂat. This is guaran teed by th e follo wing, m ore general, result: Lemma. L et R b e a d iscr ete valuation ring, and let M b e a ﬂat R - mo dule. Then any submo dule of M is again ﬂat. Pr o of. Let R b e a ring and M is an R -mod u le. Th en M is ﬂat if and only if for any ﬁnitely generated ideal I ⊂ R the n atur al map I ⊗ R M → R ⊗ R M is inj ective (see [M], Theorem 7.7). An y id eal in a discrete v a luation ring is pr incipal ([M], Theorem 11 .1); therefore ﬂatness of a mo d ule o v er a discrete v aluation ring is th e same that torsion-free (a mo dule M is called torsion-free if x 6 = 0, m 6 = 0 implies xm 6 = 0). S o now our Lemma follo ws f r om the fact that a su bmo d u le o ve r a torsion-free mo du le is torsion-free. R e mark. If R is any lo cal ring and M is a ﬁnite R -mo dule, then ﬂ atness of M imp lies that M is free ([M], Theorem 7.10). Nev ertheless, in dimens ion ≥ 2 a subm o dule of a free mo dule ma y b e not free. F or example, one can take the (lo calization of the) co ord inate ring of a cu rv e in A 2 . Com bining the Lemma ab ov e with the d iscussion of Section 1.2, we get th e f ollo wing Theorem: Theorem. L et A = ( A 0 , A 1 , I ) b e a quadr atic a lgebr a, with A 0 a c om mutative d iscr ete valuation ring, and A 1 , I ﬂat A 0 -mo dules. Then A is Koszul iﬀ A ! is, and in this c ase Ext i A − Mo d ( A 0 , A 0 ) = [( A ! ) opp ] i [ − i ] (22) and Ext i A ! − Mo d ( A 0 , A 0 ) = [ A opp ] i [ − i ] (23) for any inte ger i ≥ 0 . W e will use this Th eorem only for A 0 = k [[ ~ ]]. 14 R e mark. This theorem has an analogue for Dedekind domains. Namely , the lo calization of a D edekind domain at any prime id eal is a d iscrete v aluation ring, this is a global v ersion of it. Th e main example of a Dedekind domain is the co ordinate r ing of a non- singular aﬃne curve. Su pp ose A 0 is a Dedekind domain. W e sa y that a qu adratic algebra o v er A 0 is Koszul if its lo calization at an y prime id eal is Koszul. Then w e can pro v e the theorem analogous to th e ab ov e for A 0 a Dedekind domain. More generally , w e can sp eak ab out shea ve s of Koszul du al quadratic algebras. A t the moment the auth or do es not know an y interesti ng exa mple of suc h situation, but he do es n ot doub t that these examples exist. R e mark. Leonid P osit selski claims in [P] that he constructed the analogo us theory for an y A 0 . The argum en ts in [P] are rather complicated comparably with ours’; for the readers’ conv enience, w e ga v e here a more direct simple proof in the case of d iscrete v a luation rings. 2 T amarkin’s approac h to the Kon tsevic h formalit y Here w e o v erview the T amarkin ’s pro of of Kon tsevic h formalit y theorem. T he main references are [T1] and [H], some v ariat ions w hic h allo w to a v oid the using of the Etingof- Kazhdan quantiz ation (but replace it b y another transcendent al constru ction) w ere m ad e b y K on tsevic h in [K99]. 2.1 Kon tsevic h formality F or any asso ciativ e algebra A we denote by Ho c h q ( A ) the cohomologic al Ho c hschild com- plex of A . When A = C ∞ ( M ) is th e algebra of sm o oth fu nctions on a smo oth m an if old M , we consider some completed tensor p o w ers, or the p olydiﬀeren tial part of the usu al Ho c hsc hild complex (see, e.g., [K97]). Under this assumption, the Ho c hschild cohomolog y of A = C ∞ ( M ) is equal to smo oth p olyv ector ﬁelds T poly ( M ). More precisely , consider the follo wing Ho c hsc hild-Kostant- Rosenberg map ϕ : T poly ( M ) → Ho c h q ( C ∞ ( M )): ϕ ( γ ) = { f 1 ⊗ · · · ⊗ f k 7→ 1 k ! γ ( d f 1 ∧ · · · ∧ d f k ) } (24) for γ a k -p olyv ector ﬁ eld. Then the Ho chsc hild-Kostan t-Rosen b erg theorem is Lemma. 1. F or any p olyve ctor ﬁeld γ , the c o chain ϕ ( γ ) is a c o cycle; this gives an iso- morphism of (c o mplete d or p oly diﬀer ential) Ho chschild c oh omolo gy of A = C ∞ ( M ) with T poly ( M ) ; 2. the br acket induc e d on the H o ch schild c oho molo gy fr om the Gerstenhab er br acket c oincides, via the map ϕ , with the Schouten-Nijenhuis br acket of p olyve ctor ﬁelds. See, e.g., [K97] for d eﬁnition of the Gerstenhab er and Schouten-Nijenh uis brac k ets. 15 The second claim of th e L emm a means that [ ϕ ( γ 1 ) , ϕ ( γ 2 )] G = ϕ ([ γ 1 , γ 2 ] S N ) + d Hoch U 2 ( γ 1 , γ 2 ) (25) for some U 2 : Λ 2 ( T poly ( M )) → Ho ch q ( C ∞ ( M ))[ − 1] (we denoted b y [ ] G the Gerstenhab er brac k et and by [ , ] S N the Schouten-Nijenh uis b rac k et). In the case when M = C d M.Kon tsevic h constru cted in [K97] an L ∞ morphism U : T poly ( C d ) → Ho c h q ( S ( C d ∗ )) whose ﬁ rst T ayl or comp onen t is the Ho c hsc hild-Kostant- Rosen b erg map ϕ . (Here w e consider p olynomial p olyv ector ﬁelds, and there is no ne- cessit y to complete the Ho chsc hild complex). The second T aylo r comp onent U 2 should then satisfy (25 ), and so on. This result is called the Kontsevich’s forma lity the or em . (The resu lt for a general manifold M can b e deduced from this lo cal statemen t, see [K97], Section 7). The original Kont sevic h’s p ro of uses ideas of top ological ﬁ eld theory , n amely , the Alexandrov-Ko ntsevic h -Sc hw arz-Zaboronsky (AKSZ) mo del, see [AKSZ]. Therefore, some transcendenta l complex num b er s , the ”F eynmann integ rals” of th e theory , are in- v olv ed in to th e constr u ction. The Kontsevic h’s pro of ap p eared in 1997. One ye ar later, in 1998, D.T amarkin found in [T] another proof of the K on tsevic h formalit y for C d , using absolutel y diﬀerent tec hnique. In the rest o f this Section w e outline the T amarkin’s pro of [T], [H] in the form we u s e it in the sequ el. 2.2 The idea of the T amarkin’s proof The main id ea it to constru ct not only an L ∞ map fr om T poly ( C d ) to Ho ch q ( S ( C d ∗ )) but to in v olv e the entire structur e o n p olyv ector ﬁelds and the Ho chsc hild complex. Th is is the s tr ucture of (homotop y) Gerstenhab er algebra. F or example, on p olyv ector ﬁelds (on any m anifold) one has tw o operations: the wedge p r o duct γ 1 ∧ γ 1 of d egree 0, and the Lie b rac k et [ γ 1 , γ 2 ] S N of d egree − 1, and th ey are compatible as [ γ 1 , γ 2 ∧ γ 3 ] = [ γ 1 , γ 2 ] ∧ γ 2 ± γ 2 ∧ [ γ 1 , γ 3 ] (26) This is called a Gerstenhab er algebra. T o consider T poly ( C d ) as a Gerstenhab er algebra simpliﬁes the p roblem b ecause of th e f ollo wing Lemma: Lemma. The p olyve ctor ﬁelds T poly ( C d ) is rigid as a homoto py Gerstenhab er algebr a. Mor e pr e cisely, any Aﬀ ( C d ) -e quivariant defor mation o f T poly ( C d ) as a homoto py Ger- stenhab er algebr a is hom otopic al ly e quivalent to trivial defor mation. W e should explain what these words mean, w e d o it in the n ext Sub sections. Let us no w explain h o w it h elps to pr ov e the Kont sevic h’s formalit y th eorem. It is true, and tec hnically it is th e hardest place in the pro of, that there is a stru cture of homotopical Gerstenhab er algebra (see Section 2.3) on the Ho c hschild complex Ho c h q ( A ) of a ny asso ciativ e algebra A . It is non-trivial, b ecause the cup-pro d uct of Ho c hsc hild co c hains Ψ 1 ∪ Ψ 2 and the Gerstenhab er b rac k et [Ψ 1 , Ψ 2 ] G do not ob ey the compatibilit y 16 (26), as it should b e in a Gerstenh ab er algebra. It ob eys it only up to a homotop y , and to ﬁnd explicitly this s tructure uses also either some in tegrals like in [K99], or Drin feld’s Knizhnik-Zamolo dchik o v iterated integrals, as in [T]. W e discuss it in S ection 2.5. No w supp ose th at this structure exi sts, s uc h that the Lie a nd Commuta tiv e parts of th is structure are equiv alen t to the Gerstenh ab er brac k et and the cup-pr o duct on Ho c hsc hild co c hains. Then, as usual in homotopical algebra, there exists a h omotopical Gerstenhab er algebra stru cture on the cohomolog y , equiv ale nt to this structure on the co c hains (it is somet hing lik e ”Massey op erations” b y Merkulov and Kon tsevic h-Soib elman). This push-forwarded structur e is uniquely deﬁn ed u p to homotop y . No w w e can consider this structur e a s a form al d eformation o f the cl assical pure Gerstenhab er algebra on T poly ( C d ). Indeed, we resca le the T ayl or comp onen ts of t his structure, suc h th at the weig ht of k -linear T aylo r comp onen ts is λ k − 2 . This giv es again a homotopical Gerstenh ab er structure, wh ic h v alue at λ = 0 is the classical Gerstenhab er structure on p olyv ector ﬁelds, b ecause of the compatibilit y with Lie and Commutat ive structure, and by the Ho c hsc hild-Kostant- Rosen b erg theorem. No w we apply the Lemma ab o v e. All steps of our construction are Aﬀ ( C d )-in v a riant, therefore, th e obtained deformation can b e c hosen Aﬀ ( C d )-equiv ariant. Then the Lemma sa ys that this deform ation is trivial, and the t w o homotopical Gerstenhab er structures on T poly ( C d ) are in f act isomorphic. This implies th e Kont sevic h’s formalit y in the str onger, Gerstenhab er algebra isomorphism, form. 2.3 Koszul op erads F rom o ur p oint of view, the K oszulness of an op erad P is v ery imp ortan t b ecause in this case any P -algebra A h as ”v ery economic” r esolution whic h is f ree dg P -algebra. In the case of the op erad P = Assoc , this ”v ery economic” resolution is the Quillen’s bar- cobar construction. Thereafter, we u s e this free resolution to compute th e (truncated) deformation complex of A as P -algebra. In the case of P = Assoc this d eformation complex is the Ho c hsc hild cohomologic al complex of A without t he zero degree te rm, that is Ho ch q ( A ) / A . W e will consider only op erads of dg C -vect or spaces here, with one of th e t wo p ossible symmetric monoidal structur es. A quadr at ic op erad generated by a vec tor space E o v er C with an action of the symmetric grou p Σ 2 of tw o v ariables, w ith a Σ 3 -in v arian t space of relations R ⊂ Ind Σ 3 Σ 2 E ⊗ E (here Σ 2 acts only on the second fact or) is the quotient of the free op erad P g enerated by P (2) = E by the s pace of r elations R ⊂ P (3). The op erads Lie , C omm , As soc are q u adratic, as well as the Gerstenhab er and the P oisson op erads. See [GK], Section 2 .1 for more detail. F or a quadratic operad P deﬁne the quadratic dual op erad P ! as the quadratic op erad generated b y P ! (2) = E ∗ [1], with the sp ace of r elations R ∗ in Ind Σ 3 Σ 2 E ∗ [1] ⊗ E ∗ [1] equal to the orthogo nal complemen t to R ⊂ In d Σ 3 Σ 2 E ⊗ E . Examp le: C om ! = Lie [ − 1], Assoc ! = Assoc [ − 1], ( P ! ) ! = P . 