Cohomology theory of averaging algebras, L ∞ -structures and homotopy averaging algebras
aa r X i v : . [ m a t h . K T ] S e p COHOMOLOGY THEORY OF AVERAGING ALGEBRAS, L ∞ -STRUCTURES ANDHOMOTOPY AVERAGING ALGEBRAS KAI WANG AND GUODONG ZHOUA bstract . This paper studies averaging algebras, say, associative algebras endowed with averag-ing operators. We develop a cohomology theory for averaging algebras and justify it by interpretinglower degree cohomology groups as formal deformations and abelian extensions of averaging al-gebras. We make explicit the L ∞ -algebra structure over the cochain complex defining cohomologygroups and introduce the notion of homotopy averaging algebras as Maurer-Cartan elements of this L ∞ -algebra. C ontents
1. Introduction 12. Averaging algebras and their bimodule 23. Cohomology theory of averaging algebras 53.1. Cohomology of averaging operators 53.2. Cohomology of averaging algebras 73.3. Relationship among the cohomlogies. 84. Formal deformation of averaging algebras 95. Abelian extensions of averaging algebras 116. L ∞ -structure on the cochain complex 147. Homotopy averaging algebras 19Appendix: Proof of Theorem 6.5 20References 261. I ntroduction Throughout this paper, k denotes a field. All the vector spaces and algebras are over k and alltensor products are also taking over k .Let R be an associative algebra over field k . An averaging operator over R is a k -linear map A : R → R such that A ( x ) A ( y ) = A ( A ( x ) y ) = A ( xA ( y )for all x , y ∈ R .The research on averaging operators has a long history and it appears in various mathematicalbranches from turbulence theory to functional analysis, probability theory etc. When investigat-ing turbulence theory, Reynolds already studied implicitly the averaging operator in a famous Date : September 25, 2020.1991
Mathematics Subject Classification.
Key words and phrases. averaging algebra, averaging operator, cohomology, extension, deformation, homotopyaveraging algebra, L ∞ -algebra. paper [20] published in 1895. In a series of papers of 1930s, Kolmogoro ff and Kamp´e de F´erietintroduced explicitly the averaging operator in the context of turbulence theory and functionalanalysis [12][17]. Birkho ff continued the line of research in [3]. Moy investigated averaging op-erators from the viewpoint of conditional expectation in probability theory [18]. Kelley [13] andRota [22] studied the role of averaging operators in Banach algebras.Contrary to the above studies of analytic nature, the algebraic study on averaging operators be-gan with the Ph.D thesis of Cao [5]. He constructed explicitly free unitary commutative averagingalgebras and discovered the Lie algebra structures induced naturally from averaging operators. In-spired by the work of Loday [16] who defined a dialgebra as the enveloping algebra of a Leibnizalgebra, Aguiar associated a dissociative algebra to an averaging associative algebra [1]. In apaper [19] Guo and Pei studied averaging operators from an algebraic and combinatorial pointof view. They first construct free nonunital averaging algebras in terms of a class of bracketedwords called averaging words, related them to large Schr?der numbers and unreduced trees froman operadic point of view. Another paper by Gao and Zhang [9] contains an explicit construc-tion of free unital averaging algebras in terms of bracketed polynomials and the main tools wererewriting systems and Gr¨obner-Shirshov bases.The averaging operators attract much attention also partly because of their closely relationshipto other operators such as Reynolds operators, symmetric operators and Rota-Baxter operators[4][8][25], the latter having many applications in many other fields of mathematics [2][11].In the paper we study averaging operators from the viewpoint of deformation theory. We con-struct a cohomology theory which controls simultaneous deformations of associative multiplica-tion and the averaging operator. Because of the complexity of the problem, the cochain complexis no longer a di ff erential graded Lie algebra (DGLA) but an L ∞ -algebra. We give explicitly thehigher Lie brackets over the cochain complex. As a byproduct, we define homotopy averagingalgebras as Maurer-Cartan elements of this L ∞ -algebra.We would like to mention several papers which are somehow parallel to this paper. Motivatedfrom geometry and physics, Sheng, Tang and Zhu [23] studies embedding tensors (another nameof averaging operators in physics) for Lie algebras and they construct a cohomology theory forsuch operators on Lie algebras using derived brackets as a main tool. Another work of Chen,Ge and Xiang [6] studies embedding tensors using operadic tool and they compute explicitlythe Boardmann-Vogt resolution of the colored operad governing embedding tensors. We warmlyrecommend the comparative reading of their papers.The layout of this paper is as follows: The first section contains basic definitions and factsabout averaging algebras and bimodules over them. We define the cohomology theory of aver-aging operators and averaging algebras in Section 2. Deformation theory of averaging algebrasis developed in Section 4 and abelien extensions of averaging algebras are interpreted as the sec-ond cohomology group in Section 5. We construct explicitly the L ∞ -algebra structure over thecochain complex computing cohomology theory of averaging algebras in Section 6 and in the lastsection we introduce the notion of homotopy averaging algebras as Maurer-Cartan elements ofthis L ∞ -algebra. 2. A veraging algebras and their bimodule . OMOTOPY AVERAGING ALGEBRAS 3
In this section, we introduce averaging algebras and bimodules over them and present somebasic observations.
Definition 2.1.
Let ( R , · ) be an associative algebra over field k . If is endowed with a linearoperator A : R → R such that(1) A ( x ) A ( y ) = A ( xA ( y )) = A ( A ( x ) y ) , ∀ x , y ∈ R , then we call ( R , · , A ) an averaging algebra.Given two averaging algebras ( R , · , A ) and ( R ′ , · ′ , A ′ ), a morphism of averaging algebras from( R , · , A ) to ( R ′ , · ′ , A ′ ) is a homomorphism of algebras φ : ( R , · ) → ( S , · ′ ) satisfying φ ◦ A = A ′ ◦ φ. Definition 2.2.
Let ( R , · , A ) be an averaging algebra.(i) A bimodule over the averaging algebra ( R , · , A ) is a bimodule M over associative algebra( R , · ) endowed with an operator A M : M → M , such that for any r ∈ R , m ∈ M , thefollowing equalities hold: A ( r ) A M ( m ) = A M ( A ( a ) m ) = A M ( aA M ( m )) , (2) A M ( m ) A ( r ) = A M ( A M ( m ) a ) = A M ( mA ( a )) . (3) (ii) Given two bimodules ( M , A M ) and ( N , A N ) over averaging algebra ( R , · , A ), a morphismfrom ( M , A M ) to ( N , A N ) is a bimodule morphism f : M → N over ( R , · ) such that : f ◦ A M = A N ◦ f . The algebra R itself is naturally a bimodule over ( R , · , A ) with A R = A , called the regularbimodule.The following easy result is left as an exercise. Proposition 2.3.
Let ( M , A M ) be a bimodule over averaging algebra ( R , · , A ) . Then R ⊕ M is anaveraging algebra, where the averaging operator is A ⊕ A M and the multiplication is: ( a , m ) · ( b , n ) = ( ab , an + mb ) , for all a , b ∈ R , m , n ∈ M. Proposition 2.4.
Let ( R , · , A ) be an averaging algebra. Then the following operations:a ⋆ b : = aA ( b ) , a ⋄ b : = A ( a ) bare both associative.Conversely, let ( R , · ) be a unital algebra over k endowed with a linear operator A such thatoperations ⋆, ⋄ are associative. Then ( R , · , A ) is an averaging algebra.Proof. By definition, we have identities for any a , b , c ∈ R :( a ⋆ b ) ⋆ c = ( a · A ( b )) ⋆ c = a · A ( b ) A ( c ) = a · ( A ( b · Ac )) = a ⋆ ( b ⋆ c ) . Thus the operation ⋆ is associative. Analogously, one can check the associativity of operation ⋄ . KAI WANG AND GUODONG ZHOU
Conversely, assume that unital algebra ( R , · ) is endowed with a linear map A : R → R such thatoperations ⋆, ⋄ is associative. Then A ( a ) A ( b ) = (1 R · A ( a )) A ( b ) = (1 R ⋆ a ) ⋆ b = R ⋆ ( a ⋆ b ) = R ⋆ ( a · A ( b )) = R · A ( a · A ( b )) = A ( a · ( A ( b ))) . hold for any a , b ∈ R . Similarly, equality A ( a ) A ( b ) = A ( A ( a ) b ) also holds. Thus ( R , · , A ) is anaveraging algebra. (cid:3) Proposition 2.5.
Let ( R , · , A ) be an averaging algebra. Then ( R , ⋆, A ) , ( R , ⋄ , A ) are also averagingalgebras.Proof. For any a , b ∈ R , we have A ( a ) ⋆ A ( b ) = A ( a ) · ( A ( A ( b ))) = A ( a · A ( b )) = A ( A ( a ) · A ( b )) , and A ( a ⋆ A ( b )) = A ( a · A ( b )) , A ( A ( a ) ⋆ b ) = A ( A ( a ) · A ( b )) . Thus the relations A ( a ) ⋆ A ( b ) = A ( a ⋆ A ( b )) = A ( A ( a ) ⋆ b ) hold. So ( R , ⋆, A ) is an averagingalgebra. It is analogous to prove that ( R , ⋄ , A ) is an averaging algebra. (cid:3) Let ( R , · , A ) be an averaging algebra and ( M , A M ) be a bimodule over ( R , · , A ). Then we canmake M into a bimodule over ( R , ⋆ ) and ( R , ⋄ ). For any a ∈ R and m ∈ M , we define thefollowing actions: a ⊢ m : = A ( a ) m − A M ( am ) , m ⊣ a : = mA ( a )and a ⊲ m : = A ( a ) m , m ⊳ a : = mA ( a ) − A M ( mb ) . Proposition 2.6.
