aa r X i v : . [ m a t h . A T ] M a y INFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS
THOMAS TRADLER
Abstract.
We give a self contained introduction to A ∞ -algebras, A ∞ -bimodulesand maps between them. The case of A ∞ -bimodule-map between A and itsdual space A ∗ , which we call ∞ -inner-product, will be investigated in detail.In particular, we describe the graph complex associated to ∞ -inner-product.In a later paper, we show how ∞ -inner-products can be used to model thestring topology BV-algebra on the free loop space of a Poincar´e duality space. A ∞ -algebras were first introduced by J. Stasheff in [S1] in the study of thehomotopy associativity of H-spaces. Since then, the concept has found numerousapplications in many fields of mathematics and physics. Our interest in A ∞ -algebrasstems from applying the concept to the chain level of a Poincar´e duality space, andto ultimately obtain a model for string topology operations defined by M. Chasand D. Sullivan in [CS]. To this end, it is necessary to develop an appropriatealgebraic notion of Poincar´e duality. A detailed introduction to such a concept willbe presented in these notes.In order to describe Poincar´e duality for a topological space X , note that itscohomology H is an algebra under the cup product, and thus both H and its dual H ∗ are bimodules over H . With this, Poincar´e duality is given by a bimoduleequivalence H ( X ) → H ∗ ( X ) between the homology and cohomology of X . Asuitable chain level version of this may be obtained by considering the bimoduleconcept in a homotopy invariant way. Our approach for Poincar´e duality consistsof examining the A ∞ -algebra structure on the cochains A of X . With this, one cansee, that both A and its dual A ∗ are in fact A ∞ -bimodules in an appropriate sense,described below. Finally, we may complete the analogy by taking the chain levelPoincar´e duality to be a map between the A ∞ -bimodules A and A ∗ , which we callan ∞ -inner-product. We will review these concepts and will investigate a usefulgraph complex that is associated to A ∞ -algebras with ∞ -inner-products. We showhow the graph complex gives rise to a sequence of polyhedra, which include as aspecial case Stasheff’s associahedra coming from A ∞ -algebras.This paper is the first in a series, setting the foundation for the algebraic notationnecessary to describe Poincar´e duality at the chain level, and with this, modelingstring topology algebraically. The next step in this direction is taken in [TZS] incollaboration with M. Zeinalian, where the following theorem is proved. Theorem ([TZS] Theorem 3.1.4) . Let X be a compact triangulated Poincar´e dualityspace, in which the closure of every simplex is contractible. Then there exists asymmetric ∞ -inner-product on the cochains A of X , which on the lowest levelinduces Poincar´e duality on homology H → H ∗ . It is interesting to note, that the lowest level of this theorem consists of cappingwith the fundamental cycle of the space, which does not give a bimodule map atthe chain level, but which requires the notion of A ∞ -bimodule maps. The next step is then to model the string topology operations on the Hochschild-cochain-complex of an A ∞ -algebra A with ∞ -inner-product. Let C ∗ ( A, A ) and C ∗ ( A, A ∗ ) denote the Hochschild-cochain-complex of A with values in A and A ∗ ,respectively. The following are some well-known operations on these space, seee.g. [GJ2]. The ⌣ -product, ⌣ : C ∗ ( A, A ) ⊗ C ∗ ( A, A ) → C ∗ ( A, A ), and theGerstenhaber-bracket, [ · , · ] : C ∗ ( A, A ) ⊗ C ∗ ( A, A ) → C ∗ ( A, A ), were studied inthe deformation theory of associative algebras by M. Gerstenhaber in [G]. Onthe other hand, if A has a unit 1 ∈ A , then we may define Connes’ B -operator,which may be dualized to an operation B : C ∗ ( A, A ∗ ) → C ∗ ( A, A ∗ ). Using theseoperations, it was shown in [T], that these operations combine to give a BV-algebra. Theorem ([T] Theorem 3.1) . Let A be a unital A ∞ -algebra with symmetric andnon-degenerate ∞ -inner-product. Using the ∞ -inner-product, one has an inducedquasi-isomorphism of Hochschild-complexes C ∗ ( A, A ) → C ∗ ( A, A ∗ ) . Then, B and ⌣ assemble to give a BV-algebra on Hochschild-cohomology, such that its inducedGerstenhaber-bracket is the one given in [G] . Combining the theorems from [TZS] and [T], we obtain a BV-algebra on theHochschild-cohomology of the cochains of any Poincar´e duality space. This BV-algebra is reminiscent of the BV-algebra from string topology defined on the ho-mology of the free loop space of a compact, oriented manifold M , see [CS]. Itis an interesting and non-trivial question, if the identification of the Hochschild-cohomology with the homology of the free loop space also induces an isomorphismof the corresponding BV-algebras. A calculation by L. Menichi in [Me] shows, thatin order to recover the BV-algebra on the loop space of M , it is in general notenough to consider the cyclic A ∞ -algebra on the chain level of M , i.e. the A ∞ -structure together with the strict Poincar´e duality inner product on homology. Itis our hope, that the notion of ∞ -inner-products will be strong enough to recoverthis BV-algebra on the Hochschild-cohomology of A , but in any case will help toshed light on this question.We want to point out, that this BV-algebra is only the tip of the iceberg. In[TZ] it is shown that there is a whole PROP-action on the Hochschild-cochainsof a generalized A ∞ -algebra with ∞ -inner-product. The genus-zero part of thisPROP-action constitutes exactly a solution to the cyclic Deligne conjecture for thegeneral case of an A ∞ -algebra with ∞ -inner-product, since it lifts the above BV-algebra on Hochschild-cohomology to the Hochschild-cochain level C ∗ ( A, A ∗ ). Moregenerally, the full PROP-action in [TZ] is reminiscent of an action of the moduli-space of Riemann surfaces on the free loop space of a manifold, as it is envisionedby string topology. This shows, that ∞ -inner-products provide a suitable setup foran algebraic formulation, which captures the ideas from string topology.The structure of the paper is outlined in the following table of contents. Contents
1. A ∞ -algebras 32. A ∞ -bimodules 63. Morphisms of A ∞ -bimodules 124. ∞ -inner-products on A ∞ -algebras 16 NFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS 3
References 25Here are some general remarks on the notation in this paper. Let R denote acommutative ring with unit. All the spaces ( V , W , Z , A , M , N , ...) in this paperare always understood to be graded modules V = L i ∈ Z V i over R , all maps willalways be understood as R -module maps, and all tensor products will always beassumed over R . The degree of homogeneous elements v ∈ V i is written as | v | := i ,and the degree of maps ϕ : V i → W j is written as | ϕ | := j − i . All tensor-productsof maps and their compositions are understood in a graded way:( ϕ ⊗ ψ )( v ⊗ w ) = ( − | ψ |·| v | ( ϕ ( v )) ⊗ ( ψ ( w )) , ( ϕ ⊗ ψ ) ◦ ( χ ⊗ ̺ ) = ( − | ψ |·| χ | ( ϕ ◦ χ ) ⊗ ( ψ ◦ ̺ ) . All objects a i , v i ,... are assumed to be elements in A , V ,... respectively, if notstated otherwise.It will be necessary to look at elements of V ⊗ i ⊗ V ⊗ j . In order to distin-guish between the tensor-product in V ⊗ i and the one between V ⊗ i and V ⊗ j ,it is convenient to write the first one as a tuple ( v , ..., v i ) ∈ V ⊗ i , and then( v , ..., v i ) ⊗ ( v ′ , ..., v ′ j ) ∈ V ⊗ i ⊗ V ⊗ j . The total degree of ( v , ..., v i ) ∈ V ⊗ i isgiven by | ( v , ..., v i ) | := P ik =1 | v k | .Frequently there will be sums of the form P ni =0 ( v , ..., v i ) ⊗ ( v i +1 , ..., v n ) . Herethe convention will be used that for i = 0, one has the term 1 ⊗ ( v , ..., v n ) andfor i = n the term in the sum is ( v , ..., v n ) ⊗
1, with 1 = 1
T V ∈ T V . Similarlyfor terms P ki =0 ( v , ..., v i ) ⊗ ( v i +1 , ..., v k , w, v k +1 , ..., v n ) the expression for i = k isunderstood as ( v , ..., v k ) ⊗ ( w, v k +1 , ..., v n ). Acknowledgements.
