A relative version of the ordinary perturbation lemma
aa r X i v : . [ m a t h . K T ] F e b A RELATIVE VERSION OF THE ORDINARY PERTURBATION LEMMA
MARCO MANETTI
Abstract.
The perturbation lemma and the homotopy transfer for L ∞ -algebras is provedin a elementary way by using a relative version of the ordinary perturbation lemma for chaincomplexes and the coalgebra perturbation lemma. Introduction
Let N be a differential graded vector space and let M ⊂ N be a differential graded subspacesuch that the inclusion map ı : M → N is a quasi-isomorphism. The basic homology theoryshows that there exists a homotopy h : N → N such that Id + dh + hd : N → N is a projectiononto M . If ˜ d is a new differential on N such that ∂ = ˜ d − d is “small” in some appropriate sense,then the ordinary perturbation lemma (Theorem 3.6) gives explicit functorial formulas, in termsof ∂ and h , for a differential ˜ D on M and for an injective morphism of differential graded vectorspaces ˜ ı : ( M, ˜ D ) → ( N, ˜ d ).Has been pointed out by Huebschmann and Kadeishvili [4] that if M, N are differential graded(co)algebra, and h is a (co)algebra homotopy (Definition 2.5), then also ˜ ı is a morphism ofdifferential graded (co)algebras. This assumption are verified for instance when we consider thetensor coalgebras generated by M, N and the natural extension of h to T ( N ) (this fact is referredas tensor trick in the literature). Therefore the ordinary perturbation lemma can be easily usedto prove Kadeishvili’s theorem [10, 11] on the homotopy transfer of A ∞ structures (see also[4, 9, 13, 14, 18, 19]).If we wants to use the same strategy for L ∞ -algebras, we have to face the following problems:(1) the tensor trick breaks down for symmetric powers and coalgebra homotopies are notstable under symmetrization,(2) not every L ∞ -algebra is the symmetrization of an A ∞ -algebra.Therefore the proof of the homotopy transfer for L ∞ -algebras requires either a nontrivial addi-tional work [5, 6, 7] or a different approach, see e.g. [3, 12] and the arXiv version of [2].The aim of this paper is to show that the homotopy transfer for L ∞ -algebras (Theorem 6.1)follows easily from a slight modification (Theorem 4.3) of the ordinary perturbation lemma inwhich we assume that ˜ d is a differential when restricted to a differential graded subspace A ⊂ N satisfying suitable properties.The paper is written in a quite elementary style and we do not assume any knowledge ofhomological perturbation theory. We only assume that the reader is familiar with the basicproperties of graded tensor and graded symmetric coalgebras. The bibliography contains thedocuments that have been more useful in the writing of this paper and it is necessarily incomplete;for more complete references the reader may consult [8, 9]. I apologize in advance for everypossible misattribution of previous results.2. The category of contractions
Let R be a fixed commutative ring; by a differential graded R -module we mean a Z -graded R -module N = ⊕ i ∈ Z N i together a R -linear differential d N : N → N of degree +1. Date : February 3, 2010.2010
Mathematics Subject Classification.
Given two differential graded R -modules M, N we denote by Hom nR ( M, N ) the module of R -linear maps of degree n :Hom nR ( M, N ) = { f ∈ Hom R ( M, N ) | f ( M i ) ⊂ N i + n , ∀ i ∈ Z } . Notice that Hom R ( M, N ) are the morphisms of graded R -modules and { f ∈ Hom R ( M, N ) | d N f = f d M } is the set of cochain maps (morphisms of differential graded R -modules). Definition 2.1 (Eilenberg and Mac Lane [1, p. 81]) . A contraction is the data( M ı / / N π o o , h )where M, N are differential graded R -modules, h ∈ Hom − R ( N, N ) and ı, π are cochain mapssuch that:(1) (deformation retraction) πı = Id M , ıπ − Id N = d N h + hd N ,(2) (annihilation properties) πh = hı = h = 0. Remark . In the original definition Eilenberg and Mac Lane do not require h = 0; however,if h satisfies the remaining 4 conditions, then h ′ = hd N h satisfies also the fifth (cf. [7, Rem. 2.1]). Definition 2.3. A morphism of contractions f : ( M ı / / N π o o , h ) → ( A i / / B p o o , k )is a morphism of differential graded R -modules f : N → B such that f h = kf . Given a morphismof contractions as above we denote by ˆ f : M → A the morphism of differential graded R -modulesˆ f = pf ı .In the notation of Definition 2.3 it is easy to see that the diagrams M ˆ f / / ı (cid:15) (cid:15) A i (cid:15) (cid:15) N f / / B N f / / π (cid:15) (cid:15) B p (cid:15) (cid:15) M ˆ f / / A are commutative. In fact i ˆ f = ipf ı = f ı + ( d B kf + kd B f ) ı = f ı + f ( d N h + hd N ) ı = f ı + f ( ıπ − Id N ) ı = f ı, ˆ f π = pf ıπ = pf (Id N + d N h + hd N ) = pf + p ( d B k + kd B ) f = pf + p ( ip − Id B ) = pf. Definition 2.4.
