On the Computation of A∞-Maps

On the Computation of A∞-Maps AinhoaBerciano1,Mar´ıaJos´eJim´enez2,andPedroReal2 1Dpto. Matem´atica Aplicada, Estad´ıstica e Investigaci´on Operativa, Universidad del Pa´ıs Vasco, Barrio Sarriena s/n, 48940 Leioa (Vizcaya), Spain [email protected] 2Dpto. de Matem´atica Aplicada I, Universidad de Sevilla, Avda. Reina Mercedes s/n, 41012 Sevilla, Spain {majiro, real}@us.es Abstract. Starting from a chain contraction (a special chain homotopy equivalence) connecting a differential graded algebra Awith a differential graded module M, the so-called homological perturbation technique “tensor trick” [8] provides a family of maps, {mi}i≥1, describing an A∞- algebra structure on Mderived from the one of algebra on A.Inthis paper, taking advantage of some annihilation properties of the component morphisms of the chain contraction, we obtain a simplified version of the existing formulas of the mentioned A∞-maps, reducing the computational cost of computing mnfrom O(n!2)toO(n!). Keywords: A∞-algebra, contraction, Basic Perturbation Lemma, transference, computation. 1 Introduction Atpresent,A∞-structures (orstronghomotopystructures)findnaturalapplicationsnotonly in Algebra,TopologyandGeometrybut alsoinMathematical Physics,related totopics such as stringtheory, homologicalmirrorsymmetryor superpotentials[14,17,18]. Nevertheless,there are fewmethods forcomputingexplicitA∞-structures,being the better known technique the tensor trick [8]. This tool isusedin the contextofHomologicalPerturbation Theory. Startingfroma chain contraction c(aspecialchain homotopyequivalence,alsocalled strongdeformation retract)fromadifferentialgraded algebra Aontoadifferentialgraded moduleM,the tensortrick technique gives explicitformulas forcomputinga family ofhigher maps {mi}i≥1that provides anA∞-algebra structure on M(derived fromthealgebra structure on A). However,the associated computational costs are extremely high (see [11,12,1]). In thispaper,weareconcerned about findingamore cost-effectiveformulation ofthefamily ofmapstransferred to M.Asitisshown in section 3, the use ofannihilation properties ofthecomponentmorphisms ofthechain contraction allowstoreformulate the A∞–maps on M(which dependon the mentioned componentmorphisms). Afterwards,in Partially supported by the PAICYT research project FQM-296 and by a project of University of the Basque Country “EHU06/05”. V.G. Ganzha, E.W. Mayr, and E.V. Vorozhtsov (Eds.): CASC 2007, LNCS 4770, pp. 45–57, 2007. 46 A. Berciano, M.J. Jim´enez, and P. Real section 4wecarryout a theoreticalstudyofthetime andspaceinvested in computingmn,presentingthecomputationalsavings obtained,incomparison with the originalformulas defined bythe BasicPerturbation Lemma. The results canbe extended tothe case ofAbeinganA∞-algebra (then, another A∞-algebra structure isalsoinduced on M). We remark that such a transference canalsobe performed in the case ofcbeingageneralexplicitchain homotopyequivalence. Of course,all the results giveninthis paper canbe easily translated intothe contextofcoalgebras andA∞–coalgebras. 