Relation between Gerstenhaber bracket and Connes differential - MathOverflow

Relation between Gerstenhaber bracket and Connes differential

2011-07-08T14:15:19Z

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.

Answer by David Ben-Zvi for Relation between Gerstenhaber bracket and Connes differential

2011-07-08T14:41:27Z

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.

Answer by menichi for Relation between Gerstenhaber bracket and Connes differential

2011-07-18T07:18:32Z

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!