BV-operators and the secondary Hochschild complex

We show that the Gerstenhaber bracket on secondary Hochschild cohomology is determined by a Batalin-Vilkovisky like operator. MSC 2010 Subject Classification 16E40


Introduction
A Gerstenhaber algebra (see [3]) consists of a graded vector space W • = n≥0 W n equipped with the following two structures: (a) A dot product x · y of degree zero making W • into an associative graded commutative algebra. (b) A bracket [x, y] of degree −1 making W • into a graded Lie algebra satisfying the compatibility property that Gerstenhaber algebra structures appear in a variety of situations, from Hochschild cohomology of algebras to the exterior algebra of a Lie algebra and the algebra of differential forms on a Poisson manifold.
In [4], [5], Gerstenhaber and Voronov introduced the notion of a homotopy G-algebra, which is a brace algebra equipped with a differential of degree 1 and a dot product of degree 0 satisfying certain conditions. In particular, the cohomology groups H (V • ) of a homotopy G-algebra V • carry the structure of a Gerstenhaber algebra.
In this paper, we introduce the notion of a BV-operator ∆ = {∆ n : V n → V n−1 } n≥0 on a homotopy G-algebra V • such that the Gerstenhaber bracket on H (V • ) is determined by ∆ in a manner similar to the BV-formalism. More explicitly, for classes f ∈ H n (V • ) and g ∈ H m (V • ), we have where f ∈ Z n (V • ), g ∈ Z m (V • ) are cocycles representing f and g respectively. We note that ∆ need not be a morphism of cochain complexes and therefore may not induce any operator on H (V • ). As such, ∆ may not descend to a generator for the Gerstenhaber bracket on H (V • ). Our motivation is to introduce a BV-operator on the cochain complex defining the secondary Hochschild cohomology of a symmetric algebra A over a commutative algebra B . For a datum (A, B, ε) consisting of an algebra A, a commutative algebra B and an extension of rings ε : B → A such that ε(B ) ⊆ Z (A), the secondary Hochschild cohomology H • (A, B, ε) was introduced by Staic [9] in order to study deformations of algebras A [[t ]] having a B -algebra structure. More generally, Staic [9] introduced the secondary Hochschild complex C • ((A, B, ε); M ) with coefficients in an A-bimodule M .
In [10], Staic and Stancu showed that the secondary Hochschild complex C • (A, B, ε) := C • ((A, B, ε); A) with coefficients in A is a non-symmetric operad with multiplication, giving it the structure of a homotopy G-algebra. Hence, the secondary cohomology H • (A, B, ε) is equipped with a graded commutative cup product and a Lie bracket which makes it a Gerstenhaber algebra. For more on the secondary cohomology, the reader may see, for instance, [1], Corrigan-Salter and Staic [2], Laubacher, Staic and Stancu [8].
Let k be a field. It is well known (see Tradler [11]) that the Hochschild cohomology of a finite dimensional k-algebra A equipped with a symmetric, non-degenerate, invariant bilinear form 〈 · ,· 〉 : A × A → k carries the structure of a BV-algebra. For the terms C n ((A, B, ε)) = Hom k (A ⊗n ⊗ B ⊗ n(n−1) 2 , A) in the secondary Hochschild complex, we define the BV-operator ∆ = n+1 i =1 (−1) i n ∆ i : C n+1 (A, B, ε) → C n (A, B, ε) by the condition (see Section 3) We then show that the Gerstenhaber bracket on the secondary Hochschild cohomology of (A, B, ε) is determined by ∆ in a manner similar to the BV-formalism. From Tradler [11], we also know that the operator ∆ • : C • (A, A) → C •−1 (A, A) on usual Hochschild cochains inducing the BV-structure on H • (A, A) corresponds to the operator N s on duals of Hochschild chains, where N is the "norm operator" and s is the "extra degeneracy" (see (16)). The isomorphism between the two complexes is induced by the k-module isomorphism A * ∼ = A determined by the non-degenerate bilinear form 〈 · ,· 〉 : A × A → k. If we pass to the cohomology and take the normalized Hochschild complex which is a quasi-isomorphic subcomplex of C • (A, A), it follows that Tradler's ∆ • operator corresponds to Connes' operator on Hochschild cohomology with coefficients in A.
However, in the case of secondary cohomology, we have mentioned that the operator ∆ • defined in (1) is not a morphism of complexes and we cannot pass to cohomology. Accordingly, we show that the operator ∆ • defined in (1) fits into a commutative diagram (see Theorem 10) where B is Connes' operator. Here, C • (A, B, ε) is the normalization of the co-simplicial module C • (A, B, ε) introduced by Laubacher, Staic and Stancu [8], which is used to compute the secondary Hochschild cohomology associated to the triple (A, B, ε). It should be noted (see [8,Remark 4.7]) that despite similar names, the complex C • (A, B, ε) cannot be expressed as a secondary Hochschild complex with coefficients in some A-bimodule. The vertical morphisms in (2) are induced by composing the canonical morphisms (A ⊗ B ⊗n ) * → A * for each n ≥ 0, the isomorphism A * ∼ = A as well as the inclusion of the quasi-isomorphic subcomplex C • (A, B, ε) → C • (A, B, ε).

