User gurs - MathOverflow

smooth cohomology of Lie groups

2011-01-24T07:06:35Z

Let $G$ be a Lie group and $A$ a smooth $G$-module. Define $C^n(G,A)=\{ f: G^n \to A|~f~\text{is smooth}\}$ and $\partial^n: C^n \to C^{n+1}$ by the standard formula as used in the cohomology of abstract groups. I think this cohomology must be well studied. Can somebody provide me some references for this cohomology. A similar cohomology for topological groups has been studied by Hu and Heller using continuous cochains.

Also, can somebody tell me what kind of Lie group extensions $H^2(G,A)$ correspoind to.

Answer by gurs for cohomology theory for algebraic groups

2011-01-15T10:30:47Z

Thanks all for the valuable information.

Sections of topological group extensions

2010-12-24T08:14:32Z

Does there exists an example of an extension of topological groups $1 \to N \to E \to G \to 1$ admitting a section $s:G \to E$ which is continuous (or continuous in a neighbourhood of identity) and satisfy the propery that $s(x^{-1}) =s(x)^{-1}$. One can see that the extension $0 \to \mathbb{Z} \to \mathbb{R} \to \mathbb{S}^1 \to 1$ does not admit such a section.

For discrete groups there are many extensions admitting such a section. For example, consider the extension $0 \to \mathbb{Z} \stackrel{i}{\to} \mathbb{Z} \times \mathbb{Z}_2 \stackrel{p}{\to} \mathbb{Z}_4 \to 0$ with $i(x)= (2x, [x])$ and $p(x, [y])=[x + 2y]$. Clearly, the section $s$ defined by $s([0])=(0,[0])$, $s([1])=(-1,[1])$, $s([2])=(0,[1])$ and $s([3])=(1,[1])$ satisfy the property that $s(x^{-1})= s(x)^{-1}$. I am looking for an example in the setting of compact connected topological groups.

Comment by gurs

2011-01-24T12:48:22Z

Thanks for your comments. If $E$ is $A \times G$ as a smooth manifold, then the extension has an obvious smooth section and the proof works as per your suggestion. I am also feeeling that it should be true. Thanks again.

Comment by gurs

2011-01-24T10:56:21Z

Thanks for your comments and providing the references. I would like to add that, in the continuous cohomology for topological groups due to Hu and Heller, they showed that the second cohomology group classifies topologically split group extensions. More precisely, if $G$ is a topological group and $A$ is a topological $G$-module, then $H_{cont}^2(G,A)$ classies all extensions $1 \to A \to E \to G \to 1$ such that as a space $E$ is $A \times G$. I am guessing that the similar result is true for Lie groups when we work with smooth cohains as mentioned in my question. Is is true?

Comment by gurs

2011-01-24T09:28:16Z

Many thanks. But, what kind of Lie group extensions does $H^2(G,A)$ characterize.

Comment by gurs

2010-12-24T09:13:28Z

Let me write my question again. Does there exist an extension of topological groups $1 \to N \to E \to G \to 1$ admitting a section $s:G \to E$ which is continuous (or only continous in a neighbourhood of identity) and satisfy the property that $s(x^{-1})=s(x)^{-1}$ for all $x \in G$ but is not a homomorphism.

Comment by gurs

2010-12-24T09:04:25Z

Thanks. But I want a non-trivial extension.