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March 4, 2016

Hyperbolic Kac–Moody Groups

Posted by John Baez

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 is connected to differential geometry and physics by 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!

What comes next?

The finite-dimensional simple Lie algebras and affine Lie algebras are both special class of Kac–Moody algebras. The natural next class is the hyperbolic Kac–Moody algebras.

Here are the Dynkin diagrams of the finite-dimensional simple Lie algebras:

Here are the Dynkin diagrams of the so-called ‘untwisted’ affine Lie algebras:

and the ‘twisted’ ones:

A Dynkin diagram describes a hyperbolic Kac–Moody algebra if it’s not any of those shown above, but every proper connected subdiagram is! There are infinitely many hyperbolic Kac–Moody algebras whose Dynkin diagrams have 22 nodes, but only 238 with 3\ge 3 nodes. They’re listed here:

The ‘simply-laced’ ones were nicely drawn by Allcock and Carbone:

Concretely, the nodes in such a diagram stand for nn spacelike vectors in nn-dimensional Minkowski spacetime, where each vector has:

• inner product 22 with itself,

• inner product 1-1 with any vector corresponding to a node connected to it by an edge, and

• inner product 00 with any vector corresponding to a node not connected to it by an edge.

These are the only ways to build such arrangements of vectors in the various dimensions listed! 1010 dimensions is the maximum possible. There must be a connection to string theory, but there’s also a direct explanation. Only in hyperbolic space of dimension 9\le 9 can we find a simplex whose faces are either at 9090° or 120120° angles to each other!

The last and best of these diagrams is called E 10\mathrm{E}_{10}. It’s connected to the octonions, it’s connected to 1111-dimensional supergravity, and there are lots of interesting conjectures about its role in physics — see Allcock and Carbone’s paper for references, or this:

I’m wondering whether there is a geometrical construction of the groups corresponding to the hyperbolic Kac–Moody Lie algebras. Since after ‘particle’ and ‘string’ one naturally says ‘2-brane’, one might naively hope that this geometrical construction would be connected to 2-brane theories, or 2+1-dimensional field theories. But maybe that’s the wrong idea.

Jacques Tits found 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, Journal of Algebra 105 (1987), 542–573.

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

But this construction does not feel ‘geometric’ to me: it’s in terms of generators and relations.

Posted at March 4, 2016 6:47 PM UTC

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Re: Hyperbolic Kac–Moody Groups

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(E 10)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 (𝔽 2\mathbb{F}_2 if there is a double bond, 𝔽 2\mathbb{F}_2 or 𝔽 3\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).

Posted by: Daniel Allcock on March 5, 2016 4:50 AM | Permalink | Reply to this

Re: Hyperbolic Kac–Moody Groups

Daniel wrote:

Indeed the main frustration of the theory is that it is so far not very geometric.

Thanks for that sad news! Coming from you, it really means something.

Posted by: John Baez on March 5, 2016 6:07 AM | Permalink | Reply to this

Re: Hyperbolic Kac–Moody Groups

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.

Posted by: John Baez on March 7, 2016 6:48 PM | Permalink | Reply to this

Re: Hyperbolic Kac–Moody Groups

This paper:

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

Posted by: John Baez on March 7, 2016 6:57 PM | Permalink | Reply to this

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