I have a question about the computation of an Hochschild Cohomology. Let's consider the right action of SU(2) on its tangent algebra (the 2x2 anti hermitian matrices, denoted by $iH_{2}$) by : $$ (g,A) \mapsto Ad_{g^{-1}}A $$ I would like to calculate the second Hochschild cohomology group associated with this action. To be more precise, the calculation is about this quotient :

$$ \langle \lambda\,:\,SU(2) \to iH_{2}\,|\,\lambda(UV)-V^{-1}\lambda(U)V-\lambda(V)=0\rangle /\langle U \mapsto UAU^{-1}-A\rangle, A\in iH_{2}\rangle $$

In particular is there any chance for this to be zero using topological arguments ? What is the relation between Hochschild cohomology group and invariant cohomology ? I am pretty ignorant in that kind of things so if you have good references, that would help me a LOT!

Thank you :D Nicolas

I have a question about the computation of an Hochschild Cohomology. Or at least about a space which really looks like a cohomology space. All the functions i consider are assumed to be smooth. Let's consider the right action of SU(2) on its tangent algebra (the 2x2 anti hermitian matrices, denoted by $iH_{2}$) by : $$ (g,A) \mapsto Ad_{g^{-1}}A $$ I would like to calculate the second Hochschild cohomology group associated with this action. To be more precise, the calculation is about this quotient :

$$ \langle \lambda\,:\,SU(2) \to iH_{2}\,|\,\lambda(UV)-V^{-1}\lambda(U)V-\lambda(V)=0\rangle /\langle U \mapsto UAU^{-1}-A\rangle, A\in iH_{2}\rangle $$

In particular is there any chance for this to be zero using topological arguments ? What is the relation between Hochschild cohomology group and invariant cohomology ? I am pretty ignorant in that kind of things so if you have good references, that would help me a LOT!

Thank you :D Nicolas