DEFORMATIONS OF ASSOCIATIVE ALGEBRAS WITH INNER PRODUCTS

Homology, Homotopy nd Applitions, vol. 8(2), 2006, pp.115 131 DEFORMATIONS OF ASSOCIATIVE ALGEBRAS WITH INNER PRODUCTS JOHN TERILLA nd THOMAS TRADLER (ommunited y Jim Stsheff) Astrt We develop the deformtion theory of A lgers together with -inner produts nd identify differentil grded Lie lger tht ontrols the theory. This generlizes the deformtion theories of ssoitive lgers, A lgers, ssoitive lgers with inner produts, nd A lgers with inner produts. 1. Introdution A nturl onsidertion for n lgeri struture in topology is whether it is homotopy invrint. The C struture on the ohins of spe is lssi exmple. While mnifolds re distinguished y the inner produt fforded y Poinré dulity, n inner produt is not homotopy invrint onept. The right mening the homotopy roust onept is n -inner produt s introdued in [12]. In lgeri generlity, n -inner produt is defined in the setting of n A lger. In this pper, we desrie the deformtion theory of A lgers together with -inner produts y giving ontrolling differentil grded Lie lger. An pplition tht we hve in mind involves string topology. It is known tht if X nd Y hve the sme homotopy type, then they hve the sme string topology opertions [1]. One my ssign n A lger A X with n -inner produt I X to Poinre dulity spe X. Bsed on results in [11, 13], it is resonle to think tht if the two differentil grded Lie lgers ontrolling the deformtions of (A X, I X ) nd (A Y, I Y ) re qusi-isomorphi, then X nd Y hve the sme string topology opertions. One my speulte tht the qusi-isomorphism lss of the differentil grded Lie lger ontrolling the deformtions (A X, I X ) determines the string topology type of the spe X (muh the sme wy tht the C struture on the ohins on spe determines the rtionl homotopy type of spe; see [10]). In ny event, it would e interesting to proe this ontrolling differentil grded Lie lger for its invrints. The uthors would like to thnk Jim Stsheff, Mrtin Mrkl, nd Dennis Sullivn for mny helpful disussions. Reeived Mrh 12, 2006, revised April 7, 2006, July 20, 2006; pulished on August 16, 2006. 2000 Mthemtis Sujet Clssifition: 55P10, 16S10, 16S80. Key words nd phrses: homotopy, inner produt, deformtion theory. Copyright 2006, Interntionl Press. Permission to opy for privte use grnted.

Homology, Homotopy nd Applitions, vol. 8(2), 2006 116 Let us review the si ide of deformtion theory governed y differentil grded Lie lger [9, 3, 5, 6]. Fix ground field k of hrteristi 0. For ny differentil grded Lie lger (g = i g i, d, [, ]) over k, one n onsider deforming the differentil d in the diretion of n inner derivtion. Informlly, suh deformtion is given y n (equivlene lsses of) α mking d α := d + d(α) into differentil. The mp d α is lwys derivtion nd the ondition tht d 2 α = 0 trnsltes into the Muer Crtn eqution: dα + 1 [α, α] = 0. 2 The deformed differentil d α my involve prmeters from the mximl idel m of Z grded Artin lol ring: α (g k m) 1. If m is the mximl idel of lol Artin ring R nd α (g k m) 1 is solution to the Muer Crtn eqution, then one my ll d α deformtion of d over R. A ring mp R S will trnsport deformtion of d over R to deformtion of d over S. More formlly, one hs funtor Def g from the tegory of Z grded Artin lol rings with residue field k to the tegory of sets, ssigning to suh ring R with mximl idel m the set Def g (R) = {α (g k m) 1 : dα + 1 [α, α] = 0}/. 2 Here, is the equivlene reltion determined y the tion of the guge group, whih we now rell. Sine R is n Artin ring, m is nilpotent lger, nd (g k m) 0 g k m is nilpotent Lie lger. Therefore, there exists group G = {exp β : β (g k m) 0 }, lled the guge group, with multiplition defined y the Bker Cmpell Husdorff formul. The tion of e β G on n element α (g k m) 1 is determined y the infinitesiml tion: This tion stisfies α β α = [β, α] dβ, α (g m) 1, β (g m) 0. e d β d α e d β = d eβ α, nd preserves the set of solutions to the Murer Crtn eqution. In this pper, we work with A lgers equipped with inner produts. One hs the notion of deformtion of n A lger with n inner produt over ring R, nd there is nturl equivlene on the set of deformtions. A ring mp R S trnsports deformtions over R to deformtions over S. The ssoition R { deformtions of the A lger with the inner produt over R } /{ equivlent deformtions defines ovrint deformtion funtor. We onstrut differentil grded Lie lger (h = i h i, d, [, ]) ssoited to n A lger with n inner produt, nd prove tht the funtor desried ove is isomorphi to Def h. This is the preise mthemtil ontent of the sttement the differentil grded Lie lger (h, [, ], d) ontrols the deformtions of the A lger with n inner produt. }