The composition of contractions is defined as( M ı / / N π o o , h ) ◦ ( N i / / P p o o , k ) = ( M iı / / P πp o o , k + ihp )Given two contractions ( M ı / / N π o o , h ) and ( A i / / B p o o , k ) we define their tensor productas ( M ⊗ R A ı ⊗ i / / N ⊗ R B π ⊗ p o o , h ∗ k ) , h ∗ k = ıπ ⊗ k + h ⊗ Id B . It is straightforward to verify that the tensor product of two contractions is a contraction, itis bifunctorial and, up to the canonical isomorphism ( L ⊗ R M ) ⊗ R N ∼ = L ⊗ R ( M ⊗ R N ), it isassociative.Given a contraction ( M ı / / N π o o , h ), its tensor n th power is N nR ( M ı / / N π o o , h ) = ( M ⊗ n ı ⊗ n / / N ⊗ nπ ⊗ n o o , T n h ) , RELATIVE VERSION OF THE ORDINARY PERTURBATION LEMMA 3 where T n h = n X i =1 ( ıπ ) ⊗ i − ⊗ h ⊗ Id ⊗ n − iN . The tensor product allows to define naturally the notion of algebra and coalgebra contraction;we consider here only the case of coalgebras.
Definition 2.5.
Let N be a differential graded coalgebra over a commutative ring R withcoproduct ∆ : N → N ⊗ R N . We shall say that a contraction ( M ı / / N π o o , h ) is a coalgebracontraction if ∆ : ( M ı / / N π o o , h ) → ( M ⊗ R M ı ⊗ ı / / N ⊗ R N π ⊗ π o o , h ∗ h )is a morphism of contractions.Notice that if ∆ is a morphism of contractions then ˆ∆ is a coproduct and π, ı are morphismsof differential graded coalgebras. Conversely, a contraction ( M ı / / N π o o , h ) is a coalgebra con-traction if π, ı are morphisms of differential graded coalgebras and( ıπ ⊗ h + h ⊗ Id N ) ◦ ∆ = ∆ ◦ h. Example 2.6 (tensor trick) . Given a contraction ( M ı / / N π o o , h ) of differential graded R -modules, we can consider the reduced tensor coalgebra T ( N ) = L ∞ n =1 N nR N with the coproduct a ( x ⊗ · · · ⊗ x n ) = n − X i =1 ( x ⊗ · · · ⊗ x i ) ⊗ ( x i +1 ⊗ · · · ⊗ x n ) . We have seen that there exists a contraction( T ( M ) T ( ı ) / / T ( N ) T ( π ) o o , T h ) , where T ( ı ) = P ı ⊗ n , T ( π ) = P π ⊗ n and T h = P n T n h .We want to prove that ( T ( M ) T ( ı ) / / T ( N ) T ( π ) o o , T h ) is a coalgebra contraction, i.e. that( T ( ıπ ) ⊗ T h + T h ⊗ Id) ◦ a = a ◦ T h.
Let n be a fixed positive integer, writing T n h = n X i =1 T ni h , T ni h = ( ıπ ) ⊗ i − ⊗ h ⊗ Id ⊗ n − iN , for every i = 1 , . . . , n we have a ◦ T ni h = i − X j =1 ( ıπ ) ⊗ j ⊗ T n − ji − j h + n − X j = i T ji h ⊗ Id ⊗ n − jN . MARCO MANETTI
Therefore a ◦ T n h = n X i =1 a ◦ T ni h = n X i =1 i − X j =1 ( ıπ ) ⊗ j ⊗ T n − ji − j h + n X i =1 n − X j = i T ji h ⊗ Id ⊗ n − jN = n − X j =1 n X i = j +1 ( ıπ ) ⊗ j ⊗ T n − ji − j h + n − X j =1 j X i =1 T ji h ⊗ Id ⊗ n − jN = n − X j =1 ( ıπ ) ⊗ j ⊗ ( n − j X i =1 T n − ji h ) + n − X j =1 ( j X i =1 T ji h ) ⊗ Id ⊗ n − jN = n − X j =1 ( ıπ ) ⊗ j ⊗ T n − j h + n − X j =1 T j h ⊗ Id ⊗ n − jN . It is now sufficient to sum over n .3. Review of ordinary homological perturbation theory
Convention:
In order to simplify the notation, from now on, and unless otherwise stated, forevery contraction ( M ı / / N π o o , h ) we assume that M is a submodule of N and ı the inclusion. Given a contraction ( M ı / / N π o o , h ) of differential graded R -modules and a morphism ∂ ∈ Hom R ( N, N ), the ordinary homological perturbation theory consists is a series of statementsabout the maps(3.1) ı ∂ = X n ≥ ( h∂ ) n ı ∈ Hom R ( M, N ) , (3.2) π ∂ = X n ≥ π ( ∂h ) n ∈ Hom R ( N, M ) , (3.3) D ∂ = π∂ı ∂ = π ∂ ∂ı ∈ Hom R ( M, M ) , In order to have the above maps defined we need to impose some extra assumption. This maydone by considering filtered contractions of complete modules (as in [4]) or by imposing a sortof local nilpotency for the operators h∂, ∂h . Definition 3.1.