The action ( ⊢ , ⊣ ) (resp. ( ⊲ , ⊳ ) ) makes M become a bimodule over associativealgebra ( R , ⋆ ) (resp. ( R , ⋄ ) ).Proof. For any a , b ∈ R and m ∈ M , we have a ⊢ ( b ⊢ m ) = a ⊢ ( A ( b ) m − A M ( bm )) = A ( a ) · ( A ( b ) m − A M ( bm )) − A M ( a · A ( b ) · m − a · A M ( b · m )) = A ( a ) · A ( b ) · m − A M ( a · A ( b ) · m ) , OMOTOPY AVERAGING ALGEBRAS 5 ( a ⋆ b ) ⊢ m = ( a · A ( b )) ⊢ m = A ( a · A ( b )) · m − A M ( a · A ( b ) · m ) = a ⊢ ( b ⊢ m ) . Thus operation ⊢ makes M into a left module over ( R , ⋆ ). Analogously, operation ⊣ makes M intoa right module over ( R , ⋆ ).Moreover, we have : a ⊢ ( m ⊣ b ) = a ⊢ ( m · A ( b )) = A ( a ) · m · A ( b ) − A M ( a · m · A ( b )) = A ( a ) · m · A ( b ) − A M ( a · m ) · A ( b ) = ( A ( a ) · m − A M ( a · m )) · A ( b ) = ( a ⊢ m ) · A ( b ) = ( a ⊢ m ) ⊣ b Thus operations ( ⊢ , ⊣ ) makes M into a bimodule over associative algebra ( R , ⋆ ).Similarly, one can check that operations ( ⊲ , ⊳ ) makes M into a bimodule over ( R , ⋄ ). (cid:3)
3. C ohomology theory of averaging algebras
Let M be a bimodule over an associative algebra R. Recall that the Hochschild cohomology of R with coe ffi cients in M : ( C • Alg ( A , M ) = ∞ M n = C n Alg ( R , M ) , δ ) , where C n Alg ( R , M ) = (Hom( R ⊗ n , M ) and the di ff erential δ : C n Alg ( R , M ) → C n + ( R , M ) is given by δ ( f )( x ⊗ . . . ⊗ x n + ) = x f ( x ⊗ . . . ⊗ x n ) + n X i = ( − i f ( x ⊗ . . . ⊗ x i x i + ⊗ . . . ⊗ x n + ) + ( − n + f ( x ⊗ . . . ⊗ x n ) x n + for all f ∈ C n ( R , M ) , x , . . . , x n + ∈ R . The corresponding Hochschild cohomology is denotedHH • ( R , M ). When M = R , just denote the Hochschild cochain complex with coe ffi cients in R by C • Alg ( R ) and denote the Hochschild cohomology by HH • ( R ).3.1. Cohomology of averaging operators.
Let ( R , · , A ) be an averaging algebra and ( M , A M ) bea bimodule over ( R , · , A ). In this subsection, we define the cohomology of averaging operators.By Proposition 2.4, the averaging operator A induces two new multiplications ⋆ and ⋄ on R , andby Proposition 2.6, operations ( ⊢ , ⊣ ) (resp. ( ⊲ , ⊳ )) makes M into a bimodule over ( R , ⋆ ) (resp.( R , ⋄ )). KAI WANG AND GUODONG ZHOU
Consider the Hochschild cochain complex of ( R , ⋆ ) with the coe ffi cient in bimodule ( M , ⊢ , ⊣ ).Denote this cochain complex by C • r ( R , M ) = ∞ M i = C nr ( R , M ) , where C nr ( R , M ) = Hom( R ⊗ n , M ) and its di ff erential ∂ r : C nr ( R , M ) → C n + r ( R , M ) is given by : ∂ r ( f )( x ⊗ . . . ⊗ x n + ) = x ⊢ f ( x ⊗ . . . ⊗ x n + ) + n X i = ( − i f ( x ⊗ . . . ⊗ x i ⋆ x i + ⊗ . . . ⊗ x n + ) + ( − n + f ( x ⊗ . . . x n ) ⊣ x n + = A ( x ) f ( x ⊗ . . . ⊗ x n + ) − A M (cid:0) x f ( x ⊗ . . . ⊗ x n + ) (cid:1) + n X i = ( − i f ( x ⊗ . . . ⊗ x i A ( x i + ) ⊗ . . . x n + ) + ( − n + f ( x ⊗ . . . ⊗ x n ) A ( x n + ) . for all f ∈ C nr ( R , M ) , x , . . . , x n + ∈ R .For algebra ( R , ⋄ ), we denote its Hochschild cochain complex with coe ffi cients in bimodule( M , ⊲ , ⊳ ) by C • l ( R , M ) = ∞ M i = C nl ( R , M ) , where C nl ( R , M ) = Hom( R ⊗ n , M ) and by definition, its di ff erential ∂ l : C nl ( R , M ) → C n + l ( R , M ) isgiven by: ∂ l ( g )( x ⊗ . . . ⊗ x n + ) = x ⊲ g ( x ⊗ . . . ⊗ x n + ) + n X i = ( − i g ( x ⊗ . . . ⊗ x i ⋄ x i + ⊗ . . . ⊗ x n + ) + ( − n + g ( x ⊗ . . . ⊗ x n ) ⊳ x n + = A ( x ) g ( x ⊗ . . . ⊗ x n + ) + n X i = ( − i g ( x ⊗ . . . ⊗ A ( x i ) x i + ⊗ . . . ⊗ x n + ) + ( − n + g ( x ⊗ . . . ⊗ x n ) A ( x n + ) + ( − n A M ( g ( x ⊗ . . . ⊗ x n ) x n + )for all g ∈ C nl ( R , M ), x , . . . , x n ∈ R . When M = R , we just denote C • l ( R , R ) by C • l ( R ), and denote C • r ( R , R ) by C • r ( R ).Identifying C r ( R , M ) with C l ( R , M ), identifying C l ( R , M ) with C r ( R , M ) and taking the directsum of C nl ( R , M ) and C nr ( R , M ) for n >
2, we get the following complex:Hom( k , M ) ∂ / / Hom( R , M ) ∂ r ∂ l / / C r ( R , M ) L C l ( R , M ) ∂ r ∂ l / / · · · · · · C nr ( R , M ) L C nl ( R , M ) ∂ r ∂ l / / C n + r ( R , M ) L C n + l ( R , M ) · · · · · · , where ∂ is defined as: for f ∈ Hom( k , M ), assume that f (1) = m , then ∂ ( f )( r ) = A M ( mr ) − A M ( rm ) − mA ( r ) + A ( r ) m OMOTOPY AVERAGING ALGEBRAS 7 for any r ∈ R . One can check that ∂ r ∂ l ! ◦ ∂ = . We denote the above complex by C • AvO ( R , M ), called the cochain complex of the averagingoperator A with coe ffi cients in bimodule ( M , A M ). Its cohomology is denoted by H • AvO ( R , M ),called the cohomology of the averaging operator A of averaging algebra ( R , · , A ) with coe ffi cientsin ( M , A M ). When ( M , A M ) = ( R , A ), we just denote C • AvO ( R , R ) by C • AvO ( R ) and call it the cochaincomplex of averaging operator. We denote H • AvO ( R , M ) by H • AvO ( R ), called the cohomology ofaveraging operator A .3.2. Cohomology of averaging algebras.
In this subsection, we’ll define the cohomology ofaveraging algebras and such a cohomology theory involves both the multiplication and the aver-aging operator in an averaging algebra.Firstly, let’s build a morphism of complexes: Φ : C • Alg ( R , M ) → C • AvO ( R , M ), C ( R , M ) Φ (cid:15) (cid:15) δ / / C ( R , M ) Φ (cid:15) (cid:15) δ / / C ( R , M ) Φ (cid:15) (cid:15) C n Alg ( R , M ) Φ n (cid:15) (cid:15) δ n / / C n + ( R , M ) Φ n + (cid:15) (cid:15) Hom( k , M ) ∂ / / Hom( R , M ) ∂ r ∂ l / / C r ( R , M ) L C l ( R , M ) C nr ( R , M ) L C nl ( R , M ) ∂ nr ∂ nl / / C n + r ( R , M ) L C n + l ( R , M )Here we define Φ = id , Φ ( f ) = f ◦ A − A M ◦ f for any f ∈ Hom( R , M ), and when n > Φ n = Φ nr Φ nl ! , where Φ nr : C n ( R ) Φ nr ( f ) = f ◦ A ⊗ n − A M ◦ f ◦ (id ⊗ A ⊗ n − ) , Φ nl ( f ) = f ◦ A ⊗ n − A M ◦ f ◦ ( A ⊗ n − ⊗ id)for any f ∈ C n ( R , M ). Proposition 3.1.
The morphism Φ : C • ( R , M ) → C • AO ( R , M ) is compatible with di ff erentials.Proof. The equalities Φ ◦ δ = ∂ ◦ Φ and Φ ◦ δ = ∂ r ∂ l ! ◦ Φ are easy to verify.For n ≥
2, we need to show Φ n + r Φ n + l ! ◦ δ n = ∂ nr ∂ nl ! ◦ Φ nr Φ nl ! , that is, Φ n + r ◦ δ n = ∂ nr ◦ Φ nr KAI WANG AND GUODONG ZHOU and Φ n + l ◦ δ n = ∂ nr ◦ Φ nl . We only prove the equality the first one, the second being similar.In fact, for f ∈ Hom( R ⊗ n + , M ) and x , · · · , x n + ∈ R , we have Φ n + r ( δ n f )( x , n + ) = δ n ( f )( A ( x ) , n + ) − A M ◦ δ n ( f )( x ⊗ A ( x ) , n + ) = A ( x ) f ( A ( x ) , n + ) + P ni = ( − i f ( A ( x ) , i − ⊗ A ( x i ) A ( x i + ) ⊗ A ( x ) i + , n + ) + ( − n + f ( A ( x ) , n ) A ( a n + ) − A M ( x f ( A ( x ) , n + )) + A M ◦ f ( x A ( x ) ⊗ A ( x ) , n + ) − P ni = ( − i A M ( f ( x ⊗ A ( x ) , i − ⊗ A ( x i ) A ( x i + ) ⊗ A ( x ) i + , n + )) − ( − n + A M ◦ f ( x ⊗ A ( x ) , n ) A ( x n + ) , where we use the notations: for i ≤ j , x i , j = x i ⊗ · · · ⊗ x j and A ( x ) i , j = A ( x i ) ⊗ · · · ⊗ A ( x j ); for i > j ,they are 1 ∈ k . On the other hand, ∂ nr ◦ Φ nl ( x , n + ) = A ( a ) Φ nr ( f )( x , n + ) − A M ( x Φ nr ( f )( x , n + )) + P ni = ( − i Φ nr ( f )( x , i − ⊗ x i A ( x i + ) ⊗ x i + , n + ) + ( − n + Φ nr ( f )( x , n ) A ( x n + ) = A ( x ) f ( A ( x ) , n + ) − A ( a ) A M ◦ f ( a ⊗ A ( a ) , n + ) − A M ( x f ( A ( x ) , n + )) + A M ( x f ( x ⊗ A ( a ) , n + )) + P ni = f ( A ( a ) , i − ⊗ A ( x i A ( x i + )) ⊗ A ( x ) i + , n + ) − P ni = ( − i A M ◦ f ( x ⊗ A ( x ) , i − ⊗ A ( x i A ( x i + )) ⊗ A ( x ) i + , n + ) + A M ( f ( x A ( x ) ⊗ A ( x ) , n + ) + ( − n + f ( A ( x ) , n ) A ( x n + ) − ( − n + A M ◦ f ( x ⊗ A ( x ) , n ) A ( x n + ) . We obtain Φ n + r ( δ n f )( x , n + ) = ∂ nr ◦ Φ nl ( x , n + ), because A ( a ) A M ◦ f ( a ⊗ A ( a ) , n + ) = A M ( x f ( x ⊗ A ( a ) , n + ))and A ( x i A ( x i + ) = A ( x i ) A ( x i + ).The proof is done. (cid:3) Multiplying Φ n by ( − n , then the above commutative diagram becomes a bicomplex. Takingits totalization, we obtain a cochain complex, and denote it by C • AvA ( R , M ). Definition 3.2.