Homology, Homotopy nd Applitions, vol. 8(2), 2006 117 2. Definitions of A lgers nd inner produts We now review the onept of n inner produt on n A lger [12], [11]. The onepts of A lgers, A imodules, A imodule mps, nd A inner produts re generliztions of the usul onepts of ssoitive lgers, imodules, imodule mps, nd invrint inner produts. 2.1. A lgers Let V = j Z V j e grded module over ring S. Rell tht the suspension V [1] of V is defined to e V [1] = j Z (V [1])j with (V [1]) j := V j 1. For grded S-module A, we denote y T A the tensor lger of the suspended spe A[1], T A = S A[1] A[1] 2.... An A lger over S is defined to e pir (A, D) where A is grded S module nd D Coder(T A) of degree 1 with D 2 = 0. In ddition, we require the no homotopy unit onvention tht D hs no omponent S T A. Suppose tht (A, D) nd (A, D ) re A lgers over S. Then, n A mp from (A, D ) to (A, D) is mp λ : T A T A stisfying λ D = D λ. 2.2. A imodules Let (A, D) e n A lger over S, nd let M e grded S module. Let T M A denote the tensor iomodule T M A := k,l 0 A[1] k M[1] A[1] l of M[1] over T A. An A imodule struture on M over A is defined to e oderivtion D M Coder D (T M A, T M A) over D of degree 1 with (D M ) 2 = 0. Let (M, D M ) nd (N, D N ) e A imodules over A. Let Comp(T M A, T N A) denote the mps F : T M A T N A stisfying T M A (T A T M A) (T M A T A) M F (Id F ) (F Id) T N A (T A T M A) (T M A T A) N The spe Comp(T M A, T N A) rries differentil defined y δ M,N (F ) := D N F ( 1) F F D M. In this se, n A imodule mp from M to N is defined to e n element F Comp(T M A, T N A) of degree 0 with δ M,N (F ) = 0, i.e. D N F = F D M. 2.3. inner produts For ny f Coder(T A), there re indued oderivtions f A Coder f (T A A, T A A) nd f A Coder f (T A A, T A A), where A = hom S (A, S) denotes the dul of A.

Homology, Homotopy nd Applitions, vol. 8(2), 2006 118 One lso hs n indued mp δ f : Comp(T M A, T N A) Comp(T M A, T N A) given y δ f (F ) = f A F ( 1) f F F f A. Note tht, in prtiulr, if (A, D) is n A lger, then A nd A hve A imodule strutures given y D A nd D A. Definition 2.1. Let (A, D) e n A lger over S. We define n inner produt on A over S to e n A imodule mp I from A to A. Equivlently, n inner produt is n element I Comp(T A A, T A A) stisfying δ D (I) = D A I I D A = 0. Every inner produt, : A A S defines n element I Comp(T A A, T A A). In this se, the ondition D A I I D A = 0 is equivlent to D( 1,..., n ), n+1 = ± 1, D( 2,..., n+1 ). See the ppendix for dditionl illustrtions. 2.4. Indued mps Rell tht if λ : A A is n lger mp etween two ssoitive lgers, then every module over A is lso module over A, nd similrly for module mps. Also, λ : A A nd λ : A (A ) will e module mps over A. Here we give the orresponding homotopy generliztions. Suppose tht λ is n A mp from (A, D ) to (A, D). First, every A imodule (M, D M ) over A is lso n A imodule over A, whose struture mp is determined y the lowest omponents (whih re mps T M A M) (D M ) λ ( 1,..., k, m, k+1,..., k+l) = ±pr M D M (λ( 1,...),..., λ(..., k), m, λ( k+1,...),..., λ(..., k+l)). Here, pr M denotes the projetion onto M. The signs re given y the usul sign rule, nmely introduing sign ( 1) α β, whenever α jumps over β. The relevnt degrees re the degrees given in T M A. Also, ny A imodule mp F : T M A T N A over A indues n A imodule mp F λ : T M A T N A over A given y F λ ( 1,..., k, m, k+1,..., k+l) = ±pr N F (λ( 1,...),..., λ(..., k), m, λ( k+1,...),..., λ(..., k+l)). Furthermore, λ indues the two A imodule mps over A defined y the omponents nd λ : T A A T A A nd λ : T A A T (A ) A λ( 1,..., k+l+1) = pr A λ( 1,..., k+l+1) ( λ( 1,...,,..., k+l))( ) = ± (pr A λ( k+1,..., k+l,, 1,..., k)).

