# Is the definition of Gerstenhaber bracket related to operads?

I was reading these notes by Keller. On the page 19 he defines the Gerstenhaber bracket on the Hochschild cochain complex. But first of all he defines the operation $\bullet$ on the cochains by $$\small (f\bullet g)(a_1,\dots,a_{p+q-1})=\sum\limits_{i=0}^{p} (-1)^{i(q+1)} f(a_1,\dots,a_i,g(a_{i+1},\dots,a_{i+q}),a_{i+q+1},\dots,a_{p+q-1})$$ where $f$ is a $p$-cochain and $g$ is a $q$-cochain. After that he defines the Gerstenhaber bracket by $[f,g]_G=f\bullet g-(-1)^{(p-1)(q-1)}g\bullet f$.

I have noticed that the summands in the operation $\bullet$ look a lot like the partial composition in operads, as if $f$ was an element of arity $p$ and $g$ was an element of arity $q$ in some operad. So my question is, is it just a coincidence, or there is indeed some way to see $f,q$ as operations in some operad? Or maybe is there a way to see the operation $\bullet$ coming from some operadic context?

I am a complete novice in these things, so I am sorry if my question is silly. Anyways I would be happy if someone would explain that to me.

• No this is absolutely true, there is a connection. The book in which I was reading about this is, unfortunately, in my office! I wonder if I can make some useful comment though... – Jonathan Beardsley Aug 15 '13 at 4:25
• There's a rather interesting discussion of this in an article called "Operads and Nonassociative Deformations" by Eugen Paul. – Jonathan Beardsley Aug 15 '13 at 17:50

This example fits more generally into the following context. Let $P$ be a Koszul operad and $A$ a $P$-algebra. There is attached to $P$ a cohomology theory for $P$-algebras. The two most well known examples is that when $P = \mathsf{Lie}$ we get ordinary Lie algebra cohomology, and that when $P = \mathsf{Ass}$ we get Hochschild homology. This is in general defined by the chain complex $$\mathrm{Hom}_{\mathbb S}(P^!,\mathrm{End}(A))$$ i.e. maps of $\mathbb S$-modules from the Koszul dual co-operad of $P$ to the endomorphism operad of $A$. This chain complex sits inside $$\mathrm{Hom}_{\mathbb k}(P^!,\mathrm{End}(A)),$$ i.e. maps which are not necessarily equivariant. The latter space is itself in a natural way the sum of all components of an operad, the convolution operad. Convolution gives an operad structure on maps from any co-operad to an operad.
Now on the sum of all components of an operad there is a pre-Lie structure given by operadic composition. This is the operation you called $\bullet$, and as you say, it is just given by composition. Antisymmetrizing this gives an honest Lie bracket. This bracket on $$\mathrm{Hom}_{\mathbb k}(P^!,\mathrm{End}(A)),$$ leaves $$\mathrm{Hom}_{\mathbb S}(P^!,\mathrm{End}(A))$$ invariant, giving a Lie bracket on the chain complex for the operadic cohomology. When $P = \mathsf{Ass}$ and $A$ is an associative algebra we have defined the Gerstenhaber bracket on the Hochschild complex. This is what you are seeing.
• Just wondering, what do you mean by equivariant" here with respect to $\mathbb{S}$? – Jonathan Beardsley Aug 15 '13 at 5:56
• An $\mathbb S$-module $V$ is a collection $V(n)$ of representations of the symmetric groups $\mathbb S_n$. By an $\mathbb S$-module map $V \to W$ I mean a collection of $\mathbb S_n$-equivariant maps $V(n) \to W(n)$. – Dan Petersen Aug 15 '13 at 5:57