2 Notations and Preliminaries Webriefly recall here some basicdefinitionsin HomologicalAlgebra as well as the notationsusedthroughout the paper.See [3] or[16] forfurtherexplanations. Take a commutativeunitalringΛ.Let (M,d)be a DG-module,that is,a Λ–modulegradedon the non-negativeintegers (M=n≥0Mn)andendowed with a differentiald(ofdegree−1). An elementx∈Mnhas degree n,what will be expressed by|x|=n.Inthe case that M0=Λ,Miscalled connected andif,besides,M1=0,thenitiscalled simply connected.Givenaconnected DG–module,M,the reduced moduleMistheonewith Mn=Mnforn>1and M0=0. Wewill denote the moduleM⊗n ···⊗MbyM⊗n,with M⊗0=Λandthe morphism f⊗n ···⊗f:M⊗n→N⊗nbyf⊗n.We adhere toKoszulconvention forsigns.More concretely, givenf:M→M,h:M→M,g:N→Nand k:N→N DG–modulemorphisms,then (h⊗k)(f⊗g)=(−1)|k||f|(hf ⊗kg). Onthe other hand,iff:M⊗i→MisaDG–modulemorphism andnisa non–negativeinteger,wewill denote byf[n]:M⊗n→M⊗n−i+1 the morphism f[n]= n−i  j=0 1⊗j⊗f⊗1⊗n−i−j andthemorphism f[] :j≥iM⊗j→k≥1M⊗kwill be the one such that f[]|M⊗n=f[n]. Wewill denote by↑and↓the suspension anddesuspension operators,which shift the degree by+1and−1, respectively. A givenmorphism ofgradedmodules ofdegreek,f:M→N,induces another onebetweenthe suspended modules sf :sM →sN,givenbysf =(−1)k↑f↓. GivenaDG-module(M,d), the tensor module ofM,T(M), istheDG–module T(M)= n≥0 Tn(M)= n≥0 M⊗n On the Computation of A∞-Maps 47 whose differentialstructure isprovided byd[] M.Everymorphism ofDG-modules f:M→Ninduces another oneT(f):T(M)→T(N), such that T(f)|M⊗n= f⊗n. ADG–algebra,(A, dA,μ A), isaDG–moduleendowed with anassociativeproduct,μA,compatiblewith the differentialdAandwhich has a unitηA:Λ→A, that is,μA(ηA⊗1) = μA(1 ⊗ηA)=1.Ifthereisno confusion, subscripts will be omitted.ADG–coalgebra (C, dC,Δ C)isaDG–moduleprovided with a compatiblecoproduct andcounitξC:C→Λ(so, (ξC⊗1)ΔC=(1⊗ξC)ΔC=1). In the case ofthetensormoduleT(M), aproduct,μ,andacoproduct,Δ, canbe naturally defined on anelementa1⊗···⊗an∈Tn(M), as follows: μ((a1⊗···⊗an)⊗(an+1 ⊗···⊗an+p)) = a1⊗···⊗an+p; Δ(a1⊗···⊗an)=n i=0(a1⊗···⊗ai)⊗(ai+1 ⊗···⊗an). Therefore,T(M)acquires both structures ofDG–algebra (denoted byTa(M)) andDG–coalgebra (Tc(M)), though theyare notcompatibletoeach other (that is,(T(M),μ,Δ)isnotaHopf algebra). We recall here two equivalentdefinitionsofA∞–algebra (resp.A∞–coalgebra) [13,19]. –An A∞-algebra (respectively, A∞-coalgebra), isaDG-module(M,m1)(resp. (M,Δ1)) endowed with a family ofmaps mi:M⊗i→M(resp., Δi:M→M⊗i) ofdegreei−2 such that,forn≥1, i  n=1 i−n  k=0 (−1)n+k+nkmi−n+1(1⊗k⊗mn⊗1⊗i−n−k)=0,(1) (resp., i  n=1 i−n  k=0 (−1)n+k+nk(1⊗i−n−k⊗Δn⊗1⊗k)Δi−n+1 =0).(2) –An A∞-algebra (resp., A∞–coalgebra)is a graded moduleMendowed with amorphism ofmodules m:T(sM)→M(resp., Δ:M→T(s−1M)) such that the morphism d=−(↑mT(↓))[] (resp., d=−(T(↓)Δ↑)[] )makes Tc(sM)(resp., Ta(s−1M)) tobe a DGA–coalgebra (resp., DGA-algebra). The reduced bar construction ofaconnected DG–algebra A,¯ B(A), isaDG– coalgebra whose module structure isgivenby T(s¯ A)= n≥0 (s¯ A⊗ntimes ··· ⊗s¯ A). The totaldifferentiald¯ Bisgivenbythe sum ofthetensordifferential, dt(which isthenaturaloneon the tensorproduct)andthesimplicial differential,ds(that depends on the product on A): dt=−n−1 i=0 1⊗i⊗↑dA↓⊗1⊗n−i−1;ds=n−2 i=0 1⊗i⊗↑μA↓⊗2⊗1⊗n−i−2. 