Homology, Homotopy nd Applitions, vol. 8(2), 2006 119 3. Deformtions of A lgers nd inner produts Before we define the speifi differentil grded Lie lger (h, d, [, ]) tht ontrols the deformtions of A strutures nd inner produts, we disuss simple exmple, whih is relevnt to our setting, nd mke remrk. Exmple 3.1. Any grded ssoitive lger g eomes Lie lger y defining the rket to e the usul ommuttor. An element α g 1 stisfying α 2 = 0 is sometimes lled polriztion. With polriztion α g 1, g eomes differentil grded Lie lger y setting the differentil to e δ = d(α). With δ so defined, the Muer Crtn eqution eomes 0 = δ(γ) + 1 2 [γ, γ] = 1 [α + γ, α + γ]. 2 In other words, γ g 1 stisfies the Muer Crtn eqution if nd only if α + γ is nother polriztion. Now let S e grded ring nd onsider g defined y {( ) 0 g =, S} with the rket defined s the usul grded ommuttor of mtrix multiplition: [( ) ( )] ( ) 0 0 [, ] 0, =. d [, ] + [, d] [, ] ( ) D 0 Then, g I D 1 is polriztion if nd only if 0 = [D, D] = 2 D 2 nd 0 = [D, I] + [I, D] = 2 [D, I]. ( ) D 0 Hving hosen polriztion P =, the formul for δ = d(p ) is given y I D ( ) [( ) ( )] ( ) f 0 D 0 f 0 [D, f] 0 δ =, =. i f I D i f [D, i]+[f, I] [D, f] Now we look t the guge equivlene. First of ll, the guge group G = exp(g 0 ) is the Lie group onsisting of mtries of the form e A, for ny A g 0. The guge tion of G on g is then determined y e d(a) B = Ad(e A )(B) = e A Be A. A omputtion shows tht ( ) ( f 0 e f 0 exp = i f x e f ), where x = n 1 Then the guge equivlene summrizes s ( ) ( e A D 0 e A = I D This onludes the exmple. 1 n! k+l=n 1 f k i f l. e f De f 0 e f Ie f + [e f De f, xe f ] e f De f Remrk 3.2. Let N e grded olger over S. Then hom(n, N) will e grded ssoitive lger y omposition of liner mps nd Lie lger with the ).

Homology, Homotopy nd Applitions, vol. 8(2), 2006 120 rket defined y the grded ommuttor of omposition. The spe Coder(N) hom(n, N) is not n ssoitive sulger, ut it is Lie sulger. In prtiulr, for ny vetor spe A, Coder(T A) is grded Lie lger. An A struture on A onsists of n element D Coder(T A) stisfying D 2 = 0. Thus, one n sy tht n A struture on A is hoie of polriztion D Coder(T A). Hene, if (A, D) is n A lger, Coder(T A) rries differentil δ : Coder(T A) Coder(T A) defined y δ(f) := [D, f] = D f ( 1) f f D. The omplex (Coder(T A), δ) is lled the Hohshild ohin omplex of A. Together with the rket from hom(t A, T A), it is differentil grded Lie lger tht ontrols the deformtions of the A lger (A, D). In order to mke this sttement preise, we rell the deformtion theory of A lgers (see for exmple [2]). As first oservtion, one my note tht γ is solution to the Muer Crtn eqution in the Hohshild differentil grded Lie lger if nd only if D + γ is nother polriztion in Coder(T A); i.e., nother A struture on A. 3.1. Deformtions of A lgers Let A e grded vetor spe over field k of hrteristi zero nd let R e grded Artin lol lger with residue field k. Let m denote the mximl idel of R. We hve the deomposition R R/m m k m nd the projetion pr k : R k, hene the deomposition A R A (A m) nd the projetion pr A : A R A. For definiteness, the reder my hve the onrete exmple R = k[t]/t l+1 in mind. In this exmple, the mximl idel is m = tk[t]/t l+1, A R A + At + At 2 + + At l (with the tensor signs suppressed) nd the nturl projetion pr A mps 0 + 1 t + 2 t 2 + + l t l 0. Let (A, D) e n A lger over k. A deformtion of (A, D) over R is n A lger (A R, D ) over R with the property tht the projetion pr : T (A R) T A R T A is morphism of A lgers over k. This mens tht pr D = D pr. Suppose, tht D is deformtion of (A, D) over R. Vi ny mp R S, one n view A R s n S module nd (A R, D ) s deformtion of (A, D) over S. Let π hom(r R, R) denote the multiplition in R. Let D R denote the A struture D π on A R. The A lger (A R, D R ) is the model for trivil deformtion of (A, D). Tht is, (A R, D ) is trivil deformtion if it is isomorphi to (A R, D R ) s n A lger. This mens tht there is n utomorphism λ : T (A R) T (A R) stisfying λ D = D R λ. Two deformtions re equivlent if nd only if they differ y trivil one. 3.2. Deformtions of A lgers with inner produts Definition 3.3. Let A e grded vetor spe over field k. We define the grded Lie lger (h = i h i, [, ]) y h i = Coder(T A) i Comp(T A A, T A A) 1 i (1)