17 Let P b e a general, not necessarily quadratic, op erad. F or simplicit y , we supp ose th at all vec tor sp aces P ( n ) of an op erad P are ﬁnite-dimensional. Reca ll the construction of the bar-co mplex of P , see [GK], S ection 3.2. Denote the b ar complex of P by D ( P ). Then one has: D ( D ( P )) is quasi-isomorphic to P ([ GK], Th eorem 3.2.16). Let no w P b e a quadratic op erad. Then the b ar-complex D ( P ) is a negativ ely-graded dg op erad whose 0-th cohomology is canonically the quadratic dual op erad P ! . A quadratic op erad P is called K oszul if the bar-complex D ( P ) is a resolution of P ! . In this case D ( P ! ) give s a free r esolution of the op erad P . Example. The op erads Lie , C omm , Assoc , the Gerstenhab er and the Poisson op erads, are Koszul. See [GK], S ection 4 for a p ro of. Deﬁnition. Let P b e a quadratic Koszul op erad. A homotop y P -algebra (or P ∞ - algebra) is an algebra o v er the free dg op erad D ( P ! ). W e denote b y P ∗ the coop erad dual t o an op erad P , if all space s P ( n ) are ﬁnite- dimensional. Let P be a Koszul op erad. T hen to deﬁ ne a P ∞ -algebra s tructure on X is the same that to deﬁne a d iﬀeren tial on the free coalgebra F ∨ P ! ∗ ( X ) w hic h is a co deriv atio n of the coalgebra structure. Any P algebra is naturally a P ∞ -algebra. W e denote b y F P ( V ) the free algebra ov er the op erad P , and by F ∨ P ∗ the fr ee coalgebra o v er the co op erad P ∗ . Here we sup p ose that all s paces P ( n ) are ﬁn ite-dimensional. Recall the follo wing statemen t [GK], T hm. 4.2.5: Lemma. L et P b e a K oszul op er ad, a nd V a ve ctor sp ac e. L et X = F P ( V ) . Then the natur al pr oje ction ( F ∨ P ! ∗ ( X )) → V (27) is a quasi- isomorph ism. It follo ws from this statemen t that an y P -algebra A has the follo wing free resolution R q ( A ): R q ( A ) = ( F P ( F ∨ P ! ∗ ( A ) , Q 1 ) , Q 2 ) (28) with the n atur al diﬀerenti als Q 1 and Q 2 . No w w e deﬁne the trunc ate d deformation c omp lex of the P -algebra A as (Der( R q ( A )) , Q ) where Q comes from the d iﬀeren tial in R q ( A ). This deformation com- plex is naturally a dg Lie alg ebra with the Lie b rac k et of deriv at ions. W e ha v e the follo wing statemen t: Prop osition. The trunc a te d deformation functor asso c i ate d with this dg Lie algebr a governs the form al deform ations of A as P ∞ -algebr a. R e mark. T he w ord ”trun cated” m eans th at for the ”full” deformation fu nctor w e sh ould tak e the qu otien t mo du lo the inn er deriv ations. Although, a map X → Der( X ) is not deﬁned for an arbitrary op erad. Our truncated deformation functor looks lik e the Ho c hsc hild cohomological complex of A without th e d egree 0 term A . 18 The follo wing tric k simpliﬁ es compu tations with the deformation complex. An y coderiv ation of the coalgebra ( F ∨ P ! ∗ ( A ) , Q 1 ) can b e extended to a d eriv a tion of R q ( A ) b y the Leibniz rule. W e ha v e the follo wing th eorem: Theorem. The natur al inclusion Co der( F ∨ P ! ∗ ( A ) , Q 1 ) → Der( R q ( A )) (29) is a quasi- isomorph ism of dg Lie algebr as. It follo ws from this Th eorem and the Prop osition ab ov e that the dg Lie algebra Co der( F ∨ P ! ∗ ( A ) , Q 1 ) go v erns the formal deformation of the P ∞ -structure o n A . This, of course, can b e seen m ore directly . I n deed, a P ∞ structure on A is a diﬀeren tial on F ∨ P ! ∗ ( A ) making latter a dg coa lgebra o v er P ! . W e ha ve some distinguished diﬀeren tial Q 1 on it, arisen from the P -algebra structure on A . When w e deform it, it is replaced b y Q ~ = Q 1 + ~ d ~ suc h that ( Q 1 + ~ d ~ ) 2 = 0. In the ﬁrst order in ~ w e hav e the condition [ Q 1 , d ~ ] = 0, w here the zero square condition is the Maurer-Cartan equation in the corresp onding dg Lie algebra. 2.4 The main computation in the T amarkin’s t heory Here w e compute the deformation cohomolog y of T poly ( V ), V a complex vec tor s p ace, as Gerstenhab er algebra. W e pro v e h ere Lemma 2.2, and a more general statemen t. W e start with the follo wing Lemma: Lemma. The Gerstenhab er op er ad G is Koszul. The Koszul dual op er ad G ! is G [ − 2] . Pr o of. W e kno w th at L ie ! = C omm [ − 1] and C omm ! = Lie [ − 1]. A structur e of a Gerstenhab er algebra on W consists fr om compatible actions of C omm and Lie [1] on W . The quadratic d ual to C omm is Lie [ − 1], and the quadratic dual to Lie [1] is C omm [ − 2]. Therefore, the qu adratic dual to G is G [ − 2]. Th e Koszulity of G is pro v en in [GJ], see also [GK] an d [H]. Theorem 2.3 giv es us a wa y how to compute the deformation functor for formal deformations of T poly ( V ) as h omotop y Gerstenhab er algebra. W e tak e th e free coalge bra F ∨ G [ − 2] ∗ ( T poly ( V )) ov er the co op erad G [ − 2] ∗ cogenerate d by T poly ( V ). It is clear that F ∨ G ∗ [2] ( T poly ( V )) = S q (( F Lie T poly ( V )[1])[1]) [ − 2] (30) The pro duct ∧ : S 2 T poly ( V ) → T poly ( V ) and the Lie brac k et [ , ] : S 2 T poly ( V ) → T poly ( V )[ − 1] deﬁne t w o c o derivations of the Gerstenh ab er coalgebra structure on F ∨ G ∗ [2] ( T poly ( V )); denote them by δ Comm and δ Lie , corresp ondingly . The deform ation complex of T poly ( V ), as of Gerstenhab er algebra, is th en Co der q ( F ∨ G ∗ [2] ( T poly ( V ))) endo w ed with the d iﬀeren tial d = ad ( δ Comm ) + ad ( δ Lie ). W e denote the tw o summand s by d Comm and d Lie , corresp ondingly . 19 Theorem. The Aﬀ ( V ) -invariant sub c omplex in the “p o sitive” deformation c omplex (Co der + ( F ∨ G ∗ [2] ( T poly ( V ))) , d Comm + d Lie ) has al l vanishing c ohomolo gy. R e mark. Here the p ositive deformation complex mea ns that w e exclud e the constan t co deriv ati ons Hom( C , T poly ( V )[2]). T h e reason to consider th e p ositiv e complex is that the constant cod er iv ations do n ot app ear in the f ormal deformations of G ∞ algebra structure. Pr o of. First of all w e c ompute the cohomology of all deriv ations, then concen trate on p ositiv e Aﬀ -in v arian t sub complex. Recall in the b eginning some tautological facts. The free Gerstenh ab er algebra gen- erated b y a v ector space W is S q ( F Lie ( W [1])[ − 1]). The co free Ge rstenh ab er coal gebra cogenerate d by W is S q ( F ∨ Lie ( W [ − 1])[1]). Finally , the cofree G ∗ [2]-coa lgebra cogenerated b y W is S q ( F ∨ Lie ( W [1])[1])[ − 2]. W e deal with the coderiv ations of th e cofree coalgebra S q ( F ∨ Lie ( W [1])[1])[ − 2], and they are deﬁn ed uniquely b y their restrictions to cogenerators whic h m a y b e arbitrary . Therefore, we n eed to compute the cohomology of the complex Hom C ( S q (( F Lie ( T poly ( V )[1]))[1] ) , T poly ( V ))[2] (31) This is a bicomplex with the diﬀerential s d Comm and d Lie . W e use the sp ectral sequence of the b icomplex w hic h co mpu tes ﬁ rstly the cohomology of d Comm . W e lea v e to th e reader the simple chec k that this sp ectral sequence conv erges to the (associated graded of ) cohomology of the total complex. Compute the ﬁrst term of the sp ectral sequence. When w e tak e in the accoun t the only d iﬀeren tial d Comm , the d eformation complex Hom C ( S q ((F ree Lie ( T poly ( V )[1]))[1] ) , T poly ( V ))[2] is a direct su m of c omp lexes : Hom C ( S q ((F ree Lie ( T poly ( V )[1])) [1]) , T poly ( V ))[2] = T poly ( V )[2] M k ≥ 1 (Hom( S k ((F ree Lie T poly ( V )[1])[1]) , T poly ( V )) , d Comm )[2] (32) Lemma. L et k ≥ 1 . The c oho molo gy of the c omplex (Hom( S k ((F ree Lie T poly ( V )[1])[1]) , T poly ( V )) , d Comm )[2] is S k O (V ect ( T ∗ [ − 1] V ))[ − 2 k + 2] . Her e S k O (V ect ( T ∗ [ − 1] V )) is (the se ctions of ) the k -th symmetric p ower of the ve ctor bund le of ve ctor ﬁeld s on the sp a c e T ∗ [ − 1] V . (R e c al l th at T poly ( V ) is th e functions on T ∗ [ − 1] V ). Pr o of. W e only “explain” the statemen t, the complete pro of w ill app ear somewhere. The complex (Hom( S k ((F ree Lie T poly ( V )[1])[ 1]) , T poly ( V )) , d Comm )[2] “starts” with th e term Hom( S k ( T poly ( V )) , T poly ( V ))[ − 2 k + 2] (we tak e only the generators of the fr ee Lie 20 (co)alg ebra). The d iﬀeren tial d Comm in this term is ( d Comm Ψ)( γ 1 · · · · · γ k +1 ) = SymmΨ(( γ 1 ∧ γ 2 ) · γ 3 · · · · · γ k +1 ) ∓ Symm  γ 1 ∧ Ψ( γ 2 · · · · · γ k +1 ) ± γ 2 ∧ Ψ( γ 1 · γ 3 · · · · · γ k +1 )  (33) The k ernel of this diﬀerentia l is exactly th e answer give n in the s tatemen t of the Lemma (this is clear). Th e m ore non-trivial is to show that the “higher” cohomolo gy v anishes. Th us, in the term E 1 w e hav e L k ≥ 0 S k O (V ect ( T ∗ [ − 1] V ))[ − 2 k + 2]. No w consider the diﬀerent ial d Lie acting on E 1 . Th e S c houten b rac k et is an elemen t of S 2 O V ect ( T ∗ [ − 1] V )[ − 1]. One easily sees that the cohomology b elongs to T poly ( V )[2] ⊂ L k ≥ 0 S k O (V ect ( T ∗ [ − 1] V ))[ − 2 k + 2] (the summand f or k = 0). This cohomology is 1 - dimensional and is represented by a constant fun ction in T 0 poly ( V )[2]. W e conclude that the cohomology of the full deformation complex (31) is 1-dimensional and is concent rated in degree -2. No w consider the Aﬀ -inv arian t sub complex of the full d eform ation complex. More precisely , we compu te the cohomology of Hom C ( S q ((F ree Lie ( T poly ( V )[1])) [1]) , T poly ( V )) Aﬀ [2] (34) F rom a very general p oin t of view, the terms of our complex are Aﬀ -in v arian ts in Hom( V ⊗ a ⊗ V ∗⊗ b , V ⊗ c ⊗ V ∗⊗ d ). The Lie group GL( n ) acts in the natural wa y , and the shift x 7→ x + ( a 1 , . . . , a n ) acts trivially on all V fa ctors, and by shifts x i 7→ x i + a i on all V ∗ factors. W e kn o w all GL( n )-in v arian ts from the main theorem of in v arian t theory . They are c onstructed from t he follo wing 4 elemen tary op erations. These 4 op erations are: id : V → V , id : V ∗ → V ∗ , V ⊗ V ∗ → C , and C → V ⊗ V ∗ . F rom these 4 operations, only the last one, C → V ⊗ V ∗ , is n ot Aﬀ -in v arian t. On the other h ang, the group GL( n ) acts reductive ly , a nd the cohomology of the complex is equal to the cohomology of its GL( n )-in v arian t part. On the other hand, due to the symmetrization conditions, the inv ariant c : C → V ⊗ V ∗ can b e app lied only 0 or 1 times. So schemat ically as a v ector space th e s p ace of GL( n )- in v arian ts of (32) is K q ⊕ c · K q where K q is the space of Aﬀ -in v arian ts of (32). On e easily sees that this decomp osition agrees with the complex structur e. W e conclude that the cohomology of (34) is equal to the Aﬀ -in v aria nts of the cohomology of Hom C ( S q ((F ree Lie ( T poly ( V )[1])) [1]) , T poly ( V )) GL( n ) [2] (35) As was already mentioned ab ov e, the latter cohomology is equ al to th e Aﬀ -inv ariant s of the cohomology of (34), b ecause the group GL( n ) acts reductivel y . W e conclude that the cohomology of (35 ) is equal to C [2]. 