I would like to thank Dennis Sullivan, who pointed out thelimitations of cyclic A ∞ -algebras to me. I am also thankful to Martin Markl, JimStasheff and Mahmoud Zeinalian for many useful comments.1. A ∞ -algebras We first review the definition of A ∞ -algebras that are used in the discussion ofthis paper. Definition 1.1. A coalgebra ( C, ∆) over a ring R consists of an R -module C anda comultiplication ∆ : C → C ⊗ C of degree 0 satisfying coassociativity: C C ⊗ CC ⊗ C C ⊗ C ⊗ C ✲ ∆ ❄ ∆ ❄ ∆ ⊗ id ✲ id ⊗ ∆ Then a coderivation on C is a map f : C → C such that C C ⊗ CC C ⊗ C ✲ ∆ ❄ f ❄ f ⊗ id + id ⊗ f ✲ ∆ THOMAS TRADLER Definition 1.2.
Let V = L j ∈ Z V j be a graded module over a given ground ring R . The tensor-coalgebra of V over the ring R is given by T V := M i ≥ V ⊗ i , ∆ : T V → T V ⊗ T V, ∆( v , ..., v n ) := n X i =0 ( v , ..., v i ) ⊗ ( v i +1 , ..., v n ) . Let A = L j ∈ Z A j be a graded module over the given ground ring R . Define its suspension sA to be the graded module sA = L j ∈ Z ( sA ) j with ( sA ) j := A j − .The suspension map s : A → sA , s : a sa := a is an isomorphism of degree +1.Now the bar complex of A is given by BA := T ( sA ).An A ∞ -algebra on A is given by a coderivation D on BA of degree − D = 0.The tensor-coalgebra has the property to lift every module map f : T V → V toa coalgebra-map F : T V → T V : T VT V V ❄ projection ✲ f (cid:0)(cid:0)(cid:0)(cid:0)✒ F A similar property for coderivations on
T V will lead to give an alternative descrip-tion of A ∞ -algebras. Lemma 1.3. (a)
Let ̺ : V ⊗ n → V , with n ≥ , be a map of degree | ̺ | , whichcan be viewed as ̺ : T V → V by letting its only non-zero component beinggiven by the original ̺ on V ⊗ n . Then ̺ lifts uniquely to a coderivation ˜ ̺ : T V → T V with
T VT V V ❄ projection (cid:0)(cid:0)(cid:0)(cid:0)✒ ˜ ̺ ✲ ̺ by taking ˜ ̺ ( v , ..., v k ) := 0 , for k < n, ˜ ̺ ( v , ..., v k ) := k − n X i =0 ( − | ̺ |· ( | v | + ... + | v i | ) ( v , ..., ̺ ( v i +1 , ..., v i + n ) , ..., v k ) , for k ≥ n. Thus, ˜ ̺ | V ⊗ k : V ⊗ k → V ⊗ k − n +1 . (b) There is a one-to-one correspondence between coderivations σ : T V → T V and systems of maps { ̺ i : V ⊗ i → V } i ≥ , given by σ = P i ≥ ˜ ̺ i .Proof. (a) Denote by ˜ ̺ j the component of ˜ ̺ mapping T V → V ⊗ j . Then˜ ̺ , ..., ˜ ̺ m − uniquely determine the component ˜ ̺ m , using the coderivation NFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS 5 property of ˜ ̺ .∆(˜ ̺ ( v , ..., v k )) = (˜ ̺ ⊗ id + id ⊗ ˜ ̺ )(∆( v , ..., v k ))= k X i =0 ˜ ̺ ( v , ..., v i ) ⊗ ( v i +1 , ..., v k )+( − | ˜ ̺ |· ( | v | + ... + | v i | ) ( v , ..., v i ) ⊗ ˜ ̺ ( v i +1 , ..., v k ) . Projecting both sides to V ⊗ i ⊗ V ⊗ j ⊂ T V ⊗ T V , with i + j = m , yields∆(˜ ̺ m ( v , ..., v k )) | V ⊗ i ⊗ V ⊗ j = ˜ ̺ i ( v , ..., v k − j ) ⊗ ( v k − j +1 , ..., v k )+ ( − | ˜ ̺ |· ( | v | + ... + | v i | ) ( v , ..., v i ) ⊗ ˜ ̺ j ( v i +1 , ..., v k ) . For m = i = 1 and j = 0, this shows that ˜ ̺ = 0. ˜ ̺ = ̺ by thecondition of the Lemma, and for m ≥
2, choosing i = m − , j = 1uniquely determines ˜ ̺ m by lower components. Thus, an induction shows,that ˜ ̺ m is only nonzero on V ⊗ k for k = m + n −
1, where ˜ ̺ m ( v , ..., v m + n − )is given by P m − i =0 ( − | ̺ |· ( | v | + ... + | v i | ) ( v , ..., ̺ ( v i +1 , ..., v i + n ) , ..., v m + n − ).(b) The map α : {{ ̺ i : V ⊗ i → V } i ≥ } → Coder ( T V ) , { ̺ i : V ⊗ i → V } i ≥ X i ≥ ˜ ̺ i is well defined. Its inverse β is given by β : σ
7→ { pr V ◦ σ | V ⊗ i } i ≥ , becausethe explicit lifting property of (a) shows that β ◦ α = id , and the uniquenesspart of (a) shows that α ◦ β = id . (cid:3) Application to Definition 1.2 gives the following
Proposition 1.4.
Let ( A, D ) be an A ∞ -algebra, and let D be given by a systemof maps { D i : sA ⊗ i → sA } i ≥ , with D = 0 . Let m i : A ⊗ i → A be given by D i = s ◦ m i ◦ ( s − ) ⊗ i . Then the condition D = 0 is equivalent to the followingsystem of equations: m ( m ( a )) = 0 ,m ( m ( a , a )) − m ( m ( a ) , a ) − ( − | a | m ( a , m ( a )) = 0 ,m ( m ( a , a , a )) − m ( m ( a , a ) , a ) + m ( a , m ( a , a ))+ m ( m ( a ) , a , a ) + ( − | a | m ( a , m ( a ) , a )+( − | a | + | a | m ( a , a , m ( a )) = 0 ,... k X i =1 k − i +1 X j =0 ( − ε · m k − i +1 ( a , ..., m i ( a j , ..., a j + i − ) , ..., a k ) = 0 ,where ε = i · j − X l =1 | a l | + ( j − · ( i + 1) + k − i... Proof.
This follows from Lemma 1.3 after a careful check of the involved signs. (cid:3)
THOMAS TRADLER
Example 1.5.
Any differential graded algebra (
A, ∂, µ ) gives an A ∞ -algebra-structure on A by taking m := ∂ , m := µ and m k := 0 for k ≥
3. The equationsfrom Proposition 1.4 are the defining conditions of a differential graded algebra: ∂ ( a ) = 0 ,∂ ( a · b ) = ∂ ( a ) · b + ( − | a | a · ∂ ( b ) , ( a · b ) · c = a · ( b · c ) . There are no higher equations.
Definition 1.6.
