Cumulants, Koszul brackets and homological perturbation theory for commutative BV_\infty and IBL_\infty algebras
CCUMULANTS, KOSZUL BRACKETS AND HOMOLOGICALPERTURBATION THEORY FOR COMMUTATIVE BV ∞ AND
IBL ∞ ALGEBRAS
RUGGERO BANDIERA
Abstract.
We explore the relationship between the classical constructions of cumulants andKoszul brackets, showing that the former are an expontial version of the latter. Moreover, undersome additional technical assumptions, we prove that both constructions are compatible withstandard homological perturbation theory in an appropriate sense. As an application of theseresults, we provide new proofs for the homotopy transfer Theorem for L ∞ and IBL ∞ algebrasbased on the symmetrized tensor trick and the standard perturbation Lemma, as in the usualapproach for A ∞ algebras. Moreover, we prove a homotopy transfer Theorem for commutative BV ∞ algebras in the sense of Kravchenko which appears to be new. Along the way, we introducea new definition of morphism between commutative BV ∞ algebras. Contents
Introduction 11. (Co)cumulants and Koszul (co)brackets 51.1. Review of L ∞ [1] algebras 51.2. Cumulants and Koszul brackets 81.3. Homological perturbation theory 111.4. Cocumulants and Koszul cobrackets 161.5. Homotopy transfer for L ∞ [1]-algebras (revisited) 212. Derived BV algebras 252.1. Derived BV algebras 252.2. Morphisms of derived BV algebras 272.3. Homotopy transfer for derived BV algebras 322.4. Derived BV coalgebras 343. Homotopy transfer for IBL ∞ algebras 373.1. IBL ∞ [1] algebras 373.2. Homotopy transfer for IBL ∞ [1] algebras 40References 42 Introduction
Given a graded commutative algebra A and an endomorphism δ : A → A , the Koszul brack-ets K ( δ ) n : A (cid:12) n → A are graded symmetric maps measuring the deviation of δ from being aderivation. These were introduced by J. L. Koszul [45] in the algebraic study of differentialoperators, and have been applied to a variety of situations, from the study of the BV formal-ism in mathematical physics [1, 2, 10, 11, 12, 14] to Poisson geometry [45, 44, 28, 3]. Theiralgebraic properties have been extensively studied, see for instance [55, 56, 53, 54], and severalgeneralizations to non-commutative settings have been investigated: see [14, 15, 55, 4, 54] for a r X i v : . [ m a t h . QA ] D ec RUGGERO BANDIERA generalizations in the associative setting and [5] for a generalization in the pre-Lie setting. Seealso [68, 69] for a closely related construction of higher (derived) brackets.Given graded commutative algebras
A, B and a degree zero map f : A → B , the (classical) cumulants κ ( f ) n : A (cid:12) n → B are graded symmetric maps measuring the deviation of f from beinga morphism of graded algebras. Certain non-commutative generalizations have been consideredas well, see [49, 26, 27, 63]. These play an important role in probability theory, as well as someof its non-classical variations (such as non-commutative probability theory [49] or homotopyprobability theory [26, 27, 60]). In their algebraic formulation considered here, they have alsobeen also applied in the study of homotopy algebras and homotopy morphisms [63, 64], as theygive obstructions for a linear map between A ∞ or C ∞ algebras to be the linear part of an ∞ -morphism.In Section 1.2 we explore the relation between (classical) cumulants and Koszul brackets.Given a graded commutative algebra A , we shall denote by S ( A ) the symmetric coalgebra over A , and by E : S ( A ) → S ( A ) the coalgebra morphism which corestricts to pE : S ( A ) → A : a (cid:12) · · · (cid:12) a n → a · · · a n . Since E has linear part the identity, it is an automorphism of S ( A ),whose inverse we denote by L . It has been observed in several places that given δ : A → A ,its Koszul brackets K ( δ ) n : A (cid:12) n → A might be characterized as the Taylor coefficients ofthe coderivation K ( δ ) := L ◦ (cid:101) δ ◦ E : S ( A ) → S ( A ), where (cid:101) δ : S ( A ) → S ( A ) is the linearcoderivation extending δ , see for instance [55, 56, 62, 5]. Moreover, a similar description holdsfor the cumulants of a map f : A → B : they are the Taylor coefficients of the coalgebramorphism κ ( f ) := L ◦ S ( f ) ◦ E : S ( A ) → S ( B ), as was for instance observed in [62]. Thiscommon description is the starting point of our analysis. From it, one easily derives the basicalgebraic properties of cumulants and Koszul brackets, as well as the compatibility between thetwo constructions, see for instance Propositions 1.21 and 1.22: in particular, the latter showsthat cumulants should be considered as an exponential version of the Koszul brackets, sheddingfurther light on the relation between the two constructions.In Section 1.3, we investigate the behavior of cumulants and Koszul brackets with respectto homological perturbation theory. Given a differential d A on A (not necessarily an algebraderivation), it is well known that the associated Koszul brackets K ( d A ) n satisfy higher Jacobirelations, thus inducing a structure of L ∞ [1] algebra on A (or in other words, a structure ofstrong homotopy Lie algebra on A [ − B with a differential (not necessarily an algebra derivation) d B , together with a contraction σ : A → B , τ : B → A , h : A → A [ −
1] of A onto B , by the usual homotopy transfer Theoremfor L ∞ [1] algebras (this will be reviewd in 1.3, and then again in Section 1.5), there is aninduced L ∞ [1] algebra structure on B and an induced L ∞ [1] morphism B → A . When thecontraction satisfies certain additional technical assumptions (we say that it is a semifull algebracontraction , see Definition 1.25), we show in Proposition 1.27 that these are precisely the L ∞ [1]algebra structure on B associated with the Koszul brackets K ( d B ) n and the L ∞ [1] morphism B → A associated with the cumulants κ ( τ ) n . This result will be one of our main technical toolsin the proof of the homotopy transfer Theorems 1.45, 2.14 and 3.6. Semifull algebra contractions,as well as the dual class of semifull coalgebra contractions, were introduced by Real [66]: forour purposes, one of their more useful features will be the fact that they are closed under theStandard Perturbation Lemma 1.29, see Lemmas 1.30 and 1.39.In Section 1.4 we introduce the dual notions of cocumulants and Koszul cobrackets . Namely,given a map f : C → D between graded cocommutative coalgebras, its cocumulants are maps κ co ( f ) n : C → D (cid:12) n measuring the deviation of f from being a coalgebra morphism. Given amap δ : C → C , the Koszul cobrackets K ( δ ) n : C → C (cid:12) n are maps measuring the deviation of δ from being a coderivation. All the results from Sections 1.2 and 1.3 admit analogues in thisdual setting. UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 3
Finally, in Section 1.5 we revisit the homotopy transfer Theorem for L ∞ [1] algebras, providinga new proof in light of the results from the previous sections. Our approach is based on thesymmetrized tensor trick and the standard perturbation Lemma, in close analogy with theusual argument for A ∞ [1] algebras, see for instance [35]. We point out that if we were onlyinterested in homotopy transfer for L ∞ [1] algebras a similar but simpler argument might begiven [8]. For our purposes, this section serves more as a preparation for the similar proofs ofthe homotopy transfer Theorems 2.14, 2.17 and 3.6 for BV ∞ (co)algebras and IBL ∞ algebras.Moreover, the technical lemmas we establish along the way seem interesting in their own right.In Lemma 1.41 we give necessary and sufficient conditions for a map F : S ( U ) → S ( V ) to bea coalgebra morphism in terms of the associated cumulants κ ( F ) n , which is curious since byconstruction the latter measure the deviation of F from being a morphism of algebras, ratherthan coalgebras. Similarly, in Lemma 1.42 we give necessary and sufficient conditions for amap Q : S ( U ) → S ( U ) to be a coderivation in terms of the associated Koszul brackets K ( Q ) n (an analogous characterization of codifferential operators shall be given later, in Lemma 3.2).Finally, in Lemma 1.43 we make the key observation that the contraction of S ( U ) onto S ( V )induced by a contraction of U onto V via the symmetrized tensor trick is a semifull contractionwith respect to both the algebraic and the coalgebraic structures. Commutative BV ∞ algebras were introduced by O. Kravchenko [46] and have been applied inseveral contexts, such as deformation quantization, quantum field theory and Poisson geometry,just to name a few, see for instance [36, 20, 23, 24, 16, 65, 9, 18, 61, 48, 70] (in the references[61, 48] they are called binary QFT algebras , but in fact the two notions seem to be equivalent).More precisely, a commutative BV ∞ algebra is a graded algebra A together with a K [[ t ]]linear differential ∆ = (cid:80) n ≥ t n ∆ n : A [[ t ]] → A [[ t ]] on the algebra A [[ t ]] of formal power series:moreover, one requires that ∆ n is a differential operator of order ≤ n + 1, ∀ n ≥
0. The lastcondition might be compactly rephrased in terms of the associated Koszul brackets K (∆) n : itis equivalent to requiring K (∆ i ) n = 0 for all i < n −
1, or in other words, it is equivalent to(1) K (∆) n ≡ t n − ) , ∀ n ≥ . We should remark that commutative BV ∞ algebras in the above sense are not homotopy BV algebras in the full operadic sense [30], as we maintain strict associativity of the commutativeproduct on A , and we are only relaxing the conditions on the BV operator up to coherenthomotopies. For this reason, the name BV ∞ algebra might be misleading, as the usual machineryfor ∞ -algebras (see for instance [50, § P ∞ algebras in the sense of Cattaneo and Felder [19], which are a type of homotopy Poissonalgebras where we relax only the Lie algebra structure up to homotopy, while maintaining strictassociativity of the commutative product. To avoid this ambiguity, in the paper [7] we proposedthe name derived Poisson algebras for this kind of algebras: in line with this choice, in thebody of the paper we shall call derived BV algebras the commutative BV ∞ algebras in thesense of Kravchenko (in the rest of the introduction, however, we shall stick to the more usualterminology).In Section 2.2 we introduce morphisms between commutative BV ∞ algebras. Namely, givencommutative BV ∞ algebras ( A, ∆) and ( B, ∆ (cid:48) ), a morphism of commutative BV ∞ algebras from A to B is a degree zero K [[ t ]]-linear map f = (cid:80) n ≥ t n f n : A [[ t ]] → B [[ t ]] such that f ◦ ∆ = ∆ (cid:48) ◦ f ,and moreover the cumulants satisfy(2) κ ( f ) n ≡ t n − ) , ∀ n ≥ . We provide some justification for this definition, and in particular we show that with the abovemorphisms commutative BV ∞ algebras form indeed a category. More convincingly, we showthat the above Equation (2) is in fact an exponential version of (1). Moreover, we extend RUGGERO BANDIERA the usual correspondence from commutative BV ∞ algebras to L ∞ algebras (more precisely, toderived Poisson algebras as in [7]) to a full-fledged functor. After a first draft of this paper wasready, we learned that the same notion was considered by J.-S. Park [61]. Another definition ofmorphism was introduced by Cieliebak and Latschev [20] (and further investigated in [9]), butonly in the special case when the source algebra is free. We conclude the section by showingthat our definition is essentially equivalent with the one from [20], in the situations when thelatter applies. More precisely, in Proposition 2.13 we show that morphisms in our sense andin the sense of [20] are in bijective correspondence with each other via the exponential and thelogarithm in a certain convolution algebra.In Section 2.3 we prove that commutative BV ∞ algebra structures can be transferred alongsemifull algebra contractions via homological perturbation theory. More precisely, given a con-traction ( σ, τ, h ) of A onto B and a commutative BV ∞ algebra structure ∆ = (cid:80) n ≥ t n ∆ n on A ,we can apply the standard perturbation Lemma to the perturbation ∆ + = (cid:80) n ≥ t n ∆ n in orderto get a perturbed differential ∆ (cid:48) on B [[ t ]] and a perturbed contraction (˘ σ, ˘ τ , ˘ h ) of ( A [[ t ]] , ∆)onto ( B [[ t ]] , ∆ (cid:48) ). In the above situation, we prove in Theorem 2.14 that if ( σ, τ, h ) is a semifull al-gebra contraction, then ∆ (cid:48) is a commutative BV ∞ algebra structure on B and ˘ τ : B [[ t ]] → A [[ t ]]is a morphism of commutative BV ∞ algebras.In Section 2.4 we consider the dual category of BV ∞ cocommutative coalgebras, and extendthe previous results to this dual setting. IBL ∞ algebras (short for Involutive Lie Bialgebras up to coherent homotopies) were intro-duced in the paper [21], with applications to string topology, symplectic field theory and La-grangian Floer theory, and have been further investigated in several other papers since then, forinstance [18, 57, 25, 39, 58, 41, 59, 22, 40]We shall work with a definition of IBL ∞ algebra slightly different from, and in a certain sensedual to, the one usually appearing in the literature. More precisely, whereas IBL ∞ algebrasare usually regarded as commutative BV ∞ algebras whose underlying algebras are free (see forinstance [20, 18, 57]), we shall regard them dually as cocommutative BV ∞ coalgebras whoseunderlying coalgebras are cofree. In Remark 3.4 we shall compare the two definitions, showingthat they are essentially equivalent, aside from some minor differences regarding the degrees ofthe structure maps. For our purposes, our definition presents some technical advantages: weshall call IBL ∞ [1] algebras the IBL ∞ algebras in our sense. We also establish necessary andsufficient conditions for a map δ : S ( U )[[ t ]] → S ( U )[[ t ]] to define an IBL ∞ [1] algebra structureon U in terms of the associated Koszul brackets, see Lemma 3.2.In Section 3.2, as an application of the results from the previous sections, we shall presenta new proof of the homotopy transfer Theorem for IBL ∞ [1] algebras based on homologicalperturbation theory (different proofs can be found in [21, 41]). More precisely, given a contraction( g, f, h ) of ( U, d U ) onto ( V, d V ) there is, via the symmetrized tensor trick, an induced contractionof ( S ( U )[[ t ]] , d U ) onto ( S ( V )[[ t ]] , d V ). Given δ : S ( U )[[ t ]] → S ( U )[[ t ]] an IBL ∞ [1] algebrastructure on U , we can apply the standard perturbation Lemma to the perturbation δ + := δ − d U in order to get a perturbed differential δ (cid:48) on S ( V )[[ t ]] and a perturbed contraction( G, F, H ) of ( S ( U )[[ t ]] , δ ) onto ( S ( V )[[ t ]] , δ (cid:48) ). In the above situation, in Theorem 3.6 we shallprove that δ (cid:48) is an IBL ∞ [1] algebra structure on V , and that G : S ( U )[[ t ]] → S ( V )[[ t ]] and F : S ( V )[[ t ]] → S ( U )[[ t ]] are IBL ∞ [1] morphisms.Finally, we shall consider the behavior of Maurer-Cartan sets under homotopy transfer. Inthe L ∞ case, this is the content of a formal analog of Kuranishi’s Theorem due to Fukaya [29]and Getzler [31, 32], which will be reviewed in Theorem 1.7. With a little more work, we obtainanalogue results in the BV ∞ setting (Theorem 2.15) and IBL ∞ setting (Theorem 3.8). UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 5
Acknowledgements.
It is a pleasure to thank Marco Manetti, Niels Kowalzig, Hsuan-Yi Liaoand Luca Vitagliano for useful discussions on the subject of this paper.1. (Co)cumulants and Koszul (co)brackets
Review of L ∞ [1] algebras. We shall work over a fixed field K of characteristic zero.Given V = ⊕ i ∈ Z V i a graded space, we denote by V (cid:12) n the symmetric powers of V ( V (cid:12) := K ),and by S ( V ) = (cid:76) n ≥ V (cid:12) n the symmetric coalgebra over V . The coproduct ∆ on S ( V ) isexplicitly given by∆( v (cid:12) · · · (cid:12) v n ) = n (cid:88) i =0 (cid:88) σ ∈ S ( i,n − i ) ± K ( v σ (1) (cid:12) · · · (cid:12) v σ ( i ) ) ⊗ ( v σ ( i +1) (cid:12) · · · (cid:12) v σ ( n ) ) , where we denote by S ( i, n − i ) the set of ( i, n − i )-unshuffles and by ± K the appropriate Koszulsign. As well known, S ( V ) is the cofree object cogenerated by V in an appropriate category ofcoalgebras. In particular, given graded spaces V and W , a coalgebra morphism F : S ( V ) → S ( W ) is completely determined by its corestriction f = pF = ( f , . . . , f n , . . . ) : S ( V ) → W, f n : V (cid:12) n → W, ( p : S ( W ) → S ( W ) being the natural projection) via the formula(3) F ( v (cid:12) · · · (cid:12) v n ) = (cid:88) k,i ,...,ik ≥ i + ··· + i k = n k ! (cid:88) σ ∈ S ( i ,...,i k ) ± K f i ( v σ (1) , . . . ) (cid:12) · · · (cid:12) f i k ( . . . , v σ ( n ) ) , where ± K is the appropriate Koszul sign (in particular, for n = 0 this becomes F ( S ( V ) ) = S ( W ) ). The maps f n : V (cid:12) n → W , n ≥
1, are called the
Taylor coefficients of F .In a similar manner, every coderivation Q : S ( V ) → S ( V ) is completely determined by itscorestriction q = pQ = ( q , q , . . . , q n , . . . ) : S ( V ) → V, q n : V (cid:12) n → V, via the formula(4) Q ( v (cid:12) · · · (cid:12) v n ) = n (cid:88) i =0 (cid:88) σ ∈ S ( i,n − i ) ± K q i ( v σ (1) , . . . , v σ ( i ) ) (cid:12) v σ ( i + i ) (cid:12) · · · (cid:12) v σ ( n ) , where ± K is the appropriate Koszul sign (in particular, for n = 0 this becomes Q ( S ( V ) ) = q (1)).The maps q n : V (cid:12) n → W , n ≥
0, are called once again the
Taylor coefficients of Q .We shall denote by Coder( S ( V )) the graded Lie algebra of coderivations of S ( V ), with thecommutator bracket. In Taylor coefficients, given coderivations q = ( q , q , . . . , q n , . . . ) and r = ( r , r , . . . , r n , . . . ), their bracket is [ q, r ] = ([ q, r ] , [ q, r ] , . . . , [ q, r ] n , . . . )[ q, r ] n ( v , . . . , v n ) = n (cid:88) i =0 (cid:88) σ ∈ S ( i,n − i ) ± K (cid:16) q n − i +1 (cid:0) r i (cid:0) v σ (1) , . . . , v σ ( i ) (cid:1) , v σ ( i +1) , . . . , v σ ( n ) (cid:1) − ( − | q || r | r n − i +1 (cid:0) q i (cid:0) v σ (1) , . . . , v σ ( i ) (cid:1) , v σ ( i +1) , . . . , v σ ( n ) (cid:1) (cid:17) . Definition 1.1. An L ∞ [1] algebra structure on a graded space V is a degree one coderivation Q ∈ Coder ( S ( V )) such that Q ( S ( V ) ) = S ( V ) (i.e., q = 0) and Q = [ Q, Q ] = 0. Given L ∞ [1] algebras ( V, q , . . . , q n , . . . ) and ( W, r , . . . , r n , . . . ), an L ∞ [1] morphism between them is amorphism of DG coalgebras F : ( S ( V ) , Q ) → ( S ( W ) , R ). RUGGERO BANDIERA
Remark 1.2.
Given an L ∞ [1] algebra ( V, q , . . . , q n , . . . ), the relation [ Q, Q ] = 0 translatesinto a hierarchy of relations on the Taylor coefficients q , . . . , q n , . . . . The first few relations saythat q squares to zero and satisfies the Leibniz identity with respect to the bracket [ x, y ] :=( − | x | q ( x, y ). Furthermore, this bracket satisfies the Jacobi identity up to a homotopy givenby q . The higher Taylor coefficients q n give coherent homotopies for higher Jacobi relations. Inparticular, the shifted cohomology H ( V, q )[ −
1] is a graded Lie algebra via the induced bracket(in fact, it carries additional algebraic structure - a minimal L ∞ algebra structure - correspondingtopologically to higher Whitehead brackets). Conversely, any DG Lie algebra ( L, d L , [ − , − ]) givesrise to an L ∞ [1] algebra ( L [1] , Q ) with q ( x ) = − d L ( x ), q ( x, y ) = ( − | x | [ x, y ] and q n = 0 for n >
2. More in general, L ∞ [1] algebra structures on V correspond to strong homotopy Liealgebra structures (as in [47]) on the suspension V [ − L ∞ [1] morphism F : ( V, q , . . . , q n , . . . ) → ( W, r , . . . , r n , . . . ), theidentity F Q = RF translates into a hierarchy of relations on the Taylor coefficients of F, Q, R .The first few relations say that f : ( V, q ) → ( W, r ) is a morphism of complexes, compatiblewith the brackets (induced by) q and r up to the homotopy f . L ∞ [1] algebras are homotopy invariant algebraic structures, meaning that they can be trans-ferred along quasi-isomorphisms. In case the quasi-isomorphism fits into a contraction (seeDefinition 1.24), the transfer can be made explicit via symmetrized tensor trick and homologicalperturbation theory, which is the content of the following homotopy transfer Theorem. Theorem 1.3.
Given an L ∞ [1] algebra ( V, q , . . . , q n , . . . ) and a contraction ( g , f , h ) of ( V, q ) onto a complex ( W, r ) , there is an induced L ∞ [1] algebra structure R on W with linear part r ,together with L ∞ morphisms F : ( W, R ) → ( V, Q ) , G : ( V, Q ) → ( W, R ) with linear parts f , g respectively. Denoting by F ki the composition W (cid:12) i (cid:44) → S ( W ) F −→ S ( V ) (cid:16) V (cid:12) k , F and R aredetemined recursively by (notice that by formula (3) F ki only depends on f , . . . , f i − k +1 ) f i = i (cid:88) k =2 hq k F ki for i ≥ ,r i = i (cid:88) k =2 g q k F ki for i ≥ . It is possible to establish recursive formulas for G as well, but these are a bit more complicated.We denote by (cid:98) h : V (cid:12) i → V (cid:12) i the contracting homotopy (cid:98) h ( v (cid:12) · · · (cid:12) v i ) = 1 i ! (cid:88) σ ∈ S i i (cid:88) j =1 ± K f g ( v σ (1) ) (cid:12) · · · (cid:12) f g ( v σ ( j − ) (cid:12) h ( v σ ( j ) ) (cid:12) v σ ( j +1) (cid:12) · · · (cid:12) v σ ( i ) , (where ± K is the appropriate Koszul sign, taking into account that | h | = − ) given by thesymmetrized tensor trick. Denoting by Q ki the composition V (cid:12) i (cid:44) → S ( V ) Q −→ S ( V ) (cid:16) V (cid:12) k , the L ∞ morphism G is determined recursively by g i = i − (cid:88) k =1 g k Q ki (cid:98) h for i ≥ . Remark 1.4.
The statements about F and R are well known: for a proof we refer to [37, 29,38, 52]. The statement about G is less standard, and to the best of the author’s knowledgewas first showed by Berglund [13] (as a particular case of a much more general result): we shallpresent a different proof in subsection 1.5, where we will revisit the previous theorem in light ofthe results from the following subsections. UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 7
Remark 1.5.