The cohomology of the cochain complex C • AvA ( R ), denoted by H • AvA ( R , M ) iscalled the cohomology of the averaging algebra ( R , · , A ) with coe ffi cients in bimodule ( M , A M ).When ( M , A M ) = ( R , A ), H • AvA ( R , R ) is called the cohomology of averaging algebra ( R , · , A ) anddenoted by H • AvA ( R ).3.3. Relationship among the cohomlogies.
By the commutative diagram in the last subsection,we have a canonical short exact sequences of complexes:0 → C •− AO ( R , M ) → C • AvA ( R , M ) → C • ( R , M ) → . Then it is trivial to obtain the following result:
Theorem 3.3.
We have the following long exact sequence of cohomology groups, · · · → H n − ( R , M ) → H n AvA ( R , M ) → HH n (R , M) → H nAvO (R , M) → · · · . OMOTOPY AVERAGING ALGEBRAS 9
4. F ormal deformation of averaging algebras
Let ( R , • , A ) be an averaging algebra. Denote by µ A the multiplication • of R . Consider the1-parameterized family µ t = ∞ X i = µ i t i , µ i ∈ C ( R ) , A t = ∞ X i = A i t i , A i ∈ C ( R ) . Definition 4.1. A of an averaging algebra ( R , µ, A ) is a pair( µ t , A t ) which endows the k [[ t ]]-module ( R [[ t ]] , µ t , A t ) with the averaging algebra structure over k [[ t ]] such ( µ , A ) = ( µ, A ).Power series µ t and A t determines a 1-parameter formal deformation of the averaging algebra( R , µ, A ) if and only if for all x , y , z ∈ R , the following equations hold: µ t ( µ t ( x ⊗ y ) ⊗ z ) = µ t ( x ⊗ µ t ( y ⊗ z )) ,µ t ( A t ( x ) ⊗ A t ( y )) = A t ( µ t ( A t ( x ) ⊗ y )) ,µ t ( A t ( x ) ⊗ A t ( y )) = A t ( µ t ( x ⊗ A t ( y ))) . Expand these equations and compare the coe ffi cients of t n , we get the conditions that { µ i } i ∈ N and { A i } i ∈ N should satisfy: n X i = µ i ◦ ( µ n − i ⊗ id) = n X i = µ i ◦ (id ⊗ µ n − i ) , (4) n X i + j = µ n − i − j ◦ ( A i ⊗ A j ) = n X i + j = A n − i − j ◦ µ j ◦ ( A i ⊗ id) , (5) n X i + j = µ n − i − j ◦ ( A i ⊗ A j ) = n X i + j = A n − i − j ◦ µ j ◦ (id ⊗ A i ) . (6) Proposition 4.2.
Let ( R [[ t ]] , µ t , A t ) be a 1-parameter formal deformation of an averaging algebra ( R , µ, A ) . Then ( µ , A ) is a 2-cocycle in the cochain complex C • AvA ( R ) .Proof. Compute the equations (4) (5) (6) for n =
1, we have : µ ◦ ( µ ⊗ id) + µ ◦ ( µ ⊗ id) = µ ◦ (id ⊗ µ ) + µ ◦ (id ⊗ µ ) ,µ ◦ ( A ⊗ A ) + µ ◦ ( A ⊗ A ) + µ ◦ ( A ⊗ A ) = A ◦ µ ◦ ( A ⊗ id) + A ◦ µ ◦ ( A ⊗ id) + A ◦ µ ◦ ( A ⊗ id) ,µ ◦ ( A ⊗ A ) + µ ◦ ( A ⊗ A ) + µ ◦ ( A ⊗ A ) = A ◦ µ ◦ (id ⊗ A ) + A ◦ µ ◦ (id ⊗ A ) + A ◦ µ ◦ (id ⊗ A ) . They are equivalent to : δ ( µ ) = ,∂ r ( A ) + Φ r ( µ ) = ,∂ l ( A ) + Φ l ( µ ) = . That is, ( µ , A ) is a 2-cocycle in C • AvA ( R ). (cid:3) If µ t = µ A in the above 1-parameter formal deformation of the averaging algebra ( R , µ, A ), weget a 1-parameter formal deformation of the averaging operator A . Consequently, we have: Corollary 4.3.
Let A t be a 1-parameter formal deformation of the averaging operator A. ThenA is a 1-cocycle in the cochain complex C • AvO ( R ) . Definition 4.4.
The 2-cocycle ( µ , A ) is called the infinitesimal of the 1-parameter formal defor-mation ( R [[ t ]] , µ t , A t ) of averaging ( R , µ, A ). Definition 4.5.
Let ( R [[ t ]] , µ t , A t ) and ( R [[ t ]] , µ ′ t , A ′ t ) be two 1-parameter for deformations of av-eraging algebra ( R , µ, A ). A formal isomorphism from ( R [[ t ]] , µ ′ t , A ′ t ) to ( R [[ t ]] , µ t , A t ) is a powerseries φ t = P i > φ i t i : R [[ t ]] → R [[ t ]], where φ i : R → R , i ∈ N are linear maps with φ = id, suchthat φ t ◦ µ ′ t = µ t ◦ ( φ t ⊗ φ t ) , (7) φ t ◦ A ′ t = A t ◦ φ t . (8)Two 1-parameter formal deformations ( R [[ t ]] , µ t , A t ) and ( R [[ t ]] , µ ′ t , A ′ t ) are said to be equivalentif there exists a formal isomorphism φ t : ( R [[ t ]] , µ ′ t , A ′ t ) → ( R [[ t ]] , µ t , A t ). Theorem 4.6.
The infinitesimals of two equivalent 1-parameter formal deformations of ( R , µ, A ) are in the same cohomology class in H ( R ) . Conversely, if the infinitesimals of two 1-parameterformal deformations of ( R , µ, A ) fall into the same cohomology class, they are equivalent.Proof. Let φ t : ( R [[ t ]] , µ ′ t , A ′ t ) → ( R [[ t ]] , µ t , A t ) be a formal isomorphism. For all x , y ∈ R , wehave φ t ◦ µ ′ t ( x ⊗ y ) = µ t ◦ ( φ t ⊗ φ t )( x ⊗ y ) , (9) φ t ◦ A ′ t ( x ) = A t ◦ φ t ( x ) . (10)Expanding the identities comparing the coe ffi cients of t , we get: µ ′ ( x ⊗ y ) = µ ( x ⊗ y ) + x φ ( y ) − φ ( xy ) + φ ( x ) y , (11) A ′ ( x ) = A ( x ) + A ( φ ( x )) − φ ( A ( x )) . (12)So we have, ( µ ′ , A ′ ) − ( µ , A ) = d ( φ ) ∈ C • AvA ( R ) . That is, [( µ ′ , A ′ )] = [( µ , A )] ∈ H ( R ).Conversely, given two formal deformations ( R [[ t ]] , µ ′ t , A ′ t ) and ( R [[ t ]] , µ t , A t ) of an averagingalgebra ( R , µ, A ), suppose that ( µ , A ) and ( µ ′ , A ′ ) are in the same cohomology class of H ( R ) . (cid:3) Definition 4.7.
A 1-parameter formal deformation ( R [[ t ]] , µ t , A t ) of ( R , µ, A ) is said to be trivialif it is equivalent to be deformation ( R [[ t ]] , µ, A ), that is, there exists φ t = ∞ P i = φ i t i : R [[ t ]] → R [[ t ]],where φ : A → A are linear maps with φ = id, such that φ t ◦ µ t = µ A ◦ ( φ t ⊗ φ t ) , (13) φ t ◦ A t = A ◦ φ t . (14) Definition 4.8.
An averaging algebra ( R , µ, A ) is said to be rigid if every 1-parameter formaldeformation is trivial. Proposition 4.9.
Let ( R , µ, A ) be an averaging algebra. If H ( R ) = , ( R , µ, A ) is rigid.Proof. Let ( R [[ t ]] , µ t , A t ) be a 1-parameter formal deformation of ( R , µ, A ). By Proposition ,( µ , A ) is a 2-cocycle. By H ( R ) =
0, there exists a 1-cochain ( φ ′ , x ) ∈ C ( R , R ) ⊕ Hom( k , M )such that ( µ , A ) = − d AvA ( φ ′ , x ) , that is, µ = − δ ( φ ′ ) and A = − ∂ ( x ) − Φ ( φ ′ ). Since Φ = Id ,let φ = φ ′ + δ ( x ). Then µ = − δ ( φ ) and A = − Φ ( φ ′ ). OMOTOPY AVERAGING ALGEBRAS 11
Setting φ t = Id R + φ t , we have a deformation ( R [[ t ]] , µ t , A t ), where µ t = φ − t ◦ µ t ◦ ( φ t × φ t )and A t = φ − t ◦ A t ◦ φ t . It is not di ffi cult to see that µ t = µ + µ t + · · · , A t = A + A t + · · · . Then by repeating the argument, we can show that ( R [[ t ]] , µ t , A t ) is equivalent to the trivial exten-sion ( R [[ t ]] , µ, A ) . Thus, ( R , µ, A ) is rigid. (cid:3)
5. A belian extensions of averaging algebras
In this section, we study abelian extensions of averaging algebras and show that they are clas-sified by the second cohomology, as one would expect of a good cohomology theory.
Definition 5.1. An abelian extension of averaging algebras is a short exact sequence of homo-morphisms of averaging algebras0 −−−−−→ M i −−−−−→ ˆ R p −−−−−→ R −−−−−→ A M y ˆ A y A y −−−−−→ M i −−−−−→ ˆ R p −−−−−→ R −−−−−→ uv = u , v ∈ M . We will call ( ˆ R , ˆ A ) an abelian extension of ( R , A ) by ( M , A M ). Definition 5.2.
Let ( ˆ R , ˆ A ) and ( ˆ R , ˆ A ) be two abelian extensions of ( R , A ) by ( M , A M ). They aresaid to be isomorphic if there exists an isomorphism of averaging algebras ζ : ( ˆ R , ˆ A ) → ( ˆ R , ˆ A )such that the following commutative diagram holds:0 −−−−−→ ( M , A M ) i −−−−−→ ( ˆ R , ˆ A ) p −−−−−→ ( R , A ) −−−−−→ (cid:13)(cid:13)(cid:13)(cid:13) ζ y (cid:13)(cid:13)(cid:13)(cid:13) −−−−−→ ( M , A M ) i −−−−−→ ( ˆ R , ˆ A ) p −−−−−→ ( R , A ) −−−−−→ . A section of an abelian extension ( ˆ R , ˆ A ) of ( R , A ) by ( M , A M ) is a linear map s : R → ˆ R suchthat p ◦ s = Id R .Now for an abelian extension ( ˆ R , ˆ A ) of ( R , A ) by ( M , A M ) with a section s : R → ˆ R , we definelinear maps ρ l : R → End k ( M ) , r ( m rm ) and ρ r : R → End k ( M ) , r ( m mr )respectively by rm : = s ( r ) m , mr : = ms ( r ) , ∀ r ∈ R , m ∈ M . Proposition 5.3.
