Let $(V,\cdot)$ be an associative algebra and $W$ be a vector space endowed with a bimodule structure $\triangleright:V\otimes W\to W$ and $\triangleleft:W\otimes V\to W$ such that the following relations hold:

$(f\cdot g)\cdot h=f\cdot (g\cdot h)$

$(f\cdot g)\triangleright X=f\triangleright(g\triangleright X)$

$X\triangleleft(f\cdot g)=(X\triangleleft f)\triangleleft g$

$(f \triangleright X)\triangleleft g=f \triangleright(X \triangleleft g)$.

One defines the associated Hochshild differential $\delta:H^i(V,W)\to H^{i+1}(V,W)$ where $H^i(V,W):=Hom(V^{\otimes i+1},W)$ as: $ \delta\Phi(a_0,\ldots,a_{i+1}):=(-1)^i a_0\triangleright\Phi(a_1,\ldots, a_{i+1})$

$-(-1)^i\underset{k,l\geqslant 0}{\sum_{k+l=i}}(-1)^{k}\,\Phi(a_0,\ldots,a_{k-1}\cdot a_k,\ldots,a_{i+1})+\Phi(a_0,\ldots,a_{i})\triangleleft a_{i+1}$.

for all $\Phi\in H^i(V,W)$, $a_i\in V$.

Question: Whenever $W=V$ and $\triangleright=\triangleleft=\cdot$, the differential on $H(V,V)$ can be extended to a differential graded Lie algebra by defining the Gerstenhaber bracket, for which the Hochschild differential is a derivation.

Can one generalise this construction for an arbitrary bimodule $W$ and define a graded Lie bracket on $H(V,W)$ such that the Hochschild differential is a derivation thereof?

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    $\begingroup$ A good start would be to think what happens in degree 1. $\endgroup$ – Mariano Suárez-Álvarez Dec 12 '17 at 4:14

Not in general, no. However $H^*(V,W)$ is a representation of the Lie algebra $H^*(V,V)$, by letting, for $f \in \hom(V^{\otimes m}, W)$ and $g \in \hom(V^{\otimes n}, V)$, $$f \circ g := \sum_{i=1}^m \pm f \circ_i g,$$ just like in the definition of the Lie bracket on $H^*(V,V)$. This module structure already appeared in Gerstenhaber's 1963 paper The cohomology structure of an associative ring.


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