Sometimes we shall need to work in the complete setting, especially when con-sidering Maurer-Cartan functors. A complete space is a graded space V equipped with a com-plete descending filtration, i.e., V = F V ⊃ F V ⊃ · · · ⊃ F p V ⊃ · · · and the natural map V → lim ←− p V /F p V is an isomorphism. A morphism of complete spaces is continuous if it re-spects the filtrations. A complete L ∞ [1] algebra is an L ∞ [1] algebra ( V, q , . . . , q n , . . . ) equippedwith a complete compatible filtration F • V , i.e., q i ( F p V, . . . , F p i V ) ⊂ F p + ··· + p i V . A morphismof complete L ∞ [1] algebras F : ( V, q , . . . , q n , . . . ) → ( W, r , . . . , r n , . . . ) is continuous if so areits Taylor coefficients. If the contraction data is continuous, the previous homotopy transferTheorem can be extended to the complete setting: we refer to [6] for a precise statement. Definition 1.6.
The
Maurer-Cartan functor from complete L ∞ [1] algebras to sets sends acomplete L ∞ [1] algebra ( V, q , . . . , q n , . . . ) to its Maurer-Cartan set
MC( V ) := { x ∈ V s.t. (cid:88) n ≥ n ! q n ( x (cid:12) n ) = 0 } , and a continuous morphism F : ( V, q , . . . , q n , . . . ) → ( W, r , . . . , r n , . . . ) to the push-forwardMC( F ) : MC( V ) → MC( W ) : x → (cid:88) n ≥ n ! f n ( x (cid:12) n ) . The following formal analog of Kuranishi Theorem, due to Fukaya [29] and Getzler [31, 32],explains how Maure-Cartan sets behave under homotopy transfer. For a proof of the result asstated here, we refer to [6, Theorem 1.13].
Theorem 1.7.
In the same hypotheses as in Theorem 1.3, if we are furthermore in a completesetting, then the correspondence ρ : MC( V ) → MC( W ) × h ( V ) : x → (MC( G )( x ) , h ( x )) is bijective. The inverse ρ − admits the following recursive construction: given y ∈ MC( W ) and h ( v ) ∈ h ( V ) , we define a succession of elements x n ∈ V , n ≥ , by x = 0 and (5) x n +1 = f ( y ) − q h ( v ) + (cid:88) i ≥ i ! ( hq i − f g i ) (cid:0) x (cid:12) in (cid:1) . This succession converges (with respect to the complete topology induced by the filtration on V )to a well defined x ∈ MC( V ) , and we have ρ − ( y, h ( v )) = x . Finally, ρ − ( − , coincides with MC( F ) : MC( W ) → MC( V ) : this induces a bijective correspondence between the sets MC( W ) and Ker( h ) (cid:84) MC( V ) , whose inverse is the restriction of g . Finally, we consider the dual notion of L ∞ [ −
1] coalgebra. To motivate the definition, weobserve that given a finite dimensional L ∞ [1] algebra ( V, q , . . . , q n , . . . ), the dual maps q ∨ n : V ∨ → ( V (cid:12) n ) ∨ → ( V ∨ ) (cid:12) n assemble to a degree one map q ∨ = q ∨ + · · · + q ∨ n + · · · : V ∨ → (cid:81) n ≥ ( V ∨ ) (cid:12) n = (cid:98) S ( V ∨ ), which in turn extends uniquely to a derivation Q ∨ of the completedsymmetric algebra (cid:98) S ( V ∨ ) squaring to zero. Definition 1.8. An L ∞ [ − coalgebra structure on a graded space V is a DG algebra structure d = d + d + · · · + d n + · · · on (cid:98) S ( V ), where d n : V → V (cid:12) n . Given L ∞ [ −
1] coalgebras(
V, d = d + d + · · · + d n + · · · ), ( W, d (cid:48) = d (cid:48) + d (cid:48) + · · · + d (cid:48) n + · · · ), a morphism between themis a morphism of DG algebras f = f + f + · · · + f n + · · · : ( (cid:98) S ( V ) , d ) → ( (cid:98) S ( W ) , d (cid:48) ). Remark 1.9.
In particular, d : V → V squares to zero, and the (shifted) cobracket associatedto d satisfies the co-Jacobi identity up to the homotopy d . Hence, the shifted cohomology H ( V )[1] is a graded Lie coalgebra. RUGGERO BANDIERA
Remark 1.10.
If (
V, d = d + · · · + d n + · · · ) is an L ∞ [ −
1] coalgebra, then the dual maps d ∨ n : ( V ∨ ) (cid:12) n → ( V (cid:12) n ) ∨ → V ∨ assemble to an L ∞ [1] algebra structure ( d ∨ , . . . , d ∨ n , . . . ) on V ∨ ,even when V is infinite dimensional. Remark 1.11. L ∞ [ −
1] coalgebra structures transfer along contractions, and, as in the L ∞ [1]algebra case, this can be made explicit by combining the symmetrized tensor trick with thehomological perturbation lemma (see [13] for a proof, as well as Remark 1.47).1.2. Cumulants and Koszul brackets.
Given a graded commutative unitary algebra A , wedenote by E : S ( A ) → S ( A ) the coalgebra automorphism with Taylor coefficients pE = ( e , . . . , e n , . . . ) , e n ( a , . . . , a n ) := a · · · a n . ( p : S ( A ) → A being the natural projection). We denote by L : S ( A ) → S ( A ) the inverse of E .It is easily checked that pL = ( l , . . . , l n , . . . ) , l n ( a , . . . , a n ) = ( − n − ( n − a · · · a n . We call E and L respectively the exponential automorphism and the logarithmic automorphism of S ( A ). Remark 1.12.
In certain situations we shall need to work in a complete augmented setting.In this case, we shall assume that A has an augmentation (cid:15) : A → K and is equipped with acompatible complete filtration F A = A ⊃ F A = Ker( (cid:15) ) ⊃ · · · ⊃ F p A ⊃ · · · . In this situationthe push-forwards of a ∈ F A along E and L are given by the usual exponential and logarithmicseries E ∗ ( a ) := (cid:88) n ≥ n ! e n ( a (cid:12) n ) = (cid:88) n ≥ n ! a n = e a − A ,L ∗ ( a ) := (cid:88) n ≥ n ! l n ( a (cid:12) n ) = (cid:88) n ≥ ( − n − n a n = log( A + a ) , which justifies the names. Remark 1.13.
Here and in the rest of the paper, we shall use the following notations: given agraded unitary algebra A , we shall denote byEnd u ( A ) := { ∆ ∈ End( A ) s.t. ∆( A ) = 0 } . Given a pair of graded unitary algebras A and B , we shall denote byHom u ( A, B ) := { f ∈ Hom ( A, B ) s.t. f ( A ) = B } (notice in particular that with these notations End u ( A ) and Hom u ( A, A ) have different mean-ings).
Definition 1.14.
Given a pair of graded commutative unitary algebras A and B , togetherwith a degree zero linear map f ∈ Hom u ( A, B ), the cumulants of f are the Taylor coefficients κ ( f ) n : A (cid:12) n → B of the coalgebra morphism κ ( f ) : S ( A ) → S ( B ) defined as the composition(6) κ ( f ) := S ( A ) E −→ S ( A ) S ( f ) −−−→ S ( B ) L −→ S ( B ) , where S ( f ) : S ( A ) → S ( B ) is the coalgebra morphism S ( f )( S ( A ) ) = S ( B ) , S ( f )( a (cid:12) · · · (cid:12) a n ) = f ( a ) (cid:12) · · · (cid:12) f ( a n ) . UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 9
Remark 1.15.
The cumulants of f measure the deviation of f from being a morphism of unitaryalgebras. For instance, a direct computation shows that the first few cumulants of f are κ ( f ) ( a ) = f ( a ) , κ ( f ) ( a, b ) = f ( ab ) − f ( a ) f ( b ) ,κ ( f ) ( a, b, c ) = f ( abc ) − f ( ab ) f ( c ) − ( − | b || c | f ( ac ) f ( b ) −− ( − | a | ( | b | + | c | ) f ( bc ) f ( a ) + 2 f ( a ) f ( b ) f ( c ) . In general we have the following explicit formula for the higher cumulants (cf. for instance with[62, 64]), which again can be checked by a straightforward computation,(7) κ ( f ) n ( a , . . . , a n ) = (cid:88) ≤ k ≤ n, I ,...,Ik (cid:54) = ∅ I (cid:70) ··· (cid:70) I k = { ,...,n } ( − k − ( k − ± K (cid:89) ≤ j ≤ k f (cid:89) i ∈ I j a i . Here the sum runs over the (cid:8) nk (cid:9) non-ordered partitions of { , . . . , n } into the disjoint union of k non-empty subsets I , . . . , I k , and we denote by ± K the appropriate Koszul sign associatedwith the resulting permutation of a , . . . , a n . In particular, f is a morphism of unitary gradedalgebras if and only if κ ( f ) n = 0 for all n > Lemma 1.16.
Given graded commutative algebras
A, B, C and maps f ∈ Hom u ( A, B ) , g ∈ Hom u ( B, C ) , then κ ( g ◦ f ) = κ ( g ) ◦ κ ( f ) . Proof.
This follows immediately from the definitions κ ( f ) := L ◦ S ( f ) ◦ E , κ ( g ) := L ◦ S ( g ) ◦ E ,the fact that E : S ( B ) → S ( B ) and L : S ( B ) → S ( B ) are inverses to each other and the factthat S ( g ◦ f ) = S ( g ) ◦ S ( f ). (cid:3) Yet another way to define the higher cumulants is the following recursion: starting with κ ( f ) ( a ) = f ( a ), for all n ≥ κ ( f ) n +2 ( a , . . . , a n , b, c ) = κ ( f ) n +1 ( a , . . . , a n , bc ) −− n (cid:88) i =0 (cid:88) σ ∈ S ( i,n − i ) ± K κ ( f ) i +1 ( a σ (1) , . . . , a σ ( i ) , b ) κ ( f ) n − i +1 ( a σ ( i +1) , . . . , a σ ( n ) , c ) . Here we denote by S ( i, n − i ) the set of ( i, n − i )-unshuffles, and by ± K the Koszul sign associatedto the permutation a , . . . , a n , b, c (cid:55)→ a σ (1) , . . . , a σ ( i ) , b, a σ ( i +1) , . . . , a σ ( n ) , c . This recursion can beshown by a simple calculation, whose details are left to the reader, using the explicit formula(7) for the cumulants. Essentially, if we expand both the left and the right hand side of (8)according to (7), then κ ( f ) n +1 ( a , . . . , a n , bc ) accounts for all the terms in the expansion of κ ( f ) n +2 ( a , . . . , a n , b, c ) such that b and c belong to the same block of the partition, while − (cid:80) ni =0 (cid:80) σ ∈ S ( i,n − i ) ( ± ) κ ( f ) i +1 ( a σ (1) , . . . , a σ ( i ) , b ) κ ( f ) n − i +1 ( a σ ( i +1) , . . . , a σ ( n ) , c ) accounts for allthose terms such that b and c belong to different blocks. Remark 1.17.
It follows immediately from the definitions that in the complete setting, assum-ing furthermore that f is continuous, push-forward along κ ( f ) sends a ∈ F A to(9) κ ( f ) ∗ ( a ) := (cid:88) n ≥ n ! κ ( f ) n ( a (cid:12) n ) = log( f ( e a )) . Next we recall the classical construction of Koszul brackets (see [45, 55] for the equivalencebetween the following definitions), and explore its relation with the previous construction ofcumulants.
Definition 1.18.
Given a degree k endomorphism ∆ ∈ End ku ( A ) of A , we denote by (cid:101) ∆ : S ( A ) → S ( A ) the coderivation given in Taylor coefficients by (cid:101) ∆ = ∆ and (cid:101) ∆ n = 0 for n (cid:54) = 1. The Koszul brackets K (∆) n : A (cid:12) n → A , n ≥
0, of ∆ are the Taylor coefficients of the coderivation K (∆) : S ( A ) → S ( A ) defined as the composition(10) K (∆) := S ( A ) E −→ S ( A ) (cid:101) ∆ −→ S ( A ) L −→ S ( A ) . Remark 1.19.
The Koszul brackets of ∆ measure the deviation of ∆ from being a gradedalgebra derivation. For instance, once again by a direct computation, the first few Koszulbrackets are K (∆) ( a ) = ∆( a ) − ∆(1 A ) a, K (∆) ( a, b ) = ∆( ab ) − ∆( a ) b − ( − | a || b | ∆( b ) a + ∆(1 A ) ab, K (∆) ( a, b, c ) = ∆( abc ) − ∆( ab ) c − ( − | b || c | ∆( ac ) b − ( − | a | ( | b | + | c | ) ∆( bc ) a ++ ∆( a ) bc + ( − | a || b | ∆( b ) ac + ( − ( | a | + | b | ) | c | ∆( c ) ab − ∆(1 A ) abc. In general,(11) K (∆) n ( a , . . . , a n ) = n (cid:88) i =1 (cid:88) σ ∈ S ( i,n − i ) ± K ( − n − i ∆( a σ (1) · · · a σ ( i ) ) a σ ( i +1) · · · a σ ( n ) . Equivalently, the Koszul brackets K (∆) n might be defined by the recursion K (∆) ( a ) = ∆( a )and for n ≥ K (∆) n +2 ( a , . . . , a n , b, c ) = K (∆) n +1 ( a , . . . , a n , bc ) −− K (∆) n +1 ( a , . . . , a n , b ) c − ( − | b || c | K (∆) n +1 ( a , . . . , a n , c ) b. We have the following results, the latter two illustrating the compatibility between cumulantsand Koszul brackets.
Lemma 1.20.
If we regard
End u ( A ) and Coder( S ( A )) as graded Lie algebras via the commutatorbrackets, then the correspondence K ( u ( A ) → Coder( S ( A )) : ∆ → K (∆) is a morphismof graded Lie algebras. Proposition 1.21.
Given graded commutative unitary algebras A and B , endomorphisms ∆ ∈ End u ( A ) , ∆ (cid:48) ∈ End u ( B ) and a degree zero map f ∈ Hom u ( A, B ) , if f ◦ ∆ = ∆ (cid:48) ◦ f , then also κ ( f ) ◦ K (∆) = K (∆ (cid:48) ) ◦ κ ( f ) . Proposition 1.22.
Given a degree zero endomorphism ∆ ∈ End u ( A ) as above, such that theexponential exp(∆) ∈ Hom u ( A, A ) is well defined (i.e., the corresponding infinte sum makessense, either by nilpotency of ∆ or some other appropriate hypothesis), then the exponential exp( K (∆)) : S ( A ) → S ( A ) of the coderivation K (∆) is also well defined, and an automorphismof the symmetric coalgebra S ( A ) . Moreover, exp( K (∆)) = κ (exp(∆)) . By Definition 1.18 the proof of the first result reduces to show that End u ( A ) → Coder( S ( A )) :∆ → (cid:101) ∆ is a morpshims of graded Lie algebras, which is immediate. The second result followsstraightforwardly by comparing the Definitions 1.14 and 1.18. The same applies for the last UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 11 result, with the further observation that in the hypotheses of the enunciate the exponential ofthe coderivation (cid:101) ∆ is well defined, and is the coalgebra automorphism exp( (cid:101) ∆) = S (exp(∆)).Let us further point out the following formula, valid in the complete setting for any a ∈ F A ,(13) (cid:88) n ≥ n ! K (∆) n ( a (cid:12) n ) = e − a ∆( e a ) . In particular, (cid:80) n ≥ n ! K (∆) n ( a (cid:12) n ) = 0 if and only if ∆( e a ) = 0.We conclude this subsection by recalling the definition of differential operators on a gradedcommutative unitary algebra. Definition 1.23.
Given a graded commutative algebra A and k ≥
1, we denote byDiff u, ≤ k ( A ) := { ∆ ∈ End u ( A ) s.t. K (∆) k +1 = 0 } = { ∆ ∈ End u ( A ) s.t. K (∆) n = 0 , ∀ n > k } , where the second identity follows immediately from the recursion 12 for the Koszul brackets.In other words, ∆ ∈ Diff u, ≤ k ( A ) if ∆ is a differential operator of order ≤ k on A , such thatmoreover ∆( A ) = 0. We denote by Diff u ( A ) = (cid:83) k ≥ Diff u, ≤ k ( A ). It is well known thatDiff u, ≤ j ( A ) ◦ Diff u, ≤ k ( A ) ⊂ Diff u, ≤ ( k + j ) ( A ) and (cid:2) Diff u, ≤ j ( A ) , Diff u, ≤ k ( A ) (cid:3) ⊂ Diff u, ≤ ( k + j − ( A ),so that Diff u ( A ) ⊂ End u ( A ) is both a graded subalgebra and a graded Lie subalgebra.1.3. Homological perturbation theory.Definition 1.24.
Given a pair of DG spaces (
A, d A ) and ( B, d B ), a contraction ( σ, τ, h ) of A onto B is the datum of DG maps σ : ( A, d A ) → ( B, d B ), τ : ( B, d B ) → ( A, d A ) and a contractinghomotopy h : A → A [ −
1] such that στ = id B , hd A + d A h = τ σ − id A ,σh = 0 , hτ = 0 , h = 0 . We shall often consider the datum of a graded commutative unitary algebra ( A, · ) equippedwith a degree one differential d A squaring to zero, which might not be an algebra derivation ,and is only required to satisfy d A ( A ) = 0: we shall call this datum a graded commutative unitaryalgebra with a differential , whereas as usual if d A is an algebra derivation we call ( A, d A , · ) a DGcommutative unitary algebra . Definition 1.25.
Given graded commutative algebras A and B together with differentials d A , d B (not necessarily algebra derivations, but as explained above we do require d A ( A ) = 0, d B ( B ) = 0) and a contraction ( σ, τ, h ) of ( A, d A ) onto ( B, d B ), we say that ( σ, τ, h ) is a semifullalgebra contraction if the following identities are satisfied for all a, b ∈ A , x, y ∈ Bh ( h ( a ) h ( b )) = h ( h ( a ) τ ( x )) = h ( τ ( x ) τ ( y )) = h ( A ) = 0 ,σ ( h ( a ) h ( b )) = σ ( h ( a ) τ ( x )) = 0 , σ ( τ ( x ) τ ( y )) = xy, σ ( A ) = B . If furthermore d A is an algebra derivations, we say that ( σ, τ, h ) is a semifull DG algebra con-traction . Semifull algebra contractions were introduced by Real [66]: we refer to loc. cit. forfurther discussion on this class of contractions and several examples. Remark 1.26.
When ( σ, τ, h ) is a semifull DG algebra contraction, i.e., when d A is an algebraderivation, then the identities in the previous definition are equivalent to the seemingly stronger h ( ( − | a | +1 h ( a ) b + ah ( b ) ) = h ( a ) h ( b ) , (14) h ( aτ ( x )) = h ( a ) τ ( x ) , (15) σ ( ( − | a | +1 h ( a ) b + ah ( b ) ) = 0 , (16) σ ( aτ ( x )) = σ ( a ) x. (17) as can be seen from the following identities (18), (19), (20), (21). Moreover, the followingProposition 1.27 will imply immediately that if ( σ, τ, h ) is a semifull DG algebra contraction,then necessarily d B is an algebra derivation and τ is a morphism of graded algebras.The main technical result of this section shows that given graded commutative unitary algebraswith a differential ( A, d A ) and ( B, d B ), together with a semifull algebra contraction ( σ, τ, h ) of( A, d A ) onto ( B, d B ), the Koszul brackets K ( d B ) n on B and the ones K ( d A ) n on A are related viahomotopy transfer for L ∞ [1] algebras. More precisely, the sequence of Koszul brackets K ( d A ) n , n ≥
1, induces a (homotopy abelian) L ∞ [1] algebra structure on A . Via homotopy transferalong the contraction ( σ, τ, h ) (see Theorem 1.3), there is an induced L ∞ [1] algebra structureon B , as well as an L ∞ [1] morphism from B to A . We shall see that the former is precisely K ( d B ) : S ( B ) → S ( B ), and the latter precisely κ ( τ ) : S ( B ) → S ( A ).Before we do this, given unitary graded commutative algebras A, B with differentials (notnecessarily derivations) d A , d B and a semifull algebra contraction ( σ, τ, h ) between them, weneed to look more closely at the failure of the previous identities (14)-(17).We have h ( ah ( b )) = h (( τ σ − d A h − hd A )( a ) h ( b )) = − h ( d A h ( a ) h ( b )) , and similarly h ( h ( a ) b ) = − h ( h ( a ) d A h ( b )). Moreover h ( a ) h ( b ) = ( τ σ − d A h − hd A ) ( h ( a ) h ( b )) = − hd A ( h ( a ) h ( b )). Thus,(18) h (cid:16) ( − | a | +1 h ( a ) b + ah ( b ) (cid:17) − h ( a ) h ( b ) == h (cid:16) d A ( h ( a ) h ( b )) − d A h ( a ) h ( b ) − ( − | a | +1 h ( a ) d A h ( b ) (cid:17) == h K ( d A ) ( h ( a ) , h ( b )) . In the same way, since h ( a ) τ ( x ) = − hd A ( h ( a ) τ ( x )) and h ( aτ ( x )) = − h ( d A h ( a ) τ ( x )) = − h (cid:16) d A h ( a ) τ ( x ) + ( − | a | +1 h ( a ) τ d B ( x ) (cid:17) == − h (cid:16) d A h ( a ) τ ( x ) + ( − | a | +1 h ( a ) d A τ ( x ) (cid:17) , we see that(19) h ( aτ ( x )) − h ( a ) τ ( x ) == h (cid:16) d A ( h ( a ) τ ( x )) − d A h ( a ) τ ( x ) − ( − | a | +1 h ( a ) d A τ ( x ) (cid:17) == h K ( d A ) ( h ( a ) , τ ( x )) . The same kind of computations shows that(20) σ (cid:16) ( − | a | +1 h ( a ) b + ah ( b ) (cid:17) = σ K ( d A ) ( h ( a ) , h ( b )) . (21) σ ( aτ ( x )) − σ ( a ) x = σ K ( d A ) ( h ( a ) , τ ( x )) . Proposition 1.27.