Given a contraction ( M ı / / N π o o , h ) denote N ( N, h ) = { ∂ ∈ Hom R ( N, N ) | ∪ n ker(( h∂ ) n ı ) = M, ∪ n ker( π ( ∂h ) n ) = N } . It is plain that the maps ı ∂ , π ∂ and D ∂ are well defined for every ∂ ∈ N ( N, h ). Moreover theyare functorial in the following sense: given a morphism of contractions f : ( M ı / / N π o o , h ) → ( A i / / B p o o , k )and two elements ∂ ∈ N ( N, h ), δ ∈ N ( B, k ) such that f ∂ = δf we have f ı ∂ = X n ≥ f ( h∂ ) n ı = X n ≥ ( kδ ) n f ı = X n ≥ ( kδ ) n i ˆ f = i δ ˆ f . Similarly we have ˆ f π ∂ = p δ f , ˆ f D ∂ = D δ ˆ f . Lemma 3.2.
Let ( M ı / / N π o o , h ) be a contraction and ∂ ∈ N ( N, h ) . Then ı ∂ is injective and π ∂ ı ∂ = πı = Id M . RELATIVE VERSION OF THE ORDINARY PERTURBATION LEMMA 5
Proof.
Immediate consequence of annihilation properties. It is useful to point out that the proofof the injectivity of ı ∂ does not depend on the annihilation properties. Assume ı ∂ ( x ) = 0 and let s ≥ h∂ ) s ı ( x ) = 0. If s > h∂ ) s − ı ∂ ( x ) = ( h∂ ) s − ı ( x ) + X k ≥ s ( h∂ ) k ı ( x ) = ( h∂ ) s − ı ( x )giving a contradiction. Hence s = 0 and ı ( x ) = 0. (cid:3) Proposition 3.3.
The formula 3.1 is compatible with composition of contractions. More pre-cisely, if ( L i / / M p o o , k ) ◦ ( M ı / / N π o o , h ) = ( L ıi / / N pπ o o , h + ıkπ ) then ( ıi ) ∂ = ı ∂ i D ∂ , provided that all terms of the equation are defined.Proof. We have ı ∂ i D ∂ = X n ≥ ( h∂ ) n ı X m ≥ ( kD ∂ ) m i = X n ≥ ( h∂ ) n X m ≥ ı ( kπ∂ X s ≥ ( h∂ ) s ı ) m i = X n ≥ ( h∂ ) n X m ≥ ( ıkπ∂ X s ≥ ( h∂ ) s ) m ıi = X n ≥ ( h∂ + ıkπ∂ ) n ıi = ( ıi ) ∂ . (cid:3) Proposition 3.4.
Let ( M ı / / N π o o , h ) be a coalgebra contraction and ∂ ∈ N ( N, h ) . If ∂ is acoderivation then ı ∂ and π ∂ are morphisms of graded coalgebras and D ∂ is a coderivation.Proof. Consider the contraction( M ⊗ R M ı ⊗ ı / / N ⊗ R N π ⊗ π o o , k ) , where k = h ∗ h = ıπ ⊗ h + h ⊗ Id N , and δ = ∂ ⊗ Id N + Id N ⊗ ∂ . In order to prove that δ ∈ N ( N ⊗ R N, k ) we show that for everyinteger n ≥ kδ ) n ( ı ⊗ ı ) = X i + j = n ( h∂ ) i ı ⊗ ( h∂ ) j ı , ( π ⊗ π )( δk ) n = X i + j = n π ( ∂h ) i ⊗ π ( ∂h ) j . We prove here only the first equality by induction on n ; the second is completely similar andleft to the reader. Since kδ = h∂ ⊗ Id N + h ⊗ ∂ − ıπ∂ ⊗ h + ıπ ⊗ h∂, according to annihilation properties we have: h∂ ⊗ Id N X i + j = n ( h∂ ) i ı ⊗ ( h∂ ) j ı = X i + j = n ( h∂ ) i +1 ı ⊗ ( h∂ ) j ı,h ⊗ ∂ X i + j = n ( h∂ ) i ı ⊗ ( h∂ ) j ı = 0 , ıπ∂ ⊗ h X i + j = n ( h∂ ) i ı ⊗ ( h∂ ) j ı = 0 ,ıπ ⊗ h∂ X i + j = n ( h∂ ) i ı ⊗ ( h∂ ) j ı = ı ⊗ ( h∂ ) n +1 ı. MARCO MANETTI
Therefore ( ı ⊗ ı ) δ = X n ≥ ( kδ ) n ( ı ⊗ ı ) = X i,j ≥ ( h∂ ) i ı ⊗ ( h∂ ) j ı = ı ∂ ⊗ ı ∂ , ( π ⊗ π ) δ = X n ≥ ( π ⊗ π )( δk ) n = X i,j ≥ π ( ∂h ) i ⊗ π ( ∂h ) j = π ∂ ⊗ π ∂ . Denoting by ∆ : N → N ⊗ R N the coproduct, since ∂ is a coderivation we have δ ∆ = ∆ ∂ ;since ∆ is a morphism of contractions we have by functoriality∆ ı ∂ = ( ı ⊗ ı ) δ ˆ∆ = ( ı ∂ ⊗ ı ∂ ) ˆ∆ , ˆ∆ π ∂ = ( π ⊗ π ) δ ∆ = ( π ∂ ⊗ π ∂ )∆ , and then ı ∂ , π ∂ are morphisms of coalgebras. Finally D ∂ is a coderivation because it is thecomposition of the coderivation ∂ and the two morphisms of coalgebras ı ∂ and π . (cid:3) A proof of Proposition 3.4 is given in [4] under the unnecessary assumption that ( d + ∂ ) = 0. Definition 3.5.
