Relation between Gerstenhaber bracket and Connes differential

I'm not sure if your precise formulation appears there but I believe it should be part of the "homotopy calculus" structure studied by Tsygan and Tamarkin in various papers - see e.g. p.6 of Noncommutative differential calculus, homotopy BV algebras and formality conjectures, in which a similar relation is stated - namely that Hochschild chains with the Connes differential form a homotopy BV module over the canonical BV deformation of the homotopy Gerstenhaber algebra of Hochschild cochains.

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!