Given a semifull algebra contraction as above, the L ∞ [1] algebra structure on A associated with the Koszul brackets K ( d A ) n transfers to the one on B associated with the Koszulbrackets K ( d B ) n . Moreover, the induced L ∞ [1] morphism ( S ( B ) , K ( d B )) → ( S ( A ) , K ( d A )) isprecisely κ ( τ ) .Proof. This is a rather long computation. We need to prove that the κ ( τ ) n , K ( d B ) n satisfy thefollowing recursions: starting from κ ( τ ) = τ , K ( d B ) = d B , for all n ≥ x , . . . , x n ∈ B , UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 13 we have to show κ ( τ ) n ( x , . . . , x n ) = (cid:88) k ≥ ,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K k ! h K ( d A ) k (cid:0) κ ( τ ) i ( x σ (1) . . . ) , . . . , κ ( τ ) i k ( . . . , x σ ( n ) ) (cid:1) K ( d B ) n ( x , . . . , x n ) = (cid:88) k ≥ ,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K k ! σ K ( d A ) k (cid:0) κ ( τ ) i ( x σ (1) . . . ) , . . . , κ ( τ ) i k ( . . . , x σ ( n ) ) (cid:1) As usual, in the above formulas we denote by ± K the Koszul sign associated to the permutation x , . . . , x n (cid:55)→ x σ (1) , . . . , x σ ( n ) .We prove first that the cumulants κ ( τ ) n obey the above recursion, by induction on n . As awarm up, we begin by considering the cases n = 2 and n = 3. For n = 2, since h ( τ ( y ) τ ( z )) = 0and σ ( τ ( y ) τ ( z )) = yz , κ ( τ ) ( y, z ) = τ ( yz ) − τ ( y ) τ ( z ) = ( τ σ − id A )( τ ( y ) τ ( z )) = hd A ( τ ( y ) τ ( z )) = h K ( d A ) ( τ ( y ) , τ ( z )) , which is what we needed to show. In the last passage, we used the fact that h ( d A τ ( y ) τ ( z )) = h ( τ d B ( y ) τ ( z )) = 0, and for the same reason h ( τ ( y ) d A τ ( z )) = 0.Next, we consider the case n = 3: the following computation should already give a hint ofour inductive argument for the general case, which is based on the recursions (8) and (12) forthe cumulants and the Koszul brackets respectively, as well as the previous identities (18)-(19).For simplicity, we get rid of Koszul signs by showing the necessary relation only for degree zeroarguments x, y, z ∈ B . κ ( τ ) ( x, y, z ) = (using (8)) = κ ( τ ) ( x, yz ) − κ ( τ ) ( x, y ) τ ( z ) − κ ( τ ) ( x, z ) τ ( y ) == (using the already done n = 2 case) == h K ( d A ) ( τ ( x ) , τ ( yz )) − h K ( d A ) ( τ ( x ) , τ ( y )) τ ( z ) − h K ( d A ) ( τ ( x ) , τ ( z )) τ ( y ) == (using (19)) = h (cid:16) K ( d A ) ( τ ( x ) , τ ( y ) τ ( z )) −K ( d A ) ( τ ( x ) , τ ( y )) τ ( z ) −K ( d A ) ( τ ( x ) , τ ( z )) τ ( y ) (cid:17) ++ h K ( d A ) ( τ ( x ) , κ ( τ ) ( y, z )) + h K ( d A ) ( τ ( y ) , h K ( d A ) ( τ ( x ) , τ ( z ))) ++ h K ( d A ) ( τ ( y ) , h K ( d A ) ( τ ( x ) , τ ( z ))) = (using (12)) = h K ( d A ) ( τ ( x ) , τ ( y ) , τ ( z ))++ h K ( d A ) ( τ ( x ) , κ ( τ ) ( y, z )) + h K ( d A ) ( τ ( y ) , κ ( τ ) ( x, z )) + h K ( d A ) ( τ ( z ) , κ ( τ ) ( x, y )) . Now we do the general case. For simplicity, in the following computations we denote by κ n := κ ( τ ) n , K k := K ( d A ) k . Furthermore, as before, we shall get rid of Koszul signs by showingthe necessary relation only for x , . . . , x n , y, z ∈ B . Finally, in order to further abbreviate thefollowing equations, we shall omit the arguments x , . . . , x n , which should fill the suspensiondots in a way which should be obvious from the context. For instance, with these notationalsimplifications, the recursion (8) for the cumulants κ ( τ ) n +2 becomes κ n +2 ( . . . , y, z ) = κ n +1 ( . . . , yz ) − n (cid:88) i =0 (cid:88) σ ∈ S ( i,n − i ) κ i +1 ( . . . , y ) κ n − i +1 ( . . . , z ) . As another example, the recursion for the cumulants we need to show becomes(22) κ n +2 ( . . . , y, z ) = (cid:88) k ≥ ,i ,...,ik − ≥ ,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) k − h K k ( κ i ( . . . ) , . . . , κ i k +2 ( . . . , y, z )) ++ (cid:88) k,i ,...,ik − ≥ ,ik,ik +1 ≥ i + ··· + i k +1 = n (cid:88) σ ∈ S ( i ,...,i k +1 ) k − h K k +1 (cid:0) κ i ( . . . ) , . . . , κ i k +1 ( . . . , y ) , κ i k +1 +1 ( . . . , z ) (cid:1) . Next, by using in order the inductive hypothesis, the recursions (8) for the cumulants and theone (12) for the Koszul brackets, we see that(23) κ n +1 ( . . . , yz ) = (cid:88) k ≥ ,i ,...,ik − ≥ ,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) k − h K k ( κ i ( . . . ) , . . . , κ i k +1 ( . . . , yz )) = (cid:88) k ≥ ,i ,...,ik − ≥ ,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) k − h K k ( κ i ( . . . ) , . . . , κ i k +2 ( . . . , y, z )) + (cid:88) k ≥ ,i ,...,ik − ≥ ,ik,ik +1 ≥ i + ··· + i k +1 = n (cid:88) σ ∈ S ( i ,...,i k +1 ) k − h K k (cid:0) κ i ( . . . ) , . . . , κ i k +1 ( . . . , y ) κ i k +1 +1 ( . . . , z ) (cid:1) = (cid:88) k ≥ ,i ,...,ik − ≥ ,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) k − h K k ( κ i ( . . . ) , . . . , κ i k +2 ( . . . , y, z )) + (cid:88) k ≥ ,i ,...,ik − ≥ ,ik,ik +1 ≥ i + ··· + i k +1 = n (cid:88) σ ∈ S ( i ,...,i k +1 ) k − h K k +1 (cid:0) κ i ( . . . ) , . . . , κ i k +1 ( . . . , y ) , κ i k +1 +1 ( . . . , z ) (cid:1) + (cid:88) k ≥ ,i ,...,ik − ≥ ,ik,ik +1 ≥ i + ··· + i k +1 = n (cid:88) σ ∈ S ( i ,...,i k +1 ) k − h (cid:16) K k ( κ i ( . . . ) , . . . , κ i k +1 ( . . . , y )) κ i k +1 +1 ( . . . , z ) (cid:17) + (cid:88) k ≥ ,i ,...,ik − ≥ ,ik,ik +1 ≥ i + ··· + i k +1 = n (cid:88) σ ∈ S ( i ,...,i k +1 ) k − h (cid:16) K k (cid:0) κ i ( . . . ) , . . . , κ i k +1 +1 ( . . . , z ) (cid:1) κ i k +1 ( . . . , y ) (cid:17) We notice that the first two lines of the rightmost term of the previous equations account for allthe summands in the right hand side of (22) except for the ones(24) n (cid:88) i =0 (cid:88) σ ∈ S ( i,n − i ) h K ( κ i +1 ( . . . , y ) , κ n − i +1 ( . . . , z )) . We consider the i = n term in the above sum. Using in order the hynductive hypothesis,Equation (19) and again the inductive hypothesis, we may rewrite this as(25) h K ( κ n +1 ( . . . , y ) , τ ( z )) == (cid:88) k ≥ ,i ,...,ik − ≥ ,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) k − h K ( h K k ( κ i ( . . . ) , . . . , κ i k +1 ( . . . , y )) , τ ( z )) == − k − h K k ( κ i ( . . . ) , . . . , κ i k +1 ( . . . , y )) τ ( z )++ (cid:88) k ≥ ,i ,...,ik − ≥ ,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) k − h (cid:16) K k ( κ i ( . . . ) , . . . , κ i k +1 ( . . . , y )) τ ( z ) (cid:17) == − κ n +1 ( . . . , y ) τ ( z )+ (cid:88) k ≥ ,i ,...,ik − ≥ ,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) k − h (cid:16) K k ( κ i ( . . . ) , . . . , κ i k +1 ( . . . , y )) τ ( z ) (cid:17) UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 15
Reasoning in the same say, we may rewrite the i = 0 term in (24) as(26) h K ( τ ( y ) , κ n +1 ( . . . , z )) = − τ ( y ) κ n +1 ( . . . , z )++ (cid:88) k ≥ ,i ,...,ik − ≥ ,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) k − h (cid:16) K k ( κ i ( . . . ) , . . . , κ i k +1 ( . . . , z )) τ ( y ) (cid:17) . When 0 < i < n we can mimick the same argument, but this time using Equation (18) in placeof (19), to conclude that(27) (cid:88) σ ∈ S ( i,n − i ) h K ( κ i +1 ( . . . , y ) , κ n − i +1 ( . . . , z )) = − (cid:88) σ ∈ S ( i,n − i ) κ i +1 ( . . . , y ) κ n − i +1 ( . . . , z )++ (cid:88) k ≥ ,i ,...,ik − ≥ ,ik ≥ i + ··· + i k = i (cid:88) σ ∈ S ( i ,...,i k ,n − i ) k − h (cid:16) K k ( κ i ( . . . ) , . . . , κ i k +1 ( . . . , y )) κ n − i +1 ( . . . , z ) (cid:17) ++ (cid:88) k ≥ ,i ,...,ik − ≥ ,ik ≥ i + ··· + i k + = n − i (cid:88) σ ∈ S ( i ,...,i k ,i ) k − h (cid:16) K k ( κ i ( . . . ) , . . . , κ i k +1 ( . . . , z )) κ i +1 ( . . . , y ) (cid:17) . Putting all the above computations together, we can plug the last three Equations (25)-(27) into(23) to conclude that κ n +1 ( . . . , yz ) = n (cid:88) i =0 (cid:88) σ ∈ S ( i,n − i ) κ i +1 ( . . . , y ) κ n − i +1 ( . . . , z )++ (cid:88) k ≥ ,i ,...,ik − ≥ ,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) k − h K k ( κ i ( . . . ) , . . . , κ i k +2 ( . . . , y, z )) ++ (cid:88) k,i ,...,ik − ≥ ,ik,ik +1 ≥ i + ··· + i k +1 = n (cid:88) σ ∈ S ( i ,...,i k +1 ) k − h K k +1 (cid:0) κ i ( . . . ) , . . . , κ i k +1 ( . . . , y ) , κ i k +1 +1 ( . . . , z ) (cid:1) Finally, using the recursion (8) for the cumulants we have proved the desired identity (22).The fact that the Koszul brackets K ( d B ) n obey the necessary recursion could be shown bya similar computation, replacing the identities (18)-(19) with the ones (20)-(21) in the aboveargument. On the other hand, we observe that since κ ( τ ) = τ is injective, so is the coalgebramorphism κ ( τ ) : S ( B ) → S ( A ). In particular, there is at most one DG coalgebra strucure on S ( B ) making κ ( τ ) into a morphism of DG coalgebras from S ( B ) to ( S ( A ) , K ( d A )). Since boththe L ∞ [1] algebra structure on B induced via homotopy transfer and the one associated to K ( d B )satisfy this property, the former by the first part of the proof and the latter by Proposition 1.21,we conclude that they have to coincide. (cid:3) We conclude this subsection by reviewing the Standard Perturbation Lemma. This is a verywell known and classical result, see for instance [67, 17, 33, 34, 35], and has been applied toperform homotopical transfer of algebraic structures since the work of Kadeishvili [42, 43].
Definition 1.28.
Given a DG space (
A, d A ) and a degree one map δ A : A → A [1], we say that δ A is a perturbation of the differential d A on A if ˘ d A := d A + δ A satisfies ( ˘ d A ) = 0. Lemma 1.29.
Given a pair of DG spaces ( A, d A ) and ( B, d B ) , a contraction ( σ, τ, h ) of A onto B and a perturbation δ A of the differential on A , there is (under appropriate hypotheses ensuring convergence) an induced perturbation δ B := (cid:88) n ≥ σδ A ( hδ A ) n τ of the differential d B on B , as well as a perturbed contraction ˘ σ := (cid:88) n ≥ σ ( δ A h ) n ˘ τ := (cid:88) n ≥ ( hδ A ) n τ ˘ h := (cid:88) n ≥ ( hδ A ) n h of ( A, ˘ d A ) onto ( B, ˘ d B ) . Lemma 1.30.
The class of semifull algebra contractions is stable under arbitrary perturbations.The class of semifull DG algebra contractions is stable under perturbations by algebra derivations.Proof.
The first claim follows immediately from Definition 1.25. The second claim was provedin [66], see also [7, Proposition 2.17]. We further remark that it follows immediately fromProposition 1.27. In fact, by the first claim we know already that the perturbed contraction(˘ σ, ˘ τ , ˘ h ) is a semifull algebra contraction, and if both d A and the perturbation δ A are algebraderivations so is the perturbed differential ˘ d A (cf. also the previous Remarl 1.26). (cid:3) Cocumulants and Koszul cobrackets.
All of the previous constructions and resultsadmit dual versions in the context of coalgebras.Here and in the rest of the paper, we shall always work with coalgebras which are coassociative,cocommutative, counitary, cougmented and cocomplete. Given such a coalgebra C , we shalldenote: by ∆ C : C → C ⊗ C the coproduct; by (cid:15) C : C → K the counit; by K → C : 1 → C thecoaugmentation; by C = Ker( (cid:15) C ) the reduced coalgebra, with the reduced coproduct∆ C : C → C ⊗ C : c → ∆ C ( c ) := ∆ C ( c ) − C ⊗ c − c ⊗ C (notice in particular that C = K C ⊕ C ); by ∆ n − C : C → C ⊗ n the iterated coproducts. We saythat C is cocomplete if C = (cid:83) n ≥ Ker(∆ n − C ), or in other words if for any c ∈ C there exists N = N ( c ) (cid:29) N ( c ) = 0. Remark 1.31.
In the rest of this paper, all the coalgebras considered shall be coassociative,cocommutative, counitary, coaugmented and cocomplete, without further mention of theseproperties .We shall use the following notationEnd cu ( C ) := { δ ∈ End( C ) s.t. (cid:15) C ◦ δ = 0 , δ ( C ) = 0 } . Moreover, given a pair of counitary cocommutative of graded coalgebras C and D we shalldenote by Hom cu ( C, D ) := { f ∈ Hom ( C, D ) s.t. (cid:15) D ◦ f = (cid:15) C , f ( C ) = D } . Given f ∈ Hom cu ( C, D ), the cocumulants of f are degree zero maps κ co ( f ) n : C → D (cid:12) n measuring the deviation of f from being a morphism of graded coalgebras. Similarly, given δ ∈ End kcu ( C ), the associated Koszul cobrackets K co ( δ ) n : C → C (cid:12) n are degree k maps measuringthe deviation of δ from being a coalgebra coderivation.The most convenient way to introduce these maps is via the dual of formulas (6) and (10)respectively. More precisely, for n ≥ n − C : C → C ⊗ n the iterated UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 17 coproduct (∆ C =: id C ), by π : C ⊗ n → C (cid:12) n the projection and by E co : (cid:98) S ( C ) → (cid:98) S ( C ) theunique unitary algebra automorphism extending E co ( x ) = (cid:88) n ≥ n ! π ∆ n − C ( x ) , for x ∈ C (where we denote by (cid:98) S ( C ) = (cid:81) n ≥ C (cid:12) n the completed symmetric algebra over C ). Wedenote by L co : (cid:98) S ( C ) → (cid:98) S ( C ) the inverse of E co . It is the unique unitary algebra automorphismextending L co ( x ) = (cid:88) n ≥ ( − n − n π ∆ n − C ( x ) , x ∈ C. We call E co and L co respectively the coexponential and cologarithmic automorphisms of (cid:98) S ( C ).Given f ∈ Hom cu ( C, D ), we denote by (cid:98) κ co ( f ) : (cid:98) S ( C ) → (cid:98) S ( D ) the morphism of unitary gradedalgebras given by the composition (cid:98) κ co ( f ) : (cid:98) S ( C ) L co −−→ (cid:98) S ( C ) (cid:98) S ( f ) −−−→ (cid:98) S ( D ) E co −−→ (cid:98) S ( D ) . The cocumulants κ co ( f ) n : C → D (cid:12) n are then defined as the composition κ co ( f ) n : C (cid:44) → (cid:98) S ( C ) (cid:98) κ co ( f ) −−−−→ (cid:98) S ( D ) (cid:16) D (cid:12) n , where the left and right maps are the canonical inclusions and projections respectively.It is easy to compute explicitly the first few cocumulants. To simplify formulas, we adoptSweedler’s notation and write ∆ n − C ( x ) = (cid:80) ( x ) x (1) ⊗ · · · ⊗ x ( n ) . With these notations, we have κ co ( f ) ( x ) = f ( x ) , κ co ( f ) ( x ) = 12 (cid:88) ( f ( x )) f ( x ) (1) (cid:12) f ( x ) (2) − (cid:88) ( x ) f ( x (1) ) (cid:12) f ( x (2) ) κ co ( f ) ( x ) = 16 (cid:88) ( f ( x )) f ( x ) (1) (cid:12) f ( x ) (2) (cid:12) f ( x ) (3) − (cid:88) ( x ) , ( f ( x (1) )) f ( x (1) ) (1) (cid:12) f ( x (1) ) (2) (cid:12) f ( x (2) ) −− (cid:88) ( x ) , ( f ( x (2) )) f ( x (1) ) (cid:12) f ( x (2) ) (1) (cid:12) f ( x (2) ) (2) + 13 (cid:88) ( x ) f ( x (1) ) (cid:12) f ( x (2) ) (cid:12) f ( x (3) ) == 16 (cid:88) ( f ( x )) f ( x ) (1) (cid:12) f ( x ) (2) (cid:12) f ( x ) (3) − (cid:88) ( x ) , ( f ( x (1) )) f ( x (1) ) (1) (cid:12) f ( x (1) ) (2) (cid:12) f ( x (2) )++ 13 (cid:88) ( x ) f ( x (1) ) (cid:12) f ( x (2) ) (cid:12) f ( x (3) ) , where in the last passage we used the fact that (cid:80) ( x ) , ( f ( x (1) )) f ( x (1) ) (1) (cid:12) f ( x (1) ) (2) (cid:12) f ( x (2) ) = (cid:80) ( x ) , ( f ( x (2) )) f ( x (1) ) (cid:12) f ( x (2) ) (1) (cid:12) f ( x (2) ) (2) , due to cocommutativity of ∆ C .Notice that f ∈ Hom cu ( C, D ) is a coalgebra morphism if and only if κ co ( f ) n = 0 for all n ≥ Remark 1.32.
The cocumulants of f obey a recursion dual to (8). More precisely, define maps (cid:101) κ co ( f ) n : C → C ⊗ n , n ≥
1, recursively by (cid:101) κ co ( f ) = f and (cid:101) κ co ( f ) n +1 = (cid:0) id ⊗ n − D ⊗ ∆ D (cid:1) (cid:101) κ co ( f ) n − n − (cid:88) k =0 (cid:0) (cid:1) k,n − k − ⊗ id ⊗ D (cid:1) τ k (cid:0)(cid:101) κ co ( f ) k +1 ⊗ (cid:101) κ co ( f ) n − k (cid:1) ∆ C , where we denote by τ k : D ⊗ n +1 → D ⊗ n +1 the permutation τ k ( y ⊗ · · · ⊗ y k ⊗ y k +1 ⊗ y k +2 ⊗ · · · ⊗ y n ⊗ y n +1 ) = ± K y ⊗ · · · ⊗ y k ⊗ y k +2 ⊗ · · · ⊗ y n ⊗ y k +1 ⊗ y n +1 and by (cid:1) k,n − k − the ( k, n − k − (cid:1) k,n − k − : D ⊗ n − = D ⊗ k ⊗ D ⊗ n − k − (cid:1) −→ D ⊗ n − of the shuffle product (cid:1) . Thus for instance a straightforward computation shows that (cid:101) κ co ( f ) =∆ D f − f ⊗ ∆ C and (cid:101) κ co ( f ) = ∆ D f − (cid:0) f (cid:1) ∆ D f (cid:1) ∆ C + 2 f ⊗ ∆ C . It can be shown that the imageof (cid:101) κ co ( f ) n is contained in the S n -invariant part of U ⊗ n , and the cocumulants κ co ( f ) : C → C (cid:12) n are given by κ co ( f ) n = n ! π (cid:101) κ co ( f ) n , where we denote by π : C ⊗ n → C (cid:12) n the natural projection. Remark 1.33.
Assuming, as we are, that C and D are cocomplete, it is not hard to see that (cid:98) κ co ( f )( c ) ∈ S ( D ) ⊂ (cid:98) S ( D ) for all c ∈ C , or in other words that for all c ∈ C there exists n ( c ) (cid:29) κ co ( f ) N ( c ) = 0 for all N ≥ n ( c ). Thus (cid:98) κ co ( f ) : (cid:98) S ( C ) → (cid:98) S ( D ) restricts to a morphismof unitary graded algebras κ co ( f ) = (cid:88) n ≥ κ co ( f ) n : S ( C ) → S ( D ) . In fact, one can show a stronger statement. If f ∈ Hom cu ( C, D ), then f ( C ) = D and f restricts to f := f | C : C → D . We denote by E co : S ( D ) → S ( D ) and L co : S ( C ) → S ( C ) themorphisms of unitary graded algebras extending E co ( y ) = (cid:88) n ≥ n ! π ∆ n − D ( y ) , L co ( x ) = (cid:88) n ≥ ( − n − n π ∆ n − C ( x ) , x ∈ C, y ∈ D, where ∆ n − C : C → C ⊗ n , ∆ n − D : D → D ⊗ n are the iterated reduced coproducts, and theabove infinite sums make sense since C and D are cocomplete. Finally, we define κ co ( f ) := E co ◦ S ( f ) ◦ L co : S ( C ) → S ( D ). With these notations, it is not hard to show that the followingdiagram is commutative C (cid:15) (cid:15) κ co ( f ) (cid:47) (cid:47) S ( D ) (cid:15) (cid:15) C (cid:98) κ co ( f ) (cid:47) (cid:47) (cid:98) S ( D )where the vertical arrows are the natural inclusions. This implies the claim at the beginningof the remark, and shows moreover that we may compute the cocumulants using the reducedcoproducts in place of the unreduced ones. This applies in particular to the previous explicitformulas for κ co ( f ) n , n ≤
3, where Sweedler’s notation might be intended for the reducedcoproducts instead of the unreduced ones, and to the previous Remark 1.32, where in therecursive formula for (cid:101) κ co ( f ) n +1 we might replace ∆ C , ∆ D with ∆ C , ∆ D .Given δ ∈ End cu ( C ), we denote by (cid:98) K co ( δ ) : (cid:98) S ( C ) → (cid:98) S ( C ) the derivation defined as thecomposition (cid:98) K co ( δ ) : (cid:98) S ( C ) L co −−→ (cid:98) S ( C ) (cid:101) δ −→ (cid:98) S ( C ) E co −−→ (cid:98) S ( C ) , where as in subsection 1.2 we denote by (cid:101) δ : (cid:98) S ( C ) → (cid:98) S ( C ) the linear derivation extending δ . The Koszul cobrackets K co ( δ ) n : C → D (cid:12) n are defined as the compositions K co ( δ ) n : C (cid:44) → (cid:98) S ( C ) K co ( δ ) −−−−→ (cid:98) S ( C ) (cid:16) C (cid:12) n . UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 19
The first few Koszul cobrackets of δ ∈ End cu ( C ) are K co ( δ ) ( x ) = δ ( x ) , K co ( δ ) ( x ) = 12 (cid:88) ( δ ( x )) δ ( x ) (1) (cid:12) δ ( x ) (2) − (cid:88) ( x ) δ ( x (1) ) (cid:12) x (2) K co ( δ ) ( x ) = 16 (cid:88) ( δ ( x )) δ ( x ) (1) (cid:12) δ ( x ) (2) (cid:12) δ ( x ) (3) −− (cid:88) ( x ) , ( δ ( x (1) )) δ ( x (1) ) (1) (cid:12) δ ( x (1) ) (2) (cid:12) x (2) + 12 (cid:88) ( x ) δ ( x (1) ) (cid:12) x (2) (cid:12) x (3) . We notice that δ is a colagebra coderivation if and only if K co ( δ ) n = 0 for all n ≥ Remark 1.34.
