# 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).

• I don't know a lot about this story, but there's a survey paper by Tits that is often referenced; see here: people.mpim-bonn.mpg.de/zagier/files/doi/10.1007/BFb0084581/… He discusses various aspects of constructing these extensions. – Chuck Hague Oct 1 '13 at 16:18
• The central extension of the algebraic loop group is indeed a group object in the category of ind-varieties. See mathoverflow.net/questions/24845/… The string 2-group doesn't exist in the world of algebraic geometry. The closest thing that does exist is the central extension of G by $K_2$. – André Henriques Oct 2 '13 at 13:10
• You might have a look at §4 of Beauville-Laszlo, Commun. Math. Phys. 164, 385-419 (1994). – abx Jan 18 '14 at 10:29
• @Matthias: Why do you need simplicial sheaves? Isn't the $\mathbb A^1$-universal cover of $G$ a mere sheaf? – André Henriques Aug 5 '14 at 13:16
• @David: As for literature references, the computation of $\pi_1^{\mathbb{A}^1}(SL_n)$ can be found in Morel's book "$\mathbb{A}^1$-algebraic topology over a field", LNMA 2052, check out the section on $\mathbb{A}^1$-covering space theory. The general case for split groups is discussed in my paper "$\mathbb{A}^1$-homotopy of Chevalley groups", Journal K-theory, 5 (2010), 245-287. An even more general case of isotropic groups along with a discussion of the universal covering is in my paper with Konrad Voelkel, front.math.ucdavis.edu/1207.2364. Sorry for the advertisement. – Matthias Wendt Aug 11 '14 at 13:30

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.

• "there must be a splitting in C (not as groups) of the map to the quotient" -- not necessarily. In the case of the usual smooth central extension of $\Omega G$ there is a cocycle that is continuous and smooth only in a neighbourhood of the identity, so does not arise from a section of the underlying smooth map. This cocycle gives an element of the Segal-Mitchison-Brylinski cohomology of a topological/smooth group. – David Roberts Apr 15 '15 at 1:16
• However, thanks for the reference to Beilinson-Drinfeld. Typical that amazing people write amazing papers, and then don't bother to publish them :-) – David Roberts Apr 15 '15 at 1:17

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.

• But in the usual case of the smooth central extension of the loop group you don't get a group cocycle, just a Segal-Mitchison-Brylinski cocycle, which is is determined by a cocycle on the local group given by a neighborhood of the identity of G. Not to say that your comment doesn't apply in that setting... – David Roberts Oct 20 '13 at 10:20