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Can somebody please tell me what are the central extensions of SL2(R) by U(1), that is, what is $H^2(SL2(R), U(1)) $ ? Thank you

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    $\begingroup$ If you are considering $\mathrm{SL}_2(\mathbb{R})$ as a discrete group, the answer is rather complicated, see this question and the various answers. On the other hand if you are considering $\mathrm{SL}_2(\mathbb{R})$ as a Lie group, then any central extension is the pushforward of the universal cover by some homomorphism $\mathbb{Z}\rightarrow \mathrm{U}(1)$; this follows easily from the fact that every central extension of $\mathfrak{sl}_2(\mathbb{R})$ is trivial. $\endgroup$
    – abx
    Commented Feb 11, 2015 at 13:43
  • $\begingroup$ Just a technical comment that you should use LaTeX for mathematical symbols, to avoid ambiguity. For instance, your symbol R might just mean a ring in the context of algebraic K-theory, or might mean the field $\mathbb{R}$ of real numbers. $\endgroup$
    – Jim Humphreys
    Commented Feb 11, 2015 at 14:56

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