The Koszul cobrackets of δ obey a recursion dual to (12). More precisely, definemaps (cid:101) K co ( δ ) n : C → C ⊗ n , n ≥
1, recursively by (cid:101) K co ( δ ) = δ and (cid:101) K co ( δ ) n +1 = (cid:0) id ⊗ n − C ⊗ ∆ C (cid:1) (cid:101) K co ( δ ) n − (id ⊗ n +1 C + τ n,n +1 ) (cid:0) (cid:101) K co ( f ) n ⊗ id C (cid:1) ∆ C , where we denote by τ n,n +1 : C ⊗ n +1 → C ⊗ n +1 the transposition τ n,n +1 ( c ⊗ · · · ⊗ c n ⊗ c n +1 ) = ± K c ⊗ · · · ⊗ c n +1 ⊗ c n . Thus for instance (cid:101) K co ( δ ) = ∆ C δ − ( δ ⊗ id C + id C ⊗ δ )∆ C (cid:101) K co ( δ ) = ∆ C δ − (cid:0) id C (cid:1) ∆ C δ (cid:1) ∆ C + ( δ ⊗ id C ⊗ id C + id C ⊗ δ ⊗ id C + id C ⊗ id C ⊗ δ )∆ C . It can be shown that the image of (cid:101) K co ( δ ) n is contained in the S n -invariant part of U ⊗ n , and theKoszul cobrackets K co ( δ ) : C → C (cid:12) n are given by K co ( δ ) n = n ! π (cid:101) K co ( δ ) n , where we denote by π : C ⊗ n → C (cid:12) n the natural projection. Remark 1.35.
As in the previous remark 1.33, when C is cocomplete we can define a derivation K co ( δ ) := E co ◦ (cid:101) δ ◦ L co : S ( C ) → S ( C ), where we denote by δ the restriction δ := δ | C : C → C and by (cid:101) δ : S ( C ) → S ( C ) the linear derivatione extending it. It is not hard to check that thefollowing diagram is commutative C (cid:15) (cid:15) K co ( δ ) (cid:47) (cid:47) S ( C ) (cid:15) (cid:15) C (cid:98) K co ( δ ) (cid:47) (cid:47) (cid:98) S ( C )where the vertical arrows are the natural inclusions. In particular (cid:98) K co ( δ )( c ) ∈ S ( C ) ⊂ (cid:98) S ( C ) forall c ∈ C , and thus (cid:98) K co ( δ ) : (cid:98) S ( C ) → (cid:98) S ( C ) restricts to a derivation K co ( δ ) = (cid:88) n ≥ K co ( δ ) n : S ( C ) → S ( C ) . All the results from subsection 1.2 (namely, Lemmas 1.16, 1.20 and Propositions 1.21, 1.22)have analogues in this setting, which can be proved by the same arguments, mutatis mutandis.We shall also need to consider the analogue of Definition 1.23.
Definition 1.36.
Given a graded (cocommutative, et cet.) coalgebra C and k ≥
1, we denoteby coDiff cu, ≤ k ( C ) := { δ ∈ End cu ( C ) s.t. K co ( δ ) k +1 = 0 } (that is, the subset of End cu ( C ) consisting of codifferential operators of order ≤ k ). We denoteby coDiff cu ( C ) = (cid:83) k ≥ coDiff cu, ≤ k ( C ). It is easy to check that coDiff cu ( C ) ⊂ End cu ( C ) is botha graded subalgebra and a graded Lie subalgebra.We turn to the results from subsection 1.3. Definition 1.37.
Given a pair of graded (cocommutative, et cet.) coalgebras ( C, ∆ C , (cid:15) C ) and( D, ∆ D , (cid:15) D ) equipped with differentials d C , d D (we assume d C ∈ End cu ( C ), d D ∈ End cu ( D ), butnot require d C , d D to be coderivations), a semifull coalgebra contraction ( σ, τ, h ) of the formeronto the latter is a contraction of ( C, d C ) onto ( D, d D ) satisfying the further requirements σ ∈ Hom cu ( C, D ) , τ ∈ Hom cu ( D, C ) , h ∈ End cu ( C ) , ( h ⊗ h ) ◦ ∆ C ◦ h = ( h ⊗ σ ) ◦ ∆ C ◦ h = ( σ ⊗ σ ) ◦ ∆ C ◦ h = 0 , ( h ⊗ h ) ◦ ∆ C ◦ τ = ( h ⊗ σ ) ◦ ∆ C ◦ τ = 0 , ( σ ⊗ σ ) ◦ ∆ C ◦ τ = ∆ D . In the above hypotheses, if furthermore d C is a coalgebra coderivations we say that ( σ, τ, h ) a semifull DG coalgebra contraction . Remark 1.38.
The following Proposition 1.40 will imply immediately that if ( σ, τ, h ) is asemifull DG algebra contraction then necessarily d D is a coderivation and σ is a morphism ofgraded coalgebras. Lemma 1.39.
The class of semifull contractions of coalgebras with a differential is stable un-der arbitrary perturbations. The class of semifull DG coalgebra contractions is stable underperturbations by coalgebra coderivations.Proof.
It follows straightforwardly from the definitions. (cid:3)
Proposition 1.40.
Given graded (cocommutative, et cet.) coalgebras ( C, ∆ C , (cid:15) C ) , ( D, ∆ D , (cid:15) D ) ,together with differentials (not necessarily coderivations) d C ∈ End cu ( C ) , d D ∈ End cu ( D ) and asemifull coalgebra contraction ( σ, τ, h ) of ( C, d C ) onto ( D, d D ) , the Koszul cobrackets K co ( d D ) n and the cocumulants κ co ( σ ) n are induced via homotopy transfer from the L ∞ [ − coalgebra struc-ture on C associated with the Koszul cobrackets K co ( d C ) n .Proof. Reasoning as at the end of the proof of 1.27, it is enough to prove the statement aboutthe cocumulants.We have to show that the morphism of graded algebras κ co ( σ ) : S ( C ) → S ( D ) obeyes (and isrecursively defined from) κ co ( σ )( c ) = σ ( c ) + κ co ( σ ) (cid:16) K co ( d C ) + (cid:16) h ( c ) (cid:17)(cid:17) for all c ∈ C , where we denote by K co ( d C ) + := K co ( d C ) − (cid:102) d C : S ( C ) → S ( C ) (here as usual (cid:102) d C is the linear derivation extending d C ). These recursions are the duals of the recursions for κ ( τ ), K ( d B ), at the beginning of the proof of Proposition 1.27. They can be shown by dualizing thecomputations jn the same proof. For instance, we have to show that the first two cocumulantsof σ are κ co ( σ ) = σ (it is so by definition) and κ co ( σ ) = ( σ (cid:12) σ ) K co ( d C ) h . For the latteridentity, using the fact that ∆ D = ( σ ⊗ σ )∆ C τ and ( σ ⊗ σ )∆ C h = 0 since ( σ, τ, h ) is a semifullcoalgebra contraction, we see that (where π : C ⊗ → C (cid:12) is the natural projection) κ co ( σ ) = 12 π (cid:16) ∆ D σ − ( σ ⊗ σ )∆ C (cid:17) = 12 π ( σ ⊗ σ )∆ C ( τ σ − id C ) = 12 π ( σ ⊗ σ )∆ C d C h. On the other hand K co ( d C ) = π (cid:16) ∆ C d C − ( d C ⊗ id C + id C ⊗ d C )∆ C (cid:17) . Thus( σ (cid:12) σ ) K co ( d C ) h = 12 π ( σ ⊗ σ ) (cid:16) ∆ C d C − ( d C ⊗ id C +id C ⊗ d C )∆ C (cid:17) h = 12 π ( σ ⊗ σ )∆ C d C h = κ co ( σ ) , UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 21 where we used that ( σ ⊗ σ )( d C ⊗ id C + id C ⊗ d C )∆ C h = ( d D ⊗ id D + id D ⊗ d D )( σ ⊗ σ )∆ C h = 0.In the general case the claim might be proved by adapting the recursive computation in theproof of Proposition 1.27 to this dual setting.Alternatively, one might reason as follows. We denote by R = r + · · · + r n + · · · , r n : D → D (cid:12) n r = d D , the L ∞ [ −
1] coalgebra structure on D induced via homotopy transfer, and likewise wedenote by G = g + · · · + g n + · · · , g n : C → D (cid:12) n , g = σ , the induced L ∞ [ −
1] coalgebramorphism. To show that g n = κ co ( σ ) n , ∀ n ≥
1, it is enough to show that the dual mapscoincide, i.e., g ∨ n = κ co ( σ ) ∨ n : ( D ∨ ) (cid:12) n → ( D (cid:12) n ) ∨ → C ∨ . On the one hand, it follows directly fromthe definitions that cumulants and cocumulants are dual to each other, i.e., κ co ( σ ) ∨ n = κ ( σ ∨ ) n ,where the cumulants of σ ∨ ∈ Hom u ( D ∨ , C ∨ ) are computed with respect to the dual algebrastructures on D ∨ and C ∨ . Similarly, Koszul brackets and cobrackets are dual to each other inthe sense that K co ( d C ) ∨ n = K ( d ∨ C ) n . On the other hand, it is easy to see that homotopy transferis compatible with duality in the following sense: the L ∞ [1] algebra structure ( r ∨ , . . . , r ∨ n , . . . ) on D ∨ and the L ∞ [1] morphism ( g ∨ , . . . , g ∨ n , . . . ) : D ∨ → C ∨ are induced by transferring the dual L ∞ [1] algebra structure ( d ∨ C , . . . , K co ( d C ) ∨ n , . . . ) = ( d ∨ C , . . . , K ( d ∨ C ) n , . . . ) on C ∨ along the dualcontraction ( σ ∨ , τ ∨ , h ∨ ). Finally, putting these two observations togetether with Proposition1.27 (notice that ( σ ∨ , τ ∨ , h ∨ ) is a semifull algebra contraction) we can conclude, as desired, that g ∨ n = κ ( σ ∨ ) n = κ co ( σ ) ∨ n . (cid:3) Homotopy transfer for L ∞ [1] -algebras (revisited). Given a pair of symmetric coal-gebras S ( U ) , S ( V ), the graded space Hom( S ( U ) , S ( V )) becomes a unitary graded commutativealgebra via the convolution product (cid:63) : explicitly, F (cid:63) G = (cid:12) ◦ ( F ⊗ G ) ◦ ∆ . where ∆ is the unshuffle coproduct on S ( U ) and (cid:12) is the symmetric product on S ( V ). The unitis the map ε : S ( U ) → S ( V ) defined by ε ( S ( U ) ) = S ( V ) and ε ( x (cid:12) · · · (cid:12) x n ) = 0 for all n ≥ x , . . . , x n ∈ U .Given ϕ ∈ Hom ( S ( U ) , S ( V )) such that ϕ ( S ( U ) ) = 0, it is well defined (by cocompleteness of S ( U )) the exponential exp (cid:63) ( ϕ ) = ε + (cid:80) k ≥ k ! ϕ (cid:63)k of ϕ with respect to the convolution product.Conversely, given F ∈ Hom ( S ( U ) , S ( V )) with F ( S ( U ) ) = S ( V ) , it is well defined the logarithmlog (cid:63) ( F ) = (cid:80) k ≥ − k − k ( F − ε ) (cid:63)k of F with respect to the convolution product.By definition, given F ∈ Hom cu ( S ( U ) , S ( V )), it is a morphism of graded coalgebras if andonly if the higher cocumulants vanish, i.e., if and only if κ co ( F ) n = 0 for all n ≥
2. In thefollowing lemma we assume instead F ∈ Hom u ( S ( U ) , S ( V )) (that is, F ( S ( U ) ) = S ( V ) ), andgive an equivalent condition for F to be a morphism of graded coalgebras, this time in terms ofthe cumulants κ ( F ) n . Lemma 1.41.
Given a pair of graded spaces
U, V and F ∈ Hom u ( S ( U ) , S ( V )) , then F is amorphism of graded coalgebras if and only if κ ( F ) n ( U, . . . , U ) ⊂ V ⊂ S ( V ) for all n ≥ .If this happens, denoting by f n : U (cid:12) n → V , n ≥ , the maps defined by f n ( x , . . . , x n ) = κ ( F ) n ( x , . . . , x n ) , then F is the unique morphism of coalgebras with Taylor coefficients pF =( f , . . . , f n , . . . ) ( p : S ( V ) → V being the natural projection).Proof. A straightforward computation, using Formula (7) for the cumulants and the definitionof the convolution product (cid:63) , shows that for all n ≥ x , . . . , x n ∈ Uκ ( F ) n ( x , . . . , x n ) == (cid:88) k,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K ( − k − k F ( x σ (1) (cid:12)· · · ) (cid:12)· · ·(cid:12) F ( · · ·(cid:12) x σ ( n ) ) = log (cid:63) ( F )( x (cid:12)· · ·(cid:12) x n ) In particular, we see that F ( x (cid:12) · · · (cid:12) x n ) = exp (cid:63) (log (cid:63) ( F ))( x (cid:12) · · · (cid:12) x n ) == (cid:88) k,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K k ! κ ( F ) i ( x σ (1) , . . . ) (cid:12) · · · (cid:12) κ ( F ) i k ( . . . , x σ ( n ) ) . The above identity, together with the one (3), shows that if there exist maps f n : U (cid:12) n → V suchthat κ ( F ) n ( x , . . . , x n ) = f n ( x , . . . , x n ), then F is precisely the morphism of coalgebras withcorestriction pF = ( f , . . . , f n , . . . ). Conversely, if F is a morphism of graded coalgebras withcorestriction pF = ( f , . . . , f n , . . . ), the above identity and a straightforward induction showthat κ ( F ) n ( x , . . . , x n ) = f n ( x , . . . , x n ) ∈ V ⊂ S ( V ) for all n ≥ x , . . . , x n ∈ U , and thusthat κ ( F ) n ( U, . . . , U ) ⊂ V . (cid:3) Analagously, given Q ∈ End cu ( S ( U )), we know that it is a coderivation if and only if K co ( Q ) n =0 for all n ≥
2. In the following lemma we assume Q ∈ End u ( S ( U )) and we give an equivalentcondition for Q to be a coderivation in terms of the Koszul brackets K ( Q ) n . Lemma 1.42.
Given Q ∈ End u ( S ( U )) , then Q is a coderivation if and only if K ( Q ) n ( U, . . . , U ) ⊂ U ⊂ S ( U ) for all n ≥ . If this happens, denoting by q n : U (cid:12) n → U , n ≥ , the maps defined by q n ( x , . . . , x n ) = K ( Q ) n ( x , . . . , x n ) , then Q is the unique coalgebra coderivation with corestric-tion pQ = (0 , q , . . . , q n , . . . ) ( p : S ( U ) → U being as usual the natural projection).Proof. We notice that the identity id S ( U ) is invertible with respect to the convolution productof End( S ( U )), with inverse the map s : S ( U ) → S ( U ) defined by s ( S ( U ) ) = S ( U ) , s ( x (cid:12) · · · (cid:12) x k ) = ( − k x (cid:12) · · · (cid:12) x k for all k ≥ x , . . . , x k ∈ U (in other words, s is the antipode of the natural Hopf algebrastructure on S ( U )). The explicit formula (11) for the Koszul brackets K ( Q ) n shows that K ( Q ) n ( x , . . . , x n ) = ( Q (cid:63) s )( x (cid:12) · · · (cid:12) x n ) . In other words, denoting by i : U → S ( U ) the natural inclusion, by S ( i ) the induced inclusion S ( U ) → S ( S ( U )) : x (cid:12) · · · (cid:12) x n → i ( x ) (cid:12) · · · (cid:12) i ( x n ), and by k : S ( U ) → S ( U ) the composition k : S ( U ) S ( i ) −−→ S ( S ( U )) K ( Q ) −−−→ S ( S ( U )) p −→ S ( U ) , the above shows that Q (cid:63) s = k in the convolution algebra End( S ( U )), and thus that Q = k (cid:63) id S ( U ) , that is,(28) Q ( x (cid:12) · · · (cid:12) x n ) = n (cid:88) i =1 (cid:88) σ ∈ S ( i,n − i ) ± K K ( Q ) i ( x σ (1) , . . . , x σ ( i ) ) (cid:12) x σ ( i +1) (cid:12) · · · (cid:12) x σ ( n ) for all n ≥ x , . . . , x n ∈ U . If K ( Q ) n ( x , . . . , x n ) = q n ( x , . . . , x n ) ∈ U for certain q n : U (cid:12) n → U , then (4) shows that Q is precisely the coderivation with Taylor coefficients q =0 , q , . . . , q n , . . . . Conversely, if Q is a coderivation, the above and a straightforward inductionshow that K ( Q ) n ( x , . . . , x n ) = q n ( x , . . . , x n ) ∈ U for all n ≥ x , . . . , x n ∈ U , and thusthat K ( Q ) n ( U, . . . , U ) ⊂ U ⊂ S ( U ), as desired. (cid:3) As a final preparation, we shall show that the contraction obtained from the symmetrizedtensor trick is a semifull contraction with respect to both the algebraic and coalgebraic structures.
Lemma 1.43.
Given a pair of complexes ( U, d U ) , ( V, d V ) and a contraction ( σ, τ, h ) of ( U, d U ) onto ( V, d V ) , there is an induced contraction ( S ( σ ) , S ( τ ) , (cid:98) h ) of ( S ( U ) , (cid:102) d U ) onto ( S ( V ) , (cid:102) d V ) , where UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 23 the contracting homotopy (cid:98) h is given by the symmetrized tensor trick: explicitly, (cid:98) h ( S ( U ) ) = 0 and (cid:98) h ( x (cid:12) · · · (cid:12) x n ) = 1 n ! n (cid:88) j =1 (cid:88) σ ∈ S n ± K τ σ ( x σ (1) ) (cid:12) · · · (cid:12) τ σ ( x σ ( j − ) (cid:12) h ( x σ ( j ) ) (cid:12) x σ ( j +1) (cid:12) · · · (cid:12) x σ ( n ) , where ± K is the Koszul sign associated to h, x , . . . , x n (cid:55)→ x σ (1) , . . . , x σ ( j − , h, x σ ( j ) , . . . , x σ ( n ) (keeping in mind that | h | = − ). This is both a semifull DG algebra contraction with respect tothe symmetric products and a semifull DG coalgebra contraction with respecto to the unshufflecoproducts (in other words, we might call it a semifull DG bialgebra contraction).Proof. The fact that ( S ( σ ) , S ( τ ) , (cid:98) h ) defines a contraction of ( S ( U ) , (cid:102) d U ) onto ( S ( V ) , (cid:102) d V ) is wellknown, and in any case easy to show. Furthermore, it is clear that S ( σ ) , S ( τ ) are both algebraand coalgebra morphisms, which immediately implies some of the necessary relations. The onlyrelations to be shown which are not immediate are the following ones (cid:98) h ◦ (cid:12) ◦ ( (cid:98) h ⊗ (cid:98) h ) = 0 , (cid:98) h ◦ (cid:12) ◦ ( (cid:98) h ⊗ S ( τ )) = 0 , ( (cid:98) h ⊗ (cid:98) h ) ◦ ∆ ◦ (cid:98) h = 0 , ( (cid:98) h ⊗ S ( σ )) ◦ ∆ ◦ (cid:98) h = 0 , where (cid:12) and ∆ are respectively the symmetric product and the unshuffle coproduct on S ( U ).For the latter two, by a standard polarization argument it is enough to show them on elementsof the form x (cid:12) n , for some x ∈ U and n ≥
1. In this case, we notice that (cid:98) h (cid:16) h ( x ) (cid:12) τ σ ( x ) (cid:12) i (cid:12) x (cid:12) j (cid:17) = 0 , S ( σ ) (cid:16) h ( x ) (cid:12) τ σ ( x ) (cid:12) i (cid:12) x (cid:12) j (cid:17) = 0 , for all i, j ≥
0, since h = hτ = σh = 0 and h ( x ) (cid:12) h ( x ) = 0 by degree reasons. Thus, denotingby K either (cid:98) h or S ( σ ),( (cid:98) h ⊗ K )∆ (cid:98) h ( x (cid:12) n ) = ( (cid:98) h ⊗ K )∆ (cid:88) i + j = n − h ( x ) (cid:12) τ σ ( x ) (cid:12) i (cid:12) x (cid:12) j == (cid:88) p + q + r + s = n − (cid:18) p + rp (cid:19)(cid:18) q + sq (cid:19)(cid:98) h (cid:16) h ( x ) (cid:12) τ σ ( x ) (cid:12) p (cid:12) x (cid:12) q (cid:17) ⊗ K (cid:16) τ σ ( x ) (cid:12) r (cid:12) x (cid:12) s (cid:17) ++ (cid:18) p + rp (cid:19)(cid:18) q + sq (cid:19)(cid:98) h (cid:16) τ σ ( x ) (cid:12) p (cid:12) x (cid:12) q (cid:17) ⊗ K (cid:16) h ( x ) (cid:12) τ σ ( x ) (cid:12) r (cid:12) x (cid:12) s (cid:17) = 0 . The first two identities are shown similarly. First of all, it is enough to prove them on elementsof the form x (cid:12) m ⊗ y (cid:12) n , with x and y of degree zero. For this, we compute (cid:98) h (cid:16)(cid:98) h ( x (cid:12) m ) (cid:12) (cid:98) h ( y (cid:12) n ) (cid:17) = (cid:88) i + j = m − (cid:88) p + q = n − (cid:98) h (cid:16) h ( x ) (cid:12) h ( y ) (cid:12) τ σ ( x ) (cid:12) i (cid:12) τ σ ( y ) (cid:12) p (cid:12) x (cid:12) j (cid:12) y (cid:12) q (cid:17) = 0 , (cid:98) h (cid:16)(cid:98) h ( x (cid:12) m ) (cid:12) τ ( y ) (cid:12) n (cid:17) = (cid:88) i + j = m − (cid:98) h (cid:16) h ( x ) (cid:12) τ σ ( x ) (cid:12) i (cid:12) x (cid:12) j (cid:12) τ ( y ) (cid:12) n (cid:17) = 0 , once again, using the identities h = hτ = 0 and the fact that h ( x ) (cid:12) = h ( y ) (cid:12) = 0 by degreereasons. (cid:3) We are ready to revisit the homotopy transfer Theorem 1.3 for L ∞ [1] algebras, providing anew proof based on the results from this section. As presented here, the argument might seema bit circular, as we used Theorem 1.3 in the proof of Proposition 1.27: this issue is addressedin the following Remark 1.46. For our purposes, this will serve more as a preparation for thesimilar proofs of the homotopy transfer theorems for commutative BV ∞ and IBL ∞ algebras(Theorems 2.14 and 3.6 respectively). We point out that a simpler proof of homotopy transfer for (curved) L ∞ [1] algebras can be given in the spirit of this paper, but avoiding the use ofProposition 1.27 by replacing it with a much simpler argument: this will appear elsewhere [8].Let ( g, f, h ) be a contraction of the complex ( U, d U ) onto the one ( V, d V ), and let Q be an L ∞ [1] structure on U with linear part q = d U . There is an induced contraction ( S ( g ) , S ( f ) , (cid:98) h )of ( S ( U ) , (cid:102) d U ) onto ( S ( V ) , (cid:102) d V ) as in the previous lemma. Putting Q + = Q − (cid:102) d U , and regardingit as a perturbation of the differential (cid:102) d U on S ( U ), by the Standard Perturbation Lemma 1.29there is an induced perturbation R + = (cid:80) k ≥ S ( σ ) Q + (cid:0)(cid:98) hQ + (cid:1) k S ( τ ) of the differential (cid:102) d V on S ( V ),as well as a perturbed contraction ( G, F, H ) of ( S ( U ) , Q ) onto ( S ( V ) , R := (cid:102) d V + R + ). Remark 1.44.