Let N be a differential graded R -module. A perturbation of the differential d N is a linear map ∂ ∈ Hom R ( N, N ) such that ( d N + ∂ ) = 0. Theorem 3.6 (Ordinary perturbation lemma) . Let ( M ı / / N π o o , h ) be a contraction and let ∂ ∈ N ( N, h ) be a perturbation of the differential d N . Then D ∂ is a perturbation of d M = πd N ı and π ∂ : ( N, d N + ∂ ) → ( M, d M + D ∂ ) , ı ∂ : ( M, d M + D ∂ ) → ( N, d N + ∂ ) are morphisms of differential graded R -modules.Proof. See [4, 8] and references therein for proofs and history. We prove again this result inRemark 4.5 as a particular case of the relative perturbation lemma. (cid:3)
Remark . If ∪ n ker( h∂ ) n = N , and ∂ is a perturbation of d N , then ı ∂ is the unique morphismof graded R -modules M → N whose image is a subcomplex of ( N, d N + ∂ ) and satisfying the“gauge fixing” condition hı ∂ = 0 , πı ∂ = Id M . In fact h ( d N + ∂ ) ı ∂ = 0 and then ı ∂ = ı ∂ + hd N ı ∂ + h∂ı ∂ = ( ıπ − d N h ) ı ∂ + h∂ı ∂ = ı + ( h∂ ) ı ∂ =(Id N − h∂ ) − ı. Similarly π ∂ is the unique morphism of graded R -modules M → N whose kernel is a subcomplexof ( N, d N + ∂ ) and satisfying π ∂ h = 0 , π ∂ ı = Id M . The coalgebra perturbation lemma cited in the abstract is obtained by putting togetherProposition 3.4 and Theorem 3.6.4.
The relative perturbation lemma
Definition 4.1.
Let N be a differential graded R -module and A ⊂ N a differential gradedsubmodule. A morphism ∂ ∈ Hom R ( N, N ) is called a perturbation of d N over A if ∂ ( A ) ⊂ A and ( d N + ∂ ) ( A ) = 0 . Remark . The meaning of Definition 4.1 becomes more clear when we impose some extraassumption on ∂ . For instance, if N is a differential graded coalgebra and ∂ is a coderivation,then in general does not exist any coderivation δ of N such that δ | A = ∂ | A and ( d N + δ ) = 0.An explicit example of this phenomenon will be described in Section 5. Theorem 4.3 (Relative perturbation lemma) . Let ( M ı / / N π o o , h ) be a contraction with M ⊂ N and ı the inclusion. Let A ⊂ N be a differential graded submodule and ∂ ∈ N ( N, h ) aperturbation of d N over A . Assume moreover that: RELATIVE VERSION OF THE ORDINARY PERTURBATION LEMMA 7 (1) π ( A ) ⊂ A ∩ M . (2) ı ∂ ( A ∩ M ) ⊂ A .Then D ∂ = X n ≥ π∂ ( h∂ ) n ı = X n ≥ π ( ∂h ) n ∂ı ∈ Hom R ( M, M ) , is a perturbation of d M over A ∩ M and ı ∂ = X n ≥ ( h∂ ) n ı : ( A ∩ M, d M + D ∂ ) → ( A, d N + ∂ ) is a morphisms of differential graded R -modules.Remark . It is important to point out that we do not require that h ( A ) ⊂ A but only theweaker assumption ı ∂ ( M ∩ A ) ⊂ A . Proof.
