Let $C$ be an arbitrary algebra (more generally, a linear 1-category). The following structures are well-known:
A degree-0 product on the Hochschild cohomology $HH^*(C)$
$$
HH^*(C) \otimes HH^*(C) \to HH^*(C)
$$
$$
a \otimes b \mapsto ab
$$
A degree-0 action of Hochschild cohomology on the Hochschild homology $HH_*(C)$
$$
HH^*(C) \otimes HH_*(C) \to HH_*(C)
$$
$$
a \otimes \gamma \mapsto a\cdot \gamma
$$
A degree-1 unary operation on Hochschild homology (Connes differential)
$$
HH_*(C) \to HH_*(C)
$$
$$
\gamma \mapsto B(\gamma)
$$
A degree-1 binary operation on Hochschild cohomology (Gerstenhaber bracket)
$$
HH^*(C) \otimes HH^*(C) \to HH^*(C)
$$
$$
a \otimes b \mapsto a * b
$$
The above operations satisfy some well-known relations. (Note that I am not attempting to get the signs right.)
-
graded commutativity $ab = \pm ba$
-
more graded commutativity $a * b = \pm b * a$
-
Poisson identity $a * (bc) = (a * b)c + b(a * c)$
-
Jacobi identity $a * (b * c) + b * (c * a) + c * (a * b) = 0$
-
$B$ is a differential $B(B(\gamma)) = 0$
-
various associativities $(ab)c = a(bc)$; $(a * b) * c = a * (b * c)$; $(ab)\cdot\gamma = a\cdot(b\cdot\gamma)$
The following relation, expressing the action of a Gerstenhaber bracket on Hochschild homology in terms of the Connes differential, seems to be less well-known. At least I haven't been able to find it in the literature.
$$
(a*b)\cdot\gamma = ab\cdot B(\gamma) – a\cdot B(b\cdot \gamma) – b\cdot B(a\cdot\gamma) + B(ba\cdot\gamma)
$$
(Again, I haven't tried to get the signs right.)
Question: Is there a reference for the above relation?
Note: The above relation follows from the fact that the first homology of a certain operad space is 4-dimensional, so there must be some relation between the five degree-1 maps $HH^*(C)\otimes HH^*(C)\otimes HH_*(C)\otimes \to HH_*(C)$ which figure in the relation.
Another note: In cases where $HH^*(C) \cong HH_*(C)$ and there is a BV algebra structure, I think the relation follows from the usual definition of the Gerstenhaber bracket in terms of the BV structure. See the "Antibracket" section of this Wikipedia article.
Best Answer
Hi,
Your formula is due (without the signs!) due to Ginzburg Calabi-Yau algebras (9.3.2) as explained in Lemma 15 of my paper, Batalin-Vilkovisky algebra structures on Hochschild Cohomology, Bull. Soc. Math. France 137 (2009), no 2, 277-295 (sorry for quoting myself!)
Here is Lemma 15
Lemma 15 [17, formula (9.3.2)] Let A be a differential graded algebra. For any η, ξ ∈ HH ∗ (A, A) and c ∈ HH∗ (A, A), {ξ, η}.c = (−1)|ξ| B [(ξ ∪ η).c] − ξ.B(η.c) + (−1)(|η|+1)(|ξ|+1) η.B(ξ.c) + (−1)|η| (ξ ∪ η).B(c).
In a condensed form, this formula is
(34) $i_{\{a,b\}}=(-1)^{\vert a\vert+1}[[B,i_{a}],i_b]=[[i_{a},B],i_b].$
See formula (34) of my second paper Van Den Bergh isomorphisms in String Topology, J. Noncommut. Geom. 5 (2011), no. 1, 69-105. (sorry for quoting myself again!)
In this paper, I thought I gave a new definition of BV-algebras. But this definition appears more or less in the section "Compact formulation in terms of nested commutators." of the Wikipedia article, you quote! However, I was unable to find this definition in the bibliography quoted in the Wikipedia article.
Concerning signs, in my first paper, I made a mistake, corrected in my second paper. So (34) is correct and Lemma 15 has some signs problems.
ps: David Ben-zvi is absolutely right. This formula is a consequence of Tamarkin-tsygan calculus!