By the previous Lemma 1.43, we know that ( S ( g ) , S ( f ) , (cid:98) h ) is both a semifullDG coalgebra contraction and a semifull DG algebra contraction of ( S ( U ) , (cid:102) d U ) onto ( S ( V ) , (cid:102) d V ),and since the perturbation Q + is a coderivation of S ( U ), according to Lemmas 1.30 and 1.39we know that ( G, F, H ) is both a semifull DG coalgebra contraction and a semifull algebracontraction (not a semifull DG algebra contraction, though). For future reference, we also pointout that since both (cid:98) h and Q + preserve the subspaces S ≤ n ( U ) := ⊕ ni =0 U (cid:12) i , so does the perturbedhomotopy H = (cid:80) k ≥ ( (cid:98) hQ + ) k (cid:98) h . Finally, we notice that since the subspace U ⊂ S ( U ) is in thekernel of the perturbation Q + , we have H ( x ) = (cid:80) k ≥ ( (cid:98) hQ + ) k (cid:98) h ( x ) = h ( x ) ∈ U ⊂ S ( U ), ∀ x ∈ U . Theorem 1.45.
In the previous hypotheses, R is an L ∞ [1] algebra structure on V , and G, F are L ∞ [1] morphisms.Proof. As observed in the previous remark, (
G, F, H ) is both a semifull DG coalgebra contractionand a semifull algebra contraction. In particular, R is a coderivation and G : S ( U ) → S ( V ) isa morphism of graded coalgebras: in fact, the same computations as in the proof of Proposition1.40 show that κ co ( G ) = ( G (cid:12) G ) K co ( Q ) H = 0, hence G is a morphism of graded coalgebras,and K co ( R ) = ( G (cid:12) G ) K co ( Q ) F = 0, hence R is a coderivation.It remains to show that F : S ( V ) → S ( U ) is a morphism of graded coalgebras: this followsfrom Proposition 1.27 and the above Lemmas 1.41 and 1.42. More precisely, starting with κ ( F ) ( x ) = F ( x ) = (cid:88) k ≥ ( (cid:98) hQ + ) k S ( f ) ( x ) = (cid:88) k ≥ ( (cid:98) hQ + ) k f ( x ) = f ( x ) =: f ( x ) ∈ U ⊂ S ( U ) , using Proposition 1.27, Lemmas 1.41, 1.42 and induction on n we see that κ ( F ) n ( x , . . . , x n ) == (cid:88) k ≥ ,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K k ! H K ( Q + ) k ( κ ( F ) i ( x σ (1) , . . . ) , . . . , κ ( F ) i k ( . . . , x σ ( n ) )) == (cid:88) k ≥ ,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K k ! hq k ( f i ( x σ (1) , . . . ) , . . . , f i k ( . . . , x σ ( n ) )) ==: f n ( x , . . . , x n ) ∈ U ⊂ S ( U ) . Using Lemma 1.41 we can conclude that F is the morphism of graded coalgebras, and we alsorecovered the usual recursion for its Taylor coefficients f , . . . , f n , . . . . (cid:3) Remark 1.46.
As the previous proof depends on Proposition 1.27, and in the claim of thelatter we invoke the homotopy transfer Theorem 1.3, the whole argument might seem circular.A more careful look at the proofs shows that this is not really the case, as the only thing weactually need from Proposition 1.27 are the recursions stated at the beginning of its proof, and
UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 25 the rest of said proof is just a (rather long) direct computation to prove these recursions, anddoesn’t actually depend on Theorem 1.3.
Remark 1.47.
One might prove the homotopy transfer Theorem for L ∞ [ −
1] coalgebras alongthe same lines. In fact, given an complexes (
U, d U ) and ( V, d V ) together with a contraction( σ, τ, h ) of U onto V , there is an induced contraction of (cid:98) S ( U ) onto (cid:98) S ( V ) as in Lemma 1.43. Givenan L ∞ [ −
1] coalgebra structure on U , i.e., a DG algebra structure on (cid:98) S ( U ), we might regard it asa perturbation of the differential induced by d U , and thus via the standard perturbation Lemmathere are induced a perturbed differential on (cid:98) S ( V ) and a perturbed contraction ( F, G, H ) of (cid:98) S ( U )onto (cid:98) S ( V ). Since the perturbation was an algebra derivation, it follows at once from Lemmas1.43 and 1.30 (see also Remark 1.26) that the perturbed differential is an L ∞ [ −
1] coalgebrastructure on V , and that F : (cid:98) S ( V ) → (cid:98) S ( U ) is a morphism of L ∞ [ −
1] coalgebras. The only thingthat requires a little more work is to show that G : (cid:98) S ( U ) → (cid:98) S ( V ) is also a morphism of L ∞ [ − Derived BV algebras Commutative BV ∞ algebras were introduced by O. Kravchenko [46] and have been applied inseveral contexts, such as deformation quantization, quantum field theory and Poisson geometry,just to name a few, see for instance [36, 20, 23, 24, 16, 65, 9, 18, 61, 48, 70] (in the references[61, 48] they are called binary QFT algebras , but in fact the two notions seem to be equivalent).As explained in the introduction, these are not homotopy BV algebras in the full operadic sense[30], hence the name might be misleading. In the rest of the paper we shall call derived BValgebras the commutative BV ∞ algebras in the sense of Kravchenko, following a terminologyintroduced in [7].2.1. Derived BV algebras.Definition 2.1. Let A be a graded commutative algebra and k ∈ Z be an odd integer. Let t be a central variable of (even) degree 1 − k : we denote by A [[ t ]] the corresponding algebra offormal power series. A degree k derived BV algebra structure on A is the datum of a degree one K [[ t ]]-linear map ∆ = (cid:80) n ≥ t n ∆ n ∈ End u, K [[ t ]] ( A [[ t ]]) = K [[ t ]] ⊗ End u ( A ), | ∆ n | = 1 + n ( k − • ∆ = [∆ , ∆] = 0, which is equivalent to (cid:80) ni =0 [∆ i , ∆ n − i ] = 0 for all n ≥
0; and • K (∆) n ( a , . . . , a n ) ≡ t n − ) for all n ≥ a , . . . , a n ∈ A . Remark 2.2.
It is easy to check that the last requirement is equivalent to ∆ n ∈ Diff u, ≤ n +1 ( A )for all n ≥
0. For instance, K (∆) ( a, b ) = K (∆ ) ( a, b ) + t K (∆ ) ( a, b ) + · · · ≡ K (∆ ) ( a, b )(mod t ), thus K (∆) ( a, b ) ≡ t ) ⇔ K (∆ ) ( a, b ) = 0 ⇔ ∆ ∈ Diff u, ≤ ( A ) = Der( A ) . In general, if we assume that ∆ i ∈ Diff u, ≤ ( i +1) ( A ) for all i < n , this implies in particular that K (∆ i ) n +2 = 0 for all i < n , and thus that K (∆) n +2 ( a , . . . , a n +2 ) ≡ t n K (∆ n ) n +2 ( a , . . . , a n +2 ) (mod t n +1 ) . for all a , . . . , a n +2 ∈ A . Hence, K (∆) n +2 ≡ t n +1 ) ⇔ K (∆ n ) n +2 = 0 ⇔ ∆ n ∈ Diff u, ≤ ( n +1) ( A ) . Remark 2.3.
By definition, given a degree k derived BV algebra ( A, ∆), we have K (∆) n ( a , . . . , a n ) ≡ t n − K (∆ n − ) n ( a , . . . , a n ) (mod t n ) . According to Lemma 1.20, this shows that0 = 12 K ([∆ , ∆]) n ( a , . . . , a n )= 12 [ K (∆) , K (∆)] n ( a , . . . , a n )= n (cid:88) i =1 (cid:88) σ ∈ S ( i,n − i ) ± K K (∆) n − i +1 ( K (∆) i ( a σ (1) , . . . , a σ ( i ) ) , a σ ( i +1) , . . . , a σ ( n ) ) ≡ n (cid:88) i =1 (cid:88) σ ∈ S ( i,n − i ) ± K t n − K (∆ n − i ) n − i +1 ( K (∆ i − ) i ( a σ (1) , . . . , a σ ( i ) ) , a σ ( i +1) , . . . , a σ ( n ) ) (mod t n ) , which is equivalent to say that the degree 1 + ( n − k −
1) maps K (∆ n − ) n : A (cid:12) n → A , n ≥ L ∞ [1] algebra on A [1 − k ]. Here we denote by ± K the Koszul sign associatedto the permutation a , . . . , a n (cid:55)→ a σ (1) , . . . , a σ ( n ) : notice that, since k is supposed to be odd, thisis the same whether we consider a , . . . , a n as elements of A or of A [1 − k ].Furthermore, since by hypothesis K (∆ n − ) n +1 = 0, the recursive formula (12) for the Koszulbrackets shows that K (∆ n − ) n ( a , . . . , a n − , bc ) = K (∆ n − ) n ( a , . . . , a n − , b ) c + ( − | b || c | K (∆ n − ) n ( a , . . . , a n − , c ) b for all n ≥ , a , . . . , a n − , b, c ∈ A , i.e., the map K (∆ n − ) n is a multiderivation for all n ≥ K (∆ n − ) n , n ≥
1, defines astructure of degree k derived Poisson algebra on A . Remark 2.4.
We shall denote by BV [ k ] ( A ) ⊂ End u, K [[ t ]] ( A [[ t ]]) ∼ = End u ( A )[[ t ]] the gradedsubspace spanned by those ∆ = (cid:80) n ≥ t n ∆ n such that ∆ n ∈ Diff u, ≤ ( n +1) ( A ) , ∀ n ≥
1. It followsimmediately from the fact that (cid:2)
Diff u, ≤ j ( A ) , Diff u, ≤ k ( A ) (cid:3) ⊂ Diff u, ≤ ( k + j − ( A ) that BV [ k ] ( A ) isa graded Lie subalgebra of End u, K [[ t ]] ( A [[ t ]]). The degree k derived BV algebra structures on A are precisely the Maurer-Cartan elements of BV [ k ] ( A ), that is, the solutions ∆ ∈ BV [ k ] ( A ) ofthe equation 12 [∆ , ∆] = 0 . Given ∆ = (cid:80) n ≥ t n ∆ n ∈ BV [ k ] ( A ), we denote by P (∆) ∈ Coder( S ( A [1 − k ])) the coderivationgiven in Taylor coefficients by p P (∆) = (0 , K (∆ ) , . . . , K (∆ n − ) n , . . . ) . As in Remark 2.3, we notice that the Taylor coefficients K (∆ n − ) n of P (∆) are multiderivations.Moreover, denoting by P [ k ] ( A ) ⊂ Coder( S ( A [1 − k ])) the subspace spanned by those coderiva-tions whose Taylor coefficients are multiderivations (and vanishing on S ( A [1 − k ]) ), it is easy tocheck that P [ k ] ( A ) is closed under the commutator brackets, hence a graded Lie subalgebra ofCoder( S ( A [1 − k ])). Finally, the same kind of computations as in Remark 2.3 show more ingeneral the following fact. Proposition 2.5.
The correspondence P : BV [ k ] ( A ) → P [ k ] ( A ) : ∆ (cid:55)→ P (∆) is a morphism of graded Lie algebras. We conclude this subsection by recalling the definition of Maurer-Cartan elements of a derivedBV-algebra ( A, ∆). For this, we assume that A is equipped with a complete filtration F • A whichis compatible with both the multiplicative structure and the BV operator ∆. We call ( F • A, ∆)a complete derived BV algebra . UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 27
Definition 2.6.
Given a complete degree k derived BV algebra ( A, ∆), we say that a degreezero element a = (cid:88) i ≥− t i a i = a − t + a + ta + · · · ∈ t A [[ t ]] , a i ∈ A i ( k − , is a Maurer-Cartan element of A (cf. [20]) if∆( e a ) = 0 , or equivalently (see (13)) if (cid:88) n ≥ n ! K (∆) n ( a, . . . , a ) = 0 . Here we consider the algebras K (( t )) = ∪ n ∈ Z t n K [[ t ]], A (( t )) = ∪ n ∈ Z t n A [[ t ]] of formal Laurentseries, and we continue to denote by ∆ : A (( t )) → A (( t )) and K (∆) n : A (( t )) (cid:12) n → A (( t )) the K (( t ))-linear extensions of ∆ and its Koszul brackets.Notice that since K (∆) n ≡ t n − K (∆ n − ) n (mod t n ) the left hand side of the last identitybecomes 1 t (cid:88) n ≥ n ! K (∆ n − ) n ( a − , . . . , a − ) + terms in A [[ t ]] . In other words, a − ∈ A − k has to be a Maurer-Cartan element of the associated derived Poissonalgebra ( A, P (∆)).2.2. Morphisms of derived BV algebras.Definition 2.7. Given a pair of degree k derived BV algebras ( A, ∆) , ( B, ∆ (cid:48) ), a morphismbetween them is a degree zero map f ∈ Hom u, K [[ t ]] ( A [[ t ]] , B [[ t ]]) such that • f ◦ ∆ = ∆ (cid:48) ◦ f ; • κ ( f ) n ( a , . . . , a n ) ≡ t n − ) for all n ≥ a , . . . , a n ∈ A .Our first aim is to provide some justification for the previous definition, which, to the best ofour knowledge, hasn’t appeared before in the literature . Of course, the first thing we need toshow is the following proposition, saying that with the above definition of morphisms, degree k derived BV algebras form indeed a category. Proposition 2.8.
Given deree k derived BV algebras ( A, ∆) , ( B, ∆ (cid:48) ) and ( C, ∆ (cid:48)(cid:48) ) , together withmorphisms f : A [[ t ]] → B [[ t ]] and g : B [[ t ]] → C [[ t ]] of derived BV algebras, the composition gf : A [[ t ]] → C [[ t ]] is a morphism of derived BV algebras.Proof. This follows from Proposition 1.16, which shows κ ( gf ) n ( a , . . . , a n ) = (cid:88) k,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) k ! ± K κ ( g ) k (cid:0) κ ( f ) i ( a σ (1) , . . . ) , . . . , κ ( f ) i k ( . . . , a σ ( n ) ) (cid:1) for all a , . . . , a n ∈ A , where as usual we denote by ± K the Koszul sign associated with a , . . . , a n (cid:55)→ a σ (1) , . . . , a σ ( n ) .The above formula shows immediately that if κ ( f ) i j ( a (cid:48) , . . . , a (cid:48) i j ) ≡ t i j − ) and κ ( g ) k ( b , . . . , b k ) ≡ t k − )for all 1 ≤ j ≤ k , a (cid:48) , . . . , a (cid:48) i j ∈ A , b , . . . , b k ∈ B , then κ ( gf ) n ( a , . . . , a n ) ∼ = 0 (mod t n − ) for all a , . . . , a n ∈ A . (cid:3) Actually, after a first draft of this paper was ready, we learned that the same definition has been considered(with a different terminology) by J.-S. Park [61].
A more convincing justification comes from the following proposition (analogous to [7, Propo-sition 2.11]), which shows that (formally) the exponential group of the Lie algebra BV [ k ] ( A ) from the previous Remark 2.4 identifies with the group of K [[ t ]]-module automorphisms of A [[ t ]]satisfying the second condition from the previous Definition 2.7 (and preserving the unit A [[ t ]] ).In the framework of deformation theory (see [51]), given a derived BV algebra ( A, ∆), thisshows that the DG Lie algebra (cid:0) BV [ k ] ( A ) , [∆ , − ] , [ − , − ] (cid:1) controls the deformations of ( A, ∆) inthe category of derived BV algebras.Given ∆ ∈ End u, K [[ t ]] ( A [[ t ]]), we consider the associated formal flow exp( s ∆): to avoid con-vergence issues, we consider the parameter s as a central variable of degree 0, and the formalflow exp( s ∆) ∈ Hom u, K [[ s,t ]] ( A [[ s, t ]] , A [[ s, t ]]) as a K [[ s, t ]]-linear endomorphism of the algebraof formal power series A [[ s, t ]]. Proposition 2.9.
Given ∆ ∈ End u, K [[ t ]] ( A [[ t ]]) , we have κ (exp( s ∆)) n ( a , . . . , a n ) ≡ t n − ) for all n ≥ and a , . . . , a n ∈ A , if and only if K (∆) n ( a , . . . , a n ) ≡ t n − ) for all n ≥ and a , . . . , a n ∈ A , that is, if and only if ∆ ∈ BV [ k ] ( A ) .Proof. To simplify notations, we denote by κ ( s ) := κ (exp( s ∆)) = exp( s K (∆)), where the secondidentity follows from Proposition 1.22. Denoting as usual by p : S ( A [[ t ]]) → A [[ t ]] the naturalprojection, we see that(29) ∂ s (cid:16) κ ( s ) n ( a , . . . , a n ) (cid:17) = p (cid:16) ∂ s κ (exp( s ∆)) (cid:17) ( a (cid:12) · · · (cid:12) a n ) == p (cid:16) ∂ s exp( s K (∆)) (cid:17) ( a (cid:12) · · · (cid:12) a n ) = p (cid:16) K (∆) ◦ κ ( s ) (cid:17) ( a (cid:12) · · · (cid:12) a n ) == (cid:88) k,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K k ! K (∆) k (cid:0) κ ( s ) i ( a σ (1) , . . . ) , . . . , κ ( s ) i k ( . . . , a σ ( n ) ) (cid:1) . First we assume that κ ( s ) n ≡ t n − ), ∀ n ≥
2, and show that K (∆) n ≡ t n − ), ∀ n ≥
2, by induction on n . Suppose inductively that we have shown K (∆) i ≡ t i − ) forall i < n . Using the above Equation 29 together with the inductive hypothesis we see that0 ≡ ∂ s (cid:0) κ ( s ) n ( a , . . . , a n ) (cid:1) (mod t n − ) ≡ K (∆) n (cid:0) κ ( s ) ( a ) , . . . , κ ( s ) ( a n ) (cid:1) (mod t n − ) , for all a , . . . , a n ∈ A . Since κ ( s ) = exp( s ∆) is an isomorphism, this shows that K (∆) n ≡ t n − ), and completes the inductive step.Next we assume K (∆) n ≡ t n − ), ∀ n ≥
2, and show that κ ( s ) n ≡ t n − ), ∀ n ≥
2, once again by induction on n . Suppose inductively that κ ( s ) i ≡ t i − ) for all i < n : together with Equation (29), this implies that ∂ s (cid:0) κ ( s ) n ( a , . . . , a n ) (cid:1) ≡ K (∆) (cid:0) κ ( s ) n ( a , . . . , a n ) (cid:1) (mod t n − ) ≡ ∆ (cid:0) κ ( s ) n ( a , . . . , a n ) (cid:1) (mod t n − ) . Thus ( ∂ s ) k (cid:0) κ ( s ) n ( a , . . . , a n ) (cid:1) ≡ ∆ k (cid:0) κ ( s ) n ( a , . . . , a n ) (cid:1) (mod t n − ). Expanding κ ( s ) n ( a , . . . , a n )in formal Taylor series with respect to s , and noticing that κ (0) n = 0 for all n ≥
2, we concludethat κ ( s ) n ( a , . . . , a n ) ≡ (cid:88) k ≥ s k k ! ∆ k ( κ (0) n ( a , . . . , a n )) (mod t n − ) ≡ t n − ) . (cid:3) UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 29
In the following proposition, we promote the correspondence between derived BV algebrasand derived Poisson algebras explained in the previous Remark 2.3 to a full-fledged functor(morphisms of derived Poisson algebras were introduced in [7]). Proposition 2.10.