We first note that D ∂ = π∂ı ∂ and then D ∂ ( A ∩ M ) ⊂ A ∩ M . In order to simplify thenotation we denote d = d N and I = Id N . Setting ψ = ∂ + d∂ + ∂d ∈ Hom R ( N, N ) we have theformula(4.1) X n,m ≥ ( ∂h ) n ∂ıπ∂ ( h∂ ) m = X n,m ≥ ( ∂h ) n ψ ( h∂ ) m − X m ≥ d∂ ( h∂ ) m − X n ≥ ( ∂h ) n ∂d. In fact, since ıπ = I + hd + dh , we have ∂ıπ∂ = ∂ ( I + hd + dh ) ∂ = ∂ + ∂hd∂ + ∂dh∂ = ψ − ( I − ∂h ) d∂ − ∂d ( I − h∂ )and therefore X n,m ≥ ( ∂h ) n ∂ıπ∂ ( h∂ ) m = X n,m ≥ ( ∂h ) n ψ ( h∂ ) m − X n,m ≥ ( ∂h ) n ( I − ∂h ) d∂ ( h∂ ) m − X n,m ≥ ( ∂h ) n ∂d ( I − h∂ )( h∂ ) m = X n,m ≥ ( ∂h ) n ψ ( h∂ ) m − X m ≥ d∂ ( h∂ ) m − X n ≥ ( ∂h ) n ∂d . We have( d + ∂ ) ı ∂ = X m ≥ d ( h∂ ) m ı + X m ≥ ∂ ( h∂ ) m ı = dı + X m ≥ dh∂ ( h∂ ) m ı + X m ≥ ∂ ( h∂ ) m ı = dı + X m ≥ ( I + dh ) ∂ ( h∂ ) m ı = dı + X m ≥ ( ıπ − hd ) ∂ ( h∂ ) m ı ,ı ∂ ( d M + D ∂ ) = X n ≥ ( h∂ ) n ıd M + X n,m ≥ ( h∂ ) n ıπ∂ ( h∂ ) m ı = X n ≥ ( h∂ ) n ıd M + X m ≥ ıπ∂ ( h∂ ) m ı + h X n,m ≥ ( ∂h ) n ∂ıπ∂ ( h∂ ) m ı = X n ≥ ( h∂ ) n dı + X m ≥ ıπ∂ ( h∂ ) m ı + X n ≥ h ( ∂h ) n ψı ∂ − X m ≥ hd∂ ( h∂ ) m ı − X n ≥ h ( ∂h ) n ∂dı = dı + X m ≥ ( ıπ − hd ) ∂ ( h∂ ) m ı + X n ≥ h ( ∂h ) n ψı ∂ , and therefore ı ∂ ( d M + D ∂ ) − ( d + ∂ ) ı ∂ = X n ≥ h ( ∂h ) n ψı ∂ . In particular, for every x ∈ M ∩ A we have ψı ∂ ( x ) = 0 and then ı ∂ ( d M + D ∂ )( x ) = ( d + ∂ ) ı ∂ ( x ) . MARCO MANETTI
Now we prove that D ∂ is perturbation of d M over M ∩ A , i.e. that ( d M + D ∂ ) x = 0 for every x ∈ M ∩ A . Since πh = 0 we have πı ∂ = πı and then ı ∂ is injective. If x ∈ M ∩ A we have ı ∂ ( d M + D ∂ ) x = ( d + ∂ ) ı ∂ ( d M + D ∂ ) x = ( d + ∂ ) ı ∂ x = 0 . (cid:3) Remark . In the set-up of Theorem 4.3, if h ( A ) ⊂ A then also π ∂ : ( A, d + ∂ ) → ( A ∩ M, d M + D ∂ ) is a morphism of differential graded R -modules. In fact, under this additional assumptionwe have π ∂ ( A ) = X n ≥ π ( ∂h ) n ( A ) ⊂ A ∩ M, X n,m ≥ ( ∂h ) n ψ ( h∂ ) m h ( A ) = 0 , and therefore in A the following equalities hold: π ∂ ( d + ∂ ) = X n ≥ π ( ∂h ) n d + X n ≥ π ( ∂h ) n ∂ = πd + X n ≥ π ( ∂h ) n ∂hd + X n ≥ π ( ∂h ) n ∂ = πd + X n ≥ π ( ∂h ) n ∂ ( I + hd ) = πd + X n ≥ π ( ∂h ) n ∂ ( ıπ − dh ) . ( d + D ∂ ) π ∂ = X n ≥ πd ( ∂h ) n + X n,m ≥ π ( ∂h ) n ∂ıπ ( ∂h ) m = X n ≥ πd ( ∂h ) n + X n ≥ π ( ∂h ) n ∂ıπ + X n ≥ ,m ≥ π ( ∂h ) n ∂ıπ ( ∂h ) m = X n ≥ πd ( ∂h ) n + X n ≥ π ( ∂h ) n ∂ıπ + X n,m ≥ π ( ∂h ) n ∂ıπ∂ ( h∂ ) m h = X n ≥ πd ( ∂h ) n + X n ≥ π ( ∂h ) n ∂ıπ − X m ≥ πd∂ ( h∂ ) m h − X n ≥ π ( ∂h ) n ∂dh = X n ≥ πd ( ∂h ) n − X m ≥ πd∂ ( h∂ ) m h + X n ≥ π ( ∂h ) n ∂ıπ − X n ≥ π ( ∂h ) n ∂dh = πd + X n ≥ π ( ∂h ) n ∂ ( ıπ − dh ) . Remark . It is straightforward to verify that all the previous proofs also work for the weakernotion of contraction where the condition πı = Id M is replaced with ı is injective and ı ( M ) is adirect summand of N as graded R -module. Review of reduced symmetric coalgebras and their coderivations
From now on we assume that R = K is a field of characteristic 0. Given a graded vector space V , the twist map twtwtw : V ⊗ V → V ⊗ V, twtwtw ( v ⊗ w ) = ( − deg( v ) deg( w ) w ⊗ v, extends naturally to an action of the symmetric group Σ n on the tensor product N n V : σ twtwtw ( v ⊗ · · · ⊗ v n ) = ± v σ − (1) ⊗ · · · ⊗ v σ − ( n ) , σ ∈ Σ n . We will denote by J n V = ( N n V ) Σ n the subspace of invariant tensors. Notice that if W ⊂ V is a graded subspace, then J n W = J n V ∩ N n W . It is easy to see that the subspace S ( V ) = L ∞ n =1 J n V ⊂ L ∞ n =1 N n V = T ( V )is a graded subcoalgebra, called the reduced symmetric coalgebra generated by V . Let’s denoteby p : T ( V ) → V the projection; we will also denote by p : S ( V ) → V the restriction of the RELATIVE VERSION OF THE ORDINARY PERTURBATION LEMMA 9 projection to symmetric tensors. The following well known properties hold (for proofs see e.g.