$SU(2)$ 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.