I'm doing some constructions with the universal central extension $\widehat{\Omega G}$ of the loop group $\Omega G$ (here $G$ is a matrix group), where a priori the loops involved are just smooth, but in fact turn out to be rational functions. This got me thinking to see if everything will in fact land inside the algebraic loop group $G\left(\mathbb{C}((t))\right)$. However, I would like to know how one constructs the central extension in that case, as for some reason I can't seem to find a decent discussion of this (I'm probably being stupid in my searching...). In particular, is the central extension something like an ind-affine algebraic group? Given the cocycle describing the extension, how do you get said extension? (the method I know gives it as a quotient of a split central extension of $P\Omega G$, see links above)

Now my intended aim is to package this into something like the crossed module $\widehat{\Omega G} \to PG$ representing the String 2-group, but using more algebraic ingredients. For instance, replace the Frechet manifold $PG$ of based paths in $G$ with the space of polynomial or rational connections on the trivial $G$-bundle on $\mathbb{C}^\times$. The part I don't know is the central extension as indicated above.

EDIT: let me add that I would be most interested in knowing whether the central extension of the loop group is something like an algebraic group or if it is in some sense 'inherently transcendental' (for instance, the cocycle one uses to build it uses a residue).

simplicialsheaves? Isn't the $\mathbb A^1$-universal cover of $G$ a mere sheaf? – André Henriques Aug 5 at 13:16