Let (
A, D ) be an A ∞ -algebra. The Hochschild-cochain-complexof A is defined to be the space C ∗ ( A ) := CoDer ( BA, BA ) of coderivations on BA with the differential δ : C ∗ ( A ) → C ∗ ( A ) given by δ ( f ) := [ D, f ] = D ◦ f − ( − | f | f ◦ D . We have δ = 0, because with D of degree − D = 0, it follows that δ ( f ) = [ D, D ◦ f − ( − | f | f ◦ D ] = D ◦ D ◦ f − ( − | f | D ◦ f ◦ D − ( − | f | +1 D ◦ f ◦ D + ( − | f | + | f | +1 f ◦ D ◦ D = 0.2. A ∞ -bimodules Let (
A, D ) be an A ∞ -algebra. We now define the concept of an A ∞ -bimoduleover A , which was also considered in [GJ1] and [Ma]. This should be a generalizationof two facts. First, it is possible to define the Hochschild-cochain-complex for anyalgebra with values in a bimodule, which we would also like to do in the A ∞ case. Second, any algebra is a bimodule over itself by left- and right-multiplication,which should also hold in the A ∞ case. The following space and map are importantingredients. Definition 2.1.
For modules V and W over R , we define T W V := R ⊕ M k ≥ ,l ≥ V ⊗ k ⊗ W ⊗ V ⊗ l . Furthermore, let ∆ W : T W V → ( T V ⊗ T W V ) ⊕ ( T W V ⊗ T V ) , ∆ W ( v , ..., v k , w, v k +1 , ..., v k + l ) := k X i =0 ( v , ..., v i ) ⊗ ( v i +1 , ..., w, ..., v n )+ k + l X i = k ( v , ..., w, ..., v i ) ⊗ ( v i +1 , ..., v k + l ) . Again for modules A and M let B M A be given by T sM sA , where s is the suspensionfrom Definition 1.2.Observe that T W V is not a coalgebra, but rather a bi-comodule over T V . Weneed the definition of a coderivation from
T V to T W V . NFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS 7
Definition 2.2. A coderivation from T V to T W V is a map f : T V → T W V sothat the following diagram commutes: T V T V ⊗ T VT W V ( T V ⊗ T W V ) ⊕ ( T W V ⊗ T V ) ❄ f ✲ ∆ ❄ id ⊗ f + f ⊗ id ✲ ∆ W For modules A and M let C ∗ ( A, M ) :=
CoDer ( BA, B M A ) be the space of coderiva-tions in the above sense, called the Hochschild-cochain-complex of A withvalues in M . Lemma 2.3. (a)
Let ̺ : V ⊗ n → W be a map of degree | ̺ | , which can beviewed as a map ̺ : T V → W by letting its only non-zero component beinggiven by the original ̺ on V ⊗ n . Then ̺ lifts uniquely to a coderivation ˜ ̺ : T V → T W V with T W VT V W ❄ projection (cid:0)(cid:0)(cid:0)(cid:0)✒ ˜ ̺ ✲ ̺ by taking ˜ ̺ ( v , ..., v k ) := 0 , for k < n, ˜ ̺ ( v , ..., v k ) := k − n X i =0 ( − | ̺ |· ( | v | + ... + | v i | ) ( v , ..., ̺ ( v i +1 , ..., v i + n ) , ..., v k ) , for k ≥ n. Thus ˜ ̺ | V ⊗ k : V ⊗ k → L i + j = k − n V ⊗ i ⊗ W ⊗ V ⊗ j . (b) There is a one-to-one correspondence between coderivations σ : T V → T W V and systems of maps { ̺ i : V ⊗ i → W } i ≥ , given by σ = P i ≥ ˜ ̺ i .Proof. (a) The proof is similar to the one of Lemma 1.3 (a). Let ˜ ̺ j be thecomponent of ˜ ̺ mapping T V → L r + s = j V ⊗ r ⊗ W ⊗ V ⊗ s , and ˜ ̺ − thecomponent T V → R . The equation∆ W (˜ ̺ ( v , ..., v k )) = (˜ ̺ ⊗ id + id ⊗ ˜ ̺ )(∆( v , ..., v k ))= k X i =0 ˜ ̺ ( v , ..., v i ) ⊗ ( v i +1 , ..., v k )+( − | ˜ ̺ |· ( | v | + ... + | v i | ) ( v , ..., v i ) ⊗ ˜ ̺ ( v i +1 , ..., v k )projected to R ⊗ T V shows that ˜ ̺ − = 0. ˜ ̺ = ̺ is uniquely determinedby the statement of the Lemma, and projecting for fixed i + j = m , to the THOMAS TRADLER component M r + s = i ( V ⊗ r ⊗ W ⊗ V ⊗ s ) ⊗ V ⊗ j + V ⊗ j ⊗ M r + s = i ( V ⊗ r ⊗ W ⊗ V ⊗ s ) ⊂ T W V ⊗ T V + T V ⊗ T W V, shows that ∆ W (˜ ̺ m ( v , ..., v k )) | L r + s = i ( V r ⊗ W ⊗ V s ) ⊗ V j + V j ⊗ L r + s = i ( V r ⊗ W ⊗ V s ) is given by˜ ̺ i ( v , ..., v k − j ) ⊗ ( v k − j +1 , ..., v k )+( − | ˜ ̺ |· ( | v | + ... + | v j | ) ( v , ..., v j ) ⊗ ˜ ̺ i ( v j +1 , ..., v k ) . For m ≥
1, choosing i = m − , j = 1 uniquely determines ˜ ̺ m by lower com-ponents. Thus, an induction shows, that ˜ ̺ m is only nonzero on V ⊗ k for k = m + n −
1, where ˜ ̺ m ( v , ..., v m + n − ) is given by P m − i =0 ( − | ̺ |· ( | v | + ... + | v i | ) · ( v , ..., ̺ ( v i +1 , ..., v i + n ) , ..., v m + n − ).(b) Then maps α : {{ ̺ i : V ⊗ i → W } i ≥ } → Coder ( T V, T W V ) , { ̺ i : V ⊗ i → W } i ≥ X i ≥ ˜ ̺ i β : Coder ( T V, T W V ) → {{ ̺ i : V ⊗ i → W } i ≥ } , σ
7→ { pr W ◦ σ | V ⊗ i } i ≥ are inverse to each other by (a). (cid:3) We put a differential on C ∗ ( A, M ), similar to the one from section 1.
Proposition 2.4.
Let ( A, D ) be an A ∞ -algebra and M be a graded module. Let D M : B M A → B M A be a map of degree − . Then the induced map δ M : CoDer ( BA, B M A ) → CoDer ( BA, B M A ) , given by δ M ( f ) := D M ◦ f − ( − | f | f ◦ D ,is well-defined, (i.e. it maps coderivations to coderivations,) if and only if the fol-lowing diagram commutes: (2.1) B M A ( BA ⊗ B M A ) ⊕ ( B M A ⊗ BA ) B M A ( BA ⊗ B M A ) ⊕ ( B M A ⊗ BA ) ❄ D M ✲ ∆ M ❄ ( id ⊗ D M + D ⊗ id ) ⊕ ( D M ⊗ id + id ⊗ D ) ✲ ∆ M Proof.
