A B-infinity algebra structure of singular Hochschild complex

1. Introduction

The singularity category $sg(A)$ of a right Noetherian algebra A over a field k is defined as the Verdier quotient of the bounded derived category of finitely generated (right) A-modules by the full subcategory of complexes quasi-isomorphic to bounded complexes of finitely generated projective modules. It was introduced by Buchweitz in an unpublished paper [1] in 1986 and rediscovered, in its scheme-theoretic variant, by Orlov in 2003 [10].

Let M, N be two modules over a Gorenstein algebra G. The ith $Tate$ $cohomology$ group of M with values in N is defined as ${\underline{Ext}}_G^i(M,N):={Hom}_{sg(G)}(M,S^{i}N)$, where S is the shift functor. In [12, 13, 14], Wang defined the singular Hochschild cohomology ${\mathrm{\text{HH}}}_{sg}^{*}(A,A)$ (see Definition 3.1) for any Noetherian algebra A. It is a colimit of some Hochschild cohomology groups. In [12, 14], he showed there exists a natural isomorphism between the singular Hochschild cohomology ${\mathrm{\text{HH}}}_{sg}^{*}(A,A)$ and the Tate cohomology ${\underline{Ext}}_{A{\otimes }_k{A}^{op}}^{*}(A,A)$. At the complex level, Wang defined the singular Hochschild cochain complex $C_{sg}^{*}(A,A)$ (see Definition 3.1). It is a colimit of some cochain complexes. Wang showed that, like Hochschild cohomology, singular Hochschild cohomology carries a Gerstenhaber algebra structure and singular Hochschild cochain complex carries a $B_{\infty }$-algebra structure in [9, 12, 14].

In this paper, we calculate the low order relations of $B_{\infty }$-algebra and introduce the $bibrace$ $algebra$ (see Definition 2.7). We construct a new $B_{\infty }$-algebra structure for the singular Hochschild cochain complex $C_{sg}^{*}(A,A)$ such that the singular Hochschild cohomology carries the Gerstenhaber algebra structure in [4, 12]. Let us state the main results of this paper.

Theorem 1.1 (see Theorem 3.8). Let A be a unital associative k-algebra. There is a bibrace algebra structure on $C_{sg}^{*}(A,A)$. In particular, it lifts the Lie bracket of Gerstenhaber algebra at the homology level.

For the $B_{\infty }$-algebra $C_{sg}^{*}(A,A)$, we have the $opposite$ $B_{\infty }$-algebra $C_{sg}^{*}{(A,A)}^{\mathrm{\text{opp}}}$ (see Definition 2.13) and the $transpose$ $B_{\infty }$-algebra $C_{sg}^{*}{(A,A)}^{\mathrm{\text{tr}}}$ (see Definition 2.14).

Theorem 1.2 (see Corollary 3.9). Let A be a unital associative k-algebra. Suppose the swap map $T:C_{sg}^{*}{(A,A)}^{\mathrm{\text{tr}}}\to C_{sg}^{*}(A^{\mathrm{\text{op}}},A^{\mathrm{\text{op}}})$is a strict $A_{\infty }$-isomorphism. Then there is a $B_{\infty }$-isomorphism from the opposite $B_{\infty }$-algebra $C_{sg}^{*}{(A,A)}^{\mathrm{\text{opp}}}$to the bibrace algebra $C_{sg}^{*}(A^{\mathrm{\text{op}}},A^{\mathrm{\text{op}}})$.

Convention. In this paper, k is a field and all vector spaces are over k. For simplifying the notation, we always write ⊗ instead of ${\otimes }_k$ and write $Hom$ instead of ${Hom}_k$, when no confusion may occur. The tensor product of n copies of the space V is denoted $V^{\otimes n}$. For $v_i\in V$, the element $v_1\otimes \cdots \otimes v_n$ of $V^{\otimes n}$ is denoted by $v_1\cdots v_n$ or $(v_1,\dots ,v_n)$. The notation comes from [8]. The identity map is denoted $1$.

