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Let $G$ be a locally compact abelian group, isomorphic to $G^*$, its Pontryagin Dual. Let $T$ denote the unit circle in $\mathbb{C}$, where continuous morphisms $\chi: G\to T$ are the elements of $G^*$. How many different can $(G\times G^*)$ admit a central extension by $T$? I'm wondering how explicitly one may express such groups.

EDIT: I originally claimed this was the semi-direct product that I was interested in. I regret the mistake.

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It's hard to guess which question you have in mind. First, what is the action defining your semidirect product? Second, what do you you mean by "be realized"? –  YCor Oct 30 '12 at 18:59
    
I suppose I'm interested in the general situation, if I am not confused. However as I now understand it one can obtain one representation, the Weil representation, by first considering the projective representation of $G\times G^*$ into $L^2(G)$ and then doing some cohomology magic. I'm interested in knowing in what generality one can form such central extensions of $G\times G^*$ by $T$. Are other such semi-direct products important? –  Eric Gregor Oct 30 '12 at 23:43
    
A central extension is not a semidirect product (except direct products). So there is no semidirect product. If you mean a central extension, you should specify a 2-cocycle valued in $b:H\times H\to T$. Here $H=G\times G^*$ so you can pick $b((v,f),(v',f'))=f(v')-f'(v)$, it looks like it's a natural choice. There are probably other 2-cocycles and they are described by a cohomology group; it's unclear if you really want to consider them insofar as they can be unrelated to the fact your $H$ stands as the product of $G$ and $G^*$. –  YCor Oct 31 '12 at 0:21
    
I will retitle my question. If you want to expand your comment into an answer I will give you the bounty. –  Eric Gregor Oct 31 '12 at 2:10

1 Answer 1

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+100

In general, given a group $H$ and an abelian group $Z$, the central extensions of $G$ by the central subgroup $Z$ are classified by the 2-cohomology group $H^2(H,Z)$ (2-cocycles modulo 2-coboundaries). The zero element of this group corresponds to the direct product extension. In your case $H=G\times G^*$ (no matter whether $G$ is isomorphic to $G^*$), there is a distinguished cocycle, given by $b((v,f),(v',f'))=f(v')-f'(v)$. There are probably other 2-cocycles, but it's unclear how they can be related to the decomposition $H=G\times G^*$. Also in the topological setting, you need to restrict to the cohomology group $H^2_m(G)$ based on measurable cochains, but the latter still probably contains irrelevant elements.

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