Let f : BA → B M A be a coderivation. Then, δ M ( f ) is a coderivation, if( id ⊗ δ M ( f ) + δ M ( f ) ⊗ id ) ◦ ∆ = ∆ M ◦ δ M ( f ), i.e.( id ⊗ ( D M ◦ f ) − ( − | f | id ⊗ ( f ◦ D ) + ( D M ◦ f ) ⊗ id − ( − | f | ( f ◦ D ) ⊗ id ) ◦ ∆= ∆ M ◦ D M ◦ f − ( − | f | ∆ M ◦ f ◦ D. Using the coderivation property for f and D , we get∆ M ◦ f ◦ D = ( id ⊗ f ) ◦ ∆ ◦ D + ( f ⊗ id ) ◦ ∆ ◦ D = ( id ⊗ ( f ◦ D ) + ( − | f | D ⊗ f + f ⊗ D + ( f ◦ D ) ⊗ id ) ◦ ∆ , NFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS 9 so that the requirement for δ M ( f ) being a coderivation reduces to∆ M ◦ D M ◦ f = ( id ⊗ ( D M ◦ f ) + ( D M ◦ f ) ⊗ id + D ⊗ f + ( − | f | f ⊗ D ) ◦ ∆= ( id ⊗ D M + D ⊗ id ) ◦ ( id ⊗ f ) ◦ ∆+( D M ⊗ id + id ⊗ D ) ◦ ( f ⊗ id ) ◦ ∆= ( id ⊗ D M + D ⊗ id ) ◦ ∆ M ◦ f + ( D M ⊗ id + id ⊗ D ) ◦ ∆ M ◦ f. Thus, we get the following condition for D M ,∆ M ◦ D M ◦ f = ( id ⊗ D M + D ⊗ id + D M ⊗ id + id ⊗ D ) ◦ ∆ M ◦ f for all coderivations f : T A → T M A . With Lemma 2.3 this condition reduces to∆ M ◦ D M = ( id ⊗ D M + D ⊗ id + D M ⊗ id + id ⊗ D ) ◦ ∆ M , which is the claim. (cid:3) We can describe D M by a system of maps. Lemma 2.5. (a)
Let V be a module, and let ψ be a coderivation on T V withassociated system of maps { ψ i : V ⊗ i → V } i ≥ from Lemma 1.3. Then anymap ̺ : T W V → W given by ̺ = P k ≥ ,l ≥ ̺ k,l , with ̺ k,l : V ⊗ k ⊗ W ⊗ V ⊗ l → W , lifts uniquely to a map ˜ ̺ : T W V → T W VT W VT W V W ❄ projection ✲ ̺ (cid:0)(cid:0)(cid:0)(cid:0)✒ ˜ ̺ which makes the following diagram commute (2.2) T W V ( T V ⊗ T W V ) ⊕ ( T W V ⊗ T V ) T W V ( T V ⊗ T W V ) ⊕ ( T W V ⊗ T V ) ❄ ˜ ̺ ✲ ∆ W ❄ ( id ⊗ ˜ ̺ + ψ ⊗ id ) ⊕ (˜ ̺ ⊗ id + id ⊗ ψ ) ✲ ∆ W This map is given ˜ ̺ ( v , ..., v k , w, v k +1 , ..., v k + l ):= k X i =1 k − i +1 X j =1 ( − | ψ i | P j − r =1 | v r | ( v , ..., ψ i ( v j , ..., v i + j − ) , ..., w, ..., v k + l )+ k X i =0 l X j =0 ( − | ̺ i,j | P k − ir =1 | v r | ( v , ..., ̺ i,j ( v k − i +1 , ..., w, ..., v k + j ) , ..., v k + l )+ l X i =1 l − i +1 X j =1 ( − | ψ i | ( | w | + P k + j − r =1 | v r | ) ( v , ..., w, ..., ψ i ( v k + j , ..., v k + i + j − ) , ..., v k + l ) . (Notice that the condition of diagram (2.2) is not linear.) (b) There is a one-to-one correspondence between maps σ : T W V → T W V that make diagram (2.2) commute and maps ̺ = P ̺ k,l from (a), given by σ = ˜ ̺ . Proof. (a) As in the Lemmas 1.3 and 2.3, we denote by ˜ ̺ j , j ≥
0, the com-ponent of ˜ ̺ mapping T W V → L k + l = j V ⊗ k ⊗ W ⊗ V ⊗ l and by ˜ ̺ − thecomponent T W V → R . ψ j , for j ≥
1, denotes the component of ψ map-ping T V → V ⊗ j . Then ˜ ̺ m is uniquely determined by ˜ ̺ , ..., ˜ ̺ m − .∆ W (˜ ̺ ( v , ..., v k , w, v k +1 , ..., v k + l ))= ( id ⊗ ˜ ̺ + ψ ⊗ id + ˜ ̺ ⊗ id + id ⊗ ψ )(∆ W ( v , ..., v k , w, v k +1 , ..., v k + l ))= k X i =0 ( − | ˜ ̺ | P ir =1 | v r | ( v , ..., v i ) ⊗ ˜ ̺ ( v i +1 , ..., w, ..., v k + l )+ k X i =0 ψ ( v , ..., v i ) ⊗ ( v i +1 , ..., w, ..., v k + l )+ k + l X i = k ˜ ̺ ( v , ..., w, ..., v i ) ⊗ ( v i +1 , ..., v k + l )+ k + l X i = k ( − | ψ | ( | w | + P ir =1 | v r | ) ( v , ..., w, ..., v i ) ⊗ ψ ( v i +1 , ..., v k + l ) . Projecting both sides to R ⊗ T V shows that ˜ ̺ − = 0, and projecting forfixed i + j = m , to the component V ⊗ j ⊗ M r + s = i ( V ⊗ r ⊗ W ⊗ V ⊗ s ) + M r + s = i ( V ⊗ r ⊗ W ⊗ V ⊗ s ) ⊗ V ⊗ j ⊂ T W V ⊗ T V + T V ⊗ T W V, shows that ∆ W (˜ ̺ m ( v , ..., w, ..., v k )) | V ⊗ j ⊗ L r + s = i ( V ⊗ r ⊗ W ⊗ V ⊗ s )+ L r + s = i ( V ⊗ r ⊗ W ⊗ V ⊗ s ) ⊗ V ⊗ j is given by ± ( v , ..., v j ) ⊗ ˜ ̺ i ( v j +1 , ..., w, ..., v k + l )+ ψ j ( v , ..., v k + l − i ) ⊗ ( v k + l − i +1 , ..., w, ..., v k + l )+ ˜ ̺ i ( v , ..., w, ..., v k + l − j ) ⊗ ( v k + l − j +1 , ..., v k + l ) ± ( v , ..., w, ...v i ) ⊗ ψ j ( v i +1 , ..., v k + l ) . For m ≥
1, choosing i = m − , j = 1 uniquely determines ˜ ̺ m by lowercomponents, and the ψ j ’s. Thus, an induction shows the claim of theLemma.(b) Let X := { σ : T W V → T W V | σ makes diagram (2.2) commute } . Then α : { ̺ : T W V → W } → X, ̺ ˜ ̺,β : X → { ̺ : T W V → W } , σ pr W ◦ σ are inverse to each other by (a). (cid:3) Definition 2.6.
Let (
A, D ) be an A ∞ -algebra. Then an A ∞ -bimodule ( M, D M )consists of a graded module M together with a map D M : B M A → B M A ofdegree −
1, which makes the diagram (2.1) of Proposition 2.4 commute, and satisfies( D M ) = 0. NFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS 11
By Proposition 2.4, we may put the differential δ M : CoDer ( T A, T M A ) → CoDer ( T A, T M A ), δ ( f ) := D M ◦ f − ( − | f | f ◦ D on the Hochschild-cochain-complex. It satisfies ( δ M ) = 0, because with ( D M ) = 0, we get ( δ M ) ( f ) = D M ◦ D M ◦ f − ( − | f | D M ◦ f ◦ D − ( − | f | +1 D M ◦ f ◦ D + ( − | f | + | f | +1 f ◦ D ◦ D = 0.The definition of an A ∞ -bimodule was already stated in [GJ1] section 3 and alsoin [Ma]. Proposition 2.7.
