The n-Category Café

Thank you for highlighting my work with Lisa! Indeed the main frustration of the theory is that it is so far not very geometric. $W(\mathrm{E}_{10})$ is lovely but the Kac–Moody groups should be as geometric as Lie groups, and they are not, yet.

It may interest some people that the simplification in Tits’ description of the groups over rings holds in greater generality than those specific hyperbolic diagrams. The same presentation works for any simply laced Dynkin diagram. If you allow multiple bonds then you have to assume that that the ground ring has no tiny fields as quotients ($\mathbb{F}_2$ if there is a double bond, $\mathbb{F}_2$ or $\mathbb{F}_3$ if there is a triple bond), or that the Dynkin diagram is “3-spherical”, meaning that the any 3-node subdiagram comes from a finite-dimensional Lie algebra. See arXiv:1307.2689. The hypotheses on the ring can be omitted for affine diagrams (arXiv:1409.0176, to appear in Algebra and Number Theory).

On Mathoverflow, Lisa Carbone gave this nice answer to my question:

Thanks for the questions. The initial set-up for the Chevalley type construction (exponentiating from the Kac-Moody algebra) for all symmetrizable Kac-Moody groups (including the hyperbolic case) over arbitrary fields was worked out with Howard Garland. For Kac-Moody groups over $\mathbb{Z}$, it is summarized in my paper with Frank Wagner where we compared the Chevalley construction with the finite presentation for Tits’ group obtained in the paper with Daniel Allcock. The full Chevalley theory for Kac-Moody groups is currently being worked out. In joint work with Alex Feingold and Walter Freyn, we have constructed an action of hyperbolic Kac-Moody groups over $\mathbb{C}$ on a simplicial structure in the Lie algebra of the compact form. This simplicial complex has the structure of a Tits building, but is constructed geometrically in terms of Cartan subalgebras, rather than in terms of group cosets like the usual Tits building. This paper will also appear soon. There is an algebro-geometric construction of Kac-Moody groups by Olivier Mathieu:

• Olivier Mathieu, Construction du groupe de Kac-Moody et applications, C. R. Acad. Sci. Paris S'er I Math. 306 (1988), no. 5, 227–230.

The book of Kumar:

• Shrawan Kumar, Kac–Moody Groups, their Flag Varieties and Representation Theory, Progress in Mathematics, vol. 204, Birkhauser Boston Inc., Boston, MA, 2002.

has a lot of useful information. In spite of hard work by many people, we don’t have a uniform construction of the “overextended” hyperbolic Kac-Moody groups starting from the corresponding finite-dimensional simple Lie groups. As you must have seen, only certain of the hyperbolic Dynkin diagrams are “overextensions” of finite dimensional ones. The constructions described above are “uniform” (or rather formal) for all hyperbolic Kac-Moody groups. Individual “zoo-like” constructions could be very revealing, but there are no examples currently.

This paper:

• Anna Felikson, Pavel Tumarkin, Hyperbolic subalgebras of hyperbolic Kac–Moody algebras.

classifies subgroups relations between Weyl groups of hyperbolic Kac–Moody algebras, and shows that for every pair of a group and subgroup their exists at least one corresponding pair of algebra and subalgebra.

Here is a nice chart, taken from this paper, of inclusions among Weyl groups associated to simply-laced Dynkin diagrams:

The different kinds of arrows indicate different kinds of inclusions. For example, the arrows with hollow arrowheads are inclusions of one Kac–Moody algebra in another with the same rank.

Here you can see how $\mathrm{E}_{10}$ is the ‘biggest and best’ of the simply-laced Kac–Moody algebras: it contains all the rest!