2. $B_{\infty }$-Algebra

In this section, we assume that $V={⨁}_{p\in \mathbb{Z}}{V}^p$ is a $\mathbb{Z}$-graded k-vector space. Let $SV$ be the shift of V with ${(SV)}^p:=V^{p+1}$. The cofree tensor coalgebra over the graded vector space $SV$ is the graded tensor module:

equipped with the deconcatenation coproduct $\mathrm{\Delta }$, the counit c and the coaugmentation u in the graded case.

We denote the reduced cofree tensor coalgebra ${\overline{T}}^c(SV):=coker\mathrm{ u}$ of $SV$. We need the Koszul sign rule

where for a homogeneous map g, its degree is denoted by $|g|$ and for a homogeneous element x, its degree is denoted by $|x|$.

Definition 2.1. Let V be a $\mathbb{Z}$-graded vector space:

(i)	When the noncounital cofree tensor coalgebra ${\overline{T}}^c(SV)$ is a dg coalgebra, then V is called $A_{\infty }$-algebra.
(ii)	When the counital cofree tensor coalgebra $T^c(SV)$ is a dg bialgebra, then V is called $B_{\infty }$-algebra.

Definition 2.2 ([7]). An $A_{\infty }$-algebra over a field k is a $\mathbb{Z}$-graded vector space $V={⨁}_{p\in \mathbb{Z}}{V}^p$ endowed with graded maps $m_n:A^{\otimes n}\to A,n\ge 1$, of degree $2-n$ satisfying the following relations:

where $n=r+s+t$, $s\ge 1$ and $r,t\ge 0$.

Definition 2.3 ([7]). An $A_{\infty }$-morphism of $A_{\infty }$-algebras $f=(f_n):(V,m_n)\to (V^{\prime },m_n^{\prime })$ is given by a collection of graded maps $f_n:V^{\otimes }\to V^{\prime }$ of degree $1-n$ such that, for all $n\ge 1$, we have

where $𝜀=(r-1)(i_1-1)+(r-2)(i_2-1)+\cdots +2(i_{r-2}-1)+(i_{r-1}-1)$.

An $A_{\infty }$-morphism $f:V\to V^{\prime }$ is $strict$ if $f_i=0$ for each $i\ne 1$. The $composition$ $g{\circ }_{\infty }f$ of two $A_{\infty }$-morphisms $f={(f_n)}_{n\ge 1}:V\to V^{\prime }$ and $g={(g_n)}_{n\ge 1}:V^{\prime }\to V^{\prime \prime }$ is given by

where $𝜀$ is defined as above.

Denote by $C_{(r,s,t)}^n:=1^{\otimes r}\otimes m_s\otimes 1^{\otimes t}$ where C represents the coderivation. So the relations $({\mathcal{𝒜}}_n)$ are the abbreviation $\sum \pm m_{r+1+t}\circ C_{(r,s,t)}^n=0$.

Suppose that $T^c(SV)$ is a cofree bialgebra. The product is the map $*:T^c(SV)\otimes T^c(SV)\to T^c(SV)$ and the unit is the map $k\to T^c(SV)$. Since $T^c(SV)$ is cofree, the map * is completely determined by its value in $SV$, that is by maps:

where ${\mu }_{pq}$ is the homogenous map of degree 0.

The associative operation on $T^c(SV)$ is recovered from the ${\mu }_{pq}$ by the formula

where ${\mu }_{(\underline{p},\underline{q})}^t=({\mu }_{p_1,q_1}\cdots {\mu }_{p_t,q_t})$.