With the above notations, ( M , ρ l , ρ r , A M ) is a bimodule over the averaging alge-bra ( R , A ) . Proof.
For any x , y ∈ R , v ∈ M , since s ( xy ) − s ( x ) s ( y ) ∈ M implies s ( xy ) m = s ( x ) s ( y ) m , we have ρ l ( xy )( m ) = s ( xy ) m = s ( x ) s ( y ) m = ρ l ( x ) ◦ ρ l ( y )( m ) . Hence, ρ l is an algebra homomorphism. Similarly, ρ r is an algebra anti-homomorphism. More-over, ˆ A ( s ( r )) − s ( A ( r )) ∈ M means that ˆ A ( s ( r )) m = s ( A ( r )) m . Thus we have A ( r ) A M ( m ) = s ( A ( r )) A M ( m ) = ˆ A ( s ( r )) A M ( m ) = ˆ A ( s ( r ) A M ( m )) = ˆ A ( ˆ A ( sr ) m ) = A M ( rA ( m )) = A M ( A ( r ) m ) . It is similar to see A M ( m ) A ( r ) = A M ( A M ( m ) r ) = A M ( mA ( r )). Hence, ( M , ρ l , ρ r , A M ) is a bimoduleover ( R , A ). (cid:3) We further define linear maps ψ : R ⊗ R → M and χ : R → M respectively by ψ ( x ⊗ y ) = s ( x ) s ( y ) − s ( xy ) , ∀ x , y ∈ R ,χ ( x ) = ˆ A ( s ( x )) − s ( A ( x )) , ∀ x ∈ R . We transfer the averaging algebra structure on ˆ R to R ⊕ M by endowing R ⊕ M with a multiplication · ψ and an averaging operator A χ defined by( x , m ) · ψ ( y , n ) = ( xy , xn + my + ψ ( x , y )) , ∀ x , y ∈ R , m , n ∈ M , (15) A χ ( x , m ) = ( A ( x ) , χ ( x ) + A M ( m )) , ∀ x ∈ R , m ∈ M . (16) Proposition 5.4.
The triple ( R ⊕ M , · ψ , A χ ) is a averaging algebra if and only if ( ψ, χ ) is a 2-cocycleof the averaging algebra ( R , A ) with the coe ffi cient in ( M , A M ) .Proof. If ( A ⊕ M , · ψ , A χ ) is a averaging algebra, then the associativity of · ψ implies(17) x ψ ( y ⊗ z ) − ψ ( xy ⊗ z ) + ψ ( x ⊗ yz ) − ψ ( x ⊗ y ) z = , which means δ ( φ ) = C • ( A , M ). Since A χ is an averaging operator, for any x , y ∈ R , m , n ∈ M ,we have A χ (( x , m )) · ψ A χ (( y , n )) = A χ ( A χ ( x , m ) · ψ ( y , n )) = A χ (( x , m ) · ψ A χ ( y , n ))Then χ, ψ satisfy the following equations:[ ψ ( A ( x ) ⊗ A ( y )) − A M ( ψ ( A ( x ) ⊗ y ))] + [ A ( x ) χ ( y ) − A M ( χ ( x ) y ) − χ ( A ( x ) y ) + χ ( x ) A ( y )] = ψ ( A ( x ) ⊗ A ( y )) − A M ( ψ ( x ⊗ A ( y )))] + [ A ( x ) χ ( y ) − A M ( x χ ( y )) − χ ( xA ( y )) + χ ( x ) A ( y )] = ∂ r ( χ ) + Φ r ( ψ ) = , ∂ l ( χ ) + Φ l ( ψ ) = . Hence, ( ψ, χ ) is a 2-cocycle.Conversely, if ( ψ, χ ) is a 2-cocycle, one can easily check that ( R ⊕ M , · ψ , A χ ) is an averagingalgebra. (cid:3) Now we are ready to classify abelian extensions of an averaging algebra.
Theorem 5.5.
Let M be a vector space and A M ∈ End k ( M ) . Then abelian extensions of anaveraging algebra ( R , A ) by ( M , A M ) are classified by the second cohomology group H ( R , M ) of ( R , A ) with coe ffi cients in the bimodule ( M , A V ) . OMOTOPY AVERAGING ALGEBRAS 13
Proof.
Let ( ˆ R , ˆ A ) be an abelian extension of ( R , A ) by ( M , A M ). We choose a section s : R → ˆ R toobtain a 2-cocycle ( ψ, χ ) by Proposition 5.4. We first show that the cohomological class of ( ψ, χ )does not depend on the choice of sections. Indeed, let s and s be two distinct sections providing2-cocycles ( ψ , χ ) and ( ψ , χ ) respectively. We define φ : R → M by γ ( r ) = s ( r ) − s ( r ). Then ψ ( x , y ) = s ( x ) s ( y ) − s ( xy ) = ( s ( x ) + γ ( x ))( s ( y ) + γ ( y )) − ( s ( xy ) + γ ( xy )) = ( s ( x ) s ( y ) − s ( xy )) + s ( x ) γ ( y ) + γ ( x ) s ( y ) − γ ( xy ) = ( s ( x ) s ( y ) − s ( xy )) + x γ ( y ) + γ ( x ) y − γ ( xy ) = ψ ( x , y ) + δ ( γ )( x , y )and χ ( x ) = ˆ A ( s ( x )) − s ( A ( x )) = ˆ A ( s ( x ) + γ ( x )) − ( s ( A ( x )) + γ ( A ( x ))) = ( ˆ A ( s ( x )) − s ( A ( x ))) + ˆ A ( γ ( x )) − γ ( A ( x )) = χ ( x ) + d V ( γ ( x )) − γ ( d A ( x )) = χ ( x ) − Φ ( γ )( x ) . That is, ( ψ , χ ) = ( ψ , χ ) + d ( γ ). Thus ( ψ , χ ) and ( ψ , χ ) are in the same cohomological classin H ( R , M ).Next we prove that isomorphic abelian extensions give rise to the same element in H ( R , M ).Assume that ( ˆ R , ˆ A ) and ( ˆ R , ˆ A ) are two isomorphic abelian extensions of ( R , A ) by ( M , A M )with the associated homomorphism ζ : ( ˆ R , ˆ A ) → ( ˆ R , ˆ A ). Let s be a section of ( ˆ R , ˆ A ). As p ◦ ζ = p , we have p ◦ ( ζ ◦ s ) = p ◦ s = Id R . Therefore, ζ ◦ s is a section of ( ˆ R , ˆ A ). Denote s : = ζ ◦ s . Since ζ is a homomorphism ofdi ff erential algebras such that ζ | M = Id M , we have ψ ( x ⊗ y ) = s ( x ) s ( y ) − s ( xy ) = ζ ( s ( x )) ζ ( s ( y )) − ζ ( s ( xy )) = ζ ( s ( x ) s ( y ) − s ( xy )) = ζ ( ψ ( x , y )) = ψ ( x , y )and χ ( x ) = ˆ A ( s ( x )) − s ( A ( x )) = ˆ A ( ζ ( s ( x ))) − ζ ( s ( A ( x ))) = ζ ( ˆ A ( s ( x )) − s ( A ( x ))) = ζ ( χ ( x )) = χ ( x ) . Consequently, all isomorphic abelian extensions give rise to the same element in H ( R , M ).Conversely, given two 2-cocycles ( ψ , χ ) and ( ψ , χ ), we can construct two abelian extensions R ⊕ M , · ψ , A χ ) and ( R ⊕ M , · ψ , A χ ) via equalities (15) and (16). If they represent the samecohomological class in H ( R , M ), then there exists a linear map γ : R → M such that( ψ , χ ) = ( ψ , χ ) + ( δ ( γ ) , Φ ( γ )) . Define ζ : R ⊕ M → R ⊕ M by ζ ( r , m ) : = ( r , γ ( r ) + m ) . Then ζ is an isomorphism of these two abelian extensions. (cid:3) L ∞ - structure on the cochain complex Let’s recall the definition of L ∞ -algebras.For graded indeterminates t , . . . , t n and σ ∈ S n , the Koszul sign ǫ ( σ, t , . . . , t n ) is defined by t · t · · · · t n = ǫ ( σ, t , . . . , t n ) t σ (1) · t σ (2) · · · · t σ ( n ) , where “ · ” is the multiplication in the free graded commutative algebra k h t , . . . , t n i / ( t i t j − ( − | t i || t j | t j t i )generated by t , . . . , t n . Define also χ ( σ, t , . . . , t n ) = sgn ( σ ) ǫ ( σ, t , . . . , t n ). Definition 6.1. ([24, 15, 14]) Let L = L i ∈ Z L i be a graded space over k . Assume that L is endowedwith a family of linear operators { l n : L ⊗ n → L } n > with | l n | = n − l n ( x σ (1) ⊗ . . . ⊗ x σ ( n ) ) = χ ( σ, x , . . . , x n ) l n ( x ⊗ . . . ⊗ x n ) , ∀ σ ∈ S n , x , . . . , x n ∈ L ,(ii) n P i = P σ ∈ S ( i , n − i ) χ ( σ, x , . . . , x n )( − i ( n − i ) l n − i + ( l i ( x σ (1) ⊗ . . . ⊗ x σ ( i ) ) ⊗ x σ ( i + ⊗ . . . ⊗ x σ ( n ) ) = , where S ( i , n − i ) is the set of all ( i , n − i )-shu ffl es, i.e., S ( i , n − i ) = { σ ∈ S n , | σ (1) <σ (2) < · · · < σ ( i ) , σ ( i + < σ ( i + < · · · < σ ( n ) } ,Then ( L , { l n } n > ) is called a L ∞ -algebra.Let sL be the suspension of L , i.e., ( sL ) n = L n − . Let S • ( sL ) be the cofree cocommutativecoalgebra generated by sL . Then { l n } n > will determine a family of operators { d n } n > , where d n : ( sL ) ⊗ n → sL is defined as d n = ( − n ( n − s ◦ l n ◦ ( s − ⊗ n ). Then { d n } n > satisfies the equation X σ ∈ S ( i , n − i ) ε ( σ, sx , . . . , sx n ) d n − i + ( d i ( sx σ (1) , . . . , sx σ ( i ) ) , sx σ ( i + , . . . , sx σ ( n ) ) = . (18)Denote ∞ P i = d n : S • ( sL ) → sL by d ′ . Then d ′ can induce a coderivation d on S • ( sL ). Equation(18) ensures that the coderivation d is a di ff erential, i.e., d =
0. This can be considered as anequivalent definition of L ∞ -algebra. Definition 6.2.
Let ( L , { l n } n > ) be a L ∞ -algebra and α ∈ L − . We call α a Maurer-Cartan elementif it satisfies the following equation: ∞ X n = n ! ( − n ( n − l n ( α ⊗ n ) = . Proposition 6.3.