21 The last step is to compute the cohomology of the p ositiv e s ub complex. This is easy to do. In the term E 2 one h as ( T poly ( V ) / C )[0], and after taking of the Aﬀ -in v arian ts, we get 0. Theorem is prov en. 2.5 The ﬁnal p oin t: relation with the Etingof-Kazhda n quan tization The remaining p art of the T amarkin’s p ro of of Kontsevic h formalit y go es as follo ws. One ﬁ rstly pr o v es the Deligne c onje ctur e that there is a homotopy Gerstenhab er al- gebra stru ctur e on the Ho c hsc hild cohomolo gical complex Ho ch q ( A ) of any asso ciativ e algebra s uc h th at it induces the Sc houten b r ac k et and the w edge pro duct on the coho- mology . Th is is the only transcendental step of the construction, this s tr ucture, as it is deﬁned in [T ], d ep end s on a c hoice of Drin feld asso ciator [D]. W e apply this fact f or A = S ( V ∗ ), V a v ector sp ace o v er C . One can push-forwa rd (giv en b y the ”Ma ssey op erations”) this G ∞ structure from Ho c h q ( S ( V ∗ )) to its coho- mology T poly ( V ). Then w e get t w o G ∞ structures on T poly ( V ), the ﬁrst is giv en from the S c houten b rac k et and the w edge p r o duct of p olyv ector ﬁ elds, the second is the ab o v e pushf orward. Moreo v er, one can in tro du ce a f ormal paramete r ~ to the pushforw ard, suc h that the original one is giv en wh en ~ = 1. Then for ~ = 0 w e get the Schouten structure: it follo ws from the compatibilit y of the Deligne’s conjecture G ∞ structure with the on e on the cohomology . Thus we get a formal deformation of the classical Gersten- hab er algebra structure o n T poly ( V ). This deformation is clea rly Aﬀ ( V )-inv arian t. By Theorem 2.4, inﬁnitesimally all suc h deform ations are trivial; therefore, th ey are trivial globally . This concludes the T amarkin’s p ro of. In this Subsection we explain the T amarkin’s p ro of of th e Deligne conjecture, based on the Etingof-Kazhdan quant ization. Recall the deﬁnitions of th e dg op erads B ∞ and B Lie . By d eﬁ nition, a vec tor space X is an algebra ov er the op erad B ∞ if there is a str u cture of a d g asso ciativ e b ialgebra on the cofree coalg ebra F ∨ Asso c ( X [1]) such that the coalgebra structure coincides with the giv en one. This deﬁn ition leads to the follo wing data (see [H], Section 5 and [GJ], Section 5 for more d etail): (1) a d iﬀeren tial d : X [1] ⊗ n → X [1] ⊗ m of d egree 1, m, n ≥ 1 b eing a diﬀerential of th e free coalgebra stru cture is uniqu ely deﬁned by the pro jections to the cogenerators. W e denote them m n : X [1] ⊗ n → X [2]; (2) the algebra structure, it is also given b y the pro jection to X [1]. These are m aps m pq : X [1] ⊗ p ⊗ X [1] ⊗ q → X [1], or m pq : X ⊗ p ⊗ X ⊗ q → X [1 − p − q ]. These data should d eﬁ ne th r ee series of equations: the ﬁr st come from the asso ciativit y of { m pq } , the second come from the f act that d is a deriv at ion of the algebra stru ctur e, and the third comes from the condition d 2 = 0. These equations deﬁne a v ery complica ted op erad B ∞ . 22 It is a remark able and sur p rising r esult of Get zler-Jones [ GJ], Section 5, t hat X = Ho c h q ( A ), A an arbitrary associativ e a lgebra, is an algebra o v er the op erad B ∞ . This structure is deﬁned as follo ws: (1) m 1 is the Ho chsc hild diﬀerent ial; (2) m 2 is the cup-p r o duct on Ho c h q ( A ); (3) m i = 0 for i ≥ 3; (4) m 1 k ( f ⊗ g 1 ⊗ · · · ⊗ g k ) is the b race op eration f { g 1 , . . . , g k } deﬁned b elo w; (5) m ak = 0 for a ≥ 2. No w is the deﬁnition of the braces due to Getzler-Jones. It is b etter to describ e it graphically , as is shown in Figure 2. ..... f g g g 1 2 k Figure 1: The br ace op eration: we inser t g 1 , . . . , g k in to arguments of f , pr eserving the order of g 1 , . . . , g k , w ith the natur al sign, and tak e the sum ov er all p ossible in s ertions Let us emphasize again that it is a h ighly non-evident f act, p ro v en by a direct com- putation, that in th is wa y we m ake a B ∞ algebra structure on Ho c h q ( A ). The r ole of this construction is that the c ohomology op erad of the dg op erad B ∞ is equal to the Gerstenhab er op erad G (probably , ev en to pro v e this fac t w e need the Etingof-Kazhdan quan tization). So the idea is to prov e that there is a qu asi-isomorphism of op erads G B ∞ , and then we can consider Ho c h q ( A ) as G ∞ algebra for any asso- ciativ e alg ebra A . Th e only trancendenta l step in the T amarkin’s construction is this quasi-isomorphism of op erads G B ∞ . T ec hnically it is d one as f ollo ws. Int ro d uce some op erad B Lie as follo ws. A v ecto r space X is an algebra o v er the op erad B Lie if there i s a dg Lie bialgebra stru cture on the free L ie coalg ebra F ∨ Lie ( X [1]) generated by X [1] such th at the Lie coalgebra structure coincides with th e given one. The op erad B Lie is also quasi-isomorphic to the Gerstenhab er op erad G . Moreo ve r, there is a sim p le construction of a quasi-isomorphism G ∞ → B Lie , as follo ws. Let Y b e an algebra o v er B Lie . This means that there is a Lie dg bialgebra stru cture on the free coalge bra g = F ∨ Lie ( Y [1]). In particular, g is a Lie algebra, an d this deﬁn es a 23 diﬀeren tial on the Lie c hain complex F ∨ Com ( g [1]). Thus w e get a diﬀerent ial on the fr ee Gerstenhab er coalgebra F ∨ G ∨ ( Y [1]) cogenerated b y Y [1], w hic h by deﬁn ition means that Y is a G ∞ -algebra. This assignment is functorial, an d therefore giv es a map of op erads G ∞ → B Lie , which easily c hec ke d to b e a q u asi-isomorphism. So, the conclusion is that the op erad B Lie can b e connected to the Gerstenhab er op erad in a very simple wa y , and no w we sh ould connect th e op erad B ∞ with the op erad B Lie . This is giv en exact ly by the Etingof-Kazhdan (de)quan tizat ion (see [T1] a nd [H], Section 7 for detail). The Etingof-Kazhdan dequantiza tion is applied in a sense to F ∨ Asso c (Ho ch q ( A )[1]) which is an asso ciativ e bialgebra b y the Getzler-Jones braces’ con- struction. R e mark. Let P b e a P oisson a lgebra. Its P oisson complex P ois q ( P ) is deﬁned as th e dg space of co d eriv ati ons of the free coalge bra o v er the dual coop erad P ! ∗ [1] b y the space P [1]. Th is space of co deriv ations is naturally equip p ed with a diﬀeren tial d Po is arising from the Poisson br ac k et and the prod uct on P . The author thinks that for an y P oisson algebra P the P oisson complex P ois q ( P ) is an algebra o v er the op er ad B Lie . This structure is deﬁned exactly by some generalization of the braces constru ction. No w, if P = S ( V ∗ ) b e the P oisson alge br a with zero b rac k et, then by Getzler-Jones Ho c h q ( P ) is a B ∞ algebra, and Po is q ( P ) is a B Lie algebra. The author thin ks that some the most natural Etingof-Kazhdan d equan tization give s from the a sso ciativ e bialgebra F ∨ Asso c (Ho ch q ( P )[1]) the Lie bialgebra F ∨ Lie (P ois q ( P )[1]). So far, the author do es not kno w an y dir ect pro of of the last fact. App endix Here we explain a construction of the K on tsevic h f ormalit y morphism o v er Q . The usual construction u s es the Drinfeld’s asso ciator o v er Q and the T amarkin’s theory . This asso ciator is not giv en by an explicit form ula, it is constructed in [D] by pro ving that all asso ciators o v er Q form a torsor o v er the Grothendiec k-T eic hm ¨ uller group. The K nizhnik- Zamolo dc hik o v asso ciator giv es an example of asso ciator o ve r C ; th erefore, there exists an asso ciator o v er Q . It p ro v es that this torsor is trivial o v er Q . Then the torsor is trivial also o ve r Q , b ecause th e Grothendiec k-T eic hm ¨ uller group is pro-unip oten t. Here we prop ose a diﬀerent pro of, w hic h seems to b e more constructiv e. This a p- proac h seems to b e n ew. It follo ws from the p revious resu lts that if w e su cceed to construct a qu asi- isomorphism of op erads B ∞ → G o v er Q , we will b e done. Consider the oper ad B ∞ . It is a dg o p erad. All dg op erads form a closed model catego ry b ecause they are algebras o v er some univ ersal colored op erad. In particular, there is a homotop y operad structure on the cohomology of B ∞ , giv en by a kin d o f ”Massey op erations”. Th is homotop y op erad structure clearly is deﬁned o v er Q . T o construct it explicitly , w e sh ould ﬁrstly s plit B ∞ as a complex into a dir ect sum of its 24 cohomology and a contrac tible complex and, secondly , to con tract this complex explicitly b y a homotopy . It is clear that these t w o steps can b e p erformed o ve r Q . No w we ha v e t w o homotopy operad structures on G : one is the Gerstenhab er op erad, and anot her one is giv en by the Massey op erations. Moreo v er, there is a formal family of homotop y dg op erads dep ending on ~ w hose v alue at ~ = 1 is the ”Massey” homotop y op erad structure on G , and whose v alue at ~ = 0 is th e Gerstenhab er op er ad . W e kno w from the T amarkin’s theory describ ed ab o v e th at this deformation is trivial o v er C , b ecause the ”Massey” homotop y op erad is quasi-isomorphic to B ∞ b y constr u c- tion, whic h is quasi-isomorphic o v er C to the Gerstenhab er op erad b y Sectio n 2.5. W e are going to pro v e that this formal d eformation is trivial also o v er Q . F or this it is suﬃcien t to prov e that in ﬁnitesimally this deformation is trivial ov er Q at ea c h 0 ≤ ~ ≤ 1. Consid er a r esolution R q ( G ) of the Gerstenhab er operad o v er Q ; as G is a Koszul op erad, we can tak e its Koszul r esolution. Cons id er the truncated deformation complex D + ( G ) = Der ( R q ( G )). W e need to pro v e that all inﬁn itesimal deformations giv e trivial classes in H 1 ( D + ( G )). It is probably not true that H 1 ( D + ( G )) = 0, it w ould b e ve ry unexp ected, b ecause T amarkin imbed ed in [T3] the Grothendiec k-T eic hm ¨ uller Lie algebra into H 0 ( D + ( G )). But we kno w that a ll classes are trivial in H 1 ( D + ( G ) , C ) from the T amarkin’s th eory . As the complex D + ( G ) is deﬁn ed ov er Q , we hav e that the natural map H q ( D + ( G ) , Q ) ֒ → H q ( D + ( G ) ⊗ Q C ) = H q ( D + ( G ) , C ) (36) is an embedd ing. Therefore, all our inﬁn itesimal classes are trivial o ve r Q , and we get th at the global formal deformation is trivial o ve r Q . W e think th at this s p eculation is as exp licit as it can giv e some explicit formulas for the Kontsevi c h’s formalit y o v er Q . W e are going to describ e it in detail in the sequel. 3 Tw o inﬁnite-dimensional v arieties (and a morphism b e- t w e en them) In this Section w e asso ciate with eac h quasi-isomorphism of op er ad s Θ : G ∞ → B ∞ deﬁned up to homotopy an L ∞ morphism U (Θ) : T poly ( V ) → Ho c h q ( V ) (for an y vecto r space V ) deﬁned u p to homotop y . W e show that this L ∞ morphism is giv en by a univ ersal form ula, that is by p rediction of some weigh ts to all Kon tsevic h graphs from [K97], but our graphs may cont ain simple lo ops. T he image of th e map Θ 7→ U (Θ) gives that L ∞ morphisms for deformation q u an tizations asso ciated with w hic h we can prov e our Koszul dualit y Theorem. 