Homology, Homotopy nd Applitions, vol. 8(2), 2006 121 nd [(f, i), (g, j)] = ([f, g], δ f (j) ( 1) f g δ g (i)) = (fg ( 1) f g gf, f A j ( 1) f j jf A ( 1) f g g A i + ( 1) g ( f + i ) ig A ). (2) The skew-symmetry nd Joi identity of [, ] re strightforwrd to hek fter one noties tht δ f δ g ( 1) f g δ g δ f = δ f g ( 1) f g g f. Proposition 3.4. A pir (D, I) h is n A struture with inner produt on A if nd only if [(D, I), (D, I)] = 0. Proof. This is immedite: 0 = [(D, I), (D, I)] 0 = [D, D] = 2 D 2 nd 0 = 2 δ D (I) = 2(D A I I D A ). The ondition D 2 = 0 mens tht D defines n A struture on A nd the ondition D A I I D A = 0 mens tht I defines omptile -inner produt. Now fix n A struture together with n inner produt, whih is to sy, fix pir (D, I) h with [(D, I), (D, I)] = 0. Then, define d : h h y The triple (h, d, [, ]) is differentil grded Lie lger. d(f, i) = [(D, I), (f, i)]. (3) Definition 3.5. A deformtion of n A lger with inner produt (A, D, I) over R is n A lger over R with inner produt (A R, D, I ), suh tht the projetion pr : T (A R) T A is morphism of A lgers over k omptile with the -inner produts. Comptiility with the inner produt mens tht the following digrm of A - imodule mps over k is ommuttive: T A R (A R) T A (A R) pr k I T (A R) (A R) T A (A R) pr fpr I pr Here, the -inner produt I on A R over R indues n -inner produt on A R over k y omposing with the mp indued y the projetion hom R (A R, R) hom k (A R, k), f pr k f. There is nturl extension of I to n -inner produt I R = I π on (A R, D R ).

