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? 



A differential graded Lie algebra with the Hochschild differential
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*.