25 3.1 F ew words ab out homotop y Starting from no w, we often use the w ord ”homotop y” in th e con text like ”homotopical map of dg op erads” or ”homotopical L ∞ morphisms”. Here are some generalities on this. The Qu illen’s formalism of closed mo del categories [Q] giv es a to ol f or the inv erting of qu asi-isomorphisms in a non-ab elian case. Let O b e an op erad. Consider the category D GA ( O ) of dg algebras o v er O . W e w an t to c onstru ct a univ ersal c ategory in whic h the quasi-isomorphism s in D GA ( O ) are inv ertible. This category can b e constru cted for an y operad O and is called th e homotop y category of D GA ( O ), b ecause th e category D GA ( O ) admits a closed mo d el s tructure in wh ic h the weak equiv alences coincide w ith the quasi-isomorphisms [ H2]. There are several constructions of this category , but al l them are equiv a len t du e to th e univ ersal prop erty with resp ect to the lo calization by quasi-isomorphisms. In Section 5 we recall a very explicit construction in th e case when O is a Koszul op erad. Con trary , the d g algebras o v er a PR OP do not form a closed mod el category (the Quillen’s Axiom 0 that the categ ory admits all ﬁnite limits and colimits fails in this case; for example we do not kno w what is a free algebra o ve r a PR OP). Therefore, for dg algebras o ve r a P R OP any construction of the homotopical category (to the b est of our kno wledge) is not kno wn. On the other hand, all dg op erads form a closed mo del category as algebras o v er the unive rsal colored op erad, therefore, for dg op erads the Quillen’s construction works. In the sequel we will skip some d etails concerning that homotopical maps of op erads induce homotopical maps of dg algebras, a v oiding to enlarge this a lready rather lo ng pap er. Only what we need to kno w is that the homotopical category is unique, and in the ﬁ nal step we use a p articular construction of it for d g Lie algebras in Section 5, appropriate for our needs. 3.2 The Kontsevic h’s v ariety K The Kon tsevic h’s v ariet y K is the v a riet y of al l universal L ∞ quasi-isomorphisms T poly ( V ) → Hoch q ( S ( V ∗ )) d eﬁ ned for all vec tor sp aces V . An y suc h universal L ∞ mor- phism is by deﬁn ition given by pr ed iction of some complex w eigh ts W Γ to all Kontsevic h graphs Γ from [K97] p ossibly w ith s imple lo ops and not connected. Th ese W Γ are su b- ject to some qu adratic equations, arisin g from the L ∞ condition. Th e ﬁrst Ho c hschild- Kostan t-Rosen b erg graph h as the ﬁxed weig ht, as in the K on tsevic h’s pap er [K97]. Th is v a riet y is n ot empty as is pro v en in [K9 7]. The homotopies acts by gauge a ction (see Section 5). 26 3.3 The T amarkin’s v ariety T The T amarkin’s v a riet y in ou r strict sense is formed from all quasi-isomorphisms of op erads G ∞ → B ∞ whic h are identi t y on cohomology , mo dulo homotopies. As G ∞ is a free dg op erad, an y su c h map is u niquely deﬁned b y the generators G [ − 2]. S o, a map of op erads G ∞ → B ∞ is giv en by a map of v ector spaces G [ − 2] → B ∞ sub ject to some quadratic relations arose from th e compatibilit y with the diﬀerentia ls in the dg op erads. This v ariet y is not empty b ecause we ha v e constructed in S ection 2, follo wing [T1], such a particular qu asi-isomorphism. In a wider setting, one can consider O p ∞ maps of dg op erads G ∞ → B ∞ , b ut we do not do this. 3.4 A map X : T → K Supp ose a p oint t of t he T amarkin’s v ariet y K is ﬁxed. Then the Hoc hschild complex Ho c h q ( A ) of an y algebra A has a ﬁ xed s tructure of homotop y Gerstenhab er algebra (ﬁxed mo dulo h omotop y). Consider the case A = S ( V ∗ ) for s ome v ector sp ace V . Then we get, as is explained in Section 2, t w o structures of G ∞ algebra on T poly ( V ) whic h are sp eciﬁcations of some formal deformation at ~ = 0 and ~ = 1. Then they sh ould coincide, up to a homotop y , b ecause the ﬁrst cohomology H 1 (Co der( F ∨ G ∗ [2] ( T poly ( V )))) = T 2 poly ( V ) b y Theorem 2.4, and there is no Aﬀ ( V )-inv arian t classes (but our formal deformation is Aﬀ -in v arian t). Th us w e get a map of G ∞ algebras X 0 ( t ) : T poly ( V ) → Ho ch q ( S ( V ∗ )), where T poly ( V ) is considered with the standard Sc houten-Nijenh uis Gerstenhab er stru ctur e, and the G ∞ structure on Ho ch q ( S ( V ∗ )) dep ends on the p oin t t ∈ K . Then w e r estrict it to the Lie op erad and get an L ∞ map X ( t ) : T poly ( V ) → Ho ch q ( S ( V ∗ )) wh ic h is an L ∞ morphism for the standard Lie structur es on T poly ( V ) an d Ho c h q ( S ( V ∗ )), and this L ∞ morphism is d eﬁned up to homotop y . Th is is the construction of the map X . It is, although, n ot pro v en y et th at X ( t ) is deﬁn ed un iquely up to a h omotopy . Lemma. F or a ﬁxe d V , the L ∞ morphism X ( t ) is uniquely deﬁne d up to a homotopy. Pr o of. Su pp ose there are t w o diﬀerent G ∞ morphisms, f or the same ﬁ xed G ∞ structure on Ho ch q ( S ( V ∗ )). Then we can get d eﬁned up to a homotop y G ∞ quasi-automorphism of T poly ( V ). It has the identit y ﬁrst T a ylor comp onent b y the constructions (a p oint t ∈ T is deﬁn ed such we get th e canonical Gerstenhab er stru ctur e on the cohomology of Ho c h q ( A ) for any A ). T herefore, the logarithm of this automorph ism is w ell deﬁn ed and giv es a G ∞ deriv ati on of T poly ( V ). No w we us e the computation of Theorem 2.4 for 0-th cohomology: H 0 (Co der( F ∨ G ∗ [2] ( T poly ( V )))) = T 1 poly ( V ) is the vecto r ﬁ elds. Again, th er e are no Aﬀ ( V )-iv arian t ve ctor ﬁelds. Th erefore, o ur deriv ation is inner. But then it is zero, b ecause any inn er deriv ation acts non -trivially on the ﬁr s t T aylo r comp onent wh ic h is ﬁ xed to b e iden tit y . T h us, w e h a v e pro v ed that X ( t ) is w ell-deﬁned up to h omotop y 27 as G ∞ map, and therefore the same is true for its L ∞ part. (Compare with Lemma in the end of S ection 4.4). No w we pro ve th e f ollo wing almost eviden t corollary of the pr evious Lemma: Theorem. The L ∞ morphism X ( t ) is a universal L ∞ morphism , that is, it is given by pr e diction of some weights to al l Kontsevich gr aphs, p ossibly non-c o nne cte d and with simple lo ops , and these weights up to a homotopy do not dep end on the ve ctor sp ac e V . Pr o of. Let W ⊂ V b e a subspace. Th en w e ca n decomp ose V = W ⊕ W ⊥ , and a G ∞ structure on Ho c h q ( S ( V ∗ )) d eﬁ nes a G ∞ structure on Ho c h q ( S ( W ∗ )). C learly (b e- cause the G ∞ structures are gl ( V )-in v ariant ) it is, up to a homotop y , the structure on Ho c h q ( S ( W ∗ )) one gets from the map of operads t : G ∞ → B ∞ . W e hav e then t w o deﬁnitions of L ∞ morphisms T poly ( W ) → Ho ch q ( S ( W ∗ )): one is the direct X ( t ) W , and the second one is the restriction to W of X ( t ) V . They coincide up to a h omotop y by the Lemma ab o v e, b ecause the tw o G ∞ structures on Ho c h q ( S ( W ∗ )) are the same up to a homotop y . The remaining part of the T heorem (that the L ∞ morphism is giv e n b y a unive rsal formula though Kontsevic h graph s) f ollo ws from the gl ( V ) in v ariance of it. Is is not kno wn if th e map t 7→ X ( t ) is surjectiv e, ev en when w e allo w t to b e a n O p ∞ map of dg op erads G ∞ → B ∞ . Our Main Th eorem of this p ap er ab out the Koszul dualit y holds for the s tar-pro duct obtained fr om any L ∞ morphism in the image of X , U = X ( t ), b y the usual formula f ⋆ g = f · g + ~ U 1 ( α )( f ⊗ g ) + 1 2 ~ 2 U 2 ( α, α )( f ⊗ g ) + . . . (37) where α is a (quadratic) Poisson b iv ector ﬁ eld on V . 4 Koszul dualit y and d g categories 4.1 Some generalities on dg categories W e give here some basic d eﬁnitions on d g categories. W e deﬁn e only the things we will directly u se, see [Kel3] for muc h more detailed and sophisticated o v erview. A diﬀer ential gr ad e d (d g) c ate gory A ov e r a ﬁeld k is a cat egory , in whic h the sets of morphisms A ( X, Y ) b et w een an y t w o ob jects X, Y ∈ Ob( A ) are k -linear dg spaces (complexes of k -v ector spaces) su c h that the comp ositions are deﬁ ned as m aps A ( Y , Z ) ⊗ A ( X, Y ) → A ( X, Z ) for an y X , Y , Z ∈ Ob( A ) whic h are maps of c omp lexes . I n the last condition we regard A ( Y , Z ) ⊗ A ( X , Y ) as a complex with the diﬀerent ial deﬁ n ed by the Leibniz ru le d ( f ⊗ g ) = ( d f ) ⊗ g + ( − 1) deg f f ⊗ ( dg ) (38) 28 It is clear that a diﬀerentia l graded category w ith one ob ject is just a d iﬀeren tial graded asso ciativ e algebra. Then dg categories can b e consid ered as ”dg a lgebras with many ob jects”. F or dg algebras w e hav e a deﬁnition wh en a map F : A q → B q is a quasi-isomorphism : it means that the m ap F is a map of algebras and induces an isomorphism on cohomology . Suc h a map in general is not in v ertible, it can b e in ve rted only as an A ∞ map. What should b e a deﬁnition of a quasi-i somorph ism for dg c ate gories ? W e say that a fun ctor F : A → B b et w een t w o dg catego ries A and B is a q u asi- e quivalenc e if it is a fun ctor, wh ic h is k -linear on m orp hisms (and, as su c h, preserv es ten- sor comp ositions of morphisms ), indu ces an equ iv a lence on the leve l of cohomology , and is essenti ally sur jectiv e. The last condition means that for a dg category A we can con- sider the category H q ( A ) with the same ob jects, and w ith ( H q A )( X, Y ) = H q ( A ( X, Y )). Then a quasi-equiv alence is not in v ertible i n general, but it can b e in v erted as an A ∞ quasi-equiv a lence b et w een t w o dg catego ries. W e will not use this concept directly , and w e refer to the reader to give the deﬁn ition. No w if w e ha ve a dg algebra, w e kno w wh at is the cohomological Ho c hschild complex of it. It go v erns the A ∞ deformations of the dg algebra. It is p ossible to deﬁne the Ho chs child c ohomo lo gic al c omplex of a dg c at e gory . This will b e in a sense the main ob ject of ou r study in this pap er for some particular dg category , namely , for the B.Keller’s dg catego ry in tro duced in the next Sub s ection. Let u s giv e th e d eﬁnition of it. A t ﬁrst, it is the t otal pr o duct complex of a bicomplex. Th e v ertica l d iﬀeren tial will b e the inn er diﬀeren tial app eared from the diﬀeren tial on A ( X , Y ) for any pair X, Y ∈ Ob A . The horizon tal diﬀerent ial will an analog of th e Ho c hsc hild cohomological diﬀeren tial. The columns ha v e d egrees ≥ 0. In degree 0 we ha ve Ho c h ∗ 0 ( A ) = Y X ∈ Ob A A ( X, X ) (39) and in degree p ≥ 1 Ho c h ∗ p ( A ) = Y X 0 ,X 1 ,...,X p ∈ Ob A Hom k ( A ( X p − 1 , X p ) ⊗A ( X p − 2 , X p − 1 ) ⊗· · ·⊗A ( X 0 , X 1 ) , A ( X 0 , X p )) (40) where t he pro duct is ta ke n o v er all c hains of ob j ects X 0 , X 1 , . . . , X p ∈ Ob A of length p + 1. The Ho chsc hild d iﬀeren tial d Hoch : Ho c h ∗ p ( A ) → Ho ch ∗ ,p +1 ( A ) is deﬁn ed in th e n at- ural wa y . Let us n ote that ev en if a co chain Ψ ∈ Ho c h ∗ p ( A ) is non-zero only for a single c hain of ob jects X 0 , X 1 , . . . , X p , its Ho c hsc hild diﬀerentia l d Hoch Ψ in general is non-zero on many other chains of ob jects. Na mely , at ﬁrst it ma y b e nonzero for on an y chain X 0 , . . . , X i , Y , X i +1 , . . . , X p for 0 ≤ i ≤ p − 1 (41) suc h that th ere are nonzero comp ositions A ( Y , X i +1 ) ⊗ A ( X i , Y ) → A ( X i , X i +1 ) (this is 29 corresp onded to the regular terms in the Ho chsc hild diﬀerent ial), and on the chains Z − , X 0 , . . . , X p (42) and X 0 , . . . , X p , Z + (43) suc h that there are non-zero comp ositions A ( X 0 , X p ) ⊗ A ( Z − , X 0 ) → A ( Z − , X p ) and A ( X p , Z + ) ⊗ A ( X 0 , X p ) → A ( X 0 , Z + ) (this is corresp onded to the left and to the right b ound ary terms in the Ho chsc hild diﬀerenti al, corresp ond ingly). The Ho chsc hild cohomologic al complex of a dg category A is a dg Lie algebra with the direct generaliza tion of the Gerstenhab er br ack et . The solutions of the Maurer-Cartan equation of Ho c h q ( A ) ⊗ k [[ ~ ]] giv e the formal deformations of the dg category A as A ∞ catego ry . 