TITLE: Hochschild (co)homology and representation theory
QUESTION [7 upvotes]: Dear members of Mathoverflow,
I just discovered the notion of Hochschild (co)homology. I understand well the formalism however I am wondering about the meaning of this (co)homology for representation theory. 
I consider an algebra $\mathcal{A}$, such as $\mathcal{U}(su(n))$, a finite dimensional representation space $V$ for $\mathcal{A}$ (they are well known in the case of $\mathcal{U}(su(n))$) and the $\mathcal{A}$-bimodule $\mathcal{M}=\mathrm{End}(V)$.
In this example, what would be the interpretation of Hochschild (co)homology from the point of view of representation theory ? To what is it an obstruction ?
I can compute things in simple cases such as $\mathcal{U}(su(2))$ but I do not the global interpretation emerge in this case...
Thank you in advance,
Damien.

REPLY [2 votes]: If $A$ is a $k$-algebra and $V$, $W$ are $A$-modules, then $Hom_k(V,W)$ is an $A$-bimodule in a natural way (acting contravariantly in $V$ and covariantly in $W$).   It is well known (can be found in Cartan-Eilenberg) that $n^{th}$-Hochschild cohomology of $Hom_k(V,W)$ is $Ext_A^n(V,W)$.  Your case of $End(V)$ would then give $Ext_A(V,V)$.