Given a pair of degree k derived BV algebras ( A, ∆) , ( B, ∆ (cid:48) ) and a morphism f : A [[ t ]] → B [[ t ]] between them, the maps P ( f ) n : A (cid:12) n → B , n ≥ , defined by the identities κ ( f ) n ( a , . . . , a n ) ≡ t n − P ( f ) n ( a , . . . , a n ) (mod t n ) for all a , . . . , a n ∈ A , induce a morphism P ( f ) : ( S ( A [1 − k ]) , P (∆)) → ( S ( B [1 − k ]) , P (∆ (cid:48) )) between the associated degree k derived Poisson algebras A and B .Proof. Denoting by p : S ( B [[ t ]]) → B [[ t ]] the natural projection, we have p (cid:16) κ ( f ) ◦ K (∆) (cid:17) ( a , . . . , a n ) == n (cid:88) i =1 (cid:88) σ ∈ S ( i,n − i ) ± K κ ( f ) n − i +1 (cid:16) K (∆) i ( a σ (1) , . . . ) , . . . , a σ ( n ) (cid:17) ≡≡ n (cid:88) i =1 (cid:88) σ ∈ S ( i,n − i ) ± K t n − P ( f ) n − i +1 (cid:16) P (∆) i ( a σ (1) , . . . ) , . . . , a σ ( n ) (cid:17) (mod t n ) , where the last congruence follows directly from the definitions of P ( f ) and P (∆). Also, p (cid:16) K (∆ (cid:48) ) ◦ κ ( f ) (cid:17) ( a , . . . , a n ) == (cid:88) k,i ,...,ik ≥ i + ··· + i k = n k ! (cid:88) σ ∈ S ( i ,,...,i k ) ± K K (∆ (cid:48) ) k (cid:16) κ ( f ) i ( a σ (1) , . . . ) , . . . , κ ( f ) i k ( . . . , a σ ( n ) ) (cid:17) ≡≡ (cid:88) k,i ,...,ik ≥ i + ··· + i k = n k ! (cid:88) σ ∈ S ( i ,,...,i k ) ± K t n − P (∆ (cid:48) ) k (cid:16) P ( f ) i ( a σ (1) , . . . ) , . . . , P ( f ) i k ( . . . , a σ ( n ) ) (cid:17) (mod t n ) , Since by hypothesis f ◦ ∆ = ∆ (cid:48) ◦ f , using Proposition 1.21 we have κ ( f ) ◦ K (∆) = K (∆ (cid:48) ) ◦ κ ( f ).By the above, this shows that n (cid:88) i =1 (cid:88) σ ∈ S ( i,n − i ) ± K P ( f ) n − i +1 (cid:16) P (∆) i ( a σ (1) , . . . ) , . . . , a σ ( n ) (cid:17) == (cid:88) k,i ,...,ik ≥ i + ··· + i k = n k ! (cid:88) σ ∈ S ( i ,,...,i k ) ± K P (∆ (cid:48) ) k (cid:16) P ( f ) i ( a σ (1) , . . . ) , . . . , P ( f ) i k ( . . . , a σ ( n ) ) (cid:17) for all n ≥ a , . . . , a n ∈ A , i.e., P ( f ) : ( S ( A [1 − k ]) , P (∆)) → ( S ( B [1 − k ]) , P (∆ (cid:48) )) is an L ∞ [1] morphism.Furthermore, using the recursion (8) for the cumulants, we have0 ≡ κ ( f ) n +2 ( . . . , b, c ) (mod t n +1 ) ≡ t n P ( f ) n +1 ( . . . , bc ) − n (cid:88) i =0 (cid:88) σ ∈ S ( i,n − i ) ± K P ( f ) i +1 ( . . . , b ) P ( f ) n − i +1 ( . . . , c ) (mod t n +1 ) , that is, P ( f ) n +1 ( a , . . . , a n , bc ) = n (cid:88) i =0 (cid:88) σ ∈ S ( i,n − i ) ± K P ( f ) i +1 ( a σ (1) , . . . , a σ ( i ) , b ) P ( f ) n − i +1 ( a σ ( i +1) , . . . , a σ ( n ) , c )for all n ≥ a , . . . , a n , b, c ∈ A . This is precisely the additional requirement the Taylorcoefficients P ( f ) n have to satisfy in order for P ( f ) to define a morphism of derived Poissonalgebras, see [7, § (cid:3) Remark 2.11.
Putting f = (cid:80) n ≥ t n f n , where f n ∈ Hom n ( k − ( A, B ), f ( A ) = B , f n ( A ) = 0for n ≥
1, we can write down the conditions for f to be a morphism of derived BV algebrasmore explicitly as follows. On the one hand, the condition f ◦ ∆ = ∆ (cid:48) ◦ f translates into n (cid:88) i =0 f i ◦ ∆ n − i = n (cid:88) i =0 ∆ (cid:48) i ◦ f n − i , ∀ n ≥ . On the other hand, the second condition in Definition 2.7 translates into a hierarchy of identities,the first three being f ( ab ) = f ( a ) f ( b ) ,f ( abc ) = f ( ab ) f ( c ) + f ( ac ) f ( b ) + f ( bc ) f ( a ) − f ( a ) f ( b ) f ( c ) − f ( b ) f ( a ) f ( c ) − f ( c ) f ( a ) f ( b ) ,f ( abcd ) = f ( abc ) f ( d ) + f ( abd ) f ( c ) + f ( acd ) f ( b ) + f ( bcd ) f ( a )+ f ( ab ) f ( cd ) + f ( ac ) f ( bd ) + f ( ad ) f ( bc ) − f ( ab ) f ( c ) f ( d ) − f ( ac ) f ( b ) f ( d ) − f ( ad ) f ( b ) f ( c ) − f ( bc ) f ( a ) f ( d ) − f ( bd ) f ( a ) f ( c ) − f ( cd ) f ( a ) f ( b ) − f ( ab ) f ( c ) f ( d ) − f ( ab ) f ( d ) f ( c ) − f ( ac ) f ( b ) f ( d ) − f ( ac ) f ( d ) f ( b ) − f ( ad ) f ( b ) f ( c ) − f ( ad ) f ( c ) f ( b ) − f ( bc ) f ( a ) f ( d ) − f ( bc ) f ( d ) f ( a ) − f ( bd ) f ( a ) f ( c ) − f ( bd ) f ( c ) f ( a ) − f ( cd ) f ( a ) f ( b ) − f ( cd ) f ( b ) f ( a )+ f ( a ) f ( b ) f ( c ) f ( d ) + f ( b ) f ( a ) f ( c ) f ( d ) + f ( c ) f ( b ) f ( a ) f ( d ) + f ( d ) f ( b ) f ( c ) f ( a )+ f ( a ) f ( b ) f ( c ) f ( d ) + f ( a ) f ( c ) f ( b ) f ( d ) + f ( a ) f ( d ) f ( b ) f ( c )+ f ( b ) f ( c ) f ( a ) f ( d ) + f ( b ) f ( d ) f ( a ) f ( c ) + f ( c ) f ( d ) f ( a ) f ( b ) , · · · for all a, b, c, d ∈ A . Notice that κ ( f ) n +1 ≡ t n ) automatically implies κ ( f ) N ≡ t n ) for all N > n (by the recursion (8) and a straightforward induction), thus, given thefirst ( n −
1) identities, the condition κ ( f ) n +2 ≡ t n +1 ) translates into a single additionalidentity on f , . . . , f n . In fact, we are imposing that the coefficient of t i in the expansion of κ ( f ) n +2 ( a , . . . , a n +2 ) ∈ A [[ t ]] vanishes for all i ≤ n and a , . . . , a n +2 ∈ A : but for i < n thisalready follows from the first ( n −
1) identities, thus we only need to look at the coefficient of t n . Essentially, the n -th identity shows how to express f n ( a · · · a n +2 ) as a linear combination ofproducts of the form f i ( a σ (1) · · · a σ ( k ) ) · · · f i j ( a σ ( n − k j +3) · · · a σ ( n +2) ) with j ≥ i + · · · + i j = n , i ≥ · · · ≥ i j ≥ k + · · · + k j = n +2 , 1 ≤ k ≤ i +1 , . . . , ≤ k j ≤ i j +1, and σ ∈ S ( k , . . . , k j ).Another important property is that Maurer-Cartan elements can be pushed forward alongmorphism of derived BV algebras. Proposition 2.12.
Given a pair of complete derived BV algebras ( A, ∆) , ( B, ∆ (cid:48) ) , a continuousmorphism f : A [[ t ]] → B [[ t ]] between them and a Maurer-Cartan element a = (cid:80) k ≥− t k a k ∈ UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 311 t A [[ t ]] of A , then b := (cid:88) n ≥ n ! κ ( f ) n ( a, . . . , a ) is a Maurer-Cartan element of B (where we continue to denote by κ ( f ) n : A (( t )) (cid:12) n → B (( t )) the K (( t )) -linear extension of the cumulants of f ). Moreover, b − is the push-forward of theMaurer-Cartan element a − along the morphism P ( f ) : ( A, P (∆)) → ( B, P (∆ (cid:48) )) of derivedPoisson algebras from the previous Proposition 2.10.Proof. The Koszul brackets K (∆) n define an L ∞ [1] algebra structures on A (( t )). Denoting byMC BV ( A ) the set of Maurer-Cartan elements of the derived BV algebra A , as in Definition 2.6,and by MC L ∞ [1] ( A (( t ))) the set of Maurer-Cartan elements of the L ∞ [1] algebra A (( t )), we haveby definition MC BV ( A ) = MC L ∞ [1] ( A (( t ))) (cid:84) t A [[ t ]]. Similarly for B .According to Proposition 1.21, the cumulants κ ( f ) n : A (( t )) (cid:12) n → B (( t )) induce a morphismof L ∞ [1] algebras from A (( t )) to B (( t )). In particular, they induces a push-forwardMC( f ) : MC L ∞ [1] ( A (( t ))) → MC L ∞ [1] ( B (( t ))) : x → (cid:88) n ≥ n ! κ ( f ) n ( x, . . . , x )between the sets of Maurer-Cartan elements.Finally, given a Maurer-Cartan element a = (cid:80) i ≥− t i a i ∈ MC BV ( A ) of the derived BV algebra A , since κ ( f ) n ≡ t n − P ( f ) n (mod t n ) (using the notations of the previous Proposition 2.10),we see that MC( f )( a ) = 1 t (cid:88) n ≥ n ! P ( f ) n ( a − , . . . , a − ) + terms in B [[ t ]] . This shows b := MC( f )( a ) ∈ MC L ∞ [1] ( B (( t ))) (cid:84) t B [[ t ]] = MC BV ( B ), as well as the last claim. (cid:3) In [20] the authors introduced and studied morphisms between derived BV algebras, but onlywhen the source derived BV algebra is free as a graded commutative algebra. We close thissubsection by comparing the previous Definition 2.7 of morphism with the one from [20], in thesituations when the latter applies. We shall see that the two definitions are equivalent: moreprecisely, there is a bijective correspondence (given by the exponential and logarithm in theconvolution algebra, see below) between morphism of derived BV algebras in our sense and inthe sense of [20].Given a graded vector space U and a unitary graded commutative algebra B , the gradedcocommutative coalgebra structure on S ( U ) given by the unshuffle coproduct induces a unitarygraded commutative algebra structure on Hom( S ( U ) , B [[ t ]]), via the convolution product (cid:63) , cf.Subsection 1.5.Given derived BV algebra structures ∆ , ∆ (cid:48) on S ( U ) and B respectively, then, according tothe definition given in [20], a map ϕ ∈ Hom ( S ( U ) , B [[ t ]]) such that ϕ ( S ( U ) ) = 0 defines amorphism of derived BV algebras if the following conditions are satisfied: • exp (cid:63) ( ϕ ) ◦ ∆ = ∆ (cid:48) ◦ exp (cid:63) ( ϕ ), where we continue to denote by exp (cid:63) ( ϕ ) : S ( U )[[ t ]] → B [[ t ]]the K [[ t ]]-linear extension of exp (cid:63) ( ϕ ); • writing ϕ = (cid:80) n ≥ t n ϕ n , then ϕ n vanishes on S >n +1 ( U ) = ⊕ i ≥ n +2 U (cid:12) n for all n ≥ Proposition 2.13. . In the previous situation, ϕ defines a morphism of derived BV algebrasaccording to the definition from [20] if and only if exp (cid:63) ( ϕ ) defines a morphism of derived BV algebras according to Definition 2.7. Proof.
As already observed in the proof of Lemma 1.41, given f ∈ Hom u ( S ( U ) , B [[ t ]]), we have(30) log (cid:63) ( f )( x (cid:12) . . . (cid:12) x n ) = κ ( f ) n ( x , . . . , x n )for all n ≥ x , . . . , x n ∈ U ⊂ S ( U ), as can be seen via a direct computation using theformula (7) for the cumulants. In other words, denoting by i : U → S ( U ) and p : S ( B ) → B the natural inclusion and projection respectively (as well as their K [[ t ]]-linear extension), thefollowing diagram is commutative(31) S ( S ( U )) κ ( f ) (cid:47) (cid:47) S ( B )[[ t ]] p (cid:15) (cid:15) S ( U ) S ( i ) (cid:79) (cid:79) log (cid:63) ( f ) (cid:47) (cid:47) B [[ t ]]If exp (cid:63) ( ϕ ) is a morphism of derived BV algeras in the sense of Definition 2.7, then the aboveidentity (30) shows that ϕ ( x (cid:12) · · · (cid:12) x n ) = log (cid:63) (exp (cid:63) ( ϕ ))( x (cid:12) · · · (cid:12) x n ) = κ (exp (cid:63) ( ϕ )) n ( x , . . . , x n ) ≡ t n − ) , hence that ϕ i ( x (cid:12) · · · (cid:12) x n ) = 0 for i < n −
1, as desired.Conversely, we want to show that if ϕ is a morphism in the sense of [20], that is, it satisfies thetwo conditions stated before the proposition, then exp (cid:63) ( ϕ ) is a morphism in our sense. Since thefirst condition from Definition 2.7 is satisfied by hypothesis, we only have to show the secondcondition, that is,(32) κ (exp (cid:63) ( ϕ )) n ( X , . . . , X n ) ≡ t n − ) , ∀ n ≥ , X ∈ U (cid:12) i , . . . , X n ∈ U (cid:12) i n . The recursion (8) and a straightforward induction show that for any pair of graded commutativealgebras
A, B and f ∈ Hom u ( A, B ), we have κ ( f ) n ( a , . . . , a n ) = 0 whenever a i = A for some1 ≤ i ≤ n . In particular, we only have to show (32) for i , . . . , i n ≥
1: we shall proceed byinduction on i + · · · + i n − n . If this quantity is zero then i = · · · = i n = 1, and the samereasoning as in the first part of the proof (using (30) and the hypotheses on ϕ ) shows thedesired result. Otherwise, it is not restrictive to assume i n > X n = X (cid:48) n (cid:12) X (cid:48)(cid:48) n for some X (cid:48) n ∈ U (cid:12) j , X (cid:48)(cid:48) n ∈ U (cid:12) j , j , j ≥ , j + j = i n . Using therecursion (8) for the cumulants,(33) κ (exp (cid:63) ( ϕ )) n ( X , . . . , X n ) = κ (exp (cid:63) ( ϕ )) n +1 ( X , . . . , X (cid:48) n , X (cid:48)(cid:48) n )+ n − (cid:88) k =0 (cid:88) σ ∈ S ( k,n − k − ± K κ (exp (cid:63) ( ϕ )) k +1 ( X σ (1) , . . . , X σ ( k ) , X (cid:48) n ) κ (exp (cid:63) ( ϕ )) n − k ( X σ ( k +1) , . . . , X σ ( n − , X (cid:48)(cid:48) n ) . Since (cid:16) i σ (1) + · · · + i σ ( k ) + j − k − (cid:17) + (cid:16) i σ ( k +1) + · · · + i σ ( n − + j − n + k (cid:17) == i + · · · + i n − + j + j − n − i + · · · + i n − n − < i + · · · + i n − n, we can apply the inductive hypothesis to the right hand side of the previous identity (33) todeduce that it is congruent to zero modulo t n − , as desired. (cid:3) Homotopy transfer for derived BV algebras. Our aim in this subsection is to showthat derived BV algebra structures can be transferred along semifull DG algebra contractions(Definition 1.25). An analogous homotopy transfer theorem for derived Poisson algebras wasproved in [7, Theorem 2.18].Given a pair of DG commutative algebras ( A, d A ) and ( B, d B ) and a contraction ( σ, τ, h ) of A onto B , by abuse of notations we shall continue to denote by ( σ, τ, h ) the induced contraction UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 33 (id K [[ t ]] ⊗ σ, id K [[ t ]] ⊗ τ, id K [[ t ]] ⊗ h ) of ( A [[ t ]] , id K [[ t ]] ⊗ d A ) onto ( B [[ t ]] , id K [[ t ]] ⊗ d B ). Given a degree k derived BV algebra structure ∆ = (cid:80) n ≥ t n ∆ n on A with ∆ = d A , we write ∆ = d A + ∆ + ,where ∆ + = (cid:80) n ≥ t n ∆ n . Considering ∆ + as a perturbation of the differential d A on A [[ t ]], weare in the hypotheses of the Standard Perturbation Lemma 1.29, therefore there is induced aperturbation ∆ (cid:48) + := (cid:88) n ≥ σ ∆ + ( h ∆ + ) n τ of the differential id K [[ t ]] ⊗ d B on B [[ t ]], as well as a perturbed contraction˘ σ := (cid:88) n ≥ σ (∆ + h ) n ˘ τ := (cid:88) n ≥ ( h ∆ + ) n τ ˘ h := (cid:88) n ≥ ( h ∆ + ) n h of ( A [[ t ]] , ∆) onto ( B [[ t ]] , ∆ (cid:48) := d B + ∆ (cid:48) + ). Theorem 2.14.
In the above hypotheses, if the contraction ( σ, τ, h ) is a semifull DG algebracontraction, then ∆ (cid:48) is a degree k derived BV algebra structure on B , and ˘ τ is a morphism ofderived BV algebras from ( B, ∆ (cid:48) ) to ( A, ∆) . Furthermore, the induced degree k derived Poissonalgebra structure P (∆ (cid:48) ) on B and the induced morphism P (˘ τ ) : ( B, P (∆ (cid:48) )) → ( A, P (∆)) ofderived Poisson algebras coincide with those induced via homotopy transfer from the derivedPoisson algebra structure P (∆) on A , as in [7, Theorem 2.18] .Proof. The relations (∆ (cid:48) ) = 0, ˘ τ ◦ ∆ (cid:48) = ∆ ◦ ˘ τ , are satisfied by the standard perturbationLemma. If ( σ, τ, h ) is a semifull DG algebra contraction, according to Lemma 1.30 the perturbedcontraction (˘ σ, ˘ τ , ˘ h ) is a semifull algebra contraction of ( A [[ t ]] , ∆) onto ( B [[ t ]] , ∆ (cid:48) ) (not a semifullDG algebra contraction, though). The rest of the theorem follows easily from (the proof of)Proposition 1.27, which shows that the cumulants κ (˘ τ ) n and the Koszul brackets K (∆ (cid:48) ) n obeythe recursions κ (˘ τ ) n ( x , . . . , x n ) = (cid:88) k ≥ ,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K k ! ˘ h K (∆) k (cid:0) κ (˘ τ ) i ( x σ (1) . . . ) , . . . , κ (˘ τ ) i k ( . . . , x σ ( n ) ) (cid:1) , K (∆ (cid:48) ) n ( x , . . . , x n ) = (cid:88) k ≥ ,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K k ! ˘ σ K (∆) k (cid:0) κ (˘ τ ) i ( x σ (1) . . . ) , . . . , κ (˘ τ ) i k ( . . . , x σ ( n ) ) (cid:1) , for all x , . . . , x n ∈ B [[ t ]].The above formulas and a straightforward induction on n , together with the fact that ∆ is aderived BV algebra structure on A , show that κ (˘ τ ) n ≡ t n − ), K (∆ (cid:48) ) n ≡ t n − )for all n ≥
2, that is, ∆ (cid:48) is a derived BV algebra structure on B and ˘ τ is a morphism ofderived BV algebras. In fact, if we assume inductively to have shown that κ (˘ τ ) i ( x , . . . , x i ) ≡ t i − P (˘ τ ) i ( x , . . . , x i ) (mod t i ) (for certain maps P (˘ τ ) i : B (cid:12) i → A ) for all x , . . . , x i ∈ B and i < n , we see that κ (˘ τ ) n ( x , . . . , x n ) ≡≡ (cid:88) k ≥ ,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K t n − k ! h P (∆) k (cid:0) P (˘ τ ) i ( x σ (1) . . . ) , . . . , P (˘ τ ) i k ( . . . , x σ ( n ) ) (cid:1) (mod t n ) K (∆ (cid:48) ) n ( x , . . . , x n ) ≡≡ (cid:88) k ≥ ,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K t n − k ! σ P (∆) k (cid:0) P (˘ τ ) i ( x σ (1) . . . ) , . . . , P (˘ τ ) i k ( . . . , x σ ( n ) ) (cid:1) (mod t n )This proves the inductive step, and thus completes the proof that ∆ (cid:48) is a derived BV algebrastructure and ˘ τ : B [[ t ]] → A [[ t ]] a morphism of derived BV algebras. Furthermore, it shows thatthe Taylor coefficients of P (˘ τ ) and P (∆ (cid:48) ) obey the recursions P (˘ τ ) n ( x , . . . , x n ) = (cid:88) k ≥ ,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K k ! h P (∆) k (cid:0) P (˘ τ ) i ( x σ (1) . . . ) , . . . , P (˘ τ ) i k ( . . . , x σ ( n ) ) (cid:1) , P (∆ (cid:48) ) n ( x , . . . , x n ) = (cid:88) k ≥ ,i ,...,ik ≥ i + ··· + i k = n (cid:88) σ ∈ S ( i ,...,i k ) ± K k ! σ P (∆) k (cid:0) P (˘ τ ) i ( x σ (1) . . . ) , . . . , P (˘ τ ) i k ( . . . , x σ ( n ) ) (cid:1) . In other words, the induced L ∞ [1] algebra structure P (∆ (cid:48) ) on B [1 − k ] and the induced L ∞ [1]morphism P (˘ τ ) : ( S ( B [1 − k ]) , P (∆ (cid:48) )) → ( S ( A [1 − k ]) , P (∆)) coincide with those obtained bytransferring the L ∞ [1] algebra structure P (∆) on A [1 − k ] along the contraction ( σ, τ, h ), as inTheorem 1.3, which is precisely what we needed to show in order to prove the last claim of thetheorem (see [7]). (cid:3) In the same hypotheses as in the previous Theorem 2.14, we assume moreover that A isequipped with a complete filtration making it into a complete derived BV algebra, and that h, τ σ are continuous with respect to this filtration. Then it is easy to check that ( B, ∆ (cid:48) ) is acomplete derived BV algebra with respect to the induced filtration (i.e., the only one making σ and τ continuous) and that ˘ τ : B [[ t ]] → A [[ t ]] is continuous. In this situation we mightconsider the Maurer-Cartan sets MC BV ( A ) := MC L ∞ [1] ( A (( t ))) ∩ t A [[ t ]] and MC BV ( B ) :=MC L ∞ [1] ( B (( t ))) ∩ t B [[ t ]] (cf. the proof of Proposition 2.12 for the notations). Since (˘ σ, ˘ τ , ˘ h ) isa semifull algebra contraction of ( A (( t )) , ∆) onto ( B (( t )) , ∆ (cid:48) ), according to Proposition 1.27 theKoszul brackets K (∆ (cid:48) ) n : B (( t )) (cid:12) n → B (( t )) and the cumulants κ (˘ τ ) n : B (( t )) (cid:12) n → A (( t )) areinduced via homotopy transfer from the Koszul brackets K (∆) n : A (( t )) (cid:12) n → A (( t )) along thecontraction (˘ σ, ˘ τ , ˘ h ), which was the key point in the proof of the previous theorem. Applying theformal Kuranishi Theorem 1.7, together with Proposition 2.12, we obtain the following result. Theorem 2.15.