Let maps $M_{pq}:V^{\otimes p}\otimes V^{\otimes q}\to V$ be defined such that the following square is commutative (i.e. $M_{pq}=S^{-1}{\mu }_{pq}(S^{\otimes p}\otimes S^{\otimes q})$):

In particular, we have $M_{01}=S^{-1}{\mu }_{01}(1_k\otimes S)$ and $M_{10}=S^{-1}{\mu }_{10}(S\otimes 1_k)$. Notice that $|M_{pq}|=1-(p+q)$ since $|{\mu }_{pq}|=0$.

By the Koszul sign rule, there is an identity

For all sets of indices $(\underline{i},\underline{j}):=(i_1,\dots ,i_k;j_1,\dots ,j_k)$ such that $i_1+\cdots +i_k=p$ and $j_1+\cdots +j_k=q$, we denote the map which sends $(u_1\cdots u_p,v_1\cdots v_q)\in V^{\otimes p}\otimes V^{\otimes q}$ to

The first nontrivial associative relation of three elements ($i+j+k=3$), $(Su*Sv)*Sw=Su*(Sv*Sw)$, reads:

The following is the associative law of four elements ($i+j+k=4$):

These read, respectively,

More generally, we have the associativity of the product $*:T^c(SV)\otimes T^c(SV)\to T^c(SV)$ for any triple $(i,j,k)$ of positive integers:

The associativity of the product reads:

where $𝜖={\sum }_{r=1}^l(k+l-r)(i_r+j_r-1)+{\sum }_{r=1}^l{j}_r(i_{r+1}+\cdots +i_l)$ with $M_{(\underline{i},\underline{j})}^l=(M_{i_1,j_1}\cdots M_{i_l,j_l})$ and $\delta ={\sum }_{s=1}^m(m-s)(j_s+k_s-1)+{\sum }_{s=1}^m{k}_s(j_{s+1}+\cdots +j_l)$ with $M_{(\underline{j},\underline{k})}^m=(M_{j_1,k_1}\cdots M_{j_m,k_m})$.

Suppose that $d:T^c(SV)\to T^c(SV)$ is the differential of degree 1. The first nontrivial Leibniz rule of two elements $(j+k=2)$, $d(Su*Sv)=dSu*Sv+Su*dSv$, reads:

The following is the Leibniz rule of three elements ($j+k=3$),

These read, respectively,

More generally, we have the Leibniz rule of the product $*:T^c(SV)\otimes T^c(SV)\to T^c(SV)$ and the differential $d:T^c(SV)\to T^c(SV)$ for any tuples $(j,k)$ of positive integers:

Since $T^c(SV)$ is cofree, the maps *, d are completely determined by its value in $SV$. The Leibniz rule reads:

where $\alpha ={\sum }_{r=1}^n(n-r)(k_r+l_r-1)+{\sum }_{r=1}^n{l}_r(k_{r+1}+\cdots k_n)$, $\beta =r+s(t+l)$, $\gamma =(k+r^{\prime })+s^{\prime }{t}^{\prime }$ and $C_{(r,s,t)}^n=1^{\otimes r}\otimes m_s\otimes 1^{\otimes t}$.

Definition 2.4. A $B_{\infty }$-algebra is an $A_{\infty }$-algebra $(V,m_n:V^{\otimes n}\to V,({\mathcal{𝒜}}_n))$ equipped with operations

satisfying and two classes relations $({\mathcal{ℛ}}_{ijk})$ for any triple $(i,j,k)$ of positive integers and $({\mathcal{ℒ}}_{jk})$ for any pair $(j,k)$ of positive integers.

Remark 2.5. Let $n=i+j+k$, $i,j,k\in \mathbb{N}$. Fixing $n\in \mathbb{N}$, the number of relations $({\mathcal{ℛ}}_{ijk})$ is $(\begin{array}{c}\hfill n-1\hfill \\ \hfill 2\hfill \end{array})$. For example, if $n=6$, there are 10 relations. Precisely, $({\mathcal{ℛ}}_{114})$, $({\mathcal{ℛ}}_{411})$, $({\mathcal{ℛ}}_{123})$, $({\mathcal{ℛ}}_{321})$, $({\mathcal{ℛ}}_{132})$, $({\mathcal{ℛ}}_{231})$, $({\mathcal{ℛ}}_{141})$, $({\mathcal{ℛ}}_{213})$, $({\mathcal{ℛ}}_{312})$, $({\mathcal{ℛ}}_{222})$. Let $m=j+k$, $j,k\in \mathbb{N}$, then the number of the relations $({\mathcal{ℒ}}_{jk})$ is $m-1$.