Let ( A, D ) be an A ∞ -algebra, and let { m i : A ⊗ i → A } i ≥ be thesystem of maps associated to D by Proposition 1.4, with m = 0 . Let ( M, D M ) bean A ∞ -bimodule over A , and let { D Mk,l : sA ⊗ k ⊗ sM ⊗ sA ⊗ l → sM } k ≥ ,l ≥ be thesystem of maps associated to D M by Lemma 2.5 (b). Let b k,l : A ⊗ k ⊗ M ⊗ A ⊗ l → M be the induced maps by D Mk,l = s ◦ b k,l ◦ ( s − ) ⊗ k + l +1 . Then the condition ( D M ) = 0 is equivalent to the following system of equations: b , ( b , ( m )) = 0 ,b , ( b , ( m, a )) − b , ( b , ( m ) , a ) − ( − | m | b , ( m, m ( a )) = 0 ,b , ( b , ( a , m )) − b , ( m ( a ) , m ) − ( − | a | b , ( a , b , ( m )) = 0 ,b , ( b , ( a , m, a )) − b , ( b , ( a , m ) , a ) + b , ( a , b , ( m, a ))+ b , ( m ( a ) , m, a ) + ( − | a | b , ( a , b , ( m ) , a )+( − | a | + | m | b , ( a , m, m ( a )) = 0 ,... k X i =1 k − i +1 X j =1 ± b k − i +1 ,l ( a , ..., m i ( a j , ..., a i + j − ) , ..., m, ..., a k + l )+ k X i =0 l X j =0 ± b k − i,l − j ( a , ..., b i,j ( a k − i +1 , ..., m, ..., a k + j ) , ..., a k + l )+ l X i =1 l − i +1 X j =1 ± b k,l − i +1 ( a , ..., m, ..., m i ( a k + j , ..., a k + i + j − ) , ..., a k + l ) = 0 ... where the signs are analogous to the ones in Proposition 1.4.Proof. The result follows from Lemma 2.5, after rewriting D Mk,l and D j by b k,l and m j . (cid:3) Example 2.8.
With this, Example 1.5 may be extended in the following way. Let(
A, ∂, µ ) be a differential graded algebra with the A ∞ -algebra-structure m := ∂ , m := µ and m k := 0 for k ≥
3. Let (
M, ∂ ′ , λ, ρ ) be a differential graded bimoduleover A , where λ : A ⊗ M → M and ρ : M ⊗ A → M denote the left- and right-action,respectively. Then, M is an A ∞ -bimodule over A by taking b , := ∂ ′ , b , := λ , b , := ρ and b k,l := 0 for k + l >
1. The equations of Proposition 2.7 are the defining conditions for a differential bialgebra over A :( ∂ ′ ) ( m ) = 0 ,∂ ′ ( m.a ) = m.∂ ( a ) + ( − | m | ∂ ′ ( m ) .a,∂ ′ ( a.m ) = ∂ ( a ) .m + ( − | a | a.∂ ′ ( m ) , ( a.m ) .b = a. ( m.b ) , ( m.a ) .b = m. ( a · b ) ,a. ( b.m ) = ( a · b ) .m. There are no higher equations.For later purposes it is convenient to have the following
Lemma 2.9.
Given an A ∞ -algebra ( A, D ) and an A ∞ -bimodule ( M, D M ) , withsystem of maps { b k,l : A ⊗ k ⊗ M ⊗ A ⊗ l → M } k ≥ ,l ≥ from Proposition 2.7, thenthe dual space M ∗ := Hom R ( M, R ) has a canonical A ∞ -bimodule-structure givenby maps { b ′ k,l : A ⊗ k ⊗ M ∗ ⊗ A ⊗ l → M ∗ } k ≥ ,l ≥ , ( b ′ k,l ( a , ..., a k , m ∗ , a k +1 , ..., a k + l ))( m ) := ( − ε · m ∗ ( b l,k ( a k +1 , ..., a k + l , m, a , ..., a k )) , where ε := ( | a | + ... + | a k | ) · ( | m ∗ | + | a k +1 | + ... + | a k + l | + | m | ) + | m ∗ | · ( k + l + 1) . Proof.
To see, that ( D M ∗ ) = 0, we can use the criterion from Proposition 2.7.The top and the bottom term in the general sum of Proposition 2.7 convert to( b ′ k − i +1 ,l ( a , ..., m i ( a j , ..., a i + j − ) , ..., m ∗ , ..., a k + l ))( m )= ± m ∗ ( b l,k − i +1 ( a k +1 , ..., a k + l , m, a , ..., m i ( a j , ..., a i + j − ) , ..., a k )), and( b ′ k,l − i +1 ( a , ..., m ∗ , ..., m i ( a k + j , ..., a k + i + j − ) , ..., a k + l ))( m )= ± m ∗ ( b l − i +1 ,k ( a k +1 , ..., m i ( a k + j , ..., a k + i + j − ) , ..., a k + l , m, a , ..., a k )) . These terms come from the A ∞ -bimodule-structure of M . Similar arguments applyto the middle term:( b ′ k − i,l − j ( a , ..., b ′ i,j ( a k − i +1 , ..., m ∗ , ..., a k + j ) , ..., a k + l ))( m )= ± ( b ′ i,j ( a k − i +1 , ..., m ∗ , ..., a k + j ))( b l − j,k − i ( a k + j +1 , ..., a k + l , m, a , ..., a k − i ))= ± m ∗ ( b j,i ( a k +1 , ..., a k + j , b l − j,k − i ( a k + j +1 , ..., a k + l , m, a , ..., a k − i ) , a k − i +1 , ..., a k )) . The sum from Proposition 2.7 for the A ∞ -bimodule M ∗ contains exactly the termsof m ∗ applied the the sum for the A ∞ -bimodule M . A thorough check identifiesthe signs. (cid:3) Morphisms of A ∞ -bimodules Let (
M, D M ) and ( N, D N ) be two A ∞ -bimodules over the A ∞ -algebra ( A, D ).We next define the notion of A ∞ -bimodule-map between ( M, D M ) and ( N, D N ).Again a motivation is to have an induced map of their Hochschild-cochain-complexes. Proposition 3.1.
Let V , W and Z be modules, and let F be a map F : T W V → T Z V . Then the induced map F ♯ : CoDer ( T V, T W V ) → CoDer ( T V, T Z V ) , given NFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS 13 by F ♯ ( f ) := F ◦ f , is well-defined, (i.e. it maps coderivations to coderivations,) ifand only if the following diagram commutes: (3.1) T W V ( T V ⊗ T W V ) ⊕ ( T W V ⊗ T V ) T Z V ( T V ⊗ T Z V ) ⊕ ( T Z V ⊗ T V ) ❄ F ✲ ∆ W ❄ ( id ⊗ F ) ⊕ ( F ⊗ id ) ✲ ∆ Z Proof.