Homology, Homotopy nd Applitions, vol. 8(2), 2006 122 Definition 3.6. We sy tht (D, I ) is trivil deformtion of (D, I) provided the triple (A R, D, I ) is isomorphi to (A R, D R, I R ) s A lgers with inner produts. Tht is, if there exists n utomorphism nd omp stisfying (i) λ D = D R λ, λ : T (A R) T (A R) ρ : T A R (A R) T (A R) (A R) (ii) I λ (I R ) λ λ = D (A R) ρ + ρ D A R. It my e helpful to think of the seond ondition in Definition 3.6 s sying I equls I R under hnge of oordintes (given y λ) up to homotopy (given y ρ). Tht is, the following digrm ommutes, up to homotopy defined y ρ Comp(T A R (A R)). T A R λ (A R) T A R (A R) I (I R) λ T (A R) (A R) T (A R) (A R) Two deformtions re equivlent if nd only if they differ y trivil one. Now, the onlusion: eλ Theorem 3.7. Let (A, D) e n A lger nd let I e n -inner produt. Then the differentil grded Lie lger (h, d, [, ]) defined y equtions (1), (2) nd (3) ontrols the deformtions of the A lger with inner produt (A, D, I). Proof. The ontent of this theorem is summrized in the following two sttements. Deformtions, over R, of the (A, D, I) orrespond to solutions to the Muer Crtn eqution in h m, nd equivlent deformtions orrespond to guge equivlent solutions. First we prove the first sttement. Let α = (f, i) (h m) 1. Oserve tht dα + 1 2 [α, α] = [(D R, I R ), (f, i)] + 1 [(f, i), (f, i)] 2 = 1 2 [(D R + f, I R + i), (D R + f, I R + i)]. Then, Proposition 3.4 proves tht dα + 1 2 [α, α] = 0 if nd only if (A R, D R + f, I R + i) is deformtion of (A, D, I). It is immedite tht ny (A R, D, I ) tht is deformtion of (A, D, I) must stisfy [(D, I ), (D, I )] = 0 h R. The ft tht pr : T (A R) T A is mp of A lgers with inner produts implies tht D = D R + f nd I = I R + i for some (f, i) h m.

Homology, Homotopy nd Applitions, vol. 8(2), 2006 123 Now we prove the seond sttement. Let α = (f, i) (h m) 0. The guge tion for h eomes e d(f,i) (D R, I R ) = d(f, i) n (D R, I R ). n! n 0 It follows from δ f ( δd(f) r (D R)((δ f ) s (i)) ) = δ d(f) r+1 (D R)((δ f ) s (i)) + δ d(f) r (D R)((δ f ) s+1 (i)), tht d(f, i) n (D R, I R ) is given y ( d(f) n (D R ), (δ f ) n (I R ) k+l=n 1 1 (l+1)! (δ f ) l (i). Then for the uto- Now define λ 1 = e f = k 0 1 k! f k nd ρ = l 0 morphism λ nd the homotopy ρ, we hve d(f, i) n (D R, I R ) n! n 0 = d(f) n (D R ), (δ f ) n n! n! n 0 n 0 ( = λ 1 D R λ, λ(i ) R ) λ λ + δ λ 1 D Rλ(ρ). ) n! k!(l + 1)! δ d(f) k (D R) (δ f ) l (i). (I R )+δp d(f) k k 0 k! (D R) ( l 0 ) 1 (l + 1)! (δ f ) l (i) This proves tht e d(f,i) (D R, I R ) is trivil deformtion of (D, I). It is not hrd to see tht every trivil deformtion of (D, I) rises from n element guge equivlent to the identity. The ondition tht the A lger mp λ : T (A R) T (A R) is n utomorphism implies tht λ = e f for some f (Coder(T A) m) 0. Also, sine ρ = i 1 2 δ f (i), the mp i ρ(i) = l 0 1 (l + 1)! (δ f ) l (i) is invertile. So one n otin ny homotopy ρ, y hoosing suitle element i = m 0 m (δ f ) m (ρ) (h m) 0 with ρ(i) = ρ. 4. Moduli, infinitesiml deformtions, nd reltionship to yli ohomology Let us return riefly to generl deformtion theory in order to review the notions of infinitesiml deformtions nd moduli spe. Let (g, d, [, ]) e differentil grded Lie lger nd ssume tht Ker(d)/ Im(d) =: H(g) = m i= m Hi (g) is finite dimensionl. Consider the (grded version of the) ring of dul numers R = k[t m,..., t m ]/t i t j. Here deg(t i ) = i 1 nd the mximl idel of R is m = i t i R. From solution (γ j t j ) (g m) 1 to the Muer Crtn eqution, one my produe the mp d + t j d(γ j ) : g k[t m,..., t m ] g k[t m,..., t m ] whih

