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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.

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2 Answers

up vote 3 down vote accepted

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!

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Thanks. I'm switching the green checkmark to this answer, since it more directly addresses my question. But I would accept both answers if I could. –  Kevin Walker Jul 18 '11 at 12:25
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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.

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Thanks, that does seem relevant. In particular, the middle part of equation (0.1) of that paper (which they state for polyvector fields and differential forms) is the same as the relation I'm asking about. Presumably the generalization to arbitrary Hochschild [co]homology follows from one of their results about G-infinity structures. –  Kevin Walker Jul 8 '11 at 15:20
    
Great! I haven't tried but this must follow formally from TFT, at least for smooth proper algebras (which is presumably how you arrived at it?) - i.e., we can act with one 2-framed circle (HH^*) on another (HH_*), and ask how the product interacts with rotation of the circle.. –  David Ben-Zvi Jul 8 '11 at 15:32
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Yes, something like that. In my set-up, I consider circles divided into incoming and outgoing regions (instead of framings). HH^* corresponds to a circle composed of one incoming interval and one outgoing interval (i.e. a bigon). HH_* corresponds to the entire circle being outgoing (a 0-gon). In general one could have a circle with n incoming regions alternating with n outgoing regions (a 2n-gon). The 2-dimensional part of the TFT-ish structure is a colored operad, sort of like the little disks operad except that the circles are "colored" by 2n-gons and the surfaces could be... –  Kevin Walker Jul 8 '11 at 15:58
    
...higher genus. The surfaces also have a foliation by oriented intervals, corresponding to the direction of "time". Homology classes in the topological space of such surfaces give operations (e.g. the four operations mentioned in the question), and one can deduce all the relations I mention above from easy homology calculations for these spaces of surfaces. I don't think I need to put any restrictions on the algebra C for all this to work, but perhaps I've overlooked something? –  Kevin Walker Jul 8 '11 at 16:04
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That sounds very reasonable.. I'm thinking within the full cobordism hypothesis framework, which requires the algebra smooth proper to get a framed 2d TFT (the smoothness and properness follows from allowing pictures corresponding to the disc and the saddle). I don't have a clear picture what TFT structures are allowed without full dualizability (eg what hypotheses you need on an algebra to define a noncompact framed theory). –  David Ben-Zvi Jul 8 '11 at 16:25
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