48 A. Berciano, M.J. Jim´enez, and P. Real The coproduct Δ¯ B:¯ B(A)→¯ B(A)⊗¯ B(A)isthenaturaloneon the tensor module. In the contextofhomologicalperturbation theory, the main input data are contractions [4,9,15,7,10]: acontraction c:{N,M,f,g,φ}fromaDG-moduleN toaDG-moduleM,consists in aparticular homotopyequivalence determined bytwo DG-modulemorphisms,f:N→Mandg:M→Nandahomotopy operatorφ:N→N+1 such that fg =1 M,andφdN+dNφ+gf =1 N.Moreover, these data are alsorequired tosatisfythe anihilation properties: fφ =0,φg=0,φφ=0. GivenaDG–modulecontraction c:{N, M, f, g, φ},onecanestablish the followingones [7,8]: –The suspension contraction ofc,sc,which consists ofthesuspended DG– modules andtheinduced morphisms: sc:{sN, sM, sf, sg, sφ}, beingsf =↑f↓,sg =↑g1↓andsφ=−↑φ↓,which are briefly expressed byf,gand−φ. –The tensor module contraction,T(c), betweenthe tensormodules ofM andN: T(c):{T(N),T(M),T(f),T(g),T(φ)}, where T(φ)|Tn(N)=φ[⊗n]= n−1  i=0 1⊗i⊗φ⊗(gf)⊗n−i−1. Amorphism ofgradedmodules f:N→Niscalled pointwise nilpotent whenever forall x∈N,x=0,there exists a positiveinteger nsuch that fn(x)=0.Aperturbation of a DG-module Nconsists in amorphism ofgraded modules δ:N→Nofdegree−1, such that (dN+δ)2=0.Aperturbation datum ofthecontraction c:{N,M,f,g,φ}is a perturbation δoftheDG-moduleN satisfying that the composition φδ ispointwise nilpotent. The main tool whendealingwith contractionsistheBasic Perturbation Lemma [2,5,15], which isanalgorithm whose input isacontraction ofDG– modules c:{N,M,f,g,φ}and a perturbation datum δofcandwhose output isanewcontraction cδ:{(N,dN+δ),(M,dM+dδ),f δ,g δ,φ δ}defined bythe formulas dδ=fδΣ δ cg;fδ=f(1 −δΣδ cφ); gδ=Σδ cg;φδ=Σδ cφ; where Σδ c=i≥0(−1)i(φδ)i. The pointwise nilpotencyofthecomposition φδ guarantees that the sums are finite for each particular element. On the Computation of A∞-Maps 49 3 Transferring A∞–Algebras Via Homological Perturbation Theory A∞–algebras were first introduced byStasheffin[20]. Theyare,roughly speaking,algebras which are associative“up tohomotopy” (alsocalled strongly homotopyassociativealgebras). In the papers ofGugenheim,StasheffandLambe [6,9,8], theydescribe a technique called tensor trick bywhich,startingfromacontraction betweenaDG– algebra AandaDG–moduleM,anA∞–algebra structure isinduced on M.This transference alsoexists in the case that AisanA∞–algebra.Moreover,inthe case that a generalhomotopyequivalence isestablished betweenAandM,itis alsopossibletoderiveaformulation foranA∞–algebra structure on M.Wewill mainly focus our efforts on obtainingcomputationalimprovements in the first case. 3.1 Transference Via Contractions Let us consider the contraction c:{A, M, f, g, φ}, where Aisaconnected DG–algebra andMaDG–module.The first step consists in tensoring,inorder toobtain the underlyinggradedm oduleof the bar construction ofA, T(sc):{Tc(s¯ A),Tc(s¯ M),Tf,Tg,T(−φ)}; andthen, consideringthesimplicialdifferential, ds,which is a perturbation datum forthiscontraction, andusingtheBasicPerturbation Lemma,anewcontraction isobtained, {¯ B(A),(Tc(s¯ M), d), f,g,  φ}, where (Tc(s¯ M), d)iscalled the tilde bar construction ofM[20], denoted by  B(M). Then, the perturbed differential dinduces a family ofmapsmn:M⊗n→ Mofdegreen−2thatprovides anA∞–algebra structure on M. The transference ofanA∞–algebra structure was alsostudied byKadeishvili in [13] forthecaseM=H(A). Usingthistechnique,inthe followingtheorem,an expression ofafamily ofA∞–operationsisgivenwith regard tothe component morphisms oftheinitialcontraction. Although thisformulation isimplicitly derived fromthementioned papers [13] and[8], anexplicitproofisgivenin[12]. Theorem 1. [13,8] Let (A, dA,μ)and (M, dM)be a connected DG–algebra and a DG–module, respectively and c:{A, M, f, g, φ}a contraction between them. Then the DG–module Mis provided with an A∞–algebra structure given by the operations 50 A. Berciano, M.J. Jim´enez, and P. Real m1=−dM mn=(−1)n+1fμ (1) φ[⊗2] μ(2) ···φ[⊗n−1] μ(n−1) g⊗n,n≥2(3) where μ(k)= k−1  i=0 (−1)i+11⊗i⊗μA⊗1⊗k−i−1. As far as the computation oftheseformulas isconcerned,wecantake advantage oftheannihilation properties off,gandφtodeduce a more economical formulation formn. Theorem 2. Any composition of the kind φ[⊗s]μ(s)(s=2,...,n−1)inthe formula (3), which is given by ⎛ ⎝ s−1  j=0 1⊗j⊗φ⊗(gf)⊗s−j−1⎞ ⎠◦s−1  i=0 (−1)i+11⊗i⊗μA⊗1⊗s−i−1, can be reduced to the following sum s−1  i=0 (−1)i+11⊗i⊗φμA⊗1⊗s−i−1.(4) Moreover, given a composition of the kind (φ[⊗s−1]μ(s−1))◦(φ[⊗s]μ(s))s=3,...,n−2, for every index iin the sum (4) of φ[⊗s]μ(s),theformulaofφ[⊗s−1]μ(s−1) in such a composition can be reduced to s−2  j=i−1,j≥0 (−1)j+11⊗j⊗φμA⊗1⊗s−j−2.(5) In other words, the whole composition (φ[⊗2] μ(2))◦···◦(φ[⊗n−1] μ(n−1))in the formula of mncan be expressed by n−2  in−1=0⎛ ⎝ n−3  in−2=in−1−1···1  i2=i3−1 (φμ)(2,i2)···(φμ)(n−2,in−2)⎞ ⎠(φμ)(n−1,in−1) , where (φμ)(k,j)=(−1)j+11⊗j⊗φμA⊗1⊗k−j−1and each addend exists whenever the corresponding index ik≥0. Proof. Let us provetheformula4ofφ[⊗s]μ(s)forany s=n−1,n−2,...,2,by induction over the number k=n−soffactors ofthetype φ[⊗∗]μ(∗)that are composed,following the scheme On the Computation of A∞-Maps 51 mn=(−1)n+1fμ (1) φ[⊗2] μ(2) ···φ[⊗n−2] μ(n−2) φ[⊗n−1] μ(n−1) g⊗n  k=1  k=2  k=n−2 (6) Atthesametime,wewill provethemajor reduction oftermsgivenby(5)for s=n−2,...,2. –k=1 The composition ofmorphisms φ[⊗n−1] μ(n−1) g⊗ncanbe writtenas ⎛ ⎝ n−2  j=0 1⊗j⊗φ⊗(gf)⊗n−j−2⎞ ⎠◦n−2  i=0 (−1)i+1g⊗i⊗μAg⊗2⊗g⊗n−i−2. Now, using the facts that fg =1andφg =0,itissimpletosee that the only non null elements are those where φisapplied over μA,sothe original formulaofφ[⊗n−1] μ(n−1) issimplified to n−2  i=0 (−1)i+11⊗i⊗φμA⊗1⊗n−i−2. –k=2In thiscase,takingintoaccounttheformulaobtained fork=1, φ[⊗n−1] μ(n−1) g⊗n= n−2  i=0 (−1)i+1g⊗i⊗φμAg⊗2⊗g⊗n−i−2(7) andthatφ[⊗n−2]μ(n−2) isthecomposition ⎛ ⎝ n−3  j=0 1⊗j⊗φ⊗(gf)⊗n−j−3⎞ ⎠◦n−3  i=0 (−1)i+11⊗i⊗μA⊗1⊗n−i−3, wecanuse the anihilation properties φg =0andφ2=0,toconclude that the factorφin φ[⊗n−2] has tobe applied over μAandhence, φ[⊗n−2] μ(n−2) = n−3  j=0 (−1)j+11⊗j⊗φμA⊗(gf)⊗n−j−3.