Homology, Homotopy nd Applitions, vol. 8(2), 2006 124 stisfies ( d + t j d(γ j )) 2 = 0 modulo ti t j. One refers to γ = γ j s n infinitesiml deformtion. One n redily hek tht Def g (R) = Ker(d)/ Im(d) = H(g). Suppose Def g is prorepresentle. Tht is, there exists projetive limit of (grded) lol Artin rings O nd n equivlene of the funtors Def g ( ) hom(o, ). In the se tht O = O M is the ring of lol funtions t the se point of pointed Z grded spe M, then M is the lol moduli spe for Def g. Denote the se point of M y p. One n hek tht T p (M) hom(o M, R). It follows tht the grded tngent spe to the moduli spe t the se point is isomorphi to the ohomology of (g, d): T p (M) H(g). Now, let (A, D) e n A lger nd let I e n inner produt on (A, D). Theorem 3.7 sys tht the differentil grded Lie lger ontrolling deformtions of (A, D, I) is with rket nd differentil h = Coder(T A) Comp(T A A, T A A) [(f, i), (g, j)] = ([f, g], δ f (j) ( 1) f g δ g (i)) d(f, i) = [(D, I), (f, i)]. Thus follows the expeted infinitesiml sttement: Corollry 4.1. The grded tngent spe to the moduli spe of A strutures with inner produts is isomorphi to H(h). As finl remrk, we mention some onnetions etween the ohomology H(h) nd ouple of its ousins. If (A, D, I) is n A lger with -inner produt, we hve the Hohshild differentil grded Lie lger (Coder(T A), δ, [, ]) nd the su differentil grded Lie lger of yli Hohshild ohins Coder(T A) Cyli, defined y Coder(T A) Cyli = {f Coder(T A) : δ f (I) = 0}. If I onsists of n ordinry symmetri inner produt I =,, then the ondition δ f (I) = f A I I f A = 0 is equivlent to f( 1,..., n ), n+1 = ± 1, f( 2,..., n+1 ).

Homology, Homotopy nd Applitions, vol. 8(2), 2006 125 We hve the following mps of differentil grded Lie lgers: (Coder(T A) Cyli, δ, [, ]) (h, d, [, ]) nd (h, d, [, ]) (Coder(T A), δ, [, ]). (4) The first mp is the injetion f (f, 0) h, whih is ohin mp d(f, 0) = ([D, f], ±(f A I I f A )) = (δf, 0), euse elements of the domin re yli. The indued mp in ohomology desries sttement from [7], nmely tht the first order deformtions of D omptile with the inner produt re lssified y yli ohomology. We do not know under wht onditions the mp f (f, 0) h indues n isomorphism in ohomology. The seond mp in (4) is simply the projetion Coder(T A) Comp(T A A, T A A) Coder(T A) nd the indued mp in ohomology desries the simple sttement tht ny infinitesiml deformtion of the pir (D, I) gives n infinitesiml deformtion of D. Appendix A. Expliit formuls of δ f (i) Let f Coder(T A) nd i Comp(T A A, T A A). We wnt to desrie the term δ f (i) = f A i ( 1) f i i f A Comp(T A A, T A A) more expliitly. Here, f : k 1 A k A nd i : k,l 0 A k A A l A S hve the omponents k... 1 2 f k : A k A f k ( 1,..., k ) k... 1 2 i k,l = k,l : A k A A l A S k+1 k+l+2 k+2... k+l+1 By onvention, the inputs re lwys inserted using the ounterlokwise diretion. Then f A i ( 1) f i i f A is given y inserting f into i in ll possile omintions.

Homology, Homotopy nd Applitions, vol. 8(2), 2006 126 ± ± ± ± First, here re some exmples of how these digrms re to e red.,,, d 2,0 d,,, d, e, f, g, h, i 3,4 d i e f g h f 2 (f 2 (, ), f 2 (d, e)), f 2 (f, ) 0,0 f d e

Homology, Homotopy nd Applitions, vol. 8(2), 2006 127,, f 3 (, d, f 2 (e, f)), g, f 2 (h, i)) 1,2 i d g h e f, f 2 (d, e), f 2 (f 2 (f, g), h), i, f 4 (j, k,, ) 2,1 f e d g k h i j Here re the terms of δ f (i) = f A i ( 1) f i i f A up to sign, when they re eing pplied to elements from A k A A l A: k = 0, l = 0: f 1 (), 0,0 ±, f 1 () 0,0 ± k = 1, l = 0: f 1 (),, 1,0 ±, f 1 (), 1,0 ±,, f 1 () 1,0 ± f 2 (, ), 0,0 ±, f 2 (, ) 0,0

Homology, Homotopy nd Applitions, vol. 8(2), 2006 128 ± ± ± ± k = 0, l = 1: f 1 (),, 0,1 ±, f 1 (), 0,1 ±,, f 1 () 0,1 ± f 2 (, ), 0,0 ±, f 2 (, ) 0,0 ± ± ± ± k = 2, l = 0: f 1 (),,, d 2,0 ±, f 1 (),, d 2,0 ±,, f 1 (), d 2,0 ±,,, f 1 (d) 2,0 ± f 2 (, ),, d 1,0 ±, f 2 (, ), d 1,0 ±,, f 2 (d, ) 1,0 ± f 3 (,, ), d 0,0 ±, f 3 (d,, ) 0,0 Note tht for exmple the term,, f 2 (, d) 2,0 does not pper, euse nd d re the two speil elements of d A 2 A A 0 A, whih re put on the horizontl line of the digrm. The two speil elements from A k A A l A n never e inside ny f n.