4.2 The B.Keller’s dg category cat ( A, B , K ) In tro du ce here the main ob j ect of our story—the B. Keller’s dg category cat ( A, B , K ). Here A and B are t w o dg asso ciativ e algebras, and K is B - A -bimo d ule. The dg categ ory A = cat ( A, B , K ) has t w o ob jects, called sa y a and b , su c h th at A ( a, a ) = A , A ( b, b ) = B , A ( a, b ) = 0, A ( b, a ) = K . Only what w e n eed f rom K to deﬁn e such a dg category is a structure on K of diﬀerential grad ed B - A -bimo du le. Consider in details the Ho c hsc hild complex of the category cat ( A, B , K ). It con tains as s u bspaces Ho c h q ( A ) and Ho c h q ( B ), th e usu al Ho chsc hild cohomologi cal complexes of the algebras A and B , and also it con tains the subsp ace Ho c h q ( B , K, A ) = X m 1 ,m 2 ≥ 0 Hom( B ⊗ m 1 ⊗ K ⊗ A ⊗ m 2 , K )[ − m 1 − m 2 ] (44) As a graded space, Ho c h q ( cat ( A, B , K )) = Ho ch q ( A ) ⊕ Ho c h q ( B ) ⊕ Hoch q ( B , K, A ) (45) but certainly it is not a dir ect s um of sub c o mplexes . Namely , Ho ch q ( B , K, A ) is a sub- complex of Ho ch q ( cat ( A, B , K )), but Ho c h q ( A ) and Ho ch q ( B ) are not . There are well- deﬁned pro jections p A : Ho c h q ( cat ( A, B , K )) → Hoc h q ( A ) and Hoc h q ( cat ( A, B , K )) → Ho c h q ( B ). The Ho chsc hild comp onen t of the total diﬀeren tial acts lik e this: X 3 d K Hoch   X 1 d A Hoch 4 4 d AK Hoch = = | | | | | | | | X 2 d B Hoch j j d BK Hoch a a B B B B B B B B (46) 30 where X 1 = Ho c h q ( A ), X 2 = Ho c h q ( B ), X 3 = Ho c h q ( B , K, A ). In [Kel1], Bernh ard Keller p oses the f ollo wing qu estion: what is a suﬃ- cien t condition on the trip le ( A, B , K ) wh ic h would guarantee that the p r o jections p A : Ho c h q ( cat ( A, B , K )) → Ho c h q ( A ) and Hoch q ( cat ( A, B , K )) → Ho c h q ( B ) are quasi- isomorph isms of c ompl exes ? (They are alw a ys maps of dg Lie algebras, it is clear). The answ er is giv en as follo ws: it is enough if the follo wing conditions are satisﬁed: Consider the left act ion of B on K . It is a map of right A -modu les, and w e get a map L 0 B : B → Hom mod − A ( K, K ). W e ca n also deriv e this map to a map L B : B → RHom mod − A ( K, K ). Analogously , we deﬁne from the right A -act ion on K th e map R A : A opp → RHom B − mod ( K, K ). Deﬁnition. Let A and B b e tw o dg asso ciativ e algebras, and let K b e d g B - A -bimo d ule. W e sa y th at the trip le ( A, B , K ) is a K eller’s admissible triple if the maps L B : B → RHom mod − A ( K, K ) R A : A opp → RHom B − mod ( K, K ) (47) are quasi-isomorphisms of algebras. There are kno wn tw o examples wh en the Keller’s condition is satisﬁed: (1) A is an y dg associativ e alg ebra, and there is a m ap ϕ : B → A whic h is a quasi- isomorph ism of algebras. W e set K = A with the tautological stru cture of right A -mo dule on it, and with the left B -mo dule structure give n by the map ϕ ; (2) A is a qu ad r atic Koszul algebra, B = A ! is the K oszul d ual algebra, and K is the Koszul complex of A considered as a B - A -bimo d ule. The b oth statemen ts are pro v en in [Kel1]. The theory d ev elop ed in Section 1 makes the generalizat ion of (2) for Koszul algebras ov e r discr ete v a luation rings straight forward. The follo wing theorem w as foun d and prov en in [Kel1]: Theorem. L et ( A, B , K ) b e a Kel ler’s adm issible triple. Then th e natur al pr o je ctions p A : Hoch q ( cat ( A, B , K )) → Ho c h q ( A ) and p B : Ho ch q ( cat ( A, B , K )) → Ho c h q ( B ) ar e quasi-isomorphisms of dg Lie algebr as. Pr o of. Let t : L q → M q b e a map of complexes. Recall that its cone Cone( t ) is d eﬁned as Cone( t ) = L q [1] ⊕ M q with the d iﬀerential giv en by matrix d =  d L [1] 0 t [1] d M  T o p ro v e that the map t : L q → M q is a quasi-isomo rph ism, it is equiv alen tly than to pro v e that the cone Con e( t ) is acyclic in all degrees. 31 Let u s consider th e cone Cone( p A ) w here p A : Ho c h q ( cat ) → Ho c h q ( A ) is the n atur al pro jection. Let us prov e that if the ﬁrs t cond ition of (47) is satisﬁed, the cone Cone( p A ) is acyclic. W e can regard C one( p A ) as a bicomplex wher e the v ertical diﬀerent ials are the Ho c hsc hild diﬀerential s and the h orizon tal diﬀerential is p A [1]. This bicomplex h as t w o columns, therefore its sp ectral sequences con v erge. Compute ﬁrstly the diﬀeren tial p A [1]. Then the term E 1 is the sum of Hoch q ( B ) ⊕ Ho c h q ( B , K, A ), as a graded v ecto r space. Th ere are 3 comp onents of the d iﬀeren tial in E 1 : the Ho c hschild d iﬀeren tials in Ho c h q ( B ) an d in Ho c h q ( B , K, A ), and exactly the same diﬀerentia l d B K Hoch : Ho c h q ( B ) → Ho c h q ( B , K, A )[1], as in th e Ho c hschild complex of the category Ho c h q ( cat ( A, B , K )). Compute ﬁ r stly the cohomolog y of Hoc h q ( B , K, A ) with the only Ho c hsc hild diﬀer- en tial. One can write: Ho c h q ( B , K, A ) = Hom( T ( B ) , Hom( K ⊗ T ( A ) , K )) (48) with some diﬀeren tials, where we den ote by T ( V ) the free associativ e algebra gener- ated by V . More precisely , the term Hom C ( K ⊗ T ( A ) , K ) is equal to the complex Hom mod − A (Bar q mod − A ( K ) , K ) of maps from the bar-resolution of K in the catego ry of righ t A -mo du les to K . This is equal to RHom mod − A ( K, K ), whic h is quasi-isomorph ic to B by the ﬁ rst Keller’s condition. But this is not all w h at we n eed–w e also need to know that the left B -mo du le str u ctures on B and on RHom mod − A ( K, K ) are the same. Th is is exactly guaran teed b y the Keller’s condition, wh ic h says that th e quasi-isomorphism B → RHom mod − A ( K, K ) is induc e d by the left action of B on K . No w w e ha ve t w o complexes, which are exactly the same, and are Ho c h q ( B ), bu t there is also the comp onent d B K Hoch from one to another. I n other words, so far our complex is the cone of the iden tit y map from Ho ch q ( B ) to itself, and this cone is clearly acyclic. W e hav e pr o v ed that if the ﬁr st Keller’s condition is satisﬁed, the natural pro jection p A : Ho c h q ( cat ) → Ho c h q ( A ) is a quasi-isomorphism. If the second Keller’s condition is satisﬁed, w e conclude, analogously , that the pro jection p B : Ho c h q ( cat ) → Ho c h q ( B ) is a quasi-isomorphism. B. Keller us ed this theorem in [Kel1] to sho w that in the tw o cases listed ab ov e when the Keller’s conditions are satisﬁed, the Ho chsc hild cohomological complexes of A and B are quasi-isomorphic as dg Lie algebras. In p articular, this is true wh en A and B are Koszul du al algebras, th e case of the most in terest for us. R e mark. If A and B are Koszul dual algebras, b ut K is replaced by C , the only cohomol- ogy of the Koszul complex, w e still h a v e the quasi-isomorphism s B → RHom mod − A ( C , C ) and A opp → RHom B − mod ( C , C ), but these maps ar e not induc e d by the left (c orr esp ond- ingly, right) actions of B (c orr esp o ndingly, A ) on C . These actions deﬁne some stu pid maps wh ic h are not quasi-isomorphisms. This example shows that the Keller’s dg cate- gory in this case ma y b e n ot quasi-equiv ale nt (and it is really the case) to its homology dg category . 32 4.2.1 The Keller’s condition in the (bi)graded case As we already mentioned in Remark in Section 0.3, wh en the algebras S ( V ∗ ) and Λ( V ) are considered ju st as associativ e algebras, they are not Koszul dual. Namely , Ext Λ( V ) ( k , k ) = S [[ V ∗ ]], th e formal p ow er series instead of p olynomials. T o a vo id this problem, w e should w ork in the ca tegory of algebras with inner Z -grading and w ith c oh omolo gic al Z -grading. Finally , Λ k ( V ) should ha v e the inner grading k and the coho- mologica l grading k , wh ile S k ( V ∗ ) has the inn er grading k and the cohomologic al grading 0. Th en we sh ould switc h to the category of bigraded m o dules, and compu te Ext alge- bras in this category . In th is d eﬁnition su c h Ext algebras will b e automatically bigraded. This completely agrees with the theory of Koszul du ally d iscussed in Section 1. The only problem is that the Keller’s Theorem 4.2 wa s pro v en ab o v e for the category of graded alge bras when only the cohomologi cal grading is taken in to the acc ount. T o mak e this Theorem v alid for the bigraded case, w e should mo dify the deﬁnition of the Ho c hsc hild cohomological complex of a bigraded algebra and of a bigraded d g catego ry . W e giv e the follo wing d eﬁ nition. Let A b e a bigraded algebra (one grading is inner and another one is cohomological). W e deﬁne the graded Ho c hsc hild complex Ho c h q gr ( A ) as the d irect sum of its bigraded comp onents. The same deﬁn ition will b e d one for bigraded dg categories. In general, it is not true that the graded Ho c hsc hild complex is qu asi-isomorphic to the u sual one (whic h is graded only with resp ect to the cohomolog ical grading but not with r esp ect to the inn er). They are quasi-isomorphic for A = S ( V ∗ ), b ut for A = Λ( V ) this quasi-isomorphism fails. Indeed, the cohomolog y o f th e usu al Hochsc hild complex for A = Λ( V ) is the forma l p olyv ector ﬁ elds on V wh ile the cohomology of the graded Ho c hsc hild complex is in this case the p olyn omial p olyv ector ﬁelds on V . Th erefore, if w e need to wo rk in the bigraded category , w e should reprov e Th eorem 4.2 in this case. Let A and B b e t w o asso ciativ e bigraded algebras, and let K b e d g bigraded B − A - bimo du le. W e say th at the triple ( A, B , K ) s atisﬁes the graded K eller’s condition if the natural maps L B : B → RHom g r mod − A ( K, K ) R A : A opp → RHom B − g r mod ( K, K ) (49) are bigr ade d quasi-isomorphisms. Here the deriv ed functors are take n in the cate gories of graded mo d ules. Theorem. L et ( A, B , K ) satisﬁes the gr a de d Kel ler’s c ond ition. Then the natur al pr oje ctions p A : Ho c h q gr ( cat ( A, B , K )) → Ho c h q gr ( A ) and p B : Ho c h q gr ( cat ( A, B , K )) → Ho c h q gr ( B ) ar e qu asi- isomorph isms. The pro of is completely analogous to the u s ual case, an d w e lea v e the details to the reader. In the sequel we will omit the subscript gr with th e notation of the Ho chsc hild complex, alwa ys assuming the theory dev eloping h er e. 33 4.3 The maps p A and p B are m aps of B ∞ algebras Let A, B b e t w o associativ e algebras, and let K b e an y B − A -bimo d ule, not n eces- sarily satisfying the Keller’s condition from S ection 4.2. Th en we ha v e t w o pro jections p A : Ho c h q ( cat ( A, B , K )) → Ho c h q ( A ) and p B : Ho c h q ( cat ( A, B , K )) → Ho c h q ( B ). W e kno w from S ection 2 that the Ho c hschild co mplex Ho c h q ( A ) of an y associativ e algebra has the natur al structure of B ∞ algebra b y means of the Getzler-Jones’ braces (see Fig- ure 2). Th e s ame is true f or Ho ch q ( C ) where C is a d g category , whic h is established by the same b races’ construction. The follo wing s imple Lemma, d ue to Bernhard Keller [Kel1] , is ve ry imp ortan t for our pap er: Lemma. L et A, B b e two asso ciative algebr as, and let K b e a B − A - bimo dule. Then the natur al pr oje ctions p A : Ho ch q ( cat ( A, B , K )) → Ho c h q ( A ) and p B : Ho c h q ( cat ( A, B , K )) → Ho ch q ( B ) ar e maps of B ∞ algebr as. Pr o of. It is clear b ecause the pro jections p A and p B are compatible w ith the braces, and with the cup-pr o ducts. T hat is, they are compatible with the maps m i and m ij of the B ∞ structure, see Section 2.5. 