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:

• 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:

• 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:

• 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 $E_{10}$ [arXiv:0801.2586, doi:10.1007/s11005-007-0214-7, doi:10.1007/s11005-007-0214-7]

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.

(for much more see the references at U-duality and exceptional field theory)

• 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)

Maximal compact subalgebras

On non-trivial finite-dimensional representations of involutary (“maximal compact”) subalgebras $\mathfrak{k}$:

• Axel Kleinschmidt, Hermann Nicolai, Adriano Viganò: On spinorial representations of involutory subalgebras of Kac-Moody algebras, In: Partition Functions and Automorphic Forms, Moscow Lectures 5, Springer (2020) [arXiv:1811.11659, doi:10.1007/978-3-030-42400-8_4]

• Axel Kleinschmidt, Ralf Köhl, Robin Lautenbacher, Hermann Nicolai: Representations of involutory subalgebras of affine Kac-Moody algebras, Commun. Math. Phys. 392 (2022) 89–123 [arXiv:2102.00870, doi:10.1007/s00220-022-04342-9]

• Axel Kleinschmidt, Hermann Nicolai, Jakob Palmkvist: $K(E_9)$ from $K(E_{10})$, Journal of High Energy Physics 2007 JHEP06 (2007) [arXiv:hep-th/0611314, doi:10.1088/1126-6708/2007/06/051]

• Robin Lautenbacher, Ralf Köhl: Higher spin representations of maximal compact subalgebras of simply-laced Kac-Moody-algebras [arXiv:2409.07247]