Example 2.6. (a) If $M_{pq}=0$ for all $(p,q)$ different from $(0,1)$ and $(1,0)$, then vector space V is just the shuffle algebra. Indeed, if $(u_1\cdots u_p,v_1\cdots v_q)\in V^{\otimes p}\otimes V^{\otimes q}$, then the multiplication is

(b) If $M_{pq}=0$ for all $(p,q)$ such that $p\ge 2$, then it is a brace algebra.

Definition 2.7. A $B_{\infty }$-algebra $(V,m_n,M_{pq})$ is called a bibrace algebra, provided that $m_n=0$ for $n\ge 3$ and $M_{pq}=0$ for $p,q\ge 2$.

The underlying $A_{\infty }$-algebra structure of a bibrace algebra is just a dg algebra. For a bibrace algebra, there exists $brace$ $operations$

and $cobrace$ $operations$ for any $a,b_1,\dots ,b_p\in V$.

Remark 2.8. The Hochschild cochain complex is a brace algebra [5, 11] and the coHochschild cochain complex is a cobrace algebra.

We call two relations $({\mathcal{ℛ}}_{ijk})$ and $({\mathcal{ℛ}}_{i^{\prime }{j}^{\prime }{k}^{\prime }})$ are $symmetric$ if for each term $M_{i_1{j}_1}\cdots M_{i_n{j}_n}$ in $({\mathcal{ℛ}}_{ijk})$, there is a responding term $M_{i_n{j}_n}\cdots M_{i_1{j}_1}$ in $({\mathcal{ℛ}}_{i^{\prime }{j}^{\prime }{k}^{\prime }})$ and for each term $M_{i_1{j}_1}\cdots M_{i_n{j}_n}$ in $({\mathcal{ℛ}}_{i^{\prime }{j}^{\prime }{k}^{\prime }})$, there is also a responding term $M_{i_n{j}_n}\cdots M_{i_1{j}_1}$ in $({\mathcal{ℛ}}_{ijk})$. There are some observed properties of the relations $({\mathcal{ℛ}}_{ijk})$.

Proposition 2.9. Assume V is a $B_{\infty }$-algebra then $({\mathcal{ℛ}}_{ijk})$and $({\mathcal{ℛ}}_{kji})$are symmetric.

Proof. If there is a monomial $M_{il}(1^{\otimes i},M_{j_1{k}_1},\dots ,M_{j_l{k}_l})$ in the left side of the relations $({\mathcal{ℛ}}_{ijk})$, then there is the corresponding monomial $M_{li}(M_{k_l{j}_l}\cdots M_{k_1{j}_1},1^{\otimes i})$ in the right side of the relations $({\mathcal{ℛ}}_{kji})$ and vice versa. □

Proposition 2.10. Assume V is a bibrace algebra, then we have:

(i)	Each term is zero in $({\mathcal{ℛ}}_{ijk})$for $i,j,k\ge 2$.
(ii)	If one of $i,j,k$is $1$, then it exists non-zero terms in $({\mathcal{ℛ}}_{ijk})$.

Proof. (i) From the definition of the bibrace algebra, this is obvious.

