group theory

# Contents

## Idea

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

## Properties

### Classifying spaces

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

###### Theorem

The classifying space $B G$ of every Kac-Moody group is a homotopy colimit over classifying spaces $B G_i$, $i \in I$ of compact connected Lie groups $G_i$:

$B G \simeq hocolim_i B G_i$

in Top $\simeq$ ∞Grpd.

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

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

$\mathbf{B}G \coloneqq hocolim_i \mathbf{B}G_i$

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)