Given a Maurer-Cartan element α in L ∞ -algebra L, we can define a new L ∞ -structure { l α n } n > on L, where l α n : L ⊗ n → L is defined as:l α n ( x ⊗ . . . ⊗ x n ) = ∞ X i = i ! ( − in + i ( i + + n − P k = k P j = | x j | l n + i ( α ⊗ i ⊗ x ⊗ . . . ⊗ x n ) . Let α be a Maurer Cartan element in L ∞ algebra L . Use the equivalent definition of L ∞ -algebra,we can see s α ∈ sL satisfies the following equation on S • ( sL ): ∞ X n = n ! d n (cid:0) ( s α ) ⊗ n (cid:1) = . OMOTOPY AVERAGING ALGEBRAS 15
Define d α n ( sx ⊗ . . . sx n ) = ∞ X i = i ! d n + (cid:0) ( s α ) ⊗ i ⊗ sx ⊗ . . . ⊗ sx n (cid:1) . Then the family { d α n } n > can also induce a di ff erential on S • ( sL ).Let W be a graded space and T c ( W ) be the cofree coalgebra generated by W . For f ∈ Hom( W ⊗ n , W ) , g i ∈ Hom( W ⊗ l i , W ) , > i > m , f ¯ ◦ ( g , . . . , g m ) ∈ Hom( W ⊗ l + ··· + l m + n − m , W ) is de-fined as follows: (cid:0) f ¯ ◦ ( g , . . . , g m ) (cid:1) ( w ⊗ . . . ⊗ w l + ··· + l m + n − m ) = X i i ··· i m ( − η f (cid:16) w ⊗ . . . w i ⊗ g ( w i + ⊗ . . . ) ⊗ . . . g k ( w i k + ⊗ . . . ) ⊗ . . . ⊗ g m ( w i m + ⊗ . . . ) ⊗ . . . (cid:17) where η = m P k = | g k | ( i k P j = | w j | ).For any f , g ∈ Hom( T c ( W ) , W ), their Gerstenhaber bracket [ f , g ] G ∈ Hom( T c ( W ) , W ) is de-fined as : [ f , g ] G = f ¯ ◦ g − ( − | f || g | g ¯ ◦ f . Lemma 6.4. [10]
Let W be a graded space. Then (Hom( T c ( W ) , W ) , [ − , − ] G ) forms a gradedLie-algebra. Now, given a graded space V = L i ∈ Z V i , we will define a L ∞ -algebra C AvA ( V ). We will seethat when V is concentrated in degree 0, an averaging algebra structure on V is equivalent to aMaurer-Cartan element in this L ∞ -algebra.Firstly, the underlying graded space of C AvA ( V ) is C A ( V ) ⊕ Hom( k , V ) ⊕ Hom( sV , V ) ⊕ C AvO ( V ) > l ⊕ C AvO ( V ) > r . where C A ( V ) = Hom( T c ( sV ) , sV ) , C AvO ( V ) > l = Hom( ∞ M n = ( sV ) ⊗ n , V ) , C AvO ( V ) > r = Hom( ∞ M n = ( sV ) ⊗ n , V ) . And we define C AvO ( V ) r = Hom( k , V ) ⊕ Hom( sV , V ) ⊕ C AvO ( V ) > r C AvO ( V ) l = Hom( k , V ) ⊕ Hom( sV , V ) ⊕ C AvO ( V ) > l Now, let’s give a L ∞ -algebra structure on C AvA ( V ), i.e., we need to give a family of opera-tors { l n } n > to satisfy the conditions in Definition 6.1. Here, we identify Hom( T c ( sV ) , sV ) with s Hom( T c ( sV ) , V ).The family { l i } i > are defined by the following processes:(I) For sh ∈ Hom( k , sV ) ⊂ C A ( V ), l ( sh ) = h . And l vanishes elsewhere. (II) For sg , sh ∈ C A ( V ), l ( sg ⊗ sh ) = [ sg , sh ] G .(III) For homogeneous elements sh ∈ Hom(( sV ) ⊗ n , sV ) in C A ( V ) and g , . . . , g n ∈ C AvO ( V ) r ,define l rn + ( sh ⊗ g ⊗ . . . ⊗ g n ) ∈ C AvO ( V ) r in the following ways:(i) If g , . . . , g n are all contained in Hom( sV , V ) L C AvO ( V ) > r , then l rn + ( sh ⊗ g ⊗ . . . ⊗ g n ) = X σ ∈ S n ( − ε (cid:16) h ◦ ( sg σ (1) ⊗ . . . ⊗ sg σ ( n ) ) − ( − ( | g σ (1) | + | h | + g σ (1) ¯ ◦ ( sh ◦ (id sV ⊗ sg σ (2) ⊗ . . . ⊗ sg σ ( n ) )) (cid:17) (ii) If there exists some g i coming from Hom( k , V ), then l rn + ( sh ⊗ g ⊗ . . . ⊗ g n ) = X σ ∈ S n ( − ε (cid:16) h ◦ ( sg σ (1) ⊗ . . . ⊗ sg σ ( n ) ) − ( − (cid:0) | g σ ( p ) | + (cid:1)(cid:0) | h | + + p − P k = ( | g σ ( k ) | + (cid:1) g σ ( p ) ¯ ◦ ( sh ◦ ( g σ (1) ⊗ sg σ (2) ⊗ . . . ⊗ g σ ( p − ⊗ id sV ⊗ sg σ ( p + ⊗ . . . ⊗ sg σ ( n ) )) (cid:17) where ( − ε = χ ( σ ; g , . . . , g n ) · ( − n ( | h | + + n − P k = k P j = | g σ ( j ) | , and p is the integer such that g σ (1) , . . . , g σ ( p − ∈ Hom( k , V ) and g σ ( p ) ∈ Hom( sV , V ) ⊕ C AvO ( V ) > r .(IV) Let sh ∈ Hom(( sV ) ⊗ n , sV ) in C A ( V ), g , . . . , g n ∈ C AvO ( V ) l be homogeneous elements.Define l ln + ( sh ⊗ g ⊗ . . . ⊗ g n ) ∈ C AvO ( V ) l in the following ways:(i) If g , . . . , g n all belong to Hom( sV , V ) L C AvO ( V ) > l , then define l ln + ( sh ⊗ g ⊗ . . . ⊗ g n ) = X σ ∈ S n ( − ε (cid:16) h ◦ ( sg σ (1) ⊗ . . . ⊗ sg σ ( n ) ) − ( − ( g σ ( n ) + | h | + + n − P k = ( | g σ ( k ) | + g σ ( n ) ¯ ◦ ( sh ◦ ( sg σ (1) ⊗ . . . ⊗ sg σ ( n − ⊗ id sV )) (cid:17) , (ii) If there exists some g i ∈ Hom( k , V ), then define: l ln + ( sh ⊗ g ⊗ . . . ⊗ g n ) = X σ ∈ S n ( − ε (cid:16) h ◦ ( sg σ (1) ⊗ . . . ⊗ sg σ ( n ) ) − ( − (cid:0) | g σ ( q ) | + (cid:1)(cid:0) | h | + + q − P k = ( | g σ ( k ) | + (cid:1) g σ ( q ) ¯ ◦ ( sh ◦ ( sg σ (1) ⊗ . . . ⊗ sg σ ( q − ⊗ id sV ⊗ ⊗ g σ ( q + ⊗ . . . ⊗ sg σ ( n ) )) (cid:17) . Where q is the integer such that g σ ( q + , . . . , g σ ( n ) ∈ Hom( k , V ) and g σ ( q ) ∈ Hom( sV , V ) ⊕C AvO ( V ) > l .(V) (i) If g , . . . , g n ∈ Hom( k , V ) ⊕ Hom( sV , V ) = C AvO ( V ) l ∩ C AvO ( V ) r with at most one g i ∈ Hom( sV , V ), then the definitions of l rn + , l ln + coincide. Then we define l n + ( sh , g , . . . , g n ) : = l rn + ( sh , g , . . . , g n ) = l ln + ( sh , g , . . . , g n ) . (ii) If g , . . . , g n ∈ Hom( k , V ) ⊕ Hom( sV , V ) = C AvO ( V ) l ∩ C AvO ( V ) r with at least two g i ∈ Hom( sV , V ), then l ln + ( sh , g , . . . , g n ) ∈ C AvO ( V ) > l and l rn + ( sh , g , . . . , g n ) ∈C AvO ( V ) > r . Then we define l n + ( sh , g , . . . , g n ) = ( l rn + ( sh , g , . . . , g n ) , l ln + ( sh , g , . . . , g n )) ∈ C AvO ( V ) > r ⊕ C AvO ( V ) > l . OMOTOPY AVERAGING ALGEBRAS 17 (iii) For g , . . . , g n ∈ C AvO ( V ) r with some g i ∈ C AvO ( V ) > r , define l n + ( sh , g , . . . , g n ) : = l rn + ( sh , g , . . . , g n ) . For g , . . . , g n ∈ C AvO ( V ) l with some g i ∈ C AvO ( V ) > l , define l n + ( sh , g , . . . , g n ) : = l rn + ( sh , g , . . . , g n ) . (VI) At last we define l n + ( g ⊗ . . . ⊗ g i ⊗ sh ⊗ g i + ⊗ . . . ⊗ g n ) = ( − ( | h | + i P k = | g k | ) + i l n + ( sh ⊗ g ⊗ . . . ⊗ g n ) . And l n vanishes elsewhere. Theorem 6.5.
Let V be a graded space. Then C AvA ( V ) endowed with operations { l n } n > definedabove forms a L ∞ -algebra. Now, let’s realize averaging algebra structures as Maurer-Cartan elements in this L ∞ -algebra. Theorem 6.6.