(ii) For $({\mathcal{ℛ}}_{1jk})$, there is $M_{1j+k}(1,M_{10}\cdots M_{10}{M}_{01}\cdots M_{01})\ne 0$. Hence there are non-zero terms in the $({\mathcal{ℛ}}_{ij1})$ by Proposition 2.9. For $({\mathcal{ℛ}}_{i1k})$, there is $M_{i1}(1^{\otimes i},M_{1k})\ne 0$. □

Corollary 2.11. For a brace algebra, there are no non-zero terms in $({\mathcal{ℛ}}_{ijk})$if and only if i is greater than or equal to 2.

Proof. Notices that each term is zero in $({\mathcal{ℛ}}_{ijk})$ for $i,j,k\ge 2$ by Proposition 2.10 (i). Then we just prove that there are no nonzero terms in the relations $({\mathcal{ℛ}}_{i1k})$ and $({\mathcal{ℛ}}_{ij1})$ for $i\ge 2$. There are $M_{st}=0$ for all $s\ge 2$ by the definition of the brace algebra. For the relations $({\mathcal{ℛ}}_{i1k})$, we have

There are no nonzero terms in $({\mathcal{ℛ}}_{i1k})$ since $i\ge 2$. For the relations $({\mathcal{ℛ}}_{ij1})$, we have There are no nonzero terms in $({\mathcal{ℛ}}_{ij1})$ since $i\ge 2$. For the relations $({\mathcal{ℛ}}_{1jk})$, there is a nonzero term  □

Definition 2.12 ([2]). A $B_{\infty }$-morphism $f={(f_n)}_{n\ge 0}$ from the $B_{\infty }$-algebra $(V,m_n,M_{pq})$ to the $B_{\infty }$-algebra $(V^{\prime },m_n^{\prime },M_{pq}^{\prime })$ is an $A_{\infty }$-morphism

satisfying the following identity for any $p,q\ge 0$: where $𝜖={\sum }_{k=1}^r(r+s-k)(i_k-1)+{\sum }_{k=1}^s(s-k)(j_k-1)$ and $\delta ={\sum }_{k=1}^t(t-k)(p_k+q_k-1)+{\sum }_{k=1}^t{p}_k(q_{k+1}+\cdots +q_t)$ with $M_{(\underline{p},\underline{q})}^t=(M_{p_1,q_1}\cdots M_{p_t,q_t})$.

A $B_{\infty }$-morphism $f:V\to V^{\prime }$ is $strict$ if $f_i=0$ for each $i\ne 1$. A $B_{\infty }$-morphism $f:V\to V^{\prime }$ is a $B_{\infty }$-$isomorphism$, if there exists a $B_{\infty }$-morphism $g:V^{\prime }\to V$ such that $f{\circ }_{\infty }g=1_{A^{\prime }}$ and $g{\circ }_{\infty }f=1_A$.

Definition 2.13 ([2]). The opposite $B_{\infty }$-algebra of a $B_{\infty }$-algebra $(V,m_n,M_{pq})$ is defined to be the $B_{\infty }$-algebra $(V,m_n,M_{pq}^{\mathrm{\text{opp}}})$, where

Here we denote $𝜀=(|b_1|+\cdots +|b_q|)(|a_1|+\cdots +|a_p|)$.

Definition 2.14 ([2]). The transpose $B_{\infty }$-algebra of a $B_{\infty }$-algebra $(V,m_n,M_{pq})$ is defined to be the $B_{\infty }$-algebra $(V,m_n^{\mathrm{\text{tr}}},M_{pq}^{\mathrm{\text{tr}}})$, where

Here

Lemma 2.15 ([2]). Let $(V,m_n,M_{pq})$be a $B_{\infty }$-algebra. Then there is a natural $B_{\infty }$-isomorphism between the opposite $B_{\infty }$-algebra $V^{\mathrm{\text{opp}}}$and the transpose $B_{\infty }$-algebra $V^{\mathrm{\text{tr}}}$.

The following observation follows directly from Definition 2.12.

