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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

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  • $\begingroup$ Hochschild cohomology is defined for associative algebras. What algebra are you trying to take the Hochschild cohomology of? Do you mean continuous group cohomology instead? $\endgroup$
    – Yemon Choi
    Commented Nov 30, 2015 at 17:22
  • $\begingroup$ @YemonChoi The n-cochains i am dealing with are smooth functions of $n$ elements of $SU(2)$ with values in $iH_{2}$ with differential : $$ \delta_{n}\lambda(U_{0},...,U_{n}) = \lambda(U_{1},...,U_{n}) + \sum_{i=0}^{n-1}(-1)^{i+1}\lambda(U_{0},...,U_{i}U_{i+1},..,U_{n}) + (-1)^{n+1}Ad_{U_{n}^{-1}}\lambda(U_{0},...,U_{n-1}) $$ It seems close to the definition of Hochschild cohomology ? $\endgroup$
    – Nicolas
    Commented Nov 30, 2015 at 17:54
  • $\begingroup$ A lot of cohomology theories look similar since they come from standard bar resolutions (or, if one likes that sort of thing, suitable comonads on categories, see e.g. Chapter 8 of Weibel's book Introduction to Homological Algebra). Since I am not an expert on the history and the terminology, perhaps you could edit your question to specify that your cochains are intended to be smooth functions on products of copies of SU(2) $\endgroup$
    – Yemon Choi
    Commented Nov 30, 2015 at 19:06
  • $\begingroup$ Yep this is continuous group cohomology of su(2) with values in the antihermitian matrices. One told me that it can be calculated using classifying space of SU(2). How can it be constructed explicitely ? $\endgroup$
    – Nicolas
    Commented Dec 2, 2015 at 13:31
  • $\begingroup$ I explored an other path : One can calculate the first continuous group cohomology that i am considerings by calculating the automorphism of the extension : $$ 0\rightarrow iH \rightarrow iH \rtimes SU(2) \rightarrow 1 $$ Is there results giving the automorphisms group of a semi direct product ? $\endgroup$
    – Nicolas
    Commented Dec 3, 2015 at 15:50

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