[16]):(1) Given a morphism of graded coalgebras F : T ( V ) → T ( W ) we have F ( S ( V )) ⊂ S ( W ).(2) Given a morphism of graded vector spaces f : T ( V ) → W there exists an unique mor-phism of graded coalgebras F : T ( V ) → T ( W ) such that f = pF .(3) Given a morphism of graded vector spaces f : S ( V ) → W there exists an unique mor-phism of graded coalgebras F : S ( V ) → S ( W ) such that f = pF .Similar results hold for coderivations. More precisely for every map q ∈ Hom k ( T ( V ) , V ) thereexists an unique coderivation Q : T ( V ) → T ( V ) of degree k such that q = pQ . The coderivation Q is given by the explicit formula(5.1) Q ( a ⊗ · · · ⊗ a n ) = n X l =1 n − l X i =0 ( − k ( a + ··· + a i ) a ⊗ · · · ⊗ a i ⊗ q ( a i +1 ⊗ · · · ⊗ a i + l ) ⊗ · · · ⊗ a n , where a i = deg( a i ). Moreover Q ( S ( V )) ⊂ S ( V ) and the restriction of Q to S ( V ) depends only onthe restriction of q on S ( V ). In particular every coderivation of S ( V ) extends to a coderivationof T ( V ). Definition 5.1.
A coderivation Q of degree +1 is called a codifferential if Q = 0. Lemma 5.2.
A coderivation Q of degree +1 is a codifferential if and only if pQ = 0 .Proof. The space of coderivations of a graded coalgebra is closed under the bracket[
Q, R ] = QR − ( − deg( Q ) deg( R ) RQ and therefore if Q is a coderivation of odd degree, then its square Q = [ Q, Q ] / (cid:3) Every codifferential on T ( V ) induces by restriction a codifferential on S ( V ). Conversely it isgenerally false that a codifferential on S ( V ) extends to a codifferential on T ( V ). This is wellknown to experts; however we will give here an example of this phenomenon for the lack ofsuitable references.We restrict our attention to graded vector spaces concentrated in degree −
1, more preciselywe assume that V = L [1], where L is a vector space and [1] denotes the shifting of the degree,i.e. L [1] i = L i +1 . Under this assumption every codifferential in T ( V ) (resp.: S ( V )) is determinedby a linear map q : N V → V (resp.: q : J V → V ) of degree +1. Lemma 5.3.
In the above assumption: (1)
The map L × L → L, xy = q ( x ⊗ y ) , is an associative product if and only if q induces a codifferential in T ( V ) . (2) The map L × L → L, [ x, y ] = q ( x ⊗ y − y ⊗ x ) = xy − yx, is a Lie bracket if and only if q induces a codifferential in S ( V ) .Proof. We have seen that Q is a codifferential in T ( V ) if and only if pQ = qQ : N V → V isthe trivial map. It is sufficient to observe that qQ ( x ⊗ y ⊗ z ) = q ( q ( x ⊗ y ) ⊗ z ) − q ( x ⊗ q ( y ⊗ z )) = ( xy ) z − x ( yz ) . Similarly Q is a codifferential in S ( V ) if and only if for every x , x , x we have0 = qQ X σ ∈ Σ ( − σ x σ (1) ⊗ x σ (2) ⊗ x σ (3) ! = X σ ∈ Σ ( − σ (( x σ (1) x σ (2) ) x σ (3) − x σ (1) ( x σ (2) x σ (3) ))=[[ x , x ] , x ] + [[ x , x ] , x ] + [[ x , x ] , x ] (cid:3) Therefore every Lie bracket on L not induced by an associative product gives a codifferentialon S ( L [1]) which does not extend to a codifferential on T ( L [1]). Example 5.4.