Main Result: BV-operator on homotopy G-algebra
We begin by recalling the notion of a homotopy G-algebra from [5]. A brace algebra (see [5,Definition 1]) is a graded vector space V = n≥0 V n with a collection of multilinear operators (braces) x{x 1 , . . . , x n } satisfying the following conditions (with x{} understood to be x): For homogeneous elements x, x 1 ,. . . , x m , y 1 ,. . . ,y n , we have |y q | and |x| := deg(x) − 1.

Definition 1 (see [5, Definition 2]).
A homotopy G-algebra consists of the following data: for all m, n ≥ 0. (3) A differential d : V • → V •+1 of degree one making V into a DG-algebra with respect to the dot product.
(4) The dot product satisfies the following compatibility conditions (x 1 · x 2 ){y 1 , . . . , y n } = n k=0 (−1) k (x 1 {y 1 , . . . , y k }) · (x 2 {y k+1 , . . . , y n }) In particular, a homotopy G-algebra is equipped with a graded Lie bracket which descends to the cohomology of the corresponding cochain complex (V • , d ) (see [5]) The dot product also descends to the cohomology and the bracket with an element becomes a graded derivation for the induced dot product on H (V • ) = n≥0 H n (V • ). In other words, the cohomology (H (V • ), [ · ,· ], · ) of a homotopy G-algebra V • is canonically equipped with the structure of a Gerstenhaber algebra. We now introduce the notion of a BV-operator on a homotopy G-algebra.
If V • is a homotopy G-algebra equipped with a BV-operator ∆, we now show that the bracket on the Gerstenhaber algebra H (V • ) is determined by ∆ in a manner similar to the BV-formalism. Theorem 3. Let V • = n≥0 V n be a homotopy G-algebra equipped with a BV-operator ∆ = {∆ n : V n → V n−1 } n≥0 . Consider f ∈ H n (V • ) and g ∈ H m (V • ) and choose cocycles f ∈ Z n (V • ) and g ∈ Z m (V • ) corresponding respectively to f and g . Then, we have The Gerstenhaber bracket on the cohomology of V • is now determined by In particular, the right hand side does not depend on the choice of representatives f and g .

Application : BV-operator on secondary Hochschild cohomology
Let k be a field and A be an algebra over k. Let B be a commutative k-algebra and ε : B → A be a morphism of k-algebras such that ε(B ) ⊆ Z (A), where Z (A) denotes the center of A. Let M be an A-bimodule such that ε(b)m = mε(b) for all b ∈ B and m ∈ M . Following [9, §4.2], we consider the complex (C • ((A, B, ε); M ), δ • ) whose terms are given by An element in A ⊗n ⊗ B ⊗ n(n−1) 2 will be expressed as a "tensor matrix" of the form may be described as follows The cohomology groups of (C • ((A, B, ε); M ), δ • ) are known as the secondary Hochschild cohomologies H n ((A, B, ε); M ) of the triple (A, B, ε) with coefficients in M (see [9]). From [10, Proposition 3.1], we know that the secondary Hochschild complex C • (A, B, ε) := C • ((A, B, ε); A) carries the structure of a homotopy G-algebra. This induces a graded Lie bracket on the secondary cohomology. It follows (see [10,Corollary 3.2]) that the secondary cohomology H • (A, B, ε) carries the structure of a Gerstenhaber algebra in the sense of [3]. From now onwards, we always let A be a finite dimensional k-algebra equipped with a symmetric, non-degenerate, invariant bilinear form 〈 · ,· 〉 : A × A → k. In particular, 〈a 1 , a 2 〉 = 〈a 2 , a 1 〉, 〈a 1 a 2 , a 3 〉 = 〈a 1 , a 2 a 3 〉 for any a 1 , a 2 , a 3 ∈ A. For i ∈ {1, . . . , n + 1}, we define the maps To clarify the above operator, let us express where U (k) is a square matrix of dimension k. Then, we have where X t 12 denotes the transpose of X 12 . The operator ∆ : Following [10, §3], we know that the complex C • (A, B, ε) carries a dot product of degree 0, i.e., for f ∈ C n (A, B, ε), g ∈ C m (A, B, ε), we have f · g ∈ C m+n (A, B, ε). We also consider the operations [10, §3]. We also set Proof. This may be verified by direct computation. then, Proof. We set, for k ≥ 0, p ≥ 0: where α : We write the entire expression of (6) as where E k denotes the k-th term in the expression. We set for i , j ≥ 1 and i + j ≤ n, The first term of A i , j and that of C i , j are the same modulo a sign. Using the fact that δg = 0, the third term of A i , j and the first term of B i , j add up to give E 4 . Thus, we have It may be verified that Thus, A i , j , B i , j +1 , C i −1, j are defined for all the values of i , j with i , j ≥ 1 and i + j ≤ n + 1. Moreover, it may be verified that We also have Rearranging the terms in the above sum, and using equation (7), we get,   Proof. We consider f ∈ Z n (A, B, ε) and g ∈ Z m (A, B, ε). By definition (see [10, §3]), we know that Applying Lemma 5, we know that the cochains are coboundaries. From (10), it now follows that is a coboundary. Applying Lemma 4, it follows from (11) that is a coboundary. Using the fact that the dot product is graded commutative, we can put ∆(g ) · f = (−1) n(m−1) f · ∆(g ). The result is now clear. (A, B, ε) and g ∈ H m (A, B, ε), the Gerstenhaber bracket is determined by