Lemma 2.16. Let $(V,m_n,M_{pq})$and $(V^{\prime },m_n^{\prime },M_{pq}^{\prime })$be two bibrace algebras. Then a strict $A_{\infty }$-morphism $f:(V,m_n)\to (V^{\prime },m_n^{\prime })$is a $B_{\infty }$-morphism if and only if f is compatible with the bibrace operations,

for any $p,q\ge 1$and $a,b_1,\dots ,b_p,b,a_1,\dots ,a_q\in V$.

3. Singular Hochschild Complex

Let A be a unital associative k-algebra. Let $\overline{A}$ be the quotient k-module $A/(k\cdot 1)$ of A by the k-scalar multiplies of the unit. For convenience, we write $a_{p,p+q}:=a_p\otimes a_{p+1}\otimes \cdots \otimes a_{p+q}\in A^{\otimes q+1}$.

The $normalized$ $bar$ $resolution$ ${\mathrm{\text{Bar}}}_{*}(A)$ is the complex of A-A-bimodules with ${\mathrm{\text{Bar}}}_p(A)=A\otimes {(S\overline{A})}^p\otimes A$ for $p\in {\mathbb{Z}}_{\ge 0}$ and the differentials

For any $p\in {\mathbb{Z}}_{\ge 0}$, the kernel of $d_{p-1}$ denotes by ${\mathrm{\Omega }}_{\mathrm{\text{sy}}}^p(A)$. And ${\mathrm{\Omega }}_{\mathrm{\text{sy}}}^p(A)$ is defined in degree $-p$ as a graded A-A-bimodule. Therefore, ${\mathrm{\Omega }}_{\mathrm{\text{sy}}}^0(A)=A$. We denote

It is a graded A-A-bimodule. Let $\mu$ be the multiplication of A, so ${\mathrm{\Omega }}_{\mathrm{\text{sy}}}^1(A):=ker\{\mu :A\otimes A\to A\}$.

Let M be a graded A-A-bimodules. The $normalized$ $Hochschild$ $cochain$ $complex$ $C^{*}(A,M)$ with coefficients in M is ${Hom}_{A\otimes A^{op}}({\mathrm{\text{Bar}}}_{*}(A),M)$. Since there is a canonical isomorphism

for any $p\in {\mathbb{Z}}_{\ge 0}$. Therefore, $C^{*}(A,M)$ is the following complex: where $C^p(A,M):={\prod }_{i\in {\mathbb{Z}}_{\ge 0}}{Hom}^p({(S\overline{A})}^{\otimes i},M)$ for $p\in \mathbb{Z}$ and The differential is given by for $f\in C^p(A,M)$.

The pth $Hochschild$ $cohomology$ of A with coefficients in M, denoted by ${\mathrm{\text{HH}}}^p(A,M)$, is defined as the pth cohomology group of $C^{*}(A,M)$.

The $noncommutative$ $differential$ $forms$ [3, 6] of A is the graded k-module

It is a dg A-A-bimodule.

For any $p\in {\mathbb{Z}}_{\ge 0}$, there is an embedding of degree zero of cochain complexes:

In fact, it is a chain map. Therefore, we get an inductive system of the category of cochain complexes of k-modules:

Definition 3.1 ([14]). The singular Hochschild cochain complex of A is the colimit

The cohomology groups of $C_{sg}^{*}(A,A)$ are denoted by ${\mathrm{\text{HH}}}_{sg}^{*}(A,A)$.

Remark 3.2. In [14], Wang showed the singular Hochschild cochain complex is a brace algebra.