In the above hypotheses, there is a bijective correspondence MC BV ( B ) ∼ = MC BV ( A ) ∩ Ker(˘ h ) . In one direction it’s given by the push-forward
MC(˘ τ ) : MC BV ( B ) → MC BV ( A ) along thecontinuous morphism of complete derived BV algebras ˘ τ : ( B, ∆ (cid:48) ) → ( A, ∆) ; in the otherdirection, it sends a ∈ MC BV ( A ) ∩ Ker(˘ h ) to the Maurer-Cartan element ˘ σ ( a ) ∈ MC BV ( B ) . Derived BV coalgebras.Definition 2.16. A degree k derived BV coalgebra is a (cocommutative, et cet.) graded coal-gebra C together with a degree one K [[ t ]]-linear (where t is again a central variable of degree1 − k ) endomorphism δ ∈ End cu, K [[ t ]] ( C [[ t ]]) such that δ = 0 and K co ( δ ) n ≡ t n − ) for all n ≥ , where the last condition is equivalent to δ = (cid:80) n ≥ t n δ n and δ n ∈ coDiff cu, ≤ ( n +1) ( C ) for all n ≥ UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 35
Given a pair of degree k derived BV coalgebras ( C, δ ) and (
D, δ (cid:48) ), a morphism between themis a K [[ t ]]-linear morphism f ∈ Hom cu, K [[ t ]] ( C [[ t ]] , D [[ t ]]) such that f ◦ δ = δ (cid:48) ◦ f and κ co ( f ) n ≡ t n − ) for all n ≥ . As in Proposition 2.8, with the above definitions degree k derived BV coalgebras form acategory. Likewise, the various results from the previous subsections admit dual versions in thecontext of derived BV coalgebras.We are particularly interested in the Homotopy Transfer Theorem 2.14 and Proposition 2.13.For the former, given a derived BV coalgebra ( C, δ ) together with a semifull DG coalgebra con-traction ( σ, τ, h ) of (
C, d C := δ ) onto a DG coalgebra ( D, d D ), the perturbation δ + := (cid:80) n ≥ t n δ n of id K [[ t ]] ⊗ d C induces (via the standard Perturbation Lemma 1.29) a perturbed differential δ (cid:48) on D [[ t ]], together with a perturbed contraction (˘ σ, ˘ τ , ˘ h ) of ( C [[ t ]] , δ ) onto ( D [[ t ]] , δ (cid:48) ). Theorem 2.17.
In the above hypotheses, δ (cid:48) is a derived BV coalgebra structure on D , and ˘ σ is a morphism of derived BV coalgebras.Proof. This is shown as in the proof of Theorem 2.14, using Proposition 1.40 in place of Propo-sition 1.27. (cid:3)
Let (
C, δ ) be a derived BV coalgebra, and ( S ( V ) , δ (cid:48) ) a second derived BV coalgebra whoseunderlying graded coalgebra is cofree. The graded cocommutative coalgebra structure on C andthe symmetric algebra structure on S ( V )[[ t ]] induce a graded commutative algebra structure onHom( C, S ( V )[[ t ]]) ∼ = Hom( C, S ( V ))[[ t ]] via the convolution product (cid:63) . We denote the unit inthis algebra by ε : C → S ( V )[[ t ]]: it is the map sending C to S ( V ) and c ∈ C to zero. As insubsection 1.5, given f ∈ Hom cu ( C, S ( V )[[ t ]]) we denote by log (cid:63) ( f ) = (cid:80) n ≥ − n − n ( f − ε ) (cid:63)n .In the following lemma we give necessary and sufficient conditions for f : C [[ t ]] → S ( V )[[ t ]] tobe a morphism of derived BV coalgebras. More precisely, we show that this is equivalent to f ◦ δ = δ (cid:48) ◦ f and item (b) in the claim of the following lemma: this should be compared withProposition 2.13. When the source DG coalgebra C is also cofree, we give further equivalentconditions (items (c) and (d) in the claim of the following lemma), this time in terms of thecumulants κ ( f ) k , k ≥
1. This should be compared with Lemma 1.41, and will play a similarrole in the proof of the homotopy transfer Theorem 3.6 for
IBL ∞ algebras as the latter did inthe proof of Theorem 1.45. Lemma 2.18.
Given derived BV coalgebras ( C, δ ) , ( S ( V ) , δ (cid:48) ) , together with a degree zero map f = (cid:80) n ≥ t n f n ∈ Hom ( C, S ( V )[[ t ]]) such that f ( C ) = S ( V ) , if furthermore (the K [[ t ]] -linearextension of ) f satisfies f ◦ δ = δ (cid:48) ◦ f , then the following conditions are equivalent (where wedenote by ϕ n : C → S ( V ) , n ≥ , the maps defined by log (cid:63) ( f ) =: ϕ = (cid:80) n ≥ t n ϕ n ): (a) f is a morphism of derived BV-coalgebras; (b) Im( ϕ i ) ⊂ S ≤ ( i +1) ( V ) := ⊕ i +1 j =0 V (cid:12) j , ∀ i ≥ .If moreover C = S ( U ) is also cofree, these are further equivalent to any of the following: (c) κ ( f ) k ( U, . . . , U ) ⊂ ⊕ n ≥ t n S ≤ ( n +1) ( V ) for all k ≥ ; (d) κ ( f ) k ( S ≤ i ( U ) , . . . , S ≤ i k ( U )) ⊂ ⊕ n ≥ t n S ≤ ( i + ··· + i k − k + n +1) ( V ) for all k, i , . . . , i k ≥ .Proof. We denote by i : C → S ( C ) and p : S ( V ) → V the natural inclusion and projection(as well as their K [[ t ]]-linear extensions). We notice that ϕ := log (cid:63) ( f ) fits into the following commutative diagram(34) S ( C ) κ co ( f ) (cid:47) (cid:47) S ( S ( V ))[[ t ]] S ( p ) (cid:15) (cid:15) C i (cid:79) (cid:79) log (cid:63) ( f ) (cid:47) (cid:47) S ( V )[[ t ]]which is dual to the one (31), and can be similarly shown by a direct computation. In particular, ϕ ( c ) = log (cid:63) ( f )( c ) = S ( p ) (cid:16) κ co ( f )( c ) (cid:17) . If f is a morphism of derived BV coalgebras, then the above identity shows that the compositionof ϕ with the natural projection S ( V )[[ t ]] (cid:16) V (cid:12) n [[ t ]] vanishes modulo t n − for all n ≥
2, or inother words that the composition C ϕ i −→ S ( V ) (cid:16) V (cid:12) n vanishes whenever i < n −
1, which showsthat (a) implies (b).The converse statement can be shown by dualizing the inductive argument in the proof ofProposition 2.13, using the recursion for the cocumulants explained in Remark 1.32. Moreprecisely, denoting by p i : S ( V ) → V (cid:12) i the natural projection, then by definition f is a morphismof derived BV coalgebras if and only if κ co ( f ) n ≡ t n − ) for all n ≥ p i (cid:12) · · · (cid:12) p i n ) κ co ( f ) ≡ t n − )for all n ≥ i , . . . , i n ≥
1. If (a) holds, then by the above commutative diagram (34)equation (35) is true for i = · · · = i n = 1. To show that is true in general, one uses inductionon i + · · · + i n − n and the recursion for the cocumulants from Remark 1.32. We leave detailsto the reader, cf. also the proof of the following Lemma 3.2.This concludes the proof that (a) is equivalent to (b).Next we assume that C = S ( U ) is cofree.The equivalence between (c) and (d) can be seen by induction on i + · · · + i k − k , using therecursion (8) for the cumulants in a similar way as what we did in order to conclude the proofof Proposition 2.13.To show that (c) is equivalent to (b), we put together the previous commutative diagrams(31) and (34) to conclude that the following one is also commutative(36) S ( S ( U )) κ ( f ) (cid:47) (cid:47) S ( S ( V ))[[ t ]] p (cid:15) (cid:15) S ( U ) S ( i ) (cid:79) (cid:79) i (cid:15) (cid:15) log (cid:63) ( f ) (cid:47) (cid:47) S ( V )[[ t ]] S ( S ( U )) κ co ( f ) (cid:47) (cid:47) S ( S ( V ))[[ t ]] S ( p ) (cid:79) (cid:79) In particular κ ( f ) k ( x , . . . , x k ) = ϕ ( x (cid:12) · · · (cid:12) x k ) = (cid:88) n ≥ t n ϕ n ( x (cid:12) · · · (cid:12) x k ) , for all k ≥ x , . . . , x k ∈ U . Thus item (c) is equivalent to Im( ϕ n ) ⊂ S ≤ ( n +1) ( V ) for all n ≥
0, which is precisely item (b). (cid:3)
UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 37 Homotopy transfer for
IBL ∞ algebras IBL ∞ algebras (short for Involutive Lie Bialgebras up to coherent homotopies) were intro-duced in the paper [21], with applications to string topology, symplectic field theory and La-grangian Floer theory, and have been further investigated in several other papers since then, forinstance [18, 57, 25, 39, 58, 41, 59, 22, 40].3.1. IBL ∞ [1] algebras. We shall work with the following definition of
IBL ∞ [1] algebra, whichis slightly different from (and in a certain sense dual to) the one usually appearing in theliterature (see Remark 3.4 for a comparison). Definition 3.1. An IBL ∞ [1] algebra structure on a graded space U is a degree ( −
1) derived BV coalgebra structure δ on the symmetric coalgebra S ( U ) .Given a pair of IBL ∞ [1]-algebras ( U, δ ) and (
V, δ (cid:48) ), an
IBL ∞ [1] morphism between them isa morphism of derived BV coalgebras f : S ( U )[[ t ]] → S ( V )[[ t ]].In the following lemma, given a graded space U together with a K [[ t ]]-linear differential δ = (cid:80) n ≥ t n δ n on S ( U )[[ t ]], we give necessary and sufficient conditions in order for δ to define an IBL ∞ [1] algebra structure on U in terms of the associate Koszul brackets K ( δ n ) k , n ≥ , k ≥ s : S ( U ) → S ( U ) : x (cid:12) · · · (cid:12) x n → ( − n x (cid:12) · · · (cid:12) x n the antipode in the Hopf algebra S ( U ) (as well as its K [[ t ]]-linear extension). Lemma 3.2.
Given a graded space U and δ = (cid:80) n ≥ t n δ n ∈ End u, K [[ t ]] ( S ( U )[[ t ]]) such that δ = 0 , then the following conditions are equivalent (where we denote by δ (cid:63) s =: ϕ = (cid:80) n ≥ t n ϕ n ,and the convolution product is computed in Hom( S ( U ) , S ( U )[[ t ]]) ): (a) δ defines an IBL ∞ [1] algebra structure on U ; (b) Im( ϕ n ) ⊂ S ≤ ( n +1) ( U ) , ∀ i ≥ ; (c) K ( δ n ) k ( U, . . . , U ) ⊂ S ≤ ( n +1) ( U ) for all n ≥ , k ≥ ; (d) K ( δ n ) k ( S ≤ i ( U ) , . . . , S ≤ i k ( U )) ⊂ S ≤ ( i + ··· + i k − k + n +1) ( U ) for all n ≥ , k, i , . . . , i k ≥ .Proof. δ = (cid:80) n ≥ t n δ n defines a derived BV coalgebra structure on S ( U ) if and only if K co ( δ ) k = (cid:88) n ≥ t n K co ( δ n ) k ≡ t k − )for all n ≥ k ≥
2, that is, if and only if K co ( δ n ) k = 0 whenever n < k −
1. As in the claim of thelemma, we put δ (cid:63) s =: ϕ = (cid:80) n ≥ t n ϕ n : then ϕ n = δ n (cid:63) s . We have the following commutativediagram (which should be compared with the previous one (36))(37) S ( S ( U )) K ( δ n ) (cid:47) (cid:47) S ( S ( U )) p (cid:15) (cid:15) S ( U ) S ( i ) (cid:79) (cid:79) i (cid:15) (cid:15) δ n (cid:63)s (cid:47) (cid:47) S ( U ) S ( S ( U )) K co ( δ n ) (cid:47) (cid:47) S ( S ( U )) S ( p ) (cid:79) (cid:79) where the commutativity of the top square was already observed in the proof of Lemma 1.42,and the commutativity of the bottom square can be similarly shown by a direct computation.Using this, we can proceed as in the proof of Lemma 2.18. If δ is a derived BV-coalgebra structure, i.e., if K co ( δ n ) k = 0 for all 0 ≤ n ≤ k −
2, thenthe above shows that the composition of ϕ n = δ n (cid:63) s and the projection S ( V ) → V (cid:12) k vanisheswhenever k > n + 1: thus Im( ϕ n ) ⊂ S ≤ ( n +1) ( U ), showing that (a) implies (b).Conversely, assume that (b) holds: we need to show that K co ( δ n ) k = 0 if k > n + 1. Weshall use the recursion for the Koszul cobrackets explained in Remark 1.34. Thus, consideringthe maps (cid:101) K co ( δ n ) k : S ( U ) → S ( U ) ⊗ k introduced there, and denoting by p i : S ( U ) → U (cid:12) i thenatural projection, we need to show that(38) ( p i ⊗ · · · ⊗ p i k ) (cid:101) K co ( δ n ) k = 0for all k > n + 1 and i , . . . , i k ≥
1. We shall proceed by induction on i + · · · + i k − k . Ifthis quantity is zero, then i = · · · = i k = 1 and the thesis follows from (b) and the abovecommutative diagram (37). Otherwise, it is not restrictive to assume i k >
1: this followsfrom the fact, already observed in Remark 1.34, that the image of (cid:101) K co ( δ n ) k is contained in the S k -invariant part of S ( U ) ⊗ k . When i k >
1, Equation (38) is equivalent to0 = ( id ⊗ k − ⊗ ∆)( p i ⊗· · ·⊗ p i k ) (cid:101) K co ( δ n ) k = (cid:88) j ,j ≥ j + j = i k ( p i ⊗· · ·⊗ p i k − ⊗ p j ⊗ p j )( id ⊗ k − ⊗ ∆) (cid:101) K co ( δ n ) k , where we denote by ∆ the reduced coproduct on S ( U ). Using the recursion from Remark 1.34(cf. also Remark 1.35 and the discussion at the end of Remark 1.33), we see that( id ⊗ k − ⊗ ∆) (cid:101) K co ( δ n ) k = (cid:101) K co ( δ n ) k +1 + (id ⊗ k +1 + τ k,k +1 ) (cid:0) (cid:101) K co ( δ n ) k ⊗ id (cid:1) ∆ . Finally, using the inductive hypothesis we have ( p i ⊗ · · · ⊗ p i k − ⊗ p j ⊗ p j ) (cid:101) K co ( δ n ) k +1 = 0, aswell as( p i ⊗ · · · ⊗ p i k − ⊗ p j ⊗ p j )(id ⊗ k +1 + τ k,k +1 ) (cid:0) (cid:101) K co ( δ n ) k ⊗ id (cid:1) == ( p i ⊗ · · · ⊗ p i k − ⊗ p j ) (cid:101) K co ( δ n ) k ⊗ p j + τ k,k +1 (cid:16) ( p i ⊗ · · · ⊗ p i k − ⊗ p j ) (cid:101) K co ( δ n ) k ⊗ p j (cid:17) = 0 . This concludes the proof of the equivalence between (a) and (b).The equivalence netween (b) and (c) follows from the top square in the above commutativediagram (37), which shows that K ( δ n ) k ( x , . . . , x k ) = ϕ n ( x (cid:12) · · · (cid:12) x k ) , ∀ k ≥ , x , . . . , x k ∈ U. Finally, the equivalence between (c) and (d) can be seen by induction on i + · · · + i k − k .When i + · · · + i k − k = 0 then i = · · · = i k = 1, and (d) reduces to (c). Otherwise, given X ∈ U (cid:12) i i , . . . , X k ∈ U (cid:12) i k , it is not restrictive to assume i k > X k = X (cid:48) k (cid:12) X (cid:48)(cid:48) k for some X (cid:48) k ∈ U (cid:12) j , X (cid:48)(cid:48) k ∈ U (cid:12) j , j , j ≥ j + j = i k . Using the recursion (12) for the Koszul brackets,we see that K ( δ n ) k ( X , . . . , X k ) = K ( δ n ) k +1 ( X , . . . , X k − , X (cid:48) k , X (cid:48)(cid:48) k )++ K ( δ n ) k ( X , . . . , X k − , X (cid:48) k ) (cid:12) X (cid:48)(cid:48) k ± K K ( δ n ) k ( X , . . . , X k − , X (cid:48)(cid:48) k ) (cid:12) X (cid:48) k , and the right hand side of the above identity belongs to S ≤ i + ··· + i k − k + n +1 ( U ) by the inductivehypothesis. (cid:3) Remark 3.3.
Given a graded space U , we denote by IBL ( U ) ⊂ S ( U )[[ t ]] the subspace IBL ( U ) := ⊕ n ≥ t n S ≤ ( n +1) ( U ) . We observe that if δ = (cid:80) n ≥ t n δ n : S ( U )[[ t ]] → S ( U )[[ t ]] is an IBL ∞ [1] algebra structure on U , then the Koszul brackets K ( δ ) k preserve the subspace IBL ( U ), i.e., they restrict to maps UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 39 K ( δ ) k : IBL ( U ) (cid:12) k → IBL ( U ) inducing an L ∞ [1] algebra structure on IBL ( U ). In fact, given x , . . . , x k ∈ IBL ( U ), for j = 1 , . . . , k we write x j = (cid:80) i ≥ t i x ji , x ji ∈ S ≤ ( i +1) ( U ). Then K ( δ ) k ( x , . . . , x k ) = (cid:88) n,i ,...,i k ≥ t n + i + ··· + i k K ( δ n ) k ( x i , . . . , x ki k ) ∈ IBL ( U )since by item (d) in the previous lemma we have K ( δ n ) k ( x i , . . . , x ki k ) ∈ S ≤ n + i + ··· + i k +1 ( U ) forall n, i , . . . , i k ≥ U, δ ), (
V, δ (cid:48) ) are
IBL ∞ [1] algebras and f : S ( U )[[ t ]] → S ( V )[[ t ]] is an IBL ∞ [1] morphism,the cumulants of f restrict to maps κ ( f ) n : IBL ( U ) (cid:12) n → IBL ( V ), which are the Taylorcoefficients of an L ∞ [1] morphism κ ( f ) : ( IBL ( U ) , K ( δ )) → ( IBL ( V ) , K ( δ (cid:48) )). In fact, with thesame notations as before, κ ( f ) k ( x , . . . , x k ) = (cid:88) i ,...,i k ≥ t i + ··· + i k κ ( f ) k ( x i , . . . , x ki k ) ∈ IBL ( V )since by item (d) in Lemma 2.18 we have κ ( f ) k ( x i , . . . , x ki k ) ∈ ⊕ n ≥ t n S ≤ n + i + ··· + i k +1 ( V ) for all i , . . . , i k ≥ Remark 3.4.
Given an
IBL ∞ [1] algebra structure δ = (cid:80) n ≥ t n δ n : S ( U )[[ t ]] → S ( U )[[ t ]],according to Definition 2.16 we have δ ( S ( U ) ) = 0. Moreover, according to Lemma 3.2 we have K ( δ ) i ( x , . . . , x i ) ∈ ⊕ n ≥ t n S n +1 ( U ) for all i ≥ x , . . . , x i ∈ U . Given i ≥ n ≥
0, wedenote by p i,n : U (cid:12) i → S ≤ ( n +1) ( U ) the maps defined by the identity K ( δ ) i ( x , . . . , x i ) = (cid:88) n ≥ t n p i,n ( x , . . . , x i ) . Moreover, given i, j ≥ g ≥
0, we denote by p i,j,g : U (cid:12) i → U (cid:12) j the composition of p i,j + g − : U (cid:12) i → S ≤ j + g ( U ) and the projection S ≤ j + g ( U ) → U (cid:12) j . Thus, K ( δ ) i ( x , . . . , x i ) = (cid:88) j ≥ , g ≥ t j + g − p i,j,g ( x , . . . , x i ) . According to (28) δ ( x (cid:12) · · · (cid:12) x k ) = k (cid:88) i =1 (cid:88) σ ∈ S ( i,k − i ) ± K K ( δ ) i (cid:0) x σ (1) , . . . , x σ ( i ) (cid:1) (cid:12) · · · (cid:12) x σ ( k ) == k (cid:88) i =1 (cid:88) σ ∈ S ( i,k − i ) (cid:88) j ≥ , g ≥ ± K t j + g − p i,j,g (cid:0) x σ (1) , . . . , x σ ( i ) (cid:1) (cid:12) · · · (cid:12) x σ ( k ) In other words, δ = (cid:80) i,j ≥ , g ≥ t j + g − (cid:98) p i,j,g , where we denote by (cid:98) p i,j,g : S ( U ) → S ( U ) the mapsdefined by (cid:98) p i,j,g ( x (cid:12) · · · (cid:12) x k ) = 0 if k < i and (cid:98) p i,j,g ( x (cid:12) · · · (cid:12) x k ) := (cid:88) σ ∈ S ( i,k − i ) ± K p i,j,g (cid:0) x σ (1) , . . . , x σ ( i ) (cid:1) (cid:12) · · · (cid:12) x σ ( k ) . if k ≥ i . Finally, the condition δ = 0 translates into a hierarchy of identities on the maps p i,j,g .More precisely, arguing as in [21, Lemma 2.6] or [40, Proposition D.2.6], we see that δ = 0 ifand only if for all i, j ≥ , g ≥ g +1 (cid:88) h =1 (cid:88) i + i = i + h (cid:88) j + j = j + h (cid:88) g + g = g +1 − h p i ,j ,g ◦ h p i ,j ,g = 0 , where we denote by p i ,j ,g ◦ h p i ,j ,g “the composition of p i ,j ,g and p i ,j ,g along h matchinginputs/outputs” (we refer to [21] or [40] for a more precise definition). This shows that ourdefinition of IBL ∞ [1] algebra is essentially equivalent to the one in [21], if not for some minordifferences. Whereas our maps p i,j,g have degree | p i,j,g | = 1 − j + g − i + g − −
1: thus, when U is finite dimensional, an IBL ∞ [1]algebra structure in our sense corresponds to an IBL ∞ [1] algebra structure in the sense of [21]on the dual space U ∨ (this might also be compared with the discussion in [57, Remark 11]). Forour purposes, the previous Definition 3.1 has some technical advantages.3.2. Homotopy transfer for
IBL ∞ [1] algebras. Consider an
IBL ∞ [1] algebra ( U, δ ) withlinear part d U := p , , : U → U : in particular, by the previous identity (39) for ( i, j, g ) =(1 , , d U = 0, hence d U is a differential on U . Given a contraction f : V → U , g : U → V , h : U → U [ −
1] of (
U, d U ) onto a complex ( V, d V ), we want to show how to transferthe IBL ∞ [1] algebra structure on U along this contraction via the symmetrized tensor trick andthe standard perturbation Lemma.It is convenient to break the process into two steps.We write δ = (cid:80) n ≥ t n δ n , and observe that δ is degree one coderivation on S ( U ) such that δ ( S ( U ) ) = 0: in other words, δ is an L ∞ [1] algebra structure on U . We can transfer this L ∞ [1]algebra structure along the contraction ( g, f, h ) as explained in subsection 1.5: first we considerthe induced contraction ( S ( g ) , S ( f ) , (cid:98) h ) of ( S ( U ) , (cid:102) d U ) onto ( S ( V ) , (cid:102) d V ) as in Lemma 1.43, thenwe apply the standard perturbation Lemma 1.29 to the above contraction and the perturbation δ +0 := δ − (cid:102) d U of (cid:102) d U in order to get a transferred L ∞ [1] algebra structure δ (cid:48) on V , as well asa perturbed contraction ( G , F , H ) of ( S ( U ) , δ ) onto ( S ( V ) , δ (cid:48) ); moreover, F and G are L ∞ [1] morphisms. As observed in Remark 1.44, ( G , F , H ) is both a semifull DG coalgebracontraction and a semifull algebra contraction: moreover, the perturbed homotopy H preservesthe subspaces S ≤ n ( U ) ⊂ S ( U ).By abuse of notations, we continue to denote by δ , δ (cid:48) , ( G , F , H ) their extensions to K [[ t ]]-linear differentials on S ( U )[[ t ]] and S ( V )[[ t ]] respectively and a K [[ t ]]-linear contractionof ( S ( U )[[ t ]] , δ ) onto ( S ( V )[[ t ]] , δ (cid:48) ). Finally, we apply the standard perturbation Lemma again,this time to the contraction ( G , F , H ) of ( S ( U )[[ t ]] , δ ) onto ( S ( V )[[ t ]] , δ (cid:48) ) and the perturba-tion δ + := δ − δ = (cid:80) n ≥ t n δ n of δ , in order to get a perturbed differential δ (cid:48) on S ( V )[[ t ]] anda perturbed contraction ( G, F, H ) of ( S ( U )[[ t ]] , δ ) onto ( S ( V )[[ t ]] , δ (cid:48) ). Remark 3.5.