Let K be a field of characteristic = 2 and L a vector space of dimension 3 over K with basis A, B, H . Then does not exist any associative product on L such that AB − BA = H, HA − AH = 2 A, HB − BH = − B. We prove this fact by contradiction: assume that there exists an associative product as above,then the pair ( L, [ , ]), where [ X, Y ] = XY − Y X , is a Lie algebra isomorphic to sl ( K ). Writing H = γ A + γ B + γH we have 0 = [ H , H ] = γ [ A, H ] + γ [ B, H ]and therefore γ = γ = 0, H = γH . Possibly acting with the Lie automorphism A B, B A, H
7→ − H, it is not restrictive to assume γ = − AH, H ] = [
A, H ] H = − AH , writing AH = xA + yB + zH for some x, y, z ∈ K wehave 0 = [ AH, H ] + 2 AH = x [ A, H ] + y [ B, H ] + 2 xA + 2 yB + 2 zH = 4 yB + 2 zH giving y = z = 0 and AH = xA . Moreover 2 A = A [ H, A ] = [
AH, A ] = [ xA, A ] = 0 and then A = 0. Since 0 = A ( H ) − ( AH ) H = γAH − xAH = ( γx − x ) A we have either x = 0 or x = γ . In both cases x = − AH + HA = (2 x + 2) A = 0. Thisgives a contradiction since − AH = A ( AB − H ) = ABA = ( BA + H ) A = HA. The L ∞ -algebra perturbation lemma The bar construction gives an equivalence from the category of L ∞ -algebras and the categoryof differential graded reduced symmetric coalgebras (see e.g. [2, 3, 12]).According to Formula 5.1, every coderivation Q : T ( V ) → T ( V ) of degree +1 can be uniquelydecomposed as Q = d + ∂ , where d ( N n V ) ⊂ N n V, ∂ ( N n V ) ⊂ L n − i =1 N i V, ∀ n > . and d ( a ⊗ · · · ⊗ a n ) = n − X i =0 ( − a + ··· + a i a ⊗ · · · ⊗ a i ⊗ d ( a i +1 ) ⊗ a i +2 ⊗ · · · ⊗ a n where d = Q | V : V → V . If Q is a codifferential on T ( V ) then d ( V ) = 0, d is the naturaldifferential on the tensor powers of the complex ( V, d ) and ∂ is a perturbation of d .If Q is a codifferential on S ( V ) then d ( V ) = 0 and therefore d is the natural differential onthe symmetric powers of the complex ( V, d ) and ∂ is a perturbation of d over S ( V ). RELATIVE VERSION OF THE ORDINARY PERTURBATION LEMMA 11
Theorem 6.1.
In the above notation, let Q = d + ∂ be a coderivation of degree +1 on T ( V ) which is a codifferential on S ( V ) . Let W be a differential graded subspace of ( V, d ) and let ( W / / V o o , k ) be a contraction. Taking the tensor power as in Example 2.6, we get a coalgebracontraction ( T ( W ) ı / / T ( V ) π o o , h ) where h = T k . Setting D ∂ = X n ≥ π∂ ( h∂ ) n ı = X n ≥ π ( ∂h ) n ∂ı : S ( W ) → S ( W ) , then d + D ∂ is a codifferential in S ( W ) and ı ∂ = X n ≥ ( h∂ ) n ı : ( S ( W ) , d + D ∂ ) → ( S ( V ) , d + ∂ ) is a morphisms of differential graded coalgebras.Proof. Since h ( N n V ) ⊂ N n V and ∂ ( N n V ) ⊂ L n − i =1 N i V we have n M i =1 i O V ⊂ ker( ∂h ) n ∩ ker( h∂ ) n and therefore ∂ ∈ N ( T ( V ) , h ). According to Proposition 3.4 the maps ı ∂ : T ( W ) → T ( V ) , D ∂ : T ( W ) → T ( W )are respectively a morphism of graded coalgebras and a coderivation and then ı ∂ ( S ( W )) ⊂ S ( V ) , D ∂ ( S ( W )) ⊂ S ( W ) . The conclusion now follows from Theorem 4.3, where N = T ( V ), M = T ( W ) and A = S ( V ). (cid:3) Remark . According to Proposition 3.3 the construction of Theorem 6.1 commutes withcomposition of contractions.
