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$SU(2)$ is a Lie group, with a Lie algebra $\mathfrak{su}(2)$. I now consider the loop group $ LSU(2) = \{ \gamma: S^1 \to SU(2); \gamma \mathrm{ smooth} \} $. It is well-known that there exists non-trivial 2-cocycles on it, obtained from the invariant bilinear form on $\mathfrak{su}(2)$ (see Pressley and Segal for example).

The 2-cocycle on the Lie-algebra is very simple to write. However, the 2-cocycles on the group are rarely written explicitly. I know that it may be quite difficult for general groups but what about a group as "simple" as SU(2) ?

Does anybody an "explicit" expression of 2-cocycles on $LSU(2)$ such that, if I have an explicit expression of two loops $\gamma_1$ and $\gamma_2$ on SU(2), I can put it directly on Maple or Mathematica (or evaluate by hand) ?

EDIT: I precise my question. Consider paths in SU(2) parametrized for example by \begin{equation} \gamma( \theta) = \begin{pmatrix} cos \phi(\theta) e^{i\alpha(\theta)} & -\sin \phi(\theta) e^{-i\beta(\theta)} \\\\ \sin \phi(\theta) e^{i\beta(\theta)} & cos \phi(\theta) e^{-i\alpha(\theta)} \end{pmatrix} \end{equation} where $\alpha$, $\beta(\theta)$ and $\phi(\theta)$ are smooth $2\pi$-periodic functions. Could someone write a non-trivial 2-cocycle using integral formulas over these functions. It should be simple but I do not manage to write it down. Maybe this parametrization is not the best one; in this case, what is the most useful one ?

Thank you in advance, Damien.

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Could you please clarify what you mean by 2-cocycles on a loop group? Do you mean a smooth map $LG \times LG \to A$ satisfying the cocycle condition? What is $A$? – Konrad Waldorf Mar 7 '12 at 10:14
I mean a (smooth) map $\omega: LG\times LG \to \mathbb{C}^\times$ that satisfies the 2-co-cycle property. The Lie algebra $L\mathfrak{g}$ has an easy non-trivial 2-cocycle given by \begin{equation} \kappa(\gamma_1,\gamma_2) = \int \langle \gamma_1(\theta), \gamma'_2(\theta)\rangle d\theta \end{equation} where $\langle \cdot,\cdot\rangle$ is left-action invariant bilinear form on $L\mathfrak{g}$. The question is: is there a similar and --more important-- explicit (as much as possible) formula for the 2-co-cycle on $LG$. – Damien S. Mar 7 '12 at 16:26
The map $\omega:LG\times LG\to \mathbb S^1$ cannot be chosen continuous. There are two ways of dealing with that situation: either you take it to be (piecewise smooth) discontinuous, or you take it to be smooth but multivalued. The second definition of cocycle is somewhat more technical, but I think more powerful. – André Henriques Mar 7 '12 at 16:38
up vote 3 down vote accepted

I believe that this post contains an answer to your question.

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Thank you for your comment. I think there may be a link between the two questions but I come from analysis and not algebra: I do not understand what "tame symbol" means at all. Could you help me translate in the present case in terms of integrals, please ? Damien. – Damien S. Mar 6 '12 at 16:35
Dear André, could please consider my edit and clarify the link ? – Damien S. Mar 7 '12 at 8:37
@Damien : i think that your answer is in the question posed by André, who asks for an algebraic version of the central extension that he explicitly describes for the smooth case in the body of the question. – BS. Mar 7 '12 at 11:13
That is correct. The answer is in the question. – André Henriques Mar 7 '12 at 13:43
Damien: The stuff that MacNamara wrote is not what you should be looking at. The answer is the big expression with the two integrals. – André Henriques Mar 7 '12 at 16:35

I second André's comment that one cannot expect interesting, smooth 2-cocycles $\omega: LG \times LG \to S^1$. The situation for Lie groups is simply different to the one for Lie algebras: central Lie algebra extensions $$ 0 \to \mathfrak{a} \to \widehat{\mathfrak{g}} \to \mathfrak{g} \to 0 $$ can be classified by cocycles $\kappa: \mathfrak{g} \times \mathfrak{g} \to \mathfrak{a}$ forming the Lie algebra cohomology $H^2(\mathfrak{g},\mathfrak{a})$. Lie group extensions $$ 1 \to A \to \widehat{G} \to G \to 1 $$ that are classified by smooth cocycles $\omega:G \times G \to A$ are precisely those that are trivializable as principal $A$-bundles over $G$.

General central Lie group extensions are classified by Segal-Mitchison-Brylinski group cohomology $H^2_{SMB}(G,A)$. The 2-cocycles there involve open covers $\mathcal{U}^k$ of $G^k=G \times ...\times G$, compatible with the face maps of the simplicial manifold $G^*$, and smooth maps $$ g_{\alpha\beta}: U_{\alpha}^1 \cap U_{\beta}^1 \to A \quad\text{and}\quad h_{\alpha}: U_\alpha^2 \to A $$ satisfying various conditions. ($g_{\alpha\beta}$ are the transition functions for $\widehat{G}$ as an $A$-bundle, and $h_{\alpha}$ remembers the multiplication of the Lie group $\widehat G$).

All this is true for Lie groups $G$ just like for loop groups like $LG$. Thus, a smooth 2-cocycle $\omega: LG \times LG \to S^1$ classifies a central extension of $LG$ by the circle that is trivial as a circle bundle over $LG$. For compact, simple, simply-connected Lie groups $G$ it is well-known that such extensions are totally trivial, i.e. $\omega$ must be the coboundary of a smooth map $LG \to S^1$.

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Thank you for your answers. I understand now my mistakes. I was looking for a 2-cocycle to build the central extension of the loop group, which -- I understand it only today -- cannot be build so easily, due to topological reasons. My question changes now a little bit. How do I describe easily (i.e. in a way understandable by Maple or Mathematica) the group structure of the central extension of $\widetilde{LG}$ and the morphism $\widetilde{LG} \to LG$ ? If I want to be understandable by a machine, I cannot use the existence of homotopies between paths easily for example... – Damien S. Mar 9 '12 at 12:42

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