(8) Now, consideringthecomposition ofthesum(7)with (8), onecanobserve that,since fφ =0,foreachindexiin the sum (7), the only addends of(8) that havetobe considered forthecomposition are those j≥i−1. Onthe other hand,fg =1isalsosatisfied,so φ[⊗n−2] μ(n−2) = n−3  j=i−1 (−1)j+11⊗j⊗φμA⊗1⊗n−j−3. 52 A. Berciano, M.J. Jim´enez, and P. Real –k=mFinally, let us assume that the proposition istrueforφ[⊗n−k]μ(n−k) forall k=1,...,m−1. Now, considering,ononehand,φ[⊗n−m]μ(n−m), ⎛ ⎝ n−m−1  j=0 1⊗j⊗φ⊗(gf)⊗n−j−m−1⎞ ⎠n−m−1  i=0 (−1)i+11⊗i⊗μA⊗1⊗n−i−m−1 andthat,onthe other hand,the composition ofmorphisms φ[⊗n−m+1] μ(n−m+1) ···φ[⊗n−1] μ(n−1) g⊗n byinduction hypothesis,isasumofelements that are tensorproduct of factors ofthetype φ(something)org,using again the annihilation properties, itfollowsthat φ[⊗n−m]μ(n−m)= n−m−1  j=0 (−1)j+11⊗j⊗φμA⊗(gf)⊗n−j−m−1. Since,byinduction hypothesis, φ[⊗n−m+1] μ(n−m+1) = n−m  i=0 (−1)i+11⊗i⊗φμA⊗1⊗n−m−i, takingintoaccountthatfg =1andthefactthatfφ =0,again wecan reduce the number oftermsofφ[⊗n−m]μ(n−m)to n−m−1  j=i−1 (−1)i+11⊗i⊗φμA⊗1⊗n−m−i−1, where iistheindexcorrespondingtothe term of the precedingsumthatis beingcomposed with φ[⊗n−m]μ(n−m). Wecangeneralize the results showed abovetothe case that the “big”DGmoduleofagivencontraction isanA∞-algebra.The stabilityoftheA∞- structures with respect tothe contractionsfollowsfrom the paper [8]. In fact,it ispossibletoextract the nexttheorem as animplicitconsequence oftheresults there. Theorem 3. Given c:{A, M, f, g, φ}a contraction, where (A, m1,m 2,...)is a connected A∞-algebra and Mis a DG-module, then Minherits an A∞-algebra structure. Proof. The prooffollows the same scheme as in theorem 1(andforthatreason, wewill only sketch itslightly) , makinguseofthetensortrick andtheBasic Perturbation Lemma,with the difference that,now,the perturbation datum for the contraction On the Computation of A∞-Maps 53 T(sc):{Tc(s¯ A),T c(s¯ M),T(f),T(g)T(−φ)} istheoneinduced bythe A∞–maps dm|(s¯ A)⊗n=− n  k=2 n−k  i=0 1⊗i⊗↑mk↓⊗k⊗1⊗n−k−i. Since the family ofmaps{mi}i≥1defines anA∞-algebra structure on A, d˜ B=dt+dmisadifferentialonTc(s¯ A)(infact,(Tc(s¯ A),d˜ B)isthetilde bar construction ofA). Onthe other hand,the pointwise nilpotencyofT(−φ)dm follows because dmreduces the simplicialdimension, whileT(−φ)keeps itthe same. Thanks tothe BasicPerturbation Lemma,anewdifferentialisobtained on Tc(s¯ M), ˜ d,givenbythe formula: ˜ d=dt+T(f)dm i≥0 (−1)i(T(−φ)dm)iT(g). Thisway, ˜ dinduces a family ofmaps{mM i}i≥1on M,where mM n,up tosign, canbe expressed by fm ng⊗n+ n−2  l=1  2≤k1<...<kl≤n−1 ±fm k1(φ[⊗k1]m(k1) k2−k1+1)···(φ[⊗kl]m(kl) n−kl+1)g⊗n where m(k) n−k+1 :A⊗n→A⊗kisgivenby m(k) n−k+1 = n−k+1  i=0 1⊗i⊗mn−k+1 ⊗1⊗k−i−1. Notice that,since miisamapofdegreei−2,mM nhas degree n−2. Ifweexaminetheformulaabovein low dimensions,weobtain, up tosign: mM 2=±fm 2g⊗2; mM 3=±fm 3g⊗3±fm 2φ[⊗2] m(2) 2g⊗3; mM 4=±fm 4g⊗4±fm 2φ[⊗2] m(2) 3g⊗4±fm 3φ[⊗3] m(3) 2g⊗4 ±fm 2φ[⊗2] m(2) 2φ[⊗3] m(3) 2g⊗4. Notice that only the last addendof each map istheoneinduced in the case ofAbeinganalgebra,instead ofthe2 n−2addends generated in these cases (the number of subsets ofasetofn−2elements). At each addendofeachA∞–map, onecanobtain a reductioninnumber ofterms,ofthesamenature thanthe one showed in theorem 2.