nLab
Kac-Moody algebra

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

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Examples

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

• affine: E9

• hyperbolic: E10,

• Lorentzian: E11, …

Related concepts

• Kac-Moody group

References

General

Surveys include

• wikipedia, Kac-Moody algebra

Lecture notes include

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

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 $E$-series

• E7, E8, E9, E10, E11, …

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 $E_{10}$ (pdf)

Affine Lie algebras

As far as applications this is the most important class. See $n$Lab 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:

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