Homology, Homotopy nd Applitions, vol. 8(2), 2006 129 k = 0, l = 2: f 1 (),,, d 0,2 ±, f 1 (),, d 0,2 ±,, f 1 (), d 0,2 ±,,, f 1 (d) 0,2 ± f 2 (, ),, d 0,1 ±, f 2 (, ), d 0,1 ±,, f 2 (, d) 0,1 ± f 3 (,, ), d 0,0 ±, f 3 (,, d) 0,0 The speil elements re nd d from d A 0 A A 2 A. k = 1, l = 1: f 1 (),,, d 1,1 ±, f 1 (),, d 1,1 ±,, f 1 (), d 1,1 ±,,, f 1 (d) 1,1 ± f 2 (, ),, d 0,1 ±,, f 2 (d, ) 0,1 ±, f 2 (, ), d 1,0 ±,, f 2 (, d) 1,0 ± f 3 (,, ), d 0,0 ±, f 3 (, d, ) 0,0 The speil elements re nd d from d A 1 A A 1 A.

Homology, Homotopy nd Applitions, vol. 8(2), 2006 130 i =, 0,0 for ny k, l: Assume tht i =, 0,0 hs only lowest omponent, ut f hs ll higher omponents. We pply f A i ( 1) f i i f A to the element 1... k k+1 k+2... k+l+1 k+l+2 A k A A l A to get f( 1,..., k+l+1 ), k+l+2 0,0 ± k+1, f( k+2,..., k+l+2, 1,..., k ) 0,0 ± Referenes [1] R. Cohen, J. Klein nd D. Sullivn. The homotopy invrine of the string topology loop produt nd string rket. mth.gt/0509667, 2005. [2] A. Filowski nd M. Penkv. Deformtion theory of infinity lgers. Journl of Alger 255, 2002, 59 88. [3] W.M. Goldmn nd J.J. Millson. The deformtion theory of representtions of fundmentl groups of ompt Kähler mnifolds. Pul. Mth IHES 67, IHES, 1988, 43 96. [4] H. Kjiur. Nonommuttive homotopy lgers ssoited with open strings. mth.qa/0306332, 2003. [5] M. Kontsevih. Deformtion quntiztion of Poisson mnifolds, I. Letters in Mthemtil Physis 66, no 3, Springer, Netherlnds, 2003, 157 216. [6] M. Mnetti. Deformtion theory vi differentil grded Lie lgers. Seminri di Geometri Algeri, Suol Normle Superiore, 1999, 21 48. [7] M. Penkv. Infinity lgers, ohomology nd yli ohomology, nd infinitesiml deformtions. mth.qa/0111088, 2001.

Homology, Homotopy nd Applitions, vol. 8(2), 2006 131 [8] M. Shlessinger. Funtors of Artin rings. Trns. Am. Mth. So. 130, 1968, 208 222. [9] M. Shlessinger nd J. Stsheff. The Lie lger struture of tngent ohomology nd deformtion theory. J. Pure nd Applied Alger 38, 1985, 313 322. [10] D. Sullivn. Infinitesiml omputtions in topology. Pulitions Mthémtiques de l IHÉS 47 (1977), 269 331. [11] T. Trdler nd M. Zeinlin. Poinré dulity t the hin level. mth.at/0309455, 2002. [12] T. Trdler. Infinity inner produts on A-infinity lgers. mth.at/0108027, 2001. [13] T. Trdler. The BV lger on Hohshild ohomology indued y infinity inner produts. mth.qa/0210150, 2002. John Terill john@q.edu Deprtment of Mthemtis Queens College of the City University of New York 65-30 Kissen Blvd. Flushing, NY 11367 Thoms Trdler ttrdler@ityteh.uny.edu Deprtment of Mthemtis College of Tehnology of the City University of New York 300 Jy Street Brooklyn, NY 11201 This rtile is ville t http://intlpress.om/hha/v8/n2/7