Central extension of the algebraic loop group 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).
 A: Since this recently got bumped by the community user, I thought I'd have a go at constructing an answer.
The central extension (of G(ℂ((t))) by ℂ×) is algebraic. In particular it is a group ind-scheme, which I believe is strict and ind-affine.
However, there does not seem to be a great place in the literature to find information about this central extension (so the OP is not being stupid in his searching).
What immediately came to mind for me is Beilinson and Drinfeld's unpublished manuscript "Quantization of Hitchin's Integrable System and Hecke Eigensheaves." I can't say with truth that I am an expert on the contents within, but I believe the answers sought are contained within.
Also worth looking at, as mentioned by abx, is the paper by Beauville and Laszlo, "Conformal blocks and generalized theta functions."
On the Kac-Moody side, there is the paper "Construction d’un groupe de Kac-Moody
et applications" by Mathieu. However once this is constructed, there is the question of comparing it to the loop group, which I don't know where to find an answer to. (On the other hand, Kumar's book doesn't seem to go as far as to construct the Kac-Moody groups as ind-schemes).
Since cocycles were brought up, let me point out that for central extensions of group objects in a category C, for cocyles to exist, there must be a splitting in C (not as groups) of the map to the quotient. There will be no cocyle within the cateogry of ind-schemes. However at the level of R-points for some R, you could reasonably expect to find a cocycle. The paper "Block-compatible metaplectic cocycles" by Banks, Levi and Sepanski may be useful to some readers wanting to try their hand at certain explicit cocycles (I've found it useful myself in the context of metaplectic groups over local fields).
Hopefully some intrepid MO reader will see this and be motivated to produce a thorough, reliable and accessible exposition of this field.
A: Lie algebra cocycles are always algebraic. But if you try to integrate them up to group cocycles you meet homotopy obstacles, and you solve ODE's (twice) which give transcendental answers except in very degenerate cases.
