nLab
Kac-Moody group

Contents

Idea

A group corresponding to a Kac-Moody algebra (not necessarily affine).

Properties

Classifying spaces

For GG a topological group, write BGB G for its delooping, the classifying space for topological GG-principal bundles.

Theorem

The classifying space BGB G of every Kac-Moody group is a homotopy colimit over classifying spaces BG iB G_i, iIi \in I of compact connected Lie groups G iG_i:

BGhocolim iBG i B G \simeq hocolim_i B G_i

in Top \simeq ∞Grpd.

Moreover, the diagram II may be taken to be a sieve in the poset of subobjects of the nn-element set, for some n𝔹n \in \mathbb{B}.

This is due to Nitu Kitchloo, 1998, see for instance Kitchloo’s survey, p. 9.

Remark

This means that the classifying space of every Kac-Moody group has a smooth refinement to a smooth moduli stack given by forming

BGhocolim iBG i \mathbf{B}G \coloneqq hocolim_i \mathbf{B}G_i

in Smooth∞Grpd.

Kay moody groups appear as U-duality groups in 11-dimensional supergravity compactified to low dimensions.

References

A standard textbook is

  • Shrawan Kumar, Kac-Moody groups, their flag varieties and representation theory, Progress in Mathematics, 204. Birkhauser Boston, Inc., Boston, MA, (2002)

A survey is

  • Nitu Kitchloo, Kac-Moody groups over the last decade (pdf)

Original articles include

  • Nitu Kitchloo, On the Topology of Kac-Moody groups (arXiv), (Phd thesis)

  • Carles Broto, Nitu Kitchloo, Classifying spaces of Kac-Moody groups, Math. Z. 240,621–649 (2002) (pdf)

  • Udo Baumgartner, Jacqueline Ramagge, Bertrand Remy, Contraction groups in complete Kac-Moody groups (2008) (pdf)

  • Andreas Mars, On the topology and geometry of Kac-Moody groups, PhD thesis (2011) (web)

  • Christof Geiss, Bernard Leclerc, Jan Schröer, Kac-Moody groups and cluster algebras (arXiv:1001.3545)

Revised on May 29, 2014 12:10:35 by David Corfield (31.185.244.23)