If both f : T V → T W V and F ◦ f : T V → T Z V are coderivations, then thetop diagram and the overall diagram below commute. T V T V ⊗ T VT W V ( T V ⊗ T W V ) ⊕ ( T W V ⊗ T V ) T Z V ( T V ⊗ T Z V ) ⊕ ( T Z V ⊗ T V ) ❄ f ✲ ∆ ❄ ( id ⊗ f )+( f ⊗ id ) ❄ F ✲ ∆ W ❄ ( id ⊗ F ) ⊕ ( F ⊗ id ) ✲ ∆ Z Therefore, the lower diagram has to commute if applied to any element in Im ( f ) ⊂ T W V . By Lemma 2.3 there are enough coderivations to imply the claim. (cid:3) Again let us describe F by a system of maps. Lemma 3.2. (a)
Let V , W and Z be modules, and let ̺ : V ⊗ k ⊗ W ⊗ V ⊗ l → Z be a map, which may be viewed as a map ̺ : T W V → Z whose only nonzerocomponent is the original ̺ on V ⊗ k ⊗ W ⊗ V ⊗ l . Then ̺ lifts uniquely to amap ˜ ̺ : T W V → T Z V T Z VT W V Z ❄ projection ✲ ̺ (cid:0)(cid:0)(cid:0)(cid:0)✒ ˜ ̺ which makes the diagram (3.1) in Proposition 3.1 commute. ˜ ̺ is given by ˜ ̺ ( v , ..., v r , w, v r +1 , ..., v r + s ) := 0 , for r < k or s < l, ˜ ̺ ( v , ..., v r , w, v r +1 , ..., v r + s ):= ( − | ̺ | P r − ki =1 | v i | ( v , ..., ̺ ( v r − k +1 , ..., w, ..., v r + l ) , ..., v r + s ) , for r ≥ k and s ≥ l. Thus ˜ ̺ | V ⊗ r ⊗ W ⊗ V ⊗ s : V ⊗ r ⊗ W ⊗ V ⊗ s → V ⊗ r − k ⊗ Z ⊗ V ⊗ s − l . (b) There is a one-to-one correspondence between maps σ : T W V → T Z V making diagram (3.1) commute and systems of maps { ̺ k,l : V ⊗ k ⊗ W ⊗ V ⊗ l → Z } k ≥ ,l ≥ , given by σ = P k ≥ ,l ≥ g ̺ k,l . Proof. (a) Denote by ˜ ̺ j the component of ˜ ̺ mapping T W V → L r + s = j V ⊗ r ⊗ Z ⊗ V ⊗ s , and ˜ ̺ − the component T W V → R . Then, ˜ ̺ , ..., ˜ ̺ m − uniquelydetermine the component ˜ ̺ m .∆ Z (˜ ̺ ( v , ..., v r , w, v r +1 , ..., v r + s ))= ( id ⊗ ˜ ̺ + ˜ ̺ ⊗ id )(∆ W ( v , ..., v r , w, v r +1 , ..., v r + s ))= r X i =0 ( − | ˜ ̺ | P it =1 | v t | ( v , ..., v i ) ⊗ ˜ ̺ ( v i +1 , ..., w, ..., v r + s )+ r + s X i = r ˜ ̺ ( v , ..., w, ..., v i ) ⊗ ( v i +1 , ..., v r + s ) . Projecting both sides to R ⊗ T V shows that ˜ ̺ − = 0, and projecting forfixed i + j = m , to the component V ⊗ j ⊗ M r + s = i ( V ⊗ r ⊗ Z ⊗ V ⊗ s ) + M r + s = i ( V ⊗ r ⊗ Z ⊗ V ⊗ s ) ⊗ V ⊗ j ⊂ T Z V ⊗ T V + T V ⊗ T Z V, yields for ∆ Z (˜ ̺ m ( v , ..., w, ..., v r + s )) | V ⊗ j ⊗ L r + s = i ( V ⊗ r ⊗ Z ⊗ V ⊗ s )+ L r + s = i ( V ⊗ r ⊗ Z ⊗ V ⊗ s ) ⊗ V ⊗ j the expression ± ( v , ..., v j ) ⊗ ˜ ̺ i ( v j +1 , ..., w, ..., v r + s )+ ˜ ̺ i ( v , ..., w, ..., v r + s − j ) ⊗ ( v r + s − j +1 , ..., v r + s ) . For m ≥
1, choosing i = m − , j = 1 uniquely determines ˜ ̺ m by lowercomponents. Therefore, an induction shows that ˜ ̺ m is only nonzero on V ⊗ r ⊗ W ⊗ V ⊗ s with r − k + s − l = m , where ˜ ̺ m ( v , ..., v r , w, v r +1 , ..., v r + s )is given by ( − | ̺ | P r − ki =1 | v i | ( v , ..., ̺ ( v r − k +1 , ..., w, ..., v r + l ) , ..., v r + s ).(b) Let X := { σ : T W V → T Z V | σ makes diagram (3.1) commute } . Then α : {{ ̺ k,l : V ⊗ k ⊗ W ⊗ V ⊗ l → Z } k ≥ ,l ≥ } → X, { ̺ k,l : V ⊗ k ⊗ W ⊗ V ⊗ l → Z } k ≥ ,l ≥ X k ≥ ,l ≥ g ̺ k,l ,β : X → {{ ̺ k,l : V ⊗ k ⊗ W ⊗ V ⊗ l → Z } k ≥ ,l ≥ } ,σ
7→ { pr Z ◦ σ | V ⊗ k ⊗ W ⊗ V ⊗ l } k ≥ ,l ≥ are inverse to each other by (a). (cid:3) We may apply this to the Hochschild-complex.
Definition 3.3.
Let (
M, D M ) and ( N, D N ) be two A ∞ -bimodules over the A ∞ -algebra ( A, D ). Then a map F : B M A → B N A of degree 0 is called an A ∞ -bimodule-map , if F makes the diagram B M A ( BA ⊗ B M A ) ⊕ ( B M A ⊗ BA ) B N A ( BA ⊗ B N A ) ⊕ ( B N A ⊗ BA ) ❄ F ✲ ∆ M ❄ ( id ⊗ F ) ⊕ ( F ⊗ id ) ✲ ∆ N NFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS 15 commute, and in addition F ◦ D M = D N ◦ F .By Proposition 3.1, every A ∞ -bimodule-map induces a map F ♯ : f F ◦ f between the Hochschild-complexes, which preserves the differentials, since ( F ♯ ◦ δ M )( f ) = F ♯ ( D M ◦ f + ( − | f | f ◦ D ) = F ◦ D M ◦ f + ( − | f | F ◦ f ◦ D = D N ◦ F ◦ f + ( − | f | F ◦ f ◦ D = δ N ( F ◦ f ) = ( δ N ◦ F ♯ )( f ). Proposition 3.4.
Let ( A, D ) be an A ∞ -algebra with system of maps { m i : A ⊗ i → A } i ≥ from Proposition 1.4 associated to D , where m = 0 . Let ( M, D M ) and ( N, D N ) be A ∞ -bimodules over A with systems of maps { b k,l : A ⊗ k ⊗ M ⊗ A ⊗ l → M } k ≥ ,l ≥ and { c k,l : A ⊗ k ⊗ N ⊗ A ⊗ l → N } k ≥ ,l ≥ from Proposition 2.7 associatedto D M and D N respectively. Let F : T M A → T N A be an A ∞ -bimodule-mapbetween M and N , and let { F k,l : sA ⊗ k ⊗ sM ⊗ sA ⊗ l → sN } k ≥ ,l ≥ be a systemof maps associated to F by Lemma 3.2 (b). Rewrite the maps F k,l by f k,l : A ⊗ k ⊗ M ⊗ A ⊗ l → N by using the suspension map: F k,l = s ◦ f k,l ◦ ( s − ) ⊗ k + l +1 . Thenthe condition F ◦ D M = D N ◦ F is equivalent to the following system of equations: f , ( b , ( m )) = c , ( f , ( m )) ,f , ( b , ( m, a )) − f , ( b , ( m ) , a ) − ( − | m | f , ( m, m ( a ))= c , ( f , ( m, a )) + c , ( f , ( m ) , a ) ,f , ( b , ( a, m )) − f , ( m ( a ) , m ) − ( − | a | f , ( a, b , ( m ))= c , ( f , ( a, m )) + c , ( a, f , ( m )) ,... k X i =1 k − i +1 X j =1 ( − ε f k − i +1 ,l ( a , ..., m i ( a j , ..., a i + j − ) , ..., m, ..., a k + l +1 )+ k X j =1 k + l − j +2 X i = k − j +2 ( − ε f j,k + l − i − j +3 ( a , ..., b k − j +1 ,i + j − k − ( a j , ..., m, ..., a i + j − ) , ..., a k + l +1 )+ l X i =1 k + l − i +2 X j = k +2 ( − ε f k,l − i +1 ( a , ..., m, ..., m i ( a j , ..., a i + j − ) , ..., a k + l +1 )= k +1 X j =1 k + l − j +2 X i = k − j +2 ( − ε ′ c j,k + l − i − j +3 ( a , ..., f k − j +1 ,i + j − k − ( a j , ..., m, ..., a i + j − ) , ..., a k + l +1 ) In order to simplify notation, it is assume that in ( a , ..., a k + l +1 ) above, only thefirst k and the last l elements are elements of A and a k +1 = m ∈ M . Then thesigns are given by ε = i · j − X r =1 | a r | + ( j − · ( i + 1) + ( k + l + 1) − i, and ε ′ = ( i + 1) · ( j + 1 + j − X r =1 | a r | ) .... Proof.
