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](http://en.wikipedia.org/wiki/Batalin%2DVilkovisky_algebra).



Relation between Gerstenhaber bracket and Connes differential
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!


