nLab Kac-Moody algebra

Contents

Context

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Related topics

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

The notion of Kac-Moody Lie algebra is a generalization of that of semisimple Lie algebra to infinite dimension of the underlying vector space.

Definition

(…)

Examples

The sequence of exceptional semisimple Lie algebras E7, E7, E8 may be continued to the Kac-Moody algebras:

  • affine: E9

  • hyperbolic: E10,

  • Lorentzian: E11, …

References

General

On the Wess-Zumino-Witten model 2d CFT via Kac-Moody algebra and Virasoro algebra:

See also:

Lecture notes include

The standard textbook is

Collections of articles include

  • N. Sthanumoorty, K. Misra (eds.), Kac-Moody Lie algebras and related topics, Contemporary Mathematics 343 AMS (2002)

The EE-series

Surveys include

  • wikipedia, En

The fact that every simply laced hyperbolic Kac-Moody algebra appears as a subalgebra of E10 is in

Affine Lie algebras

As far as applications this is the most important class. See nnLab entry affine Lie algebra and

In supergravity

The following references discuss aspects of the Kac-Moody exceptional geometry of supergravity theories.

  • Hermann Nicolai, Infinite dimensional symmetries (2009) (pdf)

  • Paul Cook, Connections between Kac-Moody algebras and M-theory PhD thesis (arXiv:0711.3498)

  • Daniel Persson, Nassiba Tabti, Lectures on Kac-Moody algebras with applications in (Super-)Gravity (pdf)

Last revised on July 10, 2023 at 08:36:35. See the history of this page for a list of all contributions to it.