nLab
Kac-Moody algebra

Context

Lie theory

∞-Lie theory

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

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 higher Kac-Moody analogs of the exceptional semisimple Lie algebras E7, E7, E8 are

  • affine: E9

  • hyperbolic: E10,

  • Lorentzian: E11,

References

General

Surveys include

Lecture notes include

The standard textbook is

  • Victor Kac, Infinite dimensional Lie algebras, , Cambridge University Press (1990)

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

  • Sankaran Viswanath, Embeddings of hyperbolic Kac-Moody algebras into E 10E_{10} (pdf)

Affine Lie algebras

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

  • David Hernandez, An introduction to affine Kac-Moody algebras (pdf)

In supergravity

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

Lecture notes:

Revised on March 17, 2014 07:12:07 by Urs Schreiber (89.204.155.77)