We notice that the perturbed differential δ (cid:48) and the perturbed contraction( G, F, H ) are K [[ t ]]-linear, as such are the original contraction ( G , F , H ) and the pertur-bation δ + . As in Remark 3.3, we denote by IBL ( U ) := ⊕ n ≥ t n S ≤ n +1 ( U ) ⊂ S ( U )[[ t ]]. We noticethat both δ and H preserve IBL ( U ), as they are both K [[ t ]]-linear and preserve the subspaces S ≤ n +1 ( U ) ⊂ S ( U ) (for H this has already been observed, for the coderivation δ it follows fromthe formula (4)). Since δ : S ( U )[[ t ]] → S ( U )[[ t ]] is an IBL ∞ [1] algebra structure, it also pre-serves the subspace IBL ( U ) ⊂ S ( U )[[ t ]], this time by Remark 3.3: hence, also the perturbation δ + := δ − δ preserves IBL ( U ), as does the perturbed homotopy H = (cid:80) k ≥ ( H δ + ) k H . Theorem 3.6.
In the above hypotheses, δ (cid:48) is an IBL ∞ [1] algebra structure on V (with linearpart d V ) and F : S ( V )[[ t ]] → S ( U )[[ t ]] , G : S ( U )[[ t ]] → S ( V )[[ t ]] are IBL ∞ [1] morphisms (withlinear parts f and g respectively).Proof. We already observed that ( G , F , H ) is a semifull DG coalgebra contraction, thus wecan apply the homotopy transfer Theorem 2.17 for derived BV colagebras to deduce that δ (cid:48) isan IBL ∞ [1] algebra structure on V and G is an IBL ∞ [1] morphism.In order to conclude, we still need to show that F is an IBL ∞ [1] morphism. We shallapply Lemma 2.18, and in particular the equivalence (a) ⇔ (c). With the notations from UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 41
Remark 3.3, we denote by
IBL ( U ) := ⊕ n ≥ t n S ≤ ( n +1) ( U ) ⊂ S ( U )[[ t ]]. Thus, we need to prove κ ( F ) i ( V, . . . , V ) ⊂ IBL ( U ) for all i ≥
1. We proceed by induction on i . For the basis of theinduction, we need to check that κ ( F ) ( x ) = F ( x ) := (cid:88) k ≥ ( H δ + ) k F ( x ) = (cid:88) k ≥ ( H δ + ) k (cid:0) f ( x ) (cid:1) ∈ IBL ( U )for all x ∈ V , which follows since f ( x ) ∈ U ⊂ IBL ( U ) and H δ + preserve the subspace IBL ( U ) ⊂ S ( U )[[ t ]], according to Remark 3.5.According to Lemma 1.30, the contraction ( G, F, H ) is a ( K [[ t ]]-linear) semifull algebra con-traction of ( S ( U )[[ t ]] , δ ) onto ( S ( V )[[ t ]] , δ (cid:48) ). By Proposition 1.27, the cumulants κ ( F ) i are in-duced via homotopy transfer from the Koszul brackets K ( δ ) k , i.e., they obey the recursion κ ( F ) i ( x , . . . , x i ) = (cid:88) k ≥ ,p ,...,pk ≥ p + ··· + p k = i (cid:88) σ ∈ S ( p ,...,p k ) ± K k ! H K ( δ ) k (cid:0) κ ( F ) p ( x σ (1) . . . ) , . . . , κ ( F ) p k ( . . . , x σ ( i ) ) (cid:1) . According to Remarks 3.3 and 3.5 we have that H K ( δ ) k (cid:16) IBL ( U ) , . . . , IBL ( U ) (cid:17) ⊂ H (cid:16) IBL ( U ) (cid:17) ⊂ IBL ( U ) , which immediately implies the inductive step κ ( F ) i ( V, . . . , V ) ⊂ IBL ( U ). (cid:3) Finally, we turn our attention to the analogue of the formal Kuranishi Theorem 1.7 in thecontext of
IBL ∞ [1] algebras. Definition 3.7. An IBL ∞ [1] algebra ( U, δ ) is complete if U is a complete space and all themaps p i,j,g as in the previous Remark 3.4 are continuous (with respect to the induced filtrations).As in Remark 3.3, we denote by IBL ( U ) := ⊕ n ≥ t n S ≤ n +1 ( U ) ⊂ S ( U )[[ t ]], and by (cid:91) IBL ( U ) itscompletion with respect to the induced filtration. According to Remark 3.3, the Koszul brackets K ( δ ) i induce a complete L ∞ [1] algebra structure on (cid:91) IBL ( U ).A Maurer-Cartan element of the complete
IBL ∞ [1] algebra ( U, δ ) is by definition a Maurer-Cartan element of the corresponding L ∞ [1] algebra (cid:91) IBL ( U ). In other words, it is a degree zeroelement x = (cid:80) n ≥ t n x n , where x n is in the completion of S ≤ n +1 ( U ), such that (cid:88) n ≥ n ! K ( δ ) n ( x, . . . , x ) = 0(in particular, this implies that x ∈ U is a Maurer-Cartan element of the underlying L ∞ [1]algebra ( U, δ )). To distinguish it from others Maurer-Cartan sets we have introduced so far,we shall denote by MC IBL ∞ [1] ( U ) := MC L ∞ [1] (cid:16) (cid:91) IBL ( U ) (cid:17) the set of Maurer-Cartan elements ofthe IBL ∞ [1] algebra ( U, δ ).Given a continuous morphism of complete
IBL ∞ [1] algebras f : S ( U )[[ t ]] → S ( V )[[ t ]], ac-cording to Remark 3.3 the cumulants κ ( f ) i are the Taylor coefficients of a continuous L ∞ [1]morphism (cid:91) IBL ( U ) → (cid:91) IBL ( V ), hence there is an induced push-forwardMC( f ) : MC IBL ∞ [1] ( U ) → MC IBL ∞ [1] ( V ) : x → (cid:88) n ≥ n ! κ ( f ) n ( x, . . . , x )between the corresponding Maurer-Cartan sets.In the hypotheses of the previous Theorem 3.6, we assume moreover that U is a complete IBL ∞ [1] algebra, and that h, f g are continuous. Then it is not hard to check that ( V, δ (cid:48) )is a complete
IBL ∞ [1] algebra with respect to the induced filtration (the only one making f and g continuous) and F : ( V, δ (cid:48) ) → ( U, δ ), G : ( U, δ ) → ( V, δ (cid:48) ) are continuous
IBL ∞ [1] morphisms. Moreover, according to Remark 3.5 the contraction ( G, F, H ) of ( S ( U )[[ t ]] , δ ) onto( S ( V )[[ t ]] , δ (cid:48) ) restricts to a continuous contraction of IBL ( U ) onto IBL ( V ), hence induces acontinuous contraction of (cid:91) IBL ( U ) onto (cid:91) IBL ( V ) which we continue to denote by ( G, F, H ). Asin the proof of the previous Theorem 3.6, the L ∞ [1] algebra structure on (cid:91) IBL ( V ) associatedwith the Koszul brackets K ( δ (cid:48) ) i and the L ∞ [1] morphism (cid:91) IBL ( V ) → (cid:91) IBL ( U ) associated withthe cumulants κ ( F ) i are induced via homotopy transfer along the contraction ( G, F, H ) fromthe L ∞ [1] algebra structure on (cid:91) IBL ( U ) associated with the Koszul brackets K ( δ ) i . Although wedidn’t show that the L ∞ [1] morphism (cid:91) IBL ( U ) → (cid:91) IBL ( V ) associated with the cumulants κ ( G ) i is induced via homotopy transfer, this is not essential for the claim of Theorem 1.7 to be true,and we only need to check that the L ∞ [1] morphism κ ( G ) : (cid:91) IBL ( U ) → (cid:91) IBL ( V ) is a left inverseto κ ( F ) : (cid:91) IBL ( V ) → (cid:91) IBL ( U ), which follows from Lemma 1.16 (cf. with [DELIGNE]). Withthese preparations, we can apply the formal Kuranishi Theorem 1.7 to deduce the followingresult. Theorem 3.8.
In the above hypotheses, there are induced bijective correspondences MC IBL ∞ [1] ( U ) ∼ = MC IBL ∞ [1] ( V ) × H (cid:16) (cid:100) IBL ( U ) (cid:17) . MC IBL ∞ [1] ( V ) ∼ = MC IBL ∞ [1] ( U ) ∩ Ker( H ) , For the former, in one direction it sends x ∈ MC IBL ∞ [1] ( U ) to the pair (MC( G )( x ) , H ( x )) ∈ MC IBL ∞ [1] ( V ) × H (cid:16) (cid:100) IBL ( U ) (cid:17) : the correspondence in the other direction might be definedrecursively, as in the claim of Theorem 1.7. For the latter, in one direction it is given by MC( F ) : MC IBL ∞ [1] ( V ) → MC IBL ∞ [1] ( U ) ∩ Ker( H ) , and in the other direction it sends a Maurer-Cartan element x ∈ MC IBL ∞ [1] ( U ) ∩ Ker( H ) to the one G ( x ) = MC( G )( x ) ∈ MC IBL ∞ [1] ( V ) . References [1] J. Alfaro, P. H. Damgaard,
Non-abelian antibrackets , Phys. Lett. B (1996), 289–294; arXiv:hep-th/9511066 .[2] Alfaro J., Bering K., Damgaard P.H.,
Algebra of higher antibrackets , Nuclear Phys. B (1996), 459–503; hep-th/9604027 .[3] R. Bandiera, M. Manetti,
On coisotropic deformations of holomorphic submanifolds , Journal of MathematicalSciences (2015), Kodaira centennial issue, University of Tokyo. arXiv:1301.6000 [math.AG] .[4] R. Bandiera, Nonabelian higher derived brackets , Journal of Pure and Applied Algebra (2015); arXiv:1304.4097 [math.QA] .[5] R. Bandiera,
Formality of Kapranov’s brackets in K¨ahler geometry via pre-Lie deformation theory , Interna-tional Mathematics Research Notices (2016), no. 21, 6626-6655; arXiv:1307.8066v3 [math.QA] .[6] R. Bandiera,
Descent of Deligne-Getzler ∞ -groupoids ; arXiv:1705.02880 [math.AT] .[7] R. Bandiera, Z. Chen, M. Sti´enon, P. Xu, Shifted derived Poisson manifolds associated with Lie pairs , Com-mun. Math. Phys. (2019), 1717–1760; arXiv:1712.00665 [math.QA] .[8] R. Bandiera,
Homotopy transfer for curved L ∞ [1] algebras , in preparation.[9] D. Bashkirov, A. A. Voronov, The BV formalism for L ∞ -algebras , J. Homotopy Relat. Struct. (2017),305–327; arXiv:1410.6432 [math.QA] .[10] I. Batalin, K. Bering, P. H. Damgaard, Second class constraints in a higher-order Lagrangian formalism ,Phys. Lett. B (1997), 235–240; arXiv:hep-th/9703199 .[11] I. Batalin, R. Marnelius,
Quantum antibrackets , Phys. Lett. B (1998), 312–320; arXiv:hep-th/9805084 .[12] I. Batalin, R. Marnelius,
Dualities between Poisson brackets and antibrackets , Internat. J. Modern Phys. A (1999), no. 32, 5049–5073; arXiv:hep-th/9809210 .[13] A. Berglund, Homological perturbation theory for algebras over operads ; Algebraic & Geometric Topology (2014), 2511-2548; arXiv:0909.3485v2 [math.AT] .[14] K. Bering, Non-commutative Batalin-Vilkovisky algebras, homotopy Lie algebras and the Courant bracket ,Comm. Math. Phys. (2007), 297–341; arXiv:hep-th/0603116 . UMULANTS, KOSZUL BRACKETS AND HOMOLOGICAL PERTURBATION THEORY ... 43 [15] K. B¨orjeson, A ∞ -algebras derived from associative algebras with a non-derivation differential , J. Gen. LieTheory Appl. (2014); arXiv:1304.6231 [math.QA] .[16] C. Braun, A. Lazarev, Homotopy BV algebras in Poisson geometry , Trans. Moscow Math. Soc. (2013),217–227; arXiv:1304.6373 [math.QA] .[17] R. Brown, The twisted Eilenberg-Zilber theorem , in
Simposio di Topologia (Messina, 1964) , Edizioni Oderisi,Gubbio, 1965, pp. 33–37.[18] R. Campos, S. Merkulov, T. Willwacher,
The Frobenius properad is Koszul , Duke Math. J. (2016),2921-2989; arXiv:1402.4048 [math.QA] .[19] A. S. Cattaneo, G. Felder,
Relative formality theorem and quantisation of coisotropic submanifolds , Adv.Math. (2007), 521–548; arXiv:math/0501540 [math.QA] .[20] K. Cieliebak, J. Latschev,
The role of string topology in symplectic field theory , in
New perspectives andchallenges in symplectic field theory , 113–146, CRM Proc. Lecture Notes , Amer. Math. Soc., Providence,RI, 2009; arXiv:0706.3284 [math.SG] .[21] K. Cieliebak, K. Fukaya, J. Latschev, Homological algebra related to surfaces with boundary ; arXiv:1508.02741 [math.QA] .[22] K. Cieliebak, E. Volkokv, Eight flavours of cyclic homology ; arXiv:2003.02528 [math.AT] .[23] V. Dotsenko, S. Shadrin, B. Vallette, Givental group action on Topological Field Theories and homotopyBatalin-Vilkovisky algebras , Adv. in Math. (2013), 224-256; arXiv:1112.1432 [math.QA] .[24] V. Dotsenko, S. Shadrin, B. Vallette,
De Rham cohomology and homotopy Frobenius manifolds , J. Eur. Math.Soc. (2015), 535-547; arXiv:1203.5077 [math.KT] .[25] Martin Doubek, Branislav Jurco, Lada Peksova, Properads and Homotopy Algebras Related to Surfaces ; arXiv:1708.01195 [math.AT] .[26] G.C. Drummond-Cole, J.S. Park, J. Terilla, Homotopy probability theory I , J. Homotopy Relat. Struct. (2015), 425–435; arXiv:1302.3684 [math.PR] .[27] G.C. Drummond-Cole, J.S. Park, J. Terilla, Homotopy probability theory II , J. Homotopy Relat. Struct. (2015), 623–635; arXiv:1302.5325 [math.PR] .[28] D. Fiorenza, M. Manetti, Formality of Koszul brackets and deformations of holomorphic Poisson manifolds ,Homology, Homotopy and Applications (2012), pp.63-75; arXiv:1109.4309 [math.QA] [29] K. Fukaya, Deformation theory, homological algebra and mirror symmetry , in
Geometry and physics of branes(Como, 2001) , Ser. High Energy Phys. Cosmol. Gravit., IOP, Bristol, 2003, pp. 121–209[30] I. Galvez-Carrillo, A. Tonks, B. Vallette,
Homotopy Batalin-Vilkovisky algebras , J. Noncomm. Geom. (2012),539-602; arXiv:0907.2246 [math.QA] .[31] E. Getzler, Lie theory for nilpotent L ∞ -algebras , Ann. of Math. , no. 1 (2009), 271-301; arXiv:math/0404003v4 .[32] E. Getzler, Maurer-Cartan elements and homotopical perturbation theory ; arXiv:1802.06736 [math.KT] .[33] V. K. A. M. Gugenheim, On the chain-complex of a fibration , Illinois J. Math. (1972), 398–414.[34] V. K. A. M. Gugenheim, L. A. Lambe, and J. D. Stasheff, Perturbation theory in differential homologicalalgebra, II , Illinois J. Math. (1991), 357–373.[35] J. Huebschmann, T. Kadeishvili, Small models for chain algebras , Math. Z. (1991), 245–280.[36] J. Huebschmann,
Higher homotopies and Maurer-Cartan algebras: quasi-Lie-Rinehart, Gerstenhaber, andBatalin-Vilkovisky algebras , in
The breadth of symplectic and Poisson geometry , Progr. Math., vol. ,Birkh¨auser, Boston, 2005, pp. 237–302.[37] J. Huebschmann, J. D. Stasheff,
Formal solution of the master equation via HPT and deformation theory ,Forum Math. (2002), 847–868. MR 1932522[38] J. Huebschmann, The sh-Lie algebra perturbation lemma , Forum Math. (2011); arXiv:0710.2070 .[39] P. H´ajek, Twisted IBL-infinity-algebra and string topology: First look and examples ; arXiv:1811.05281[math-ph] .[40] P. H´ajek, IBL-Infinity Model of String Topology from Perturbative Chern-Simons Theory , Ph.D. Thesis,University of Augsburg, October 2019; arXiv:2003.07933 [math-ph] .[41] E. Hoffbeck, J. Leray, B. Vallette,
Properadic homotopical calculus , Int. Math. Res. Not., rnaa091, https://doi.org/10.1093/imrn/rnaa091 ; arXiv:1910.05027 [math.QA] .[42] T. Kadeishvili, On the homology theory of fibre spaces , Uspekhi Mat. Nauk. (1980) (Russian), englishversion: arXiv:math/0504437 .[43] T. Kadeishvili, The algebraic structure in the cohomology of A( ∞ )-algebras , Soobshch. Akad. Nauk Gruzin.SSR (1982), 249-252.[44] Y. Kosmann-Schwarzbach Y., Graded Poisson Brackets and Field Theory , in
Modern Group TheoreticalMethods in Physics , Mathematical Physics Studies, vol , 189-196. [45] J.-L. Koszul, Crochet de Schouten-Nijenhuis et cohomologie , Ast´erisque, n ◦ hors series, Soc. Math. Fr., 1985,pp. 257-271.[46] O. Kravchenko, Deformations of Batalin–Vilkovisky algebras , in
Poisson geometry (Warsaw, 1998) , vol. of Banach Center Publ., Polish Acad. Sci., Warsaw (2000), 131-139; arXiv:math/9903191 [47] T. Lada, J. Stasheff, Introduction to SH Lie algebras for physicists , Internat. J. Theoret. Phys. (1993),1087–1103; hep-th/9209099 .[48] R. Lawrence, N. Ranade, D. Sullivan, Quantitative towers in finite difference calculus approximating thecontinuum ; arXiv:2011.07505 [math.NA] [49] F. Lehner, Cumulants in noncommutative probability theory I. Noncommutative exchangeability systems ,Math. Zeit. (2004), 67–100; arXiv:math/0210442 [math.CO] [50] J.-L. Loday, B. Vallette,
Algebraic operads , Grundlehren der Mathematischen Wissenschaften ,SpringerVerlag, Berlin, 2012.[51] M. Manetti,
Differential graded Lie algebras and formal deformation theory , in
Algebraic Geometry: Seattle2005 , Proc. Sympos. Pure Math. , 785-810, 2009.[52] M. Manetti, A relative version of the ordinary perturbation lemma , Rend. Mat. Appl. (2010), 221-238; arXiv:1002.0683 [math.KT] [53] M. Manetti, G. Ricciardi, Universal Lie Formulas for Higher Antibrackets , SIGMA (2016), 053, 20 pages; arXiv:1509.09032 [math.QA] .[54] M. Manetti, Uniqueness and intrinsic properties of non-commutative Koszul brackets , J. Homotopy Relat.Struct. (2017), 487-509; arXiv:1512.05480 [math.QA] .[55] M. Markl, On the origin of higher braces and higher-order derivations , J. Homotopy Relat. Struct. (2015),637–667; arXiv:1309.7744 [math.KT] .[56] M. Markl, Higher braces via formal (non)commutative geometry , in
Geometric Methods in Physics. Trendsin Mathematics , Birkh¨auser, 2015, 67-81; arXiv:1411.6964 [math.AT] .[57] M. Markl, A. A. Voronov,
The MV formalism for IBL ∞ and BV ∞ algebras , Lett. Mat. Phys. (2017),1515-1543; arXiv:1511.01591 [math.QA] .[58] S. Merkulov, Prop of ribbon hypergraphs and strongly homotopy involutive Lie bialgebras ; arXiv:1812.04913[math.QA] .[59] F. Naef, T. Willwacher, String topology and configuration spaces of two points ; arXiv:1911.06202 [math.QA] [60] J.-S. Park, Homotopy theory of probability spaces I: classical independence and homotopy Lie algebras ; arXiv:1510.08289 [math.PR] .[61] J.S. Park, Homotopical Computations in Quantum Field Theory ; arXiv:1810.09100 [math.QA] .[62] N. Ranade, D. Sullivan, The cumulant bijection and differential forms ; arXiv:1407.0422v2 .[63] N. Ranade, Topological perspective on Statistical Quantities I ; arXiv:1707.02900 [math.AT] .[64] N. Ranade, Topological perspective on Statistical Quantities II ; .[65] L. Vitagliano, Representations of homotopy Lie-Rinehart algebras , Math. Proc. Camb. Phil. Soc. (2015),155-191;[66] P. Real,
Homological perturbation theory and associativity , Homology Homotopy Appl. (2000), 51-88.[67] W. Shih, Homologie des espaces fibr´es , Inst. Hautes ´Etudes Sci. Publ. Math. (1962), no. 13, 88.[68] T. Voronov,
Higher derived brackets and homotopy algebras , J. Pure Appl. Algebra (2005), 133-153; arXiv:0304038 [math.QA] .[69] T. Voronov,
Higher derived brackets for arbitrary derivations , Travaux math´ematiques, fasc. XVI, Univ.Luxemb., Luxembourg (2005), 163-186; arXiv:0412202 [math.QA] .[70] A. A. Voronov,
Quantizing deformation theory II ; Pure and Applied Mathematics Quarterly (2020),125-152; arXiv:1806.11197 [math.QA] . Universit`a degli studi di Roma “La Sapienza”
Email address ::