Theorem 7. For secondary cohomology classes f ∈ H n
Here f and g are any cocycles representing the classes f and g respectively.
Proof. This follows directly by applying Theorem 3 and Proposition 6.

Relation with extra degeneracy and norm operator
We continue with A being a finite dimensional k-algebra equipped with a symmetric, nondegenerate and invariant bilinear form 〈 · ,· 〉 : A × A → k and B a commutative k-algebra with a morphism of k-algebras ε : B → A such that ε(B ) ⊆ Z (A). In particular, the non-degenerate pairing on A induces mutually inverse isomorphisms  A) is the isomorphism induced by φ : A * ∼ = → A, while s and N respectively are the usual extra degeneracy and norm operators given by a 1 , . . . , a n ) := f (a 1 , . . . , a n , 1) N := 1 + λ + · · · + λ n : C n (A) −→ C n (A) (λ · f )(a 0 , . . . , a n ) := (−1) n f (a n , a 0 , . . . , a n−1 ) If we pass to the normalized Hochschild complex which is a quasi-isomorphic subcomplex of C • (A, A * ), then (16) induces the following commutative diagram In the case of secondary Hochschild cohomology, we have shown in Section 3 that ∆ • is not in general a morphism of complexes, i.e., it does not descend to cohomology. We will now show that the operator ∆ • on secondary Hochschild cohomology H • (A, B, ε) fits into a commutative diagram similar to (18).
In [8], Laubacher, Staic and Stancu have introduced a co-simplicial module C • (A, B, ε) which is used to compute the secondary Hochschild cohomology associated to the triple (A, B, ε). The terms of this co-simplicial module are given by It is important to note (see [8,Remark 4.7]) that despite the similar names, the complex C • (A, B, ε) cannot be expressed as the secondary Hochschild complex of (A, B, ε) with coefficients in some A-bimodule. This is because the "coefficient module" Hom(A ⊗ B n , k) appearing in (19) varies with n.
In addition, the cosimplicial module C • (A, B, ε) is equipped with a cyclic operator, which can be used to compute the secondary cyclic cohomology associated to the triple (A, B, ε). Using the isomorphisms given by we first transfer the cyclic operator from [8] to a complex C • (A, B, ε) whose terms are given by for any f ∈ C n (A, B, ε), a i ∈ A and b i , j ∈ B .
Proof. This is clear from the definition in [8, §4.2] and the isomorphisms in (20).
We also let Φ Proposition 9. Let A be a finite dimensional k-algebra equipped with a symmetric, nondegenerate and invariant bilinear form 〈 · ,· 〉 : A × A → k and B be a commutative k-algebra with a morphism of k-algebras ε : B → A such that ε(B ) ⊆ Z (A). Then, the following diagram commutes: Proof. We will show that for any n ≥ 0, the following digram is commutative: Hom A ⊗n ⊗ B ⊗ n(n−1) C. R. Mathématique, 2020, 358, n 11-12, 1239-1258 Since Φ n−1 is an isomorphism, it suffices to check that this diagram is commutative when composed with Φ n−1 : , k). Then, for a i ∈ A and b i , j ∈ B , we have On the other hand, let g := (N s)( f ). Then, we have This proves the result.
We now let C • (A, B, ε) denote the normalized complex associated to the cosimplicial module C • (A, B, ε) given in (19). Again, using the isomorphisms in (20), the complex C • (A, B, ε) becomes isomorphic to the normalized complex C • (A, B, ε) whose terms are given by where B is Connes' operator.
Proof. The commutativity of the lower square has already been shown in Proposition 9. By definition, Connes' operator on the complex C • (A, B, ε) is given by B = N s(1 − λ) which reduces to N s on the normalized complex C • (A, B, ε). Hence, it may be directly verified that the upper diagram commutes.