We know that the projective representation of a group $G$ is classified by $H_{grp}^2(G,R/Z) = H^3(BG,Z)$, where $H^*_{grp}$ is the group-cohomology class. Then do we have a classification of the projective representations of diffeomorphism group of $S^2$ (or more generally, of $S^n$).
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2$\begingroup$ Do you mean infinite dimensional representations? Then you have to be much more specific what the target group is. $\endgroup$– MishaFeb 21, 2014 at 20:00
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1$\begingroup$ Note that every homomorphism from the full diffeomorphism group to a finite dimensional connected Lie group has image of order at most 2. Is this what you are looking for? $\endgroup$– MishaFeb 21, 2014 at 21:15
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First of all, I would like to comment on the sentence:
"We know that the projective representations of a group G are classified by $H^2_{grp}(G,\mathbb R/\mathbb Z)=H^3(BG,\mathbb Z)$, where $H_{grp}$ is group-cohomology."
That is only true for a certain class of groups: those are the groups for which $H^3_{grp}(G,\mathbb R)=H^2_{grp}(G,\mathbb R)=0$. This includes all compact Lie groups (in particular finite groups), but also loop groups of compact Lie groups.
The vanishing of $H^3_{grp}(G,\mathbb R)$ is a somewhat obscure condition, but the vanishing of $H^2_{grp}(G,\mathbb R)$ is very concrete: no central extensions by $\mathbb R$. Here are some examples of groups with non-vanising $H^2_{grp}(G,\mathbb R)$: $G=\mathbb R^2$; $G=\mathit{Diff}(S^1)$; and many more.
Concerning the question of $\mathit{Diff}(S^2)$, I don't think that it has any interesting representations -- at least I've never heard that being mentionned anywhere.
answered May 9, 2014 at 4:45
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$\begingroup$ Thanks! Then how about the the Lie algebra for $Diff(S^2)$. Does the Lie algebra has a centre extension? $\endgroup$ May 9, 2014 at 4:57
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$\begingroup$ I don't really know. I would guess not. What I do know is that that's called Gelfand-Fuschs cohomology. So the Gelfand-Fuschs cohomology of $S^2$ is the Lie algebra cohomology of the Lie algebra of vector fields on $S^2$. $\endgroup$ May 9, 2014 at 5:01