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.

guess(but I am not sure this is true in detail) is that the vanishing of higher cohomology groups corresponds to the Lie algebra being semisimple, so all submodules of a given module split off as module summands. Presumably if one replaces su(n) by something solvable then not every indecomposable module is irreducible, and my instinct is that this should correspond to some non-trivial $H^1$ – Yemon Choi Oct 4 '12 at 8:48