4.4 W e form ul ate a new v ersion of the Main Theorem Let A = S ( V ∗ ), B = Λ( V ), and K = K q ( S ( V ∗ )). W e know from Secti on 1.1 the the algebra S ( V ∗ ) is Koszul, and its Koszul dual A ! = Λ( V ). Th us, we can apply Theorem 4.2 to the trip le ( S ( V ∗ ) , Λ( V ) , K q ( S ( V ∗ ))). W e ha v e constructed a B ∞ algebra Ho c h q ( cat ( A, B , K )) for A, B , K a s ab o v e, and the diagram Ho c h q ( cat ( A, B , K )) p A u u l l l l l l l l l l l l l p B ) ) R R R R R R R R R R R R R Ho c h q ( A ) Ho c h q ( B ) (50) where the t w o righ t maps are maps of B ∞ algebras. Let no w t : G ∞ → B ∞ b e a p oint of the T amarkin’s man if old, see S ection 3.3. T hen th e diagram (50) is a diagram of maps of G ∞ algebras, dep ending on t ∈ T . Let now U = X 0 ( t ) b e the u niv ersal G ∞ morphism G V : T poly ( V ) → Ho c h q ( S ( V ∗ )) deﬁned for all ﬁnite-dimensional (graded) ve ctor spaces V , see Sectio n 3.4. It dep end s on the p oin t t ∈ T and is deﬁn ed up to a homotop y . Denote b y G S ( t ) and G Λ ( t ) the sp ecializatio ns of this univ ersal G ∞ morphism fo r the v ector spaces V and V ∗ [1], corresp ondin gly . Identify T poly ( V ) with T poly ( V ∗ [1]) as in Section 0.1 of the I n tro du ction. Then we hav e the follo wing d iagram of G ∞ maps: 34 Ho c h q ( A ) T poly ( V ) G S ( t ) 8 8 q q q q q q q q q q G Λ ( t ) & & M M M M M M M M M M Ho c h q ( cat ( A, B , K )) p A i i R R R R R R R R R R R R R p B u u l l l l l l l l l l l l l Ho c h q ( B ) (51) dep end ing on t ∈ T . Here and in Sections 6 w e pr o v e the f ollo wing statemen t: Theorem. F or any ﬁxe d t ∈ T , the diagr am (51) is homotopic al ly c o mmutative, that is, it is c ommutative in the Q uil len ’s homotopic al c ate gory. No w restrict ourselv es with the L ∞ comp onent of t he G ∞ maps. Then clearly the diagram remains to b e homotopically commutativ e. W e hav e the follo wing Corollary . (A new version of the Main Theorem) F or any t ∈ T , the diagr am (51) deﬁnes a homotopic al ly c ommutative diagr am of L ∞ maps. W e explai n in Section 7 in detail wh y this Corollary implies the Main Theorem in our previous version, f or Koszul du alit y in deformation qu an tization. No w let us b egin to pr o v e the Theorem ab ov e . Pr o of of The or em (b e ginning) : The p r o of is based on the follo wing K ey-Lemma: Key-Lemma. F or any t ∈ T , the diagr amm (51) deﬁnes a c ommuta tive diagr am of isomorph isms maps on c ohomolo gy. W e pro v e this Lemma in Section 6, and it will tak e some w ork. No w let us explain ho w the T h eorem follo ws f rom the Key-Lemma. The d iagram (51) is a diagram of G ∞ quasi-isomorphisms (the tw o left arrows clearly are quasi-isomorphisms, and the t w o right ones are by the Keller’s Theorem pro v en in Section 4.2). W e can u niquely u p to a homotop y inv ert a G ∞ quasi-isomorphism. Th en the map G ( t ) = ( G Λ ( t )) − 1 ◦ p B ◦ p − 1 A ◦ G S ( t ) (52) is un iquely deﬁned, up to a homotop y , G ∞ quasi-automorphism of T poly ( V ). Now, by the K ey-Lemma, its ﬁrst T a ylor comp onen t is th e iden tit y map. Th en w e can take the logarithm D = log( G ) (53) whic h is a G ∞ deriv ati on of T poly ( V ). W e are in the situation of the follo wing lemma: Lemma. L et G b e an Aﬀ ( V ) -e quivariant G ∞ automorp hism of the Gerstenhab e r algebr a T poly ( V ) with the standar d Gerstenhab er structur e, whose ﬁrst c omp onent i s the identity map. Then the G ∞ automorp hism G is the identity. 35 Pr o of. As ab o v e, we can tak e D = log G , then D is an Aﬀ ( V )-in v ariant G ∞ deriv ati on of T poly ( V ). By Theorem 2.4, this G ∞ deriv ati on is 0. Therefore, G = exp D is th e iden tit y . The T h eorem is now prov en mo d out the Key-Lemma which we pr o v e in Section 6. 5 The homotopical category of dg algebras o v er a Koszul op erad Here w e give , follo wing [Sh 3], a construction of the h omotop y category , appropr iate f or our needs in the n ext Sections of this p ap er. Our emph asis h ere is ho w the homotop y relation reﬂects in the gauge equiv alence condition for deformation quan tizat ion. W e restrict ourselv es with the case of the op erad of Lie algebras b ecause this is the only case w e will use. The constructions for general K oszul op erad are an alogous. Here we use the constru ction of Quillen homotopical category giv en in [Sh3]. In a sense, it is ”the righ t cylinder homotop y relation”. Recall h ere th e deﬁn ition. 5.1 The homotop y relation from [Sh3] Let g 1 , g 2 b e t w o d g Lie algebras. Then th er e is a dg Lie algebra k ( g 1 , g 2 ) whic h is pro-nilp otent and suc h that the solutions of the Maurer-Cartan equati on in k ( g 1 , g 2 ) 1 are exactly the L ∞ morphisms from g 1 to g 2 . Then the zero degree comp onen t k ( g 1 , g 2 ) 0 acts on the Maurer-Cartan solutions, as usu al in deformation theory (the dg Lie algebra k ( g 1 , g 2 ) is pro-nilp oten t), and this action giv es a homotop y relation. The d g Lie algebra k ( g 1 , g 2 ) is constructed as f ollo ws. As a dg v ector s pace, it is k ( g 1 , g 2 ) = Hom( C + ( g 1 , C ) , g 2 ) (54) Here C ( g 1 , C ) is the c hain complex of the d g Lie algebra g 1 , it is n aturally a coun ital dg coalge bra, and C + ( g 1 , C ) is the k ernel of the counit map. Deﬁne no w a Lie brac k et on k ( g 1 , g 2 ). Let θ 1 , θ 2 ∈ k ( g 1 , g 2 ) b e tw o elemen ts. T h eir brac k et [ θ 1 , θ 2 ] is deﬁn ed (up to a sign) as C + ( g 1 , C ) ∆ − → C + ( g 1 , C ) ⊗ 2 θ 1 ⊗ θ 2 − − − − → g 2 ⊗ g 2 [ , ] − → g 2 (55) where ∆ is the copro duct in C + ( g 1 , C ) and [ , ] is the Lie brac ke t in g 2 . It follo ws from the co comm utativit y of ∆ that in this w a y we get a Lie algebra. An elemen t F of degree 1 in k ( g 1 , g 2 ) is a collection of maps F 1 : g 1 → g 2 F 2 : Λ 2 ( g 1 ) → g 2 [ − 1] F 3 : Λ 3 ( g 1 ) → g 2 [ − 2] . . . (56) 36 and the Maurer-Cartan equation d k F + 1 2 [ F , F ] k = 0 is the s ame that the collection { F i } are t he T a ylor components of an L ∞ map which w e d enote also b y F . Note that the diﬀeren tial in k ( g 1 , g 2 ) comes f rom 3 diﬀerentia ls: the b oth inner diﬀeren tials in g 1 and g 2 , an d from the c hain diﬀerent ial in C + ( g 1 , C ). No w th e solutions of the Maurer-Cartan equation f orm a q u adric in g 1 , and for any pro-nilp otent dg Lie algebra g , the comp onent g 0 acts on (the p ro-nilp otent completion of ) this q u adric by v ector ﬁ elds. Namely , eac h X ∈ g 0 deﬁnes a v ector ﬁeld dF dt = − dX + [ X, F ] (57) It can b e d irectly c hec k ed that this v ector ﬁ eld indeed pr eserv es the quadr ic. In our case, this vec tor ﬁ eld can b e exp onen tiated to an action on the pr o-nilp oten t completion on k . This action giv es our homotopy relation on L ∞ morphisms. 5.2 Application to deformation qua n tization Let g 1 , g 2 b e tw o d g Lie algebras, and let F 1 , F 2 : g 1 → g 2 b e tw o h omotopic in the sense of Section 2.4.1 L ∞ morphisms. Let α b e a solution of the Maurer -C artan equation in g 1 . An y L ∞ morphism F : g 1 → g 2 giv es a solution F ∗ α of th e Maur er-Cartan equation in g 2 , by formula F ∗ α = F 1 ( α ) + 1 2 F 2 ( α ∧ α ) + 1 6 F 3 ( α ∧ α ∧ α ) + . . . (58) (supp ose that this inﬁnite sum make s sense). Then in our situation w e ha v e t wo solutions F 1 ∗ α and F 2 ∗ α of the Maur er-Cartan equation in g 2 . Lemma. Supp o se that al l inﬁnite sums (exp onents) we ne e d make sense i n our situation. Supp ose two L ∞ morphism s F 1 , F 2 : g 1 → g 2 ar e homotopic in the sense of Se ction 2.4.1, and supp ose that α is a so lution of the Maur er-Carta n e quation in g 1 . Then th e two solutions F 1 ∗ α and F 2 ∗ α of the M aur er-Cart an e quation in g 2 ar e gauge e quivalent. Pr o of. Let X ∈ k ( g 1 , g 2 ) 0 b e the generator of the homotop y b et wee n F 1 and F 2 . Deﬁne X ∗ α = X ( α ) + 1 2 X ( α ∧ α ) + 1 6 X ( α ∧ α ∧ α ) + . . . (59) Then X ∗ α ∈ ( g 2 ) 0 . Consider the vec tor ﬁeld on ( g 2 ) 1 : dg dt = − d ( X ∗ α ) + [ X ∗ α, g ] (60) Then the exp onen t of this v ector ﬁ eld maps F 1 ∗ α to F 2 ∗ α . 37 6 The main computation Here w e prov e the Key-Lemma 4.4 whic h is only remains to conclude the pro of of Th eorem 4.4. 6.1 W e are going to constru ct ”the Ho chsc hild-Kostan t-Rosen b erg map” ϕ cat H K R : T poly ( V ) → Ho c h q ( cat ( A, B , K )) w here A = S ( V ∗ ), B = Λ( V ), and K is th e Koszul complex of S ( V ∗ ). At the ﬁ nal step of the compuation, w e norm alize the Koszul d iﬀeren tial by dim V , as follo ws: d norm Koszul = 1 dim V dim V X a =1 x a ∂ ∂ ξ a (61) Ho w ev er, in the computation b elo w we su pp ose that the Koszul complex is not n ormal- ized. Th e normalized Koszul complex deﬁnes the equiv alen t Keller’s categ ory , so it is irrelev a nt. Our Ho c hschild-Kosta nt-Ro senberg map ϕ cat H K R will make the follo wing d iagram com- m utativ e (up to a sign) on the cohomology: Ho c h q ( A ) T poly ( V ) G S ( t ) 8 8 q q q q q q q q q q G Λ ( t ) & & M M M M M M M M M M ϕ cat H K R / / Ho c h q ( cat ( A, B , K )) p A i i R R R R R R R R R R R R R p B u u l l l l l l l l l l l l l Ho c h q ( B ) (62) W e did n ot sp ecify the sign, b u t it do es n ot mak e any p roblem. In the co mpu tation b elo w w e use the graphical represen tation of the co chains in Hom(Λ( V ) ⊗ m ⊗ K ⊗ S ( V ∗ ) , K ). The r eader familiar with the Kon tsevic h’s p ap er [K97] will immediately understand our graph ical represen tation. (But for other readers, w e deﬁne our co chains b y the exp licit formulas, see (63)-(65) b elo w). In our graphical co chains, we consider a circle w ith tw o ﬁ x ed p oin ts, 0 and ∞ . The arguments from Λ( V ) are plac ed on the left h alf of the circle, and the argumen ts from S ( V ∗ ) a re placed on the righ t half. An y arro w is the operator P dim V a =1 ∂ ∂ ξ a · ∂ ∂ x a . In our con v en tion, whic h coincides with the one in [K97], the start-p oint of a ny arrow ”diﬀeren tiates” the o dd argument s, w hile the end-p oint diﬀeren tiates the ev en argumen ts. W e hav e one p oin t inside the d isc b ounded b y the circle, where we place our p olyv ector ﬁeld γ . W e use the notati on γ = γ S ⊗ γ Λ (where γ S and γ Λ are the ev en and the odd co ordinates of γ ) and sup p ose that γ is h omogeneous in the b oth x i ’s and ξ j ’s co ordinates. After this general r emarks , let us start. 