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

• affine: E9

• hyperbolic: E10,

• Lorentzian: E11, …

Related concepts

• Kac-Moody group

• higher Kac-Moody algebra

• Borcherds algebra

References

General

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

• Peter Goddard, David Olive, Kac-Moody and Virasoro algebras in relation to quantum physics, International Journal of Modern Physics A 01 02 (1986) 303-414 [doi:10.1142/S0217751X86000149, spire:18583]

See also:

• wikipedia, Kac-Moody algebra

Lecture notes include

• Antony Wassermann, Kac-Moody and Virasoro algebras, course notes (2011) (pdf)

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

The standard textbook is

• Victor G. Kac, Infinite Dimensional Lie Algebras, Progress in Mathematics 44 Springer 1983 [doi:10.1007/978-1-4757-1382-4], Cambridge University Press (1990) [doi:10.1017/CBO9780511626234]

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 $E_{10}$, Letters in Mathematical Physics 83 (2008) Issue 2, pp 139-148, doi:10.1007/s11005-007-0214-7, arXiv:0801.2586.

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.

• 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)