Just as the theory of finite-dimensional simple Lie algebras is connected to differential geometry and physics via the theory of simple Lie groups, the theory of affine Lie algebras was connected to differential geometry and physics via the realization that these are the Lie algebras of central extensions of loop groups:

Andrew Pressley and Graeme Segal,

*Loop Groups*, Oxford U. Press, Oxford, 1988.Graeme Segal, Loop groups.

Indeed it's not much of an exaggeration to say that central extensions of loop groups are to strings as simple Lie groups are to particles.

The finite-dimensional simple Lie algebras and affine Lie algebras are both special cases of Kac--Moody algebras. The next class of Kac--Moody algebras are the hyperbolic Kac--Moody algebras, and these have been completely classified:

- Lisa Carbone, Sjuvon Chung, Leigh Cobbs, Robert McRae, Debajyoti Nandi, Yusra Naqvi and Diego Penta, Classification of hyperbolic Dynkin diagrams, root lengths and Weyl group orbits.

My question is whether a geometrical construction of any of the corresponding hyperbolic Kac--Moody groups is known. Since after 'particle' and 'string' one naturally says '2-brane', one might naively hope that these are connected to 2-brane theories, or perhaps 2+1-dimensional field theories. But maybe that's the wrong idea.

Jacques Tits famously gave a way to construct Kac--Moody groups, not only over the real and complex numbers but over arbitrary commutative rings:

- Jacques Tits, Uniqueness and presentation of Kac–Moody groups over fields,
*J. Algebra***105**(1987), 542–573.

This has been simplified for a certain class of hyperbolic Kac--Moody groups, namely the simply-laced ones:

- Daniel Alcock, Lisa Carbone, Presentation of hyperbolic Kac-Moody groups over rings.

This construction does not feel 'geometric' to me: it's in terms of generators and relations.

Just for fun, here are the Dynkin diagrams of the finite-dimensional simple Lie algebras:

Here are the Dynkin diagrams of the untwisted affine Lie algebras:

and the twisted ones:

A Dynkin diagram describes a hyperbolic Kac--Moody algebra if it's not among those shown above, but every proper connected subdiagram is. There are infinitely many hyperbolic Kac--Moody algebras whose Dynkin diagrams have $2$ nodes, but only 238 with $\ge 3$ nodes. The simply-laced ones were nicely drawn by Allcock and Carbone:

The last of these diagrams is called $\mathrm{E}_{10}$, and there are lots of interesting conjectures about its role in physics --- see Allcock and Carbone's paper for references. These could be clues to a geometric construction of the corresponding group.

uniformgeometric construction of these 238 groups? The only uniform constructions I know of Lie groups of type A-G are by Chevalley/Steinberg -- by generators and relations, like Tits for Kac-Moody groups. Otherwise, one finds these groups in a non-uniform manner by geometry -- e.g. type G_2 from octonions, etc.. So wouldn't one expect a large zoo of geometric constructions for the hyperbolic Kac-Moody groups? $\endgroup$ – Marty Mar 5 '16 at 1:00uniformgeometric construction. A zoo would also be delightful. In fact, a geometric construction ofany onehyperbolic Kac-Moody group would be interesting to me , since I don't know any at all! I'll reword my question to make that clear. $\endgroup$ – John Baez Mar 5 '16 at 1:50