38 6.2 Some graph-complex The p roblem of a construction of quasi-iso morph ism ϕ cat H K R : T poly ( V ) → Ho ch q ( cat ) is rather non-trivial. Indeed, the usual Ho c hsc hild-Kostan t-Rosen b erg co chains ϕ S H K R ( γ ) ∈ H och ( S ( V ∗ )) and ϕ Λ H K R ( γ ) ∈ Ho c h q (Λ( V )) are not co cycles w hen co nsid ered as co c hains in Ho c h q ( cat ). Indeed, their b oundaries hav e comp onents wh ic h b elong in Hom( K q ⊗ S ( V ∗ ) ⊗ m 1 , K q ) and in Hom(Λ( V ) ⊗ m 1 ⊗ K q , K q ), corresp ondingly . Our map ϕ cat H K R con tain as sum mand the b oth co chains ϕ S H K R and ϕ Λ H K R , and man y other su m- mands. These other summands are the co chains associated with the graphs F 0 0 ,m 2 and F ∞ m 1 , 0 sho wn in Figure b elow. 0 0 0 infinity infinity infinity F F G infinity m m m m m m 0 1 1 1 2 2 2 Figure 2: Th e co c hains F ∞ m 1 ,m 2 , F 0 m 1 ,m 2 , an d G m 1 ,m 2 for m 1 = 3 , m 2 = 4 It is instru ctiv e to formulate th e follo wing Prop osition in a bit m ore generalit y than w e really n eed, for all graphs F 0 m 1 ,m 2 and F ∞ m 1 ,m 2 . Denote the corresp onding maps Φ Γ in Hom(Λ( V ) ⊗ m 1 ⊗ K q ⊗ S ( V ∗ ) ⊗ m 2 , K q ) by F ∞ m 1 ,m 2 ( γ ), F 0 m 1 ,m 2 ( γ ), and G m 1 ,m 2 ( γ ), wh ere γ ∈ T poly ( V ). Supp ose that γ is h omogeneous in b oth x i ’s and ξ ’s. As maps T poly ( V ) → Ho c h q ( cat ) th e maps F 0 m 1 ,m 2 and F ∞ m 1 ,m 2 ha v e degree 0, and the map G m 1 ,m 2 has degree 1. W e ha v e the follo wing explicit formulas for these map s : G m 1 ,m 2 ( γ )( λ ) = 1 n ! 1 m ! dim V X i 1 ,..,i m 1 =1 dim V X j 1 ,...,j m 2 =1 ± k  λ ∧ ∂ ξ i 1 ( λ 1 ) ∧ · · · ∧ ∂ ξ i m 1 ( λ m 1 ) ∧ ( ∂ ξ j 1 ◦ · · · ◦ ∂ ξ j m 2 ( γ Λ ))  × × ( ∂ x i 1 ◦ · · · ◦ ∂ x i m 1 )( γ S ) · ∂ x j 1 ( f 1 ) . . . ∂ x j m 2 ( f m 2 ) (63) 39 F 0 m 1 ,m 2 ( γ )( λ ) = 1 m 1 ! ( m 2 + 1)! dim V X i 1 ,..,i m 1 =1 dim V X a,j 1 ,...,j m 2 =1 ± ∂ x a ( k )  λ ∧ ∂ ξ i 1 ( λ 1 ) ∧ · · · ∧ ∂ ξ i m 1 ( λ m 1 ) ∧ ( ∂ ξ a ◦ ∂ ξ j 1 ◦ · · · ◦ ∂ ξ j m 2 ( γ Λ ))  × × ( ∂ x i 1 ◦ · · · ◦ ∂ x i m 1 )( γ S ) · ∂ x j 1 ( f 1 ) . . . ∂ x j m 2 ( f m 2 ) (64) F ∞ m 1 ,m 2 ( γ )( λ ) = 1 ( m 1 + 1)! 1 m 2 ! dim V X b,i 1 ,..,i m 1 =1 dim V X j 1 ,...,j m 2 =1 ± k  ∂ ξ b  λ ∧ ∂ ξ i 1 ( λ 1 ) ∧ · · · ∧ ∂ ξ i m 1 ( λ m 1 ) ∧ ( ∂ ξ j 1 ◦ · · · ◦ ∂ ξ j m 2 ( γ Λ ))  × × ( ∂ x b ◦ ∂ x i 1 ◦ · · · ◦ ∂ x i m 1 )( γ S ) · ∂ x j 1 ( f 1 ) . . . ∂ x j m 2 ( f m 2 ) (65) Here, as usual, w e denote by { x i } some basis in V ∗ , and b y { ξ i } the d ual b asis in V [ − 1]. Let γ b e a p olynomial p olyv ector ﬁeld in T poly ( V ), homogeneous in b oth x ’s and ξ ’s. Denote deg S γ and deg Λ γ the corresp onding homogeneit y degrees. (W e ha v e the Lie degree d eg γ = d eg Λ γ − 1). Denote d Hoch and d Koszul the Hochsc hild and Koszul components of the diﬀerentia l acting on Ho ch q (Λ( V ) , K q , S ( V ∗ )) ⊂ Ho c h q ( cat ). Prop osition. Supp ose ♯I n Γ ( v ) ≤ d eg S γ and ♯S tar ( v ) ≤ deg Λ γ for e ach sep ar ate gr aph Γ in the claims b elow, wher e v is the only vertex of the ﬁrst typ e. Supp o se that F 0 m,n ( γ ) etc. me ans the sum over al l or derings of the sets S tar ( v ) and I n ( v ) (se e (10) and (11) in the deﬁnitio n of an admissible gr a ph), tha t is, over al l admissible gr ap hs which ar e the same ge ometric al ly. (The sum should b e taken with the appr o priate signs dep ending natur al ly on the or derings). Then we have: (i) d Hoch F 0 m,n ( γ ) = ± G m,n +1 ( γ ) , (ii) d Koszul F 0 m,n ( γ ) = ± d im V · (deg Λ ( γ ) − n ) · G m,n ( γ ) , (iii) d Hoch F ∞ m,n ( γ ) = ± G m +1 ,n ( γ ) , (iv) d Koszul F ∞ m,n ( γ ) = ± d im V · (deg S ( γ ) − m ) · G m,n ( γ ) . Pr o of. The p ro of of Pr op osition is ju st a straigh tforwa rd computation. F or conv enience of the r eader, w e pr esen t it here in all details. W e giv e the pro ofs of (i) and (ii); the pro ofs of the second tw o statemen ts are analo- gous. Pro v e (i). It w ould b e instructiv e f or the reader to recall b efore the pro of the pro of that the clas- sical Ho c hschild-Kosta nt-Ro senberg ϕ S H K R ( γ ) is a Hochsc hild cocycle in Ho c h q ( S ( V ∗ )) 40 for an y γ ∈ T poly ( V ). It goes as follo ws: w e associate with a k -p olyv ector ﬁeld γ the co c hain ϕ S H K R ( γ ) ∈ Hom( S ( V ∗ ) ⊗ k , S ( V ∗ )) deﬁned as ϕ S H K R ( γ )( f 1 ⊗ · · · ⊗ f k ) = dim V X i 1 ,...,i k =1 ± γ ( dx i 1 ∧ · · · ∧ dx i k ) ∂ x i 1 ( f 1 ) . . . ∂ x i k ( f k ) (66) The only n on zero terms ma y app ear when all i 1 , . . . , i k are diﬀeren t,and the sign ± is the s ign of the p erm utation (1 , 2 , . . . , k ) 7→ ( i 1 , i 2 , . . . , i k ). The p ro of th at ϕ S H K R ( γ ) is a Ho c hsc hild cocycle just u ses the Leibniz formula ∂ x a ( f i f i +1 ) = ∂ x a ( f i ) f i +1 + f i ∂ x a ( f i +1 ) and the Ho chsc hild cob oun dary f orm ula d Hoch (Ψ)( f 1 ⊗ · · · ⊗ f k +1 ) = f 1 Ψ( f 2 ⊗ f 2 ⊗ · · · ⊗ f k +1 ) − − Ψ(( f 1 f 2 ) ⊗ f 3 ⊗ . . . ) + Ψ( f 1 ⊗ ( f 2 f 3 ) ⊗ . . . ) ∓ . . . + ( − 1) k +1 Ψ( f 1 ⊗ · · · ⊗ f k ) f k +1 (67) W e see that the all terms will b e m utually ca nceled. Now let u s see when this kind of phenomenon ma y b e destro y ed in the cob oundary of F 0 m,n ( γ ). It is clear that any problem place is the marked p oin t 0 at the b oundary of the circle. Consider the su m of t w o ”problematic” summand s. This is ± F 0 m,n ( γ )( λ 1 ⊗· · ·⊗ λ m ⊗ ( λ m +1 ( k )) ⊗ f n ⊗· · ·⊗ f 1 ) ∓ F 0 m,n ( γ )( λ 1 ⊗· · ·⊗ λ m ⊗ (( k ) f n +1 ) ⊗ f n ⊗· · ·⊗ f 1 ) (68) Here w e use the notati on λ ( k ) and ( k ) f for the l eft action of Λ( V ) and for th e righ t action of S ( V ∗ ), corresp ond ingly . These t wo s ummands giv e fr om (64) ± ∂ x a ( λ m +1 ( k )) = ± λ m +1 ( ∂ x a k ) (69) whic h clearly is canceled with (a part of ) the previous summ an d , ∓ F 0 m,n ( γ )( λ 1 ⊗ · · · ⊗ ( λ m λ m +1 ) ⊗ k ⊗ f n ⊗ · · · ⊗ f 1 ) (70 ) So the ﬁrst su mmand in (68) do es not con tribute to the answer. Contrary , the second summand giv es the term ∂ x a ( k · f n +1 ) = ∂ x a ( k ) · f n +1 + k · ∂ x a ( f n +1 ) (71) The ﬁrst summ and in (71) is canceled with the one of t wo summand s in F 0 m,n ( γ )( λ 1 ⊗ · · · ⊗ λ m ⊗ k ⊗ ( f n +1 · f n ) ⊗ · · · ⊗ f 1 ). The second s u mmand in (71) is not canceled with an other summand, a nd it giv es the only term whic h con tributes to the answer. This term clearly giv es G m,n +1 ( γ ). Pro v e (ii). 41 W e need to compute dim V X i 1 ,..,i m 1 =1 dim V X a,j 1 ,...,j m 2 =1 ± d Koszul { ∂ x a ( k )  λ ∧ ∂ ξ i 1 ( λ 1 ) ∧ · · · ∧ ∂ ξ i m 1 ( λ m 1 ) ∧ ( ∂ ξ a ◦ ∂ ξ j 1 ◦ · · · ◦ ∂ ξ j m 2 ( γ Λ ))  × × ( ∂ x i 1 ◦ · · · ◦ ∂ x i m 1 )( γ S ) · ∂ x j 1 ( f 1 ) . . . ∂ x j m 2 ( f m 2 ) } ∓ dim V X i 1 ,..,i m 1 =1 dim V X a,j 1 ,...,j m 2 =1 ± ∂ x a ( d Koszul k )  λ ∧ ∂ ξ i 1 ( λ 1 ) ∧ · · · ∧ ∂ ξ i m 1 ( λ m 1 ) ∧ ( ∂ ξ a ◦ ∂ ξ j 1 ◦ · · · ◦ ∂ ξ j m 2 ( γ Λ ))  × × ( ∂ x i 1 ◦ · · · ◦ ∂ x i m 1 )( γ S ) · ∂ x j 1 ( f 1 ) . . . ∂ x j m 2 ( f m 2 ) (72) W e hav e: d Koszul k = dim V X p =1 x p ∂ ξ p (73) Then (72) is equ al to dim V X i 1 ,..,i m 1 =1 dim V X a,j 1 ,...,j m 2 =1 ± x p ∂ ξ p { ∂ x a ( k )  λ ∧ ∂ ξ i 1 ( λ 1 ) ∧ · · · ∧ ∂ ξ i m 1 ( λ m 1 ) ∧ ( ∂ ξ a ◦ ∂ ξ j 1 ◦ · · · ◦ ∂ ξ j m 2 ( γ Λ ))  × × ( ∂ x i 1 ◦ · · · ◦ ∂ x i m 1 )( γ S ) · ∂ x j 1 ( f 1 ) . . . ∂ x j m 2 ( f m 2 ) } ∓ dim V X i 1 ,..,i m 1 =1 dim V X a,j 1 ,...,j m 2 =1 ± ∂ x a ( x p ∂ ξ p k )  λ ∧ ∂ ξ i 1 ( λ 1 ) ∧ · · · ∧ ∂ ξ i m 1 ( λ m 1 ) ∧ ( ∂ ξ a ◦ ∂ ξ j 1 ◦ · · · ◦ ∂ ξ j m 2 ( γ Λ ))  × × ( ∂ x i 1 ◦ · · · ◦ ∂ x i m 1 )( γ S ) · ∂ x j 1 ( f 1 ) . . . ∂ x j m 2 ( f m 2 ) (74) 42 where the su mmation o v er p is assumed. C learly (74) is A + B wh er e A = dim V X i 1 ,..,i m 1 =1 dim V X a,j 1 ,...,j m 2 =1 ± x p [ ∂ ξ p { ∂ x a ( k )  λ ∧ ∂ ξ i 1 ( λ 1 ) ∧ · · · ∧ ∂ ξ i m 1 ( λ m 1 ) ∧ ( ∂ ξ a ◦ ∂ ξ j 1 ◦ · · · ◦ ∂ ξ j m 2 ( γ Λ ))  × × ( ∂ x i 1 ◦ · · · ◦ ∂ x i m 1 )( γ S ) · ∂ x j 1 ( f 1 ) . . . ∂ x j m 2 ( f m 2 ) } ∓ dim V X i 1 ,..,i m 1 =1 dim V X a,j 1 ,...,j m 2 =1 ± ( ∂ ξ p ∂ x a k )  λ ∧ ∂ ξ i 1 ( λ 1 ) ∧ · · · ∧ ∂ ξ i m 1 ( λ m 1 ) ∧ ( ∂ ξ a ◦ ∂ ξ j 1 ◦ · · · ◦ ∂ ξ j m 2 ( γ Λ ))  × × ( ∂ x i 1 ◦ · · · ◦ ∂ x i m 1 )( γ S ) · ∂ x j 1 ( f 1 ) . . . ∂ x j m 2 ( f m 2 )] (75) and B = ∓ dim V dim V X i 1 ,..,i m 1 =1 dim V X a,j 1 ,...,j m 2 =1 ± ( ∂ ξ a k )  λ ∧ ∂ ξ i 1 ( λ 1 ) ∧ · · · ∧ ∂ ξ i m 1 ( λ m 1 ) ∧ ( ∂ ξ a ◦ ∂ ξ j 1 ◦ · · · ◦ ∂ ξ j m 2 ( γ Λ ))  × × ( ∂ x i 1 ◦ · · · ◦ ∂ x i m 1 )( γ S ) · ∂ x j 1 ( f 1 ) . . . ∂ x j m 2 ( f m 2 ) (76) In the last equatio n the sym b ol δ ap app ears when we take the comm utator [ ∂ x a , x p ] in the second summand of (74), which gives the factor dim V and summation only o v er a in B . W e con tin ue for A an d B separately . 43 Let u s start with B . W e hav e: B = ∓ dim V dim V X i 1 ,..,i m 1 =1 dim V X a,j 1 ,...,j m 2 =1 ± k  λ ∧ ∂ ξ i 1 ( λ 1 ) ∧ · · · ∧ ∂ ξ i m 1 ( λ m 1 ) ∧ ( ξ a ∂ ξ a ◦ ∂ ξ j 1 ◦ · · · ◦ ∂ ξ j m 2 ( γ Λ ))  × × ( ∂ x i 1 ◦ · · · ◦ ∂ x i m 1 )( γ S ) · ∂ x j 1 ( f 1 ) . . . ∂ x j m 2 ( f m 2 ) = ∓ dim V · (deg Λ ( γ ) − m 2 ) × × dim V X i 1 ,..,i m 1 =1 dim V X a,j 1 ,...,j m 2 =1 ± k  λ ∧ ∂ ξ i 1 ( λ 1 ) ∧ · · · ∧ ∂ ξ i m 1 ( λ m 1 ) ∧ ∂ ξ j 1 ◦ · · · ◦ ∂ ξ j m 2 ( γ Λ ))  × × ( ∂ x i 1 ◦ · · · ◦ ∂ x i m 1 )( γ S ) · ∂ x j 1 ( f 1 ) . . . ∂ x j m 2 ( f m 2 ) = = ± dim V · (deg Λ ( γ ) − m 2 ) · G m,n ( γ ) (77) No w turn bac k to the computation of A . Clearly (up to the sign, bu t the signs alw a ys wo rk for u s) that A = 0. Indeed, schemat ically the formula (75) for A lo oks lik e ∂ ξ p ( k ( λ ∧ T )) − ( ∂ ξ p ( k ))( λ ∧ T ) for some T ∈ Λ( V ). If w e deﬁne k ′ ( λ ) = k ( λ ∧ T ) we need to compu te ( ∂ ξ p ( k ′ ))( λ ) − ( ∂ ξ p ( k ))( λ ∧ T ) (78) But ( ∂ ξ p ( k ′ ))( λ ) = k ′ ( ξ p ∧ λ ) = k ( ξ p ∧ λ ∧ T ). No w w e see that the t wo su mmands in (78) are equal. W e ha v e prov ed th e statemen ts (i) and (ii) of the Prop osition. The p r o ofs of (iii) and (iv) are analogous. 6.3 Construction of the H o chs chi ld-Kostan t-Rosen b erg map ϕ cat H K R No w we hav e everything we need to construct the m ap ϕ cat H K R : T poly ( V ) → Ho c h q ( cat ). Of course, it w ould b e b etter to sp ecify th e signs in the Prop osition ab o v e; how ev er, we will see that the construction b elo w do es not dep end seriously on these signs. Supp ose that deg S γ = m , d eg Λ γ = n . Start with e ϕ S H K R ( γ ) ∈ Ho ch q ( S ( V ∗ )), whic h is by deﬁnition the Ho chsc hild-Kopstan t-Rosen b erg co chain without division by the n !. It total diﬀeren tial in Ho c h q ( cat ) is d tot e ϕ S H K R ( γ ) = ( ± ) G 0 ,n ( γ ). F rom no w on, w e will supp ose that the all signs in Prop osition ab o v e are ” + ”, if some of them are ” − ”, the form ula will b e the same up to some signs. So supp ose that d tot e ϕ S H K R ( γ ) = G 0 ,n ( γ ) w ith sign +. W e know from statemen t (i) of the Pr op osition that d Hoch F 0 0 ,n − 1 ( γ ) = G ,n ( γ ), the same co chain. Therefore, d Hoch ( e ϕ S H K R ( γ ) − F 0 0 ,n − 1 ( γ )) = 0. But t hen e ϕ S H K R ( γ ) − F 0 0 ,n − 1 ( γ ) has a non-trivial Koszul diﬀeren tial whic h can b e found by Prop osition (ii). W e ha v e: d Koszul ( e ϕ S H K R ( γ ) − F 0 0 ,n − 1 ( γ )) = d Koszul ( F 0 0 ,n − 1 ( γ )) = dim V · G 0 ,n − 1 ( γ ). No w 44 w e wan t to kill this cob oun dary b y the Ho c hschild d iﬀeren tial. W e ha v e: d Hoch (dim V · F 0 0 ,n − 2 ( γ )) = dim V · G 0 ,n − 1 ( γ ). Contin uing in this w a y , we ﬁ nd that (we omit γ at eac h term): d tot ( e ϕ S H K R − F 0 0 ,n − 1 + dim V · F 0 0 ,n − 2 − · · · + · · · + ( − 1) n ( n − 1)! d im n − 1 V · F 0 0 , 0 ) = ( − 1) n n ! dim n V G 0 , 0 (79) But we can start also w ith e ϕ Λ H K R ( γ ), a nd ﬁnally get also G 0 , 0 with some m ultiplicit y . More pr ecisely , we h a v e: d tot ( e ϕ Λ H K R − F ∞ m − 1 , 0 + dim V · F ∞ m − 2 , 0 − · · · + ( − 1) m ( m − 1)! dim m − 1 V · F ∞ 0 , 0 ) = ( − 1) m m ! dim m V G 0 , 0 (80) W e ﬁnally set: ϕ cat H K R =( − 1) n 1 n ! dim n V e ϕ S H K R + n X i =1 ( − 1) i ( i − 1)! d im i − 1 V F 0 0 ,n − i ! − ( − 1) m 1 m ! dim m V   e ϕ Λ H K R + m X j =1 ( − 1) j ( j − 1)! dim j − 1 V F ∞ m − j, 0   (81) It is a co cycle in the Ho c hsc hild cohomological complex Ho ch q ( cat ): d tot ϕ cat H K R ( γ ) = 0 (82) for any γ ∈ T poly ( V ). W e can prov e the follo wing Theorem. The map ϕ cat H K R : T poly ( V ) → Ho ch q ( cat ) is a quasi-isomorphism of c o m- plexes. When we use the normalize d Koszul diﬀer ential inste ad of the usual one (so, it has the same eﬀe ct as to set dim V = 1 in the formula ab ove), the map ϕ cat H K R makes the diagr am (62) c ommutative (up to a non-essential sign) on the level of c oh omolo gy. Pr o of. The second statemen t is clear. The ﬁrst one (that ϕ cat H K R is a qu asi-isomorphism of complexes) follo ws from t he seco nd one and from T heorem 4.2 whic h sa ys that the maps p A and p B are quasi-isomorphisms in our case. Key-Lemma 4.4 is prov en. Theorem 4.4 is p ro v en. 7 Pro of of the M ain Theorem First of all, we form ulate the Main Theorem exactly in the form we will p ro v e it here. 45 7.1 The ﬁnal form ulation of the Ma in Theorem Theorem. ( Main Theorem, ﬁnal form) Supp ose t : G ∞ → B ∞ is a quasi- isomorph ism of op er ads, and let U V = X ( t ) V : T poly ( V ) → Hoc h q ( S ( V ∗ )) b e the c o rr e- sp onding L ∞ map, deﬁne d uniquely up to homotopy (se e Se ction 3). L et α b e a quadr atic Poisson bive ctor on V , and let D ( α ) b e the c orr esp onding quadr atic Poisson bive ctor on V ∗ [1] . Denote b y S ( V ∗ ) ~ and Λ( V ) ~ the c orr esp onding deform ation quantizations of S ( V ∗ ) ⊗ C [[ ~ ]] and Λ( V ) ⊗ C [[ ~ ]] gi v en by f ⋆ g = f · g + ~ · U 1 ( α )( f ⊗ g ) + 1 2 ~ 2 · U 2 ( α ∧ α )( f ⊗ g ) + . . . (83) Then the algebr a s S ( V ∗ ) ~ and Λ( V ) ~ ar e gr ade d (wher e d eg ~ = 0 , deg x i = 1 for al l i ) and quadr atic. Also, they ar e Koszul as algebr as over the discr ete v aluation ring C [[ ~ ]] , se e Se ction 1. Mor e over, they ar e Koszul dual to e ach other. W e pro v e the Theorem throughou t th is Section. 7.2 An elemen tary Lemma W e start with the follo wing simp le statemen t: Lemma. (1) Su pp ose K ~ is a fr e e C [[ ~ ]] -mo dule, which is also a left (or right) C [[ ~ ]] - line a r mo dule over an algebr a A ~ which is supp o se d to b e also fr e e as C [[ ~ ]] -mo d ule. Then if the sp e cialization K ~ =0 is a fr e e mo dule over the sp e cialization A ~ =0 , K ~ is a fr e e left (right) A ~ -mo dule; (2) supp ose K q ~ is a c omplex of fr e e C [[ ~ ]] -mo dules ( deg ~ = 0 ) with C [[ ~ ]] -line ar dif- fer ential . Supp ose that the i -th c oh omolo gy (for some i ) of the sp e cialization K q ~ =0 is zer o . Then the i -th c ohomolo gy of K q ~ is also zer o. Pr o of. The b oth statemen ts are stand ard; let us r ecall the pro ofs for conv enience of the reader. (1): Sup p ose th e con trary , then f or s ome k i ( ~ ) ∈ K ~ and some a i ( ~ ) ∈ A ~ one has P i a i ( ~ ) · k i ( ~ ) = 0. Let N b e the minimal p o w er of ~ in the equation. T hen w e can divide the equ ation o v er ~ N and the equation still holds, b ecause the b oth A ~ and K ~ are fr ee C [[ ~ ]]-mo du les. Then we r educe o v er ~ and get a nont rivial linear equ ation for the A ~ =0 -mo dule K ~ =0 whic h cont radicts to the assu mption. (2): Let k i ( ~ ) b e an i -cicycle in K q ~ , we should pro ve that it is a cob ound ary . S upp ose ~ N is the minimal p o w er of ~ in k i ( ~ ), then we divide o v er ~ N . W e get again a co cycle, b ecause the diﬀeren tial is C [[ ~ ]]-linear and K q ~ is a free C [[ ~ ]]-mo du le. Denote this new co cycle again by k i ( ~ ). Then its zero degree in ~ term is a co cycle in the reduced complex K ~ =0 and we can kill it by some cob oundary . Th en su bstract and divide o v er minimal p ow er of ~ , ans so on. 46 7.3 The algebras S ( V ∗ ) ~ and Λ( V ) ~ are Koszul W e start to p ro v e the T h eorem. Prov e ﬁr stly th at the alge bras S ( V ∗ ) ~ and Λ( V ) ~ are graded quadratic and Koszul. The ﬁ rst statemen t is pro v en analogo usly to the sp ecu- lation in Section 0.2. The diﬀerence that here in a universal deformation quant ization w e may ha v e more general graphs than in the K on tsevic h’s qu an tization, namely non- connected graph s and graphs with simp le lo ops. But it d o es not change the pro of. Let us pro ve th at these algebras are Koszul. Consider the case of S ( V ∗ ) ~ , th e pro of for Λ( V ) ~ is analogous. By Lemma 1.2.5, it is necessary to p ro v e that the Koszul complex K q ~ =  S ( V ∗ ) ~ ⊗ C [[ ~ ]] Hom C [[ ~ ]] ( S ( V ∗ ) ! , C [[ ~ ]]) , d Koszul  is acyclic in all degrees except degree 0. Th e complex K q ~ is clearly a complex of free C [[ ~ ]]-mo dules with a C [[ ~ ]]-linear dif- feren tial. W e are in situation of Lemma 7.2(2), b ecause the sp ecializa tion at ~ = 0 gives clearly the Koszul complex for the usual algebra S ( V ∗ ) wh ich is k no wn to b e acyclic. W e are d one. 7.4 W e con tin ue to prov e the Main Theorem No w w e pro ve the only n on-trivial part of the Theorem, that the alge bras S ( V ∗ ) ~ and Λ( V ) ~ are Koszul d ual. Consider the diagram (51 ). It is a diagram of G ∞ quasi-isomorphisms w hic h is kno wn to b e homotopically commutativ e , see Theorem 4.4. Then w e can construct a G ∞ quasi-isomorphism F : T poly ( V ) → Ho c h q ( cat ( A, B , K )) dividin g the diagram int o t wo comm utativ e triangles. Restrict F to its L ∞ part. Then w e get an L ∞ quasi-isomorphism F : T poly ( V ) : Ho ch q ( cat ( A, B , K )). Here A = S ( V ∗ ) ⊗ C [[ ~ ]], B = Λ( V ) ⊗ C [[ ~ ]], etc. Then this L ∞ map F attac hes to the Maurer-Cartan solution α ∈ T poly ( V ) (our quadratic P oisson biv ector ﬁeld) a solution of the Maurer-Cartan equation in Ho c h q ( cat ( A, B , K )), b y formula F ∗ ( α ) = ~ F 1 ( α ) + 1 2 ~ 2 F 2 ( α ∧ α ) + . . . (84) What a solution of the Maurer-Cartan equati on in Ho ch q ( cat ( A, B , K )) means im more direct terms ? It consists from the follo wing data: (i) A deformation quantiza tion A ~ of the algebra A = S ( V ∗ ) ⊗ C [[ ~ ]]; (ii) a d eformation quantiz ation B ~ of the algebra B = Λ( V ) ⊗ C [[ ~ ]]; (iii) a deformed diﬀerent ial on the Koszul complex K q ( S ( V ∗ )) ⊗ C [[ ~ ]], we denote th e deformed complex by K q ~ ; (iv) a structure of a B ~ - A ~ -bimo dule on K q ~ . 47 The cru cial p oin t is the follo wing Lemma: Lemma. The algebr a A ~ is gauge e quivalent (and ther efor e isomo rphic) to the algebr a S ( V ∗ ) ~ fr o m Se ction 7.1, and the algebr a B ~ is gauge e quivalent to Λ( V ) ~ . Pr o of. It follo ws from the comm utativit y of the diagram (51), and fr om Lemma 5.2. 7.5 W e ﬁn ish to prov e the Main Theorem F rom Lemm a 7.4, it is enough to pro v e that the quadratic graded algebras A ~ and B ~ are Koszul du al to eac h other. F or this (b ecause the b oth algebras are Koszul) it is en ough to pr o v e that B ~ = A ! ~ . Let us p ro v e it. The complex K ~ is a complex of B ~ - A ~ mo dules. As co mplex of A ~ -mo dules, it is free b y Lemma 7.2(1). By Lemma 7.2(2), it is a fr ee A ~ -resolution of the mo du le C [[ ~ ]]. Therefore, we can us e K ~ for the computation of the Koszul d ual algebra: ( A ~ ) ! = RHom M od − A ~ ( K ~ , K ~ ) ( 85) On th e other hand, from the bimo d ule structure (see (iv) in the list in Section 7.4), we ha v e an algebr a h omomorphism B ~ → RHom M od − A ~ ( K ~ , K ~ ) (86) W e only need to pr o v e that it is an isomorp h ism. It again follo ws from the facts that th e b oth sides are free C [[ ~ ]]-mo du les (for the l.h.s. it is clear, for the r .h .s. it follo w s from (85)), and th at the sp ecialization of (86) at ~ = 0 is an isomorphism. Theorem 7.1 is p ro v en. References [AKSZ] M. Alexandrov, M. Kontsevic h , A. Sc h w arz, O. Z ab oronsky , The Geome- try of the Master Equation and T op ologica l Quan tum Field Theory , p eprin t hep-th/950201 0 , Intern. Journal. M o dern. Phys. A12 (1997) 1405-14 30, [AM] M.F. A tiy ah and I.G. Macdonald, Intr o duction to c ommutative algebr a , Addison- W esley , 1969, [BGS] A. Beilinson, V. Ginzburg, W. S o ergel, K oszul du alit y patterns in rep resen tation theory , Journal of the AM S , 9(2), 1996, 473-527 , [D] V. Drin feld, On qu asitriangular quasi-Hopf algebras and a group closely connected with Gal( Q / Q ), L eningr ad Math. J. , V ol. 2 (1991), no. 4, 829-860, [GJ] E. Getzl er, J.D.S. Jones, Op erad s , homotop y alg ebra and iterated in tegrals for double lo op spaces, h ep-th.94030 55, 48 [H] V. Hinic h, T amarkin’s pr o of of Kont sevic h formalit y theorem, math.0003052; [H2] V. Hinic h, Closed mo d el structures, Homo logical alge bra of homoto py algebras, math.97020 15, [EK] P . Etingof, D. Kazhdan, Quantiza tion of Lie bialgebras, I,I I, Sele cta Math. , 2(196 ), no.1, 1-41, and 4(1998), no.2, 233-269 , [K97] M. K ontsevic h, Deformation quan tization of P oisson manifolds I, prepr in t q-alg/9 709040 , [K99] M. Kont sevic h, Op erads and Motiv es in deformation quantiza tion, preprint q-alg/9 904055 , [Kel1] B. Keller, Deriv ed in v ariance of higher structures on the Ho c hschild co mplex, a v a ilable at www .math.jussieu.fr / ˜ kelle r, [Kel2] B. Keller, Hochsc hild cohomology and deriv ed Picard group, preprint arXiv:math/031 0221 , [Kel3] B. Keller, On diﬀeren tial graded catego ries, p reprint arXiv:math/06011 85 , [L] J.-L. Lo day , Cyc lic Homol o gy , S pringer-V erlag Series of Comp rehensive S tudies in Mathematics, V ol. 301, 2nd Edition, Sp ringer, 1998, [M] H. Matsum ura, Commutative Ring The ory , C am bridge Unive rsity Press, 1989, [P] L. Po sitselski, Nonhomogeneous Koszul dualit y and D -Ω, unpub lished manuscript, [PP] A. P olishc h uk, L. Po sitselski, Quadr atic al gebr as , AMS Univ ersit y Lecture Series, V ol. 37, 2005, [Q] D. Quillen, Homotopic al Algebr a , S p ringer L NM.43, 1967, [Sh1] B. Shoikhet, Kontsevic h formalit y and PBW algebras, preprint arXiv:0708.16 34 , [Sh2] B. S hoikhet, A p ro of of the Tsygan formalit y conjecture for chains, p reprint arXiv:math/001 0321 , A dvanc es in Math. 179 (2003), 7-37, [Sh3] B. S hoikhet, An explicit construction of the Q uillen homotopical cate gory of L ∞ algebras, preprint arXiv:0706.13 33 , [T1] D. T amarkin, A p ro of of M.Kont sevic h formalit y theorem, math.9803025, [T2] D. T amarkin, Deformation complex of a d -algebra is a ( d + 1)-algebra, math.00100 72, 49 [1] [T3] D. T amarkin, Action of the Grothendiec k-T eic hm ueller g roup on the op erad of Gerstenh ab er algebras, math.020203 9. F acult y of Science, T ec hnology and Communicatio n, Camp us Limp ertsb erg, Un iv ersit y of Luxemb ourg, 162A av en ue de la F aiencerie, L-1511 LUXEMBOURG e-mail : bor ya por t@yahoo. com 50

Koszul duality in deformation quantization and Tamarkins approach to Kontsevich formality

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