Let R be a vector space. Then an averaging structure on R is equivalent to aMaurer-Cartan elements in C AvA ( R ) .Proof. Consider R as a graded vector space concentrated in degree 0. Then the degree -1 part of C AvA ( R ) is C AvA ( R ) − = Hom(( sV ) ⊗ , sV ) M Hom( sV , V ) . Assume that α = ( m , τ ) be a Maurer-Cartan element in C AvA ( R ), that is, α satisfy the equation: ∞ X k = k ! ( − k ( k − l k ( α ⊗ k ) = . (19)By the definietion of { l n } n > ,12! ( − l ( α ⊗ α ) = −
12 [ m , m ] G ,
13! ( − l ( α ⊗ ) = − (cid:16) l r ( m ⊗ τ ⊗ τ ) + l r ( τ ⊗ m ⊗ τ ) + l r ( τ ⊗ τ ⊗ m ) , l l ( m ⊗ τ ⊗ τ ) + l l ( τ ⊗ m ⊗ τ ) + l l ( τ ⊗ τ ⊗ m ) (cid:17) = − (cid:16) l r ( m ⊗ τ ⊗ τ ) , l l ( m ⊗ τ ⊗ τ ) (cid:17) = − (cid:16) s − m ◦ ( s τ ⊗ s τ ) − τ ◦ ( sm ◦ (id ⊗ s τ )) , s − m ◦ ( s τ ⊗ s τ ) − τ ◦ ( sm ◦ ( s τ ⊗ id)) (cid:17) Then Equation (19) implies that [ m , m ] G = s − m ◦ (( s τ ) ⊗ ) − τ ¯ ◦ ( sm ◦ (id ⊗ s τ )) = s − m ◦ (( s τ ) ⊗ ) − τ ¯ ◦ ( sm ◦ ( s τ ⊗ id)) = . (22)Define µ = s − ◦ m ◦ s ⊗ : A ⊗ → A , A = τ ◦ s : A → A . Then Equation (20) is equivalent to µ ◦ (id ⊗ µ ) = µ ◦ ( µ ⊗ id) . That is, µ is associative. Equations (21) (22) imply µ ◦ ( A ⊗ A ) = A ◦ ( µ ◦ (id ⊗ A )) , µ ◦ ( A ⊗ A ) = A ◦ ( µ ◦ ( A ⊗ id)) , which means that A is an averaging operator on associative algebra ( R , µ ). Conversely, let ( R , µ, A )be an averaging algebra. Define m : ( sR ) ⊗ → sR , τ : sR → R to be m ( sa ⊗ sb ) = s µ ( a ⊗ b ), τ ( sa ) = A ( a ). Then ( m , τ ) is a Maurer-Cartan element in C AvA ( R ). (cid:3) Proposition 6.7.
Let ( R , µ, A ) be an averaging algebra and α = ( m , τ ) be the correspondingMaurer-Cartan element in C AvA ( R ) . Then the underground complex of L ∞ -algebra ( C AvA ( R ) , { l α } ) is exactly the cochain complex sC • AvA ( R ) defined in Section 3.Proof. Identify C • ( R ) with C A ( R ) and identify C • AvO with Hom( k , R ) ⊕ Hom( sR , R ) ⊕ C AvO ( R ) > r ⊕C AvO ( R ) > l . Let s f ∈ Hom(( sR ) ⊗ n , sR ). Then l α ( s f ) = ∞ X k = k ! ( − k ( k + + k l k + ( α ⊗ k ⊗ s f ) = l ( m ⊗ s f ) + n ! ( − n ( n − l n + ( τ ⊗ n ⊗ s f ) , where l ( m ⊗ s f ) = [ m , s f ] G , it’s just the di ff erential δ of Hochschild cochain complex. We have1 n ! ( − n ( n − l n + ( τ ⊗ n ⊗ s f ) = n ! ( − n ( n − + n ( l rn + ( s f ⊗ τ ⊗ n ) , l ln + ( s f ⊗ τ ⊗ n )) . Then by definition,1 n ! ( − n ( n − + n l rn + ( s f ⊗ τ ⊗ n ) = ( − n ( f ◦ (( s τ ) ⊗ n ) − τ ◦ ( s f ◦ (id ⊗ ( s τ ) ⊗ n − ))) , which is exactly the same as ( − n Φ nr defined in Section 3.2. Similarly, we can see l ln + ( s f ⊗ τ ⊗ n )is just ( − n Φ nl . In particular, for s f ∈ Hom( sR , sR ), l ( s f ⊗ τ ) = l r ( s f ⊗ τ ) = l l ( s f ⊗ τ ) = − ( f ◦ s τ − τ ◦ ( s f )), it is the same as − Φ .For g ∈ C AvO ( R ) r , we have ∞ X k = k ! ( − k ( k + + k l k + ( α ⊗ k ⊗ g ) = − (cid:0) l ( m ⊗ τ ⊗ g ) + l ( τ ⊗ m ⊗ g ) (cid:1) = − l ( m ⊗ τ ⊗ g ) = − (cid:16)(cid:0) − s − m ◦ ( s τ ⊗ sg ) + τ ◦ ( m ◦ (id ⊗ sg )) (cid:1) − (cid:0) s − m ◦ ( sg ⊗ s τ ) − g ¯ ◦ ( m ◦ (id ⊗ s τ )) (cid:1)(cid:17) = s − m ◦ ( s τ ⊗ sg ) − τ ◦ ( m ◦ (id ⊗ sg )) − g ¯ ◦ ( m ◦ (id ⊗ τ )) + s − m ◦ ( sg ⊗ s τ ) , and it is the same as ∂ r defined in Section 3.1. For g ∈ C AvO ( R ) l , it is all the same. Especially,when g ∈ Hom( k , R ), then by definition, we have ∞ X k = k ! ( − k ( k + + k l k + ( α ⊗ k ⊗ g ) = − (cid:0) l ( m ⊗ τ ⊗ g ) + l ( τ ⊗ m ⊗ g ) (cid:1) = − l ( m ⊗ τ ⊗ g ) = − (cid:16) − s − m ◦ ( s τ ◦ sg ) + τ ◦ ( m ◦ (id ⊗ sg )) − ( s − m ◦ ( sg ⊗ s τ ) − τ ◦ ( m ◦ ( sg ⊗ id))) (cid:17) And it is the same as the operator ∂ defined in Section 3.1. OMOTOPY AVERAGING ALGEBRAS 19 (cid:3)
Proposition 6.8.
Let V be a graded space. Then a Maurer-Cartan element α in C AvA ( V ) willinduce a L ∞ -algebra structure on C AvO ( V ) . In particular, for an averaging algebra ( R , µ, A ) , thecochain complex C AvO ( R ) is a di ff erential graded Lie-algebra.Proof. Given a Maurer-Cartan element α , α will induce L ∞ -algebra structure { l α n } n > on C AvA ( V ).And notice that all operations l α n can be restricted to C AvO ( V ). Thus C AvO ( V ) forms a L ∞ -subalgebraof ( C AvA ( V ) , { l α n } n > ).Let ( R , µ, A ) be an averaging algebra and ( m , τ ) be the corresponding Maurer-Cartan elementin C AvA ( R ). Identify C AvO ( R ) with C AvO ( R ). Since m = − s ◦ µ ◦ ( s − ) ⊗ ∈ Hom(( sR ) ⊗ , sR ) in C A ( R ), the restriction of l α n on C AvO ( R ) is zero for n >
3. Thus C AvO ( R ) is just a di ff erential gradedLie-algebra. (cid:3)
7. H omotopy averaging algebras
In this subsection, we’ll define homotopy averaging algebras.Recall that an A ∞ -algebra structure is equivalent to a Maurer-Cartan element in the gradedLie-algebra C A ( V ) : = (Hom( T c ( sV ) , sV ) , [ − , − ] G ) where T c ( sV ) = ∞ L n = ( sV ) ⊗ n . We can definehomotopy averaging algebra structure in similar way. For a graded vector space V , consider thefollowing subspace of C AvA ( V ): C AvA ( V ) : = C A ( V ) M Hom( sV , V ) M C AvO ( V ) > r M C AvO ( V ) > l . Obviously, C AvA ( V ) is a L ∞ -subalgebra of C AvA ( V ) with the operations { l n } n > restricted on C AvA ( V ).Then we have the following definition: Definition 7.1.
Let V be a graded space. A homotopy averaging algebra( Av ∞ ) structure on V isa Maurer-Cartan element in the L ∞ -algebra C AvA ( V ).Let’s describe this structure explicitly.A Maurer-Cartan element α in C AvA ( V ) corresponds to a family of operators: { b n } n > ∪ { c } ∪ { c rn } n > ∪ { c ln } n > where b n : ( sV ) ⊗ n → sV , c : sV → V belongs C AvA ( V ), Hom( sV , V ) respectively, and c rn :( sV ) ⊗ n → V , c ln : ( sV ) ⊗ n → V belongs to C AvO ( V )( V ) > r , C AvO ( V ) > l respectively. All these oper-ators are of degree −
1. Then α satisfying the Maurer-Cartan equation implies that the family ofoperators satisfies the following equations: X i + j = n b n − j + ◦ (id ⊗ i ⊗ b j ⊗ id ⊗ n − i − j ) = , n X m = X l + ··· + l m = n (cid:16) s − b m ◦ ( sc rl ⊗ . . . ⊗ sc rl m ) − c rl ¯ ◦ ( b m ◦ (id ⊗ sc rl ⊗ . . . ⊗ sc rl m )) (cid:17) = , n X m = X l + ··· + l m = n (cid:16) s − b m ◦ ( sc ll ⊗ . . . ⊗ sc ll m ) − c ll m ¯ ◦ ( b m ◦ ( sc ll ⊗ . . . ⊗ sc ll m − ⊗ id)) (cid:17) = . for any n >
1, where c r = c l = c . Then we define m n = s − ◦ b n ◦ s ⊗ n : V ⊗ n → V , A = c ◦ s : V → V , A rn = c rn ◦ s ⊗ n : V ⊗ n → V and A ln = c ln ◦ s ⊗ n : V ⊗ n → V for all n >
1, where | m n | = n − , | A | = , | A rn | = | A ln | = n − { m n } n > ∪ { A } ∪ { A rn } n > ∪ { A ln } n > satisfies the following identities:(i) X i + j = n ( − i + jk m n − j + ◦ (id ⊗ i ⊗ m j ⊗ id ⊗ n − i − j ) = , (ii) n X k = X l + ··· + l k = n ( − ε n m k ◦ ( A rl ⊗ . . . ⊗ A rl k ) − X p + q + = l ( − q · k P j = ( l j − + kp + q A rl ◦ (id ⊗ p ⊗ m k ◦ (id ⊗ A rl ⊗ . . . ⊗ A rl k ) ⊗ id ⊗ q ) o = , (iii) n X k = X l + ··· + l k = n ( − ε n m k ◦ ( A ll ⊗ . . . ⊗ A ll k ) − X p + q + = l k ( − p · k − P j = ( l j − + ( l k − · k − P j = l j + kp + q A ll k ◦ (id ⊗ p ⊗ m k ◦ ( A ll ⊗ . . . ⊗ A ll k − ⊗ id) ⊗ id ⊗ q ) o = A l = A r = A , and ε = k P j = l j ( l j − + k ( k − + k P j = ( l j − l + · · · + l j − ) = n ( n − + k ( k − + k P j = ( k − j + l j .The equation (i) is just the definition of A ∞ -algebras.Let’s compute the equation (ii)(iii) for n = , n = , − m ◦ A + A ◦ m = n = , m ◦ ( A ⊗ A ) − A ◦ ( m ◦ (id ⊗ A )) = − m ◦ A r + A r ◦ ( m ⊗ id) + A r ◦ (id ⊗ m ) , m ◦ ( A ⊗ A ) − A ◦ ( m ◦ ( A ⊗ id)) = − m ◦ A l + A l ◦ ( m ⊗ id) + A l ◦ (id ⊗ m ) . The equation (1) above implies that A is compatible with the di ff erential, so operator A can beinduced on the homology of the complex ( V , m ). The equation (2) says that A is an averagingoperator up to homotopy with respect to m on V and the obstructions are A r and A l .So we get another definition of homotopy averaging algebra. Definition 7.2.