The formula follows immediately from the explicit lifting properties in Lemma2.5 (a) and Lemma 3.2 (a). (cid:3)
Example 3.5.
Examples 1.5 and 2.8 can be extended in the following way. Let(
A, ∂, µ ) be a differential graded algebra with the A ∞ -algebra-structure m := ∂ , m := µ and m k := 0 for k ≥
3. Let (
M, ∂ M , λ M , ρ M ) and ( N, ∂ N , λ N , ρ N ) bedifferential graded bimodules over A , with the A ∞ -bialgebra-structures given by b , := ∂ M , b , := λ M , b , := ρ M and b k,l := 0 for k + l >
1, and c , := ∂ N , c , := λ N , c , := ρ N and c k,l := 0 for k + l >
1. Finally, let f : M → N bea bialgebra map of degree 0. Then f becomes a map of A ∞ -bialgebras by taking f , := f and f k,l := 0 for k + l >
0. The equations from Proposition 3.4 are thedefining conditions of a differential bialgebra map from M to N : f ◦ ∂ M ( m ) = ∂ N ◦ f ( m ) f ( m.a ) = f ( m ) .af ( a.m ) = a.f ( m )There are no higher equations.4. ∞ -inner-products on A ∞ -algebras There are canonical A ∞ -bialgebra-structures on a given A ∞ -algebra A and onits dual space A ∗ . We will define ∞ -inner products as A ∞ -bialgebra-maps from A to A ∗ . Lemma 4.1.
Let ( A, D ) be an A ∞ -algebra., and let D be given by the system ofmaps { m i : A ⊗ i → A } i ≥ from Proposition 1.4. (a) There is a canonical A ∞ -bimodule-structure on A given by b k,l : A ⊗ k ⊗ A ⊗ A ⊗ l → A, b k,l := m k + l +1 . (b) There is a canonical A ∞ -bimodule-structure on A ∗ given by b k,l : A ⊗ k ⊗ A ∗ ⊗ A ⊗ l → A ∗ , ( b k,l ( a , ..., a k , a ∗ , a k +1 , ..., a k + l ))( a ):= ± a ∗ ( m k + l +1 ( a k +1 , ..., a k + l , a, a , ..., a k )) , with the signs from Lemma 2.9.Proof. (a) The A ∞ -bialgebra extension described in Lemma 2.5 (a) becomesthe extension by coderivation described in Lemma 1.3 (a). Equations ofProposition 2.7 become the equations of Proposition 1.4 and the diagram(2.1) from Proposition 2.4 becomes the usual coderivation diagram for D .(b) This follows from (a) and Lemma 2.9. (cid:3) Example 4.2.
For a differential algebra (
A, ∂, µ ), the above A ∞ -bialgebra struc-ture on A is exactly the bialgebra structure given by left- and right-multiplication,since b , ( a ⊗ b ) = m ( a ⊗ b ) = a · b and b , ( a ⊗ b ) = m ( a ⊗ b ) = a · b , for a, b ∈ A .Similarly the A ∞ -bialgebra structure on A ∗ is given by right- and left-multiplicationin the arguments: b , ( a ⊗ b ∗ )( c ) = b ∗ ( m ( c ⊗ a )) = b ∗ ( c · a ) and b , ( a ∗ ⊗ b )( c ) = a ∗ ( m ( b ⊗ c )) = a ∗ ( b · c ), for a, b, c ∈ A , and a ∗ , b ∗ ∈ A ∗ . Definition 4.3.
Let (
A, D ) be an A ∞ -algebra. Then, we call any A ∞ -bimodule-map F from A to A ∗ an ∞ -inner-product on A . NFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS 17
Proposition 4.4.
Let ( A, D ) be an A ∞ -algebra. Then, specifying an ∞ -inner prod-uct on A is equivalent to specifying a system of inner-products on A , { < ., ., ... > k,l : A ⊗ k + l +2 → R } k ≥ ,l ≥ which satisfy the following relations: k + l +2 X i =1 ( − P i − j =1 | a j | < a , ..., ∂ ( a i ) , ..., a k + l +2 > k,l = X i,j,n ± < a i , ..., m j ( a n , ... ) , ... > r,s , where in the sum on the right side, there is exactly one multiplication m j ( j ≥ )inside the inner-product < ... > r,s and this sum is taken over all i, j, n subject tothe following conditions: (i) The cyclic order of the ( a , ..., a k + l +2 ) is preserved. (ii) a k + l +2 is always in the last slot of < ... > r,s . (iii) a k + l +2 could be inside some m j . By (ii), this is only the case, when the firstargument in the inner product is a i = a , as for example in the expression < a i , ..., m j ( a n , ..., a k + l +2 , a , ..., a i − ) > r,s for i > . (iv) The special arguments a k +1 and a k + l +2 are never multiplied by m j at thesame time. (v) The numbers r and s are uniquely determined by the position of the element a k +1 in the inner-product < ... > r,s . More precisely, a k +1 is in ( r + 1) -thspot of < ... > r,s , and s is determined by the inner product < ... > r,s having r + s + 2 arguments. A graphical representation of the above conditions is given in Definition 4.5 andExample 4.6 below.
Proof.
We use the description from Proposition 3.4 for A ∞ -bimodule-maps. AnA ∞ -bimodule-map from A to A ∗ is given by maps f k,l : A ⊗ k ⊗ A ⊗ A ⊗ l → A ∗ ,for k, l ≥
0. These are interpreted as maps A ⊗ k ⊗ A ⊗ A ⊗ l ⊗ A → R , which wedenoted by the inner-product-symbol < ... > k,l : < a , ..., a k + l +1 , a ′ > k,l := ( − | a ′ | ( f k,l ( a , ..., a k + l +1 ))( a ′ )The A ∞ -bimodule-map condition from Proposition 3.4 becomes(4.1) X ± f k,l ( ..., m i ( ... ) , ..., a, ... ) + X ± f k,l ( ..., b i,j ( ..., a, ... ) , ... )+ X ± f k,l ( ..., a, ..., m i ( ... ) , ... ) = X ± c i,j ( ..., f k,l ( ..., a, ... ) , ... ) . Here a ∈ A is the ( k + 1)-th entry of an element in A ⊗ k ⊗ A ⊗ A ⊗ l , so that it comesfrom the A ∞ -bimodule A , instead of the A ∞ -algebra A .Now, by Lemma 4.1 (a), b i,j = m i + j +1 is one of the multiplications from the A ∞ -structure, and therefore the left side of the equation is f k,l applied to all possiblemultiplications m i . As f k,l maps into A ∗ , we may apply the left hand side of (4.1)to an element a ′ ∈ A and use the notation < ... > k,l :(4.2) X ± ( f k,l ( ..., m i ( ... ) , ... ))( a ′ ) = X ± < ..., m i ( ... ) , ..., a ′ > k,l . Next, use Lemma 4.1 (b) to rewrite the right hand side of (4.1):(4.3) X ± ( c i,j ( a , ..., f k,l ( ..., a, ... ) , ..., a k + l +1 ))( a ′ )= X ± ( f k,l ( ..., a, ... ))( m r ( ..., a k + l +1 , a ′ , a , ... ))= X ± < ..., a, ..., m r ( ..., a k + l +1 , a ′ , a , ... ) > k,l Equations (4.2) and (4.3) show that we take a sum over all possibilities of applyingone multiplication to the arguments of the inner-product subject to the conditions(i)-(iv). This is the statement of the Proposition, after isolating the ∂ -terms on theleft. For condition (v), notice that the extensions of D and D A from Lemma 1.3(a) and Lemma 2.5 (a) record the special entry a in the A ∞ -bimodule A . Thus,the A ∞ -bimodule element a determines the index k , and l is determined by thenumber of arguments of < ... > k,l .An explicit check shows the correctness of the signs. (cid:3) There is a diagrammatic way of picturing Proposition 4.4.
