2-cocycle on LSU(2)

Question

$SU(2)$ 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.

Answer 1

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

Answer 2

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 $LG \to S^1$.