Let V be a graded space. Assume that V is endowed with a family of operators { m n : V ⊗ n → V } n > ∪ { A : V → V } ∪ { A rn : V ⊗ n → V } n > ∪ { A ln : V ⊗ n → V } n > with | m n | = n − | A | = | A rn | = | A ln | = n −
1. If these operators satisfy the equations (i) (ii) (iii) above, we call V ahomotopy averaging algebra. A ppendix : P roof of T heorem Lemma 7.3.
Let n > , i n − . Then for any π ∈ S n , there exists a unique triple ( δ, σ, τ ) with σ ∈ S ( i , n − i ) , δ ∈ S i , τ ∈ S n − i such that π ( l ) = σδ ( l ) for l i, and π ( i + m ) = σ ( i + τ ( m )) for m n − i. OMOTOPY AVERAGING ALGEBRAS 21
Let V be a graded space. The operation “¯ ◦ ” on Hom( T c ( V ) , V ) satisfying the Pre-Jacobi iden-tity: Lemma 7.4.
For any homogeneous elements f ; g , . . . , g m ; h , . . . , h n in Hom( T c ( V ) , V ) , the fol-lowing identity holds: (cid:16) f ¯ ◦ ( g , . . . , g m ) (cid:17) ¯ ◦ (cid:16) h , . . . , h n (cid:17) = X i i ··· i m n ( − m P k = | g k | ( ik P j = | h j | ) f ¯ ◦ (cid:16) h , . . . , h i , g ¯ ◦ ( h i + , . . . ) , . . . , h i m , g m ¯ ◦ ( h i m + , . . . ) . . . (cid:17) Now, let’s prove the theorem 6.5.
Theorem 6.5.
Let V be a graded space. Then C AvA ( V ) endowed with operations { l n } n > definedabove forms a L ∞ -algebra.Proof. By the definition of L ∞ -algebras, we need to check the equation in C AvA ( V ) :(6 . X i + j = n X σ ∈ S(i , n − i) ( − i ( n − i ) χ ( σ, x , . . . , x n ) l n − i + (cid:16) l i ( x σ (1) , . . . , x σ ( i ) ) , x σ ( i + , . . . , x σ ( n ) (cid:17) = . If all x i are contained in C A ( V ) = Hom( T c ( sV ) , sV ), then equation 6.1 is just the Jacobiidentity on the graded Lie algebra C A ( A ). Apart from this, by the definition of l n , the term l n − i + (cid:16) l i ( x σ (1) , . . . , x σ ( i ) ) , x σ ( i + , . . . , x σ ( n ) (cid:17) is trivial unless there are exactly two elements in { x , . . . , x n } coming from C A ( V ) and all other n − C AvO ( V ) l or C AvO ( V ) r . We assumethat x = sh ∈ Hom(( sV ) ⊗ n , sV ) and x = sh ∈ Hom(( sV ) ⊗ n , sV ). Then n must be n + n + x i = g i − ∈ C AvO ( V ) r . When g i ∈ Hom( k , V ) ⊕ Hom( sV , V ) ⊕ Hom( T c ( sV ) > , V ) l ,the proof is all the same.For simplicity, we assume that all g i are not in Hom( s , V ). If there is some g i ∈ Hom( k , V ), thecalculation is similar.So now, we are going to check the equation 6.1 for x = sh , x = sh , x = g , . . . , x n + n + = g n + n − with g i ∈ Hom( sV , V ) ⊕ Hom( k , V ) ⊕ Hom( sV , V ) ⊕ Hom( T c ( sV ) > , V ) r . Then the term l n − i + (cid:16) l i ( x σ (1) , . . . , x σ ( i ) ) , x σ ( i + vanishes unless i = , n + , n +
1. Let’s compute them one byone.(A) When i =
2, the (2 , n − ffl e is identity map, we have l n + n − ( l ( sh , sh ) , g , . . . , g n + n − ) = l n + n − ([ sh , sh ] G , g , . . . , g n + n − ) = X π ∈ S n + n − ( − ε n s − ( sh ¯ ◦ sh ) ◦ ( sg π (1) , . . . , sg π ( n + n − ) − ( − ( | g π (1) | + | h | + | h | ) g π (1) ¯ ◦ (cid:16) ( sh ¯ ◦ sh ) ◦ (id sV ⊗ sg π (2) ⊗ . . . ⊗ sg π ( n ) ) (cid:17)o + X π ∈ S n + n − ( − ε ( − ( | h | + | h | + + n s − ( sh ¯ ◦ sh ) ◦ ( sg π (1) , . . . , sg π ( n + n − ) − ( − ( | g π (1) | + h + h ) g π (1) ¯ ◦ (cid:0) ( sh ¯ ◦ sh ) ◦ (id sV ⊗ sg π (2) ⊗ . . . ⊗ sg π ( n + n − ) (cid:1)o = X π ∈ S n + n − ( − ε n n − X i = ( − ( | h | + i P j = ( | g π ( j ) | + h ◦ ( sg π (1) , . . . , sg π ( i ) , sh ◦ ( sg π ( i + , . . . , sg π ( i + n ) ) , . . . , sg π ( n + n − ) o + X π ∈ S n + n − ( − ε ( − + ( | g π (1) | + | h | + | h | ) g π (1) ¯ ◦ n sh ◦ (cid:16) sh ◦ (id sV ⊗ sg π (2) ⊗ . . . sg π ( n ) ) ⊗ . . . ⊗ sg π ( n + n − (cid:17) + n − X i = ( − ( | h | + i P j = ( | g π ( j ) | + sh ◦ (cid:16) id sV ⊗ sg π (2) ⊗ . . . ⊗ sg π ( i ) ⊗ sh ◦ ( sg π ( i + , . . . , sg π ( i + n ) ) ⊗ . . . ⊗ sg π ( n + n − (cid:17)o + X π ∈ S n + n − ( − ε ( − ( | h | + | h | + + n − X i = ( − ( | h | + i P j = ( | g π ( j ) | + • n h ◦ (cid:16) sg π (1) ⊗ . . . ⊗ sg π ( i ) ⊗ sh ◦ ( sg π ( i + ⊗ . . . ⊗ sg π ( i + n ) ) ⊗ . . . ⊗ sg π ( n + n − (cid:17)o + X π ∈ S n + n − ( − ε ( − ( | h | + | h | + + ( | g π (1) | + | h | + | h | ) • g π (1) ¯ ◦ n sh ◦ (cid:16) sh ◦ (cid:0) id sV ⊗ sg π (2) ⊗ . . . ⊗ sg π ( n ) (cid:1) ⊗ . . . ⊗ sg π ( n + n − (cid:17) + n − X i = ( − ( | h | + i P j = ( | g π ( j ) | + sh ◦ (cid:16) id sV , sg π (2) , . . . , sg π ( i ) , sh ◦ (cid:0) sg π ( i + , . . . , sg π ( i + n ) (cid:1) , . . . sg π ( n + n − (cid:17)o . where ( − ε = χ ( π, g , . . . , g n + n − )( − ( n + n − h + h ) + n + n − P k = k P j = | g π ( j ) | .(B) When i = n + X ρ ∈ S ( n + , n ) ( − ( n + n χ ( ρ, x , . . . , x n + n + ) l n + (cid:0) l n + ( x ρ (1) , . . . , x ρ ( n + ) , . . . , x ρ ( n + n + (cid:1) = X σ ∈ S ( n , n − ( − ( n + n χ ( σ, g , . . . , g n + n − )( − ( | h | + | h | + + n P j = | g σ ( j ) | ) + n + • l n + (cid:16) l n + (cid:0) sh , g σ (1) , . . . , g σ ( n ) (cid:1) , sh , g σ ( n + , . . . , g σ ( n + n − (cid:17) = X σ ∈ S ( n , n − ( − ( n + n χ ( σ, g , . . . , g n + n − )( − ( | h | + | h | + + n P j = | g σ ( j ) | ) + n + • ( − ( | h | + | h | + + n P j = | g σ ( j ) | + n − + l n + (cid:16) sh , l n + ( sh , g σ (1) , . . . , g σ ( n ) ) | {z } : = b h , g σ ( n + , . . . , g σ ( n + n − (cid:17) = X σ ∈ S ( n , n − χ ( σ, g , . . . , g n + n − )( − ( | h | + n − + ( n + n + n l n + (cid:16) sh , b h , g σ ( n + , . . . , g σ ( n + n − (cid:17) = X σ ∈ S ( n , n − X τ ∈ S n − ( − θ n h ◦ (cid:16) s b h ⊗ sg σ ( n + τ (1)) ⊗ . . . ⊗ sg σ ( n + τ ( n − (cid:17)| {z } B − ( − ( | b h | + | h + | ) b h ¯ ◦ (cid:16) sh ◦ (cid:0) id sV ⊗ sg σ ( n + τ (1)) ⊗ . . . ⊗ sg σ ( n + τ ( n − (cid:1)(cid:17)| {z } B o OMOTOPY AVERAGING ALGEBRAS 23 + X σ ∈ S ( n , n − X τ ∈ S ( n − n − X i = ( − θ n h ◦ (cid:16) sg σ ( n + τ (1)) ⊗ . . . ⊗ sg σ ( n + τ ( i )) ⊗ s b h ⊗ . . . ⊗ sg σ ( n + τ ( n − (cid:17)| {z } B − ( − ( | h | + | g σ ( n + τ (1)) | + g σ ( n + τ (1)) ¯ ◦ (cid:16) sh ◦ (cid:0) id sV ⊗ sg σ ( n + τ (2)) ⊗ . . . ⊗ sg σ ( n + τ ( i )) ⊗ s b h ⊗ . . . ⊗ sg σ ( n + τ ( n − (cid:1)(cid:17)| {z } B o where( − θ = χ ( σ, g , . . . , g n + n − ) χ ( τ, g σ ( n + , . . . , g σ ( n + n − ) · ( − ( | h | + n − + ( n + n + n ( − n ( | h | + + ( n − | b h | ) + n − P k = k P j = | g σ ( n + τ ( j )) | . ( − θ = χ ( σ, g , . . . , g n + n − ) χ ( τ, g σ ( n + , . . . , g σ ( n + n − ) · ( − ( | h | + n − + ( n + n + n · ( − n ( | h | + + n − P k = k P j = | g σ ( n + τ ( j )) | + i P j = | g σ ( n + τ ( j )) | + ( n − i − | b h | + ( | b h | )( i P j = | g σ ( n + τ ( j )) | ) + i Let’s compute the terms B , B , B , B together with their coe ffi cients. B = X σ ∈ S ( n , n − X τ ∈ S n − ( − θ h ◦ ( s b h ⊗ . . . ⊗ sg σ ( n + τ (1)) ⊗ . . . ⊗ sg σ ( n + τ ( n − ) = X σ ∈ S ( n , n − X τ ∈ S n − ( − θ h ◦ (cid:16) sl n + ( sh , g σ (1) , . . . , g σ ( n ) ) , sg σ ( n + τ (1)) ⊗ . . . ⊗ sg σ ( n + τ ( n − (cid:17) = X σ ∈ S ( n , n − X τ ∈ S n − ( − θ X δ ∈ S n χ ( δ, g σ (1) , . . . , g σ ( n ) )( − n ( | h | + + n − P k = k P j = | g σ ( δ ( j )) | • h ◦ n(cid:16) sh ◦ ( sg σδ (1) , . . . , sg σδ ( n ) ) ⊗ sg σ ( n + τ (1)) ⊗ . . . ⊗ sg σ ( n + τ ( n − (cid:17) − ( − ( | g σδ (1) | + | h | + • (cid:16) sg σδ (1) ¯ ◦ (cid:0) sh ◦ (id sV ⊗ sg σδ (2) ⊗ . . . ⊗ sg σδ ( n ) ) (cid:1) ⊗ sg σ ( n + τ (1)) ⊗ . . . ⊗ sg σ ( n + τ ( n − (cid:17)o = X π ∈ S ( n + n − ( − η h ◦ n(cid:16) sh ◦ (cid:0) sg π (1) ⊗ . . . ⊗ sg π ( n ) (cid:1) ⊗ sg π ( n + ⊗ . . . ⊗ sg π ( n + n − (cid:17) − ( − ( | g π (1) | + | h | + (cid:16) sg π (1) ¯ ◦ (cid:0) sh ◦ (id sV ⊗ sg π (2) ⊗ . . . ⊗ sg π ( n ) ) (cid:1) ⊗ sg π ( n + ⊗ . . . ⊗ sg π ( n + n − (cid:17)o where ( − η = ( − θ χ ( δ, g σ (1) , . . . , g σ n )( − n ( | h | + + n − P k = k P j = | g σ ( δ ( j )) | = χ ( π, g , . . . , g n + n − )( − ( | h | + | h | )( n + n − + + n + n − P k = k P j = | g π ( j ) | . Notice that, in the last equality above, we use lemma 7.3 to replace the triple ( δ, σ, τ ) by itscorresponding permutation π ∈ S n + n − and we use the fact χ ( π, g , . . . , g n + n − ) = χ ( σ, g , . . . , g n + n − ) χ ( τ, g σ ( n + , . . . , g σ ( n + n − ) χ ( δ, g σ (1) , . . . , g σ ( n ) ) . Similarly, we have B = X σ ∈ S ( n , n − X τ ∈ S n − ( − θ ( − + ( | b h | + | h | + b h ¯ ◦ (cid:16) sh ◦ (id sV ⊗ sg σ ( n + τ (1)) ⊗ . . . ⊗ sg σ ( n + τ ( n − ) (cid:17) = X π ∈ S ( n + n − ( − η n h ◦ (cid:16) sg π (1) ⊗ . . . ⊗ sg π ( n ) (cid:17) − ( − ( | h | + | g π (1) | + • (cid:16) g π (1) ¯ ◦ (cid:0) sh ◦ (id sV ⊗ sg π (2) ⊗ . . . ⊗ sg π ( n ) ) (cid:1)(cid:17)o ¯ ◦ (cid:16) sh ◦ (cid:0) id sV ⊗ sg π ( n + ⊗ . . . ⊗ sg π ( n + n − (cid:1)(cid:17)| {z } : = e h = X π ∈ S ( n + n − ( − η n n X k = ( − ( n P j = k + ( | g π ( j ) | + | e h | ) h ◦ (cid:16) sg π (1) ⊗ . . . ⊗ sg π ( k ) ¯ ◦ e h ⊗ sg π ( k + ⊗ . . . ⊗ sg π ( n ) (cid:17)o + ( − η ( − + ( | h | + | g π (1) | + n g π (1) ¯ ◦ (cid:16) sh ◦ (id sV ⊗ sg π (2) ⊗ . . . ⊗ sg π ( n ) ) , e h (cid:17) + ( − | e h | (cid:0) n P j = ( | g π ( j ) | + (cid:1) g π (1) ¯ ◦ (cid:16) sh ◦ (cid:0) e h ⊗ sg π (2) ⊗ . . . ⊗ sg π ( n ) (cid:1)(cid:17) + n X k = ( − | e h | (cid:0) n P j = k + ( | g π ( j ) | + (cid:1) g π (1) ¯ ◦ (cid:16) sh ◦ (cid:0) id sV ⊗ sg π (2) ⊗ . . . ⊗ sg π ( k ) ¯ ◦ e h ⊗ sg π ( k + ⊗ . . .. . . ⊗ sg π ( n ) (cid:1)(cid:17) + ( − | e h | (cid:0) n P j = ( | g π ( j ) | + + | h | + (cid:1) g π (1) ¯ ◦ (cid:16) e h , sh ◦ (id sV ⊗ sg π (2) ⊗ . . . ⊗ sg π ( n ) ) (cid:17)o where ( − η = ( − η ( − + ( | b h | + | h | + = χ ( π, g , . . . , g n + n − )( − ( | h | + | h | )( n + n − + + n + n − P k = k P j = | g π ( j ) | + + ( | b h | + | h | + = χ ( π, g , . . . , g n + n − )( − ( | h | + | h | )( n + n − + n + n − P k = k P j = | g π ( j ) | + (cid:0) | h | + n − + n P j = | g π ( j ) | (cid:1)(cid:0) | h | + (cid:1) B = X σ ∈ S ( n , n − X τ ∈ S ( n − n − X i = ( − θ h ◦ n sg σ ( n + τ (1)) ⊗ . . . ⊗ sg σ n + τ ( i ) ⊗ s b h ⊗ . . . ⊗ sg σ ( n + τ ( n − o = X π ∈ S ( n + n − n − X i = ( − η h ◦ n sg π ( n + ⊗ . . . ⊗ sg π ( n + i ) ⊗ sh ◦ ( sg π (1) ⊗ . . . ⊗ sg π ( n ) ) ⊗ . . . ⊗ sg π ( n + n − − ( − ( | h | + | g π (1) | + sg π ( n + ⊗ . . . ⊗ sg π ( n + i ) ⊗ sg π (1) ¯ ◦ (cid:16) sh ◦ (id sV ⊗ sg π (2) ⊗ . . . ⊗ sg π ( n )) (cid:17) ⊗ . . .. . . ⊗ sg π ( n + n − o where( − η = ( − θ χ ( δ, g σ (1) , . . . , g σ ( n ) )( − n ( | h | + + n − P k = | g π ( j ) | = χ ( σ, g , . . . , g n + n − ) χ ( τ, g σ ( n + , . . . , g σ ( n + n − ) · ( − ( | h | + n − + ( n + n + n OMOTOPY AVERAGING ALGEBRAS 25 · ( − n ( | h | + + n − P k = k P j = | g ( σ + τ ( j )) | + i P j = | g σ ( n + τ ( j )) | + ( n − i − | b h | + | b h | (cid:0) n + i P j = n + | g π ( j ) | (cid:1) + i · ( − n ( | h | + + n − P k = | g π ( j ) | χ ( δ, g σ (1) , . . . , g σ ( n ) ) = χ ( π, g , . . . , g n + n − )( − ( | h | + | h | )( n + n − + + n + n − P k = k P j = | g π ( j ) | + (1 + | h | + n + n P j = | g π ( j ) | )( i + i P j = | g π ( n + j ) | ) B = X π ∈ S ( n + n − n − X i = ( − η g π ( n + ¯ ◦ n sh ◦ (cid:16) id sV ⊗ sg π ( n + ⊗ . . . ⊗ sg π ( n + i ) ⊗ sh ◦ ( sg π (1) ⊗ . . . sg π ( n ) ) ⊗ . . . ⊗ sg π ( n + n − (cid:17) − ( − ( | h | + | g π (1) | + sh ◦ (cid:16) id sV ⊗ sg π ( n + ⊗ . . . ⊗ sg π ( n + i ) ⊗ sg π (1) ¯ ◦ (cid:0) sh ◦ (ı sV ⊗ sg π (2) ⊗ . . .. . . sg π ( n ) ) (cid:1) ⊗ . . . ⊗ sg π ( n + n − (cid:17)o where ( − η = ( − η ( − + ( | h | + | g π ( n + | + . (C) When i = n + X ρ ∈ S ( n + , n ) ( − ( n + n χ ( ρ, x , . . . , x n + n + ) l n + (cid:0) l n + ( x ρ (1) , . . . , x ρ ( n + ) , . . . , x ρ ( n + n + (cid:1) = X σ ∈ S ( n , n − ( − ( n + n ( − n + ( | h | + n P j = g σ ( j ) ) χ ( σ, g , g , . . . , g n + n − ) • l n + ( l ( sh , g σ (1) , . . . , g σ ( n ) ) , sh , g σ ( n + , . . . , g σ ( n + n − ) = X σ ∈ S ( n , n − ( − ( n + n ( − n + ( | h | + n P j = g σ ( j ) ) ( − ( | h | + n P j = g σ ( j ) + | h | + n ) + χ ( σ, x , . . . , x n + n + ) • l n + ( sh , l n + ( sh , g σ (1) , . . . , g σ ( n )) , g σ ( n + , . . . , g σ ( n + n − ) = X σ ∈ S ( n , n − ( − ( n + n + + ( | h | + | h | + n ) χ ( σ, g , . . . , g n ) • l n + ( sh , l n + ( sh , g σ (1) , . . . , g σ ( n )) , g σ ( n + , . . . , g σ ( n + n − )Switch the roles of h , n and h , n in the expansion of l n + (cid:16) sh , l n + ( sh , g σ (1) , . . . , g σ ( n ) ) , g σ ( n + , . . . , g σ ( n + n − (cid:17) , which has been computed in case (B), then we will get the expansion of l n + ( sh , l n + ( sh , g σ (1) , . . . , g σ ( n )) , g σ ( n + , . . . , g σ ( n + n − ) . 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