Definition 4.5.
Let (
A, D ) be an A ∞ -algebra with ∞ -inner-product { < ., ., ... > k,l : A ⊗ k + l +2 → R } k ≥ ,l ≥ . To the inner-product < ... > k,l , we associate the symbol bc ...kk + 1 k + l + 2 k + 2 ... k + l + 1More generally, to any inner-product which has (possibly iterated) multiplications m , m , m , ... (but without differential ∂ = m ), such as < a i , ..., m j ( ... ) , ..., m p ( ..., m q ( ... ) , ... ) , ... > k,l , we associate a diagram like above, by the following rules:(i) To every multiplication m j , associate a tree with j inputs and one output. b m j The symbol for the multiplication may also occur in a rotated way.(ii) To the inner product < ... > r,s , associate the open circle: bc ...rr + 1 r + s + 2 r + 2 ... r + s + 1 NFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS 19
There are r elements attached at the top of the circle, and s elements atthe bottom of the circle, and the two (special) inputs ( r + 1) and ( r + s + 2)are attached on the left and right. This gives a total of r + s + 2 inputs.(iii) The inputs a i , for i = 1 , . . . , r + s + 2, will be attached counterclockwise,where the last element a r + s +2 is in the far right slot. For the multiplications m j of the graph, we use the counterclockwise orientation of the plane tofind the correct order of the arguments a i in m j .We call these diagrams inner-product-diagrams . Example 4.6.
Let a, b, c, d, e, f, g, h, i, j, k ∈ A . • < a, b, c, d > , , ( deg = 2): bc abc d • < a, b, c, d, e, f, g, h, i > , , ( deg = 7): bc b acd ie f g h • < m ( m ( b, c ) , m ( d, e )) , m ( f, a ) > , , ( deg = 0): bbb bbc c fabd e • < a, b, m ( c, d, m ( e, f )) , g, m ( h, i )) > , , ( deg = 4): bc bbb ab g ihcd e f • < c, m ( d, e ) , m ( m ( f, g ) , h ) , i, m ( j, k, a, b ) > , , ( deg = 5): bb bb bc g kbfh ai jcde Definition 4.7.
We define a chain-complex associated to inner-product-diagrams.We define the degree of the inner-product-diagram associated to < ... > k,l withmultiplications m i , ..., m i n to be k + l + P nj =1 ( i j − n ≥
0, let C n be the space generated by inner-product-diagrams of degree n .Then let C := L n ≥ C n .As for the differential d on C , we use the composition with the operator ˜ ∂ := P i id ⊗ ... ⊗ id ⊗ ∂ ⊗ id ⊗ ... ⊗ id , where ∂ = m is at the i -th spot:( d ( < ..., m ( ..., m ( ... ) , ... ) , ... > ))( a , ..., a s ):= ( < ..., m ( ..., m ( ... ) , ... ) , ... > )( s X i =1 ( − P i − j =1 | a j | ( a , ..., ∂ ( a i ) , ..., a s ))Some remarks and interpretations of this expression are in order. First, considerthe inner-product < ... > k,l without any multiplications. By Proposition 4.4, thedifferential applies one multiplication into the inner-product-diagram in all possiblespots, such that the two lines on the far left and on the far right are not beingmultiplied; compare Proposition 4.4 (iv).In the case, that multiplications are applied to the inner-product, one can observefrom Proposition 1.3, that P i m n ◦ ( id ⊗ ... ⊗ ∂ ⊗ ... ⊗ id ) is given by the two terms(4.4) X i m n ◦ ( id ⊗ ... ⊗ ∂ ⊗ ... ⊗ id ) = n − X k =2 X i m n +1 − k ◦ ( id ⊗ ... ⊗ m k ⊗ ... ⊗ id )+ ∂ ◦ m n . NFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS 21
The sum over i on both sides of the above equation applies m k to the i -th spot.The first term on the right hand side of (4.4) transforms the multiplication m n intoa sum of all possible sompositions of m n +1 − k and m k : b m n = ⇒ b m k b m n − k +1 The last term (4.4) is used for an inductive argument of the above. One gets aterm ∂ ( m n ( ... )) attached to the inner-product or possibly another multiplication,that has arguments with ˜ ∂ applied, so that the above discussion can be continuedinductively.We conclude, that the differential applies exactly one multiplication in all possi-ble spots, without multiplying the given far left and far right inputs. Examples aregiven in Example 4.8 below.It is d : C n → C n − , and d = 0. Proof.
According to the definition of the degrees above, a multiplication m n with n inputs contributes by n −
2. Taking the differential applies one more multiplicationin all possible ways. If we attach m n to the diagram, then it replaces n argumentswith one argument in the higher level. Therefore,new degree = (old degree) − n + 1 + ( n − − . We can prove d = 0 in two ways: • Algebraically:The definition of d on the inner-products is given by composition withthe operator ˜ ∂ = P i id ⊗ ... ⊗ id ⊗ ∂ ⊗ id ⊗ ... ⊗ id , where ∂ is in the i -thspot. Thus d is composition with˜ ∂ = X i,j ± id ⊗ ... ⊗ ∂ ⊗ ... ⊗ ∂ ⊗ ... ⊗ id = 0 . This vanishes, since the sum has two terms, where ∂ occurs at the i -th andthe j -th spot. This is obtained, by either first applying ∂ to the i -th andthen to the j -th spot, or vice versa. These two possibilities cancel as ∂ isof degree − ∂ either has to move over the second ∂ , by whichan additional minus sign is introduced, or not. • Diagrammatically (without signs): d applies one new multiplication to the inner-product-diagram, so that d applies two new multiplications. For two multiplications, we have thefollowing two possibilities. (i) In the first case, the multiplications are on different outputs. b bbb bb The above figure shows that the final terms are obtained in two differ-ent ways, which in fact cancel each other.(ii) The other possibility is to have multiplications on the same output.Again, these terms may be obtained in two ways that cancel eachother: b bbb bb (cid:3)
Example 4.8.
Let a, b, c ∈ A . • k = 0, l = 0: d ( < a, b > , ) = 0 bc a b d( ) = 0 NFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS 23 • k = 1, l = 0: d ( < a, b, c > , ) = < a · b, c > , ± < b, c · a > , bc b ca d( ) = b bc b ca ± bbc b ca • k = 0, l = 1: d ( < a, b, c > , ) = < a · b, c > , ± < a, b · c > , bc a cb d( ) = b bc a cb ± bbc a cb In the following three figures, where k + l = 2, the righthand side is understoodto be a sum over the five, or respectively six, inner-product-diagrams. Then, as d = 0, the terms may be arranged according to their boundaries. We obtain thepolyhedra associated to the inner-products < ... > k,l . • k = 2, l = 0: bc d( ) = bbcb bc bc bb bc bc b • k = 1, l = 1: bc d( ) = b bcbc bb bc bc bb bc bc b • k = 0, l = 2: bc d( ) = bbcb bc bc bb bc bc b Finally, we graph the polyhedra in the case k + l = 3. In general, the polyhedronassociated to < ... > k,l is isomorphic to the one from < ... > l,k . Furthermore, thepolyhedra for < ... > n, and < ... > ,n are the ones known as Stasheff’s associahe-dra. • The polyhedron for k = 3 , l = 0 and for k = 0 , l = 3: NFINITY-INNER-PRODUCTS ON A-INFINITY-ALGEBRAS 25 • The polyhedron for k = 2 , l = 1 and for k = 1 , l = 2: References [CJ] R.L. Cohen, J.D.S. Jones
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Thomas Tradler, Department of Mathematics, College of Technology of the CityUniversity of New York, 300 Jay Street, Brooklyn, NY 11201, USA
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