Lemma 3.3 ([13]). We have an isomorphism of dg A-A-

For $f\in C^{m-p}(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^p(A))=Hom({(S\overline{A})}^{\otimes m},{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^p)$, we denote

and

For any $r\ge 0$, we denote

Lemma 3.4. For $f\in C^{m-p}(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^p(A))=Hom({(S\overline{A})}^{\otimes m},{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^p(A))$, then

(i)	${\mathrm{\Omega }}_{\mathrm{\text{sy}}}^r({\mathrm{\Omega }}_{\mathrm{\text{sy}}}^s(f))={\mathrm{\Omega }}_{\mathrm{\text{sy}}}^{r+s}(f)$for any $s,r\ge 0$.
(ii)	${\mathrm{\Omega }}_{\mathrm{\text{sy}}}^1:C^{*}(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^p(A))\to C^{*}(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^{p+1}(A))$, $f\mapsto {\mathrm{\Omega }}_{\mathrm{\text{sy}}}^1(f)$is a chain map for any $p\ge 0$.

Proof. On the one hand, the map ${\mathrm{\Omega }}_{\mathrm{\text{sy}}}^r({\mathrm{\Omega }}_{\mathrm{\text{sy}}}^s(f))$ sends $a_0\otimes {\overline{a}}_{1,m+r+s}\otimes a_{m+r+s+1}$ to

If we set $f({\overline{a}}_{1,m})={\sum }_i{b}_0^i\otimes {\overline{b}}_{1,p-1}^i\otimes b_p^i$, then it exists the only nonzero term in ${\alpha }_{p+s}^{-1}({\mathrm{\Omega }}_{\mathrm{\text{sy}}}^s(f)(1\otimes {\overline{a}}_{1,m+s}\otimes 1)$. On the other hand, the map ${\mathrm{\Omega }}_{\mathrm{\text{sy}}}^{s+r}(f)$ sends $a_0\otimes {\overline{a}}_{1,m+r+s}\otimes a_{m+r+s+1}$ to It identifies with This proves the first conclusion.

For any $m,p\ge 0$, there is the following commutative diagram:

where ${\delta }_{p+1}^{m-p}$ is the Hochschild differential in $C^{*}(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^{p+1}(A))$. We check that the following diagram is commutative:

for $f\in C^{m-p}(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^p(A))$ and $a_0\otimes {\overline{a}}_{1,m+2}\otimes a_{m+3}\in {\mathrm{\text{Bar}}}_{m+2}(A)$, on the one hand, we have that On the other hand, we have that Note that and we prove the second conclusion. □

Remark 3.5. By Lemma 3.4, we get an inductive system in the category of complexes of k-modules:

At the homology level, there exists an inductive system [13]

The colimit of this inductive system exactly is Hence we have The colimit of this inductive system is a Gerstenhaber algebra. We will lift it at the complex level.

We will construct a new $B_{\infty }$-structure for $C_{sg}^{*}(A,A)$. For $f\in C^{m-p}(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^p(A))$ and $g_k\in C^{n_k-q_k}(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^{q_k}(A))$, $k=1,\dots ,l$, we construct the bibrace operations:

where $𝜀=p+q_1+q_l$ and ${\sum }_{k=1}^l{r}_k+{\sum }_{k=1}^{l-1}{q}_k+l\le m$, $0\le t\le q_1$, ${\sum }_{k=1}^{l-1}{s}_k+{\sum }_{k=2}^{l-1}{q}_k+l\le m-t$ (cf. Fig. 1); where $\eta =p-{\sum }_{k=2}^{l-1}{n}_k+{\sum }_{k=1}^l{q}_k+l-1$ and ${\sum }_{k=1}^l{r}_k+{\sum }_{k=1}^{l-1}{n}_k+l\le p$, $0\le t\le n_1-1$, ${\sum }_{k=1}^{l-1}{s}_k+{\sum }_{k=2}^{l-1}{n}_k\le p+1-t$ (cf. Fig. 2).

Example 3.6. Figure 3 illustrates $f\{g_1,g_2\}$ for $f\in C^2(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^2(A))$, $g_1\in C^0(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^2(A))$, $g_2\in C^1(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^2(A))$.