Remark . In the notation of Theorem 6.1, if S n k : J n V → J n V, S n k = 1 n ! X σ ∈ Σ n σ twtwtw ◦ T n k ◦ σ − twtwtw , is the symmetrization of T n k and Sk = P S n k , then ( S ( W ) ı / / S ( V ) π o o , Sk ) is a contractionbut in general it is not a coalgebra contraction.In the set-up of Theorem 6.1 the map π ∂ : T ( V ) → T ( W ) is a morphism of graded coalgebrasand then induces a morphism of graded coalgebras π ∂ : S ( V ) → S ( W ) such that π ∂ ı ∂ is theidentity on S ( W ). Unfortunately our proof does not imply that π ∂ is a morphism of complexes(unless ( d + ∂ ) = 0 in T ( V ) or D ∂ = 0). However it follows from the homotopy classification of L ∞ -algebras [12] that a morphism of differential graded coalgebras Π : S ( V ) → S ( W ) such thatΠ ı ∂ = Id always exists.We have proved that the map ı ∂ : T ( W ) → T ( V ) satisfies the equation ı ∂ = ı + ( h∂ ) ı ∂ andthen ı ∂ : S ( W ) → S ( V ) is the unique morphism of symmetric graded coalgebras satisfying therecursive formula(6.1) pı ∂ = pı + kp∂ı ∂ (where p : S ( V ) → V is the projection) . It is possible to prove that the validity of the Equation 6.1 gives a combinatorial description of ı ∂ as sum over rooted trees [2, 3] and assures that ı ∂ : ( S ( W ) , d + π∂ı ∂ ) → ( S ( V ) , d + ∂ ) is amorphism of differential graded coalgebras (see e.g. the arXiv version of [2]). References [1] S. Eilenberg and S. Mac Lane:
On the groups H( π, n ) , I. Ann. of Math. (1953), 55-106.[2] D. Fiorenza and M. Manetti: L ∞ structures on mapping cones. Algebra Number Theory (2007) 301-330, arXiv:math.QA/0601312 .[3] K. Fukaya: Deformation theory, homological algebra and mirror symmetry.
Geometry and physics of branes(Como, 2001), Ser. High Energy Phys. Cosmol. Gravit., IOP Bristol (2003) 121-209. Electronic versionavailable at .[4] J. Huebschmann and T. Kadeishvili:
Small models for chain algebras.
Math. Z. (1991) 245-280.[5] J. Huebschmann and J. Stasheff:
Formal solution of the master equation via HPT and deformation theory.
Forum Math. (2002) 847-868, arXiv:math.AG/9906036v2 .[6] J. Huebschmann: The Lie algebra perturbation lemma.
In: Festschrift in honor of M. Gerstenhaber’s 80-thand Jim Stasheff’s 70-th birthday, Progress in Math. (to appear), arXiv:0708.3977 .[7] J. Huebschmann:
The sh-Lie algebra perturbation lemma. arXiv:0710.2070 .[8] J. Huebschmann:
Origins and breadth of the theory of higher homotopies.
In: Festschrift in honor of M.Gerstenhaber’s 80-th and Jim Stasheff’s 70-th birthday, Progress in Math. (to appear), arXiv:0710.2645 .[9] J. Huebschmann:
On the construction of A ∞ -structures. arXiv:0809.4791 .[10] T. Kadeishvili: On the homology theory of fibre spaces. (Russian).
Uspekhi Mat. Nauk. (1980), englishversion arXiv:math/0504437 .[11] T. V. Kadeishvili:
The algebraic structure in the cohomology of A ( ∞ ) -algebras. Soobshch. Akad. NaukGruzin. SSR (1982), 249-252.[12] M. Kontsevich:
Deformation quantization of Poisson manifolds, I.
Letters in Mathematical Physics (2003) 157-216, arXiv:q-alg/9709040 .[13] M. Kontsevich, Y. Soibelman: Deformations of algebras over operads and Deligne’s conjecture.
In: G. Ditoand D. Sternheimer (eds)
Conf´erence Mosh´e Flato 1999, Vol. I (Dijon 1999) , Kluwer Acad. Publ., Dordrecht(2000) 255-307, arXiv:math.QA/0001151 .[14] M. Kontsevich, Y. Soibelman:
Homological mirror symmetry and torus fibrations.
K. Fukaya, (ed.)et al., Symplectic geometry and mirror symmetry. Proceedings of the 4th KIAS annual interna-tional conference, Seoul, South Korea, August 14-18, 2000. Singapore: World Scientific. (2001) 203-263, arXiv:math.SG/0011041 .[15] S. Mac Lane:
Categories for the working mathematician.
Springer-Verlag, New York, 1971.[16] M. Manetti:
Lectures on deformations on complex manifolds.
Rend. Mat. Appl. (7) (2004) 1-183, arXiv:math.AG/0507286 .[17] M. Markl: Ideal perturbation lemma.
Comm. Algebra (2001), 5209-5232, arXiv:math.AT/0002130v2 .[18] M. Markl:
Transferring A ∞ (strongly homotopy associative) structures. arXiv:math.AT/0401007v3 (2009).[19] S.A. Merkulov: Strong homotopy algebras of a K¨ahler manifold.
Intern. Math. Res. Notices (1999), 153-164, arXiv:math.AG/9809172arXiv:math.AG/9809172