Remark 3.7. For $f\in C^{m-p}(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^p(A))$ and $g\in C^{n-q}(A,{\mathrm{\Omega }}_{\mathrm{\text{sy}}}^q(A))$, we recall the cup product [13]

We note that ∪is compatible with the map ${\mathrm{\Omega }}_{\mathrm{\text{sy}}}^1$ and the Hochschild differential $\delta$. In addition, we define $m_1=[\mu ,-]$, $m_2=\cup$ and $m_k=0$ for $k\ge 3$ at the $A_{\infty }$-structure level.

Theorem 3.8. There is a bibrace algebra structure on $C_{sg}^{*}(A,A)$. In particular, it lifts the Lie bracket of Gerstenhaber algebra at the homology level.

Proof. Let us denote:

•	$m_1:=[\mu ,-]$,
•	$m_2:=\cup$,
•	$m_k:=0$ $k\ge 3$,
•	$M_{10}=M_{01}=1$,
•	$M_{1n}:=$ the brace operation $-\{-,\dots ,-\}$,
•	$M_{n1}:=$ the cobrace operation $\{-,\dots ,-\}-$,
•	$M_{mn}:=0$ otherwise $n,m\in \mathbb{N}$.

The relations $({\mathcal{ℛ}}_{ijk})$ and $({\mathcal{ℒ}}_{jk})$ can be done directly by graphic presentations.

Notice that $f\{g\}=\{f\}g$, we define the Lie bracket

This proves the theorem. □

We consider the $swap$ $isomorphism$

which sends $f\in C_{sg}^{*}(A,A)$ to for any ${\overline{a}}_1,\dots ,{\overline{a}}_m\in S\overline{A}$, where $𝜀=|f|+{\sum }_{i=1}^{m-1}|{\overline{a}}_i|(|{\overline{a}}_{i+1}|+\cdots +|{\overline{a}}_m|)$.

The following proof of corollary is similar to the proof of [2, Proposition 6.4].

Corollary 3.9. Suppose the swap map $T:C_{sg}^{*}(A,A)\to C_{sg}^{*}(A^{\mathrm{\text{op}}},A^{\mathrm{\text{op}}})$is a strict $A_{\infty }$-isomorphism. Then there is a $B_{\infty }$-isomorphism from the opposite $B_{\infty }$-algebra $C_{sg}^{*}{(A,A)}^{\mathrm{\text{opp}}}$to the bibrace algebra $C_{sg}^{*}(A^{\mathrm{\text{op}}},A^{\mathrm{\text{op}}})$.

Proof. There is a $B_{\infty }$-isomorphism between the transpose $B_{\infty }$-algebra $C_{sg}^{*}{(A,A)}^{\mathrm{\text{tr}}}$ and the opposite $B_{\infty }$-algebra $C_{sg}^{*}{(A,A)}^{\mathrm{\text{opp}}}$ by Lemma 2.15.

By definition,

where $r=\frac{q(q+1)}{2}+{\sum }_{i=1}^{q-1}|g_i|(|g_{i+1}|+\cdots +|g_q|)$ and $s=\frac{p(p+1)}{2}+{\sum }_{i=1}^{p-1}|f_i|(|f_{i+1}|+\cdots +|f_p|)$. Moreover, we have the following two identities: Hence, we have It is showed that T is a strict $B_{\infty }$-isomorphism from $C_{sg}^{*}{(A,A)}^{\mathrm{\text{tr}}}$ to $C_{sg}(A^{\mathrm{\text{op}}},A^{\mathrm{\text{op}}})$ by Lemma 2.16. Thus there is a $B_{\infty }$-isomorphism from the opposite $B_{\infty }$-algebra $C_{sg}{(A,A)}^{\mathrm{\text{opp}}}$ to the bibrace algebra $C_{sg}(A^{\mathrm{\text{op}}},A^{\mathrm{\text{op}}})$. □

Acknowledgments

The authors are very grateful to Prof. Nanqing Ding and Prof. Bernhard Keller. We thank the referee for his careful reading of the paper and for many helpful remarks.