# Kac–Moody algebra

A **Kac–Moody algebra** is a [Lie algebra](https://www.edgepedia.org/lie-algebra), usually infinite-dimensional, defined by generators and relations through a generalized [Cartan matrix](https://www.edgepedia.org/cartan-matrix).

| Key facts | |
|---|---|
| Named for | Victor Kac and Robert Moody, who independently began the systematic study of these algebras<sup>[1](https://encyclopediaofmath.org/wiki/Kac-Moody_algebra)</sup> |
| Defining data | An \(n \times n\) generalized Cartan matrix; the algebra g(A) is a quotient of a free Lie algebra on generators (\(h_{i}\), \(e_{i}\), \(f_{i}\)) by the Chevalley and Serre relations<sup>[2](https://ncatlab.org/nlab/show/Kac-Moody%20algebra)</sup> |
| Dimension | g(A) is finite dimensional if and only if A is positive definite; otherwise the algebra is infinite dimensional<sup>[1](https://encyclopediaofmath.org/wiki/Kac-Moody_algebra)</sup> |
| Classification | For symmetrizable indecomposable matrices: positive definite gives finite-dimensional simple Lie algebras, positive semidefinite gives affine type, indefinite gives indefinite type<sup>[6](https://perso.uclouvain.be/timothee.marquis/papers/OverviewKM.pdf)</sup> |
| Central theorem | The Weyl–Kac character formula, the basic result of integrable highest-weight representation theory<sup>[1](https://encyclopediaofmath.org/wiki/Kac-Moody_algebra)</sup> |
| Applications | Affine Kac–Moody representation theory in conformal field theory, exactly solvable models and string theory<sup>[1](https://encyclopediaofmath.org/wiki/Kac-Moody_algebra)</sup> |

## Definition by generators and relations

The starting point is a generalized Cartan matrix: an \(n \times n\) matrix whose diagonal entries are equal to 2 and whose off-diagonal entries are non-positive integers, with \(a_{ij} = 0\) exactly when \(a_{ji} = 0\). From such a matrix \(A\) one builds a Lie algebra \(g(A)\) as a quotient of the free Lie algebra on generators \((h_i, e_i, f_i)\) by the Lie ideal generated by the Chevalley relations and the Serre relations.<sup>[2](https://ncatlab.org/nlab/show/Kac-Moody%20algebra)</sup> The elements \(e_i\) and \(f_i\) are called the Chevalley generators, and they generate the derived subalgebra \(g^{0}(A) = [g(A), g(A)]\); the subalgebra spanned by the \(h_i\) plays the role of a [Cartan subalgebra](https://www.edgepedia.org/cartan-subalgebra), in analogy with the finite-dimensional classical case.<sup>[3](https://math.berkeley.edu/~barrett/resources/km.pdf)</sup>

The construction generalizes the presentation of finite-dimensional simple Lie algebras by generators and relations, which [Jean-Pierre Serre](https://www.edgepedia.org/jean-pierre-serre) showed in 1966 follows from relations of [Claude Chevalley](https://www.edgepedia.org/claude-chevalley) and [Harish-Chandra](https://www.edgepedia.org/harish-chandra) with simplifications by Nathan Jacobson.

## Root-space decomposition

The Cartan subalgebra h serves as the analog of a Cartan subalgebra for g(A).<sup>[3](https://math.berkeley.edu/~barrett/resources/km.pdf)</sup> A nonzero element α of the dual space h* is a root if a root space of nonzero vectors x satisfying [h, x] = α(h)x for all h in h exists.

## Types of Kac–Moody algebras

An important subclass corresponds to symmetrizable generalized Cartan matrices, which can be written as DS with D diagonal and positive and S symmetric.

- A positive definite matrix S gives rise to a finite-dimensional simple Lie algebra. Equivalently, g(A) is finite dimensional if and only if the symmetrization DS is positive definite, so Kac–Moody algebras are infinite-dimensional analogs of finite-dimensional semisimple Lie algebras.<sup>[1](https://encyclopediaofmath.org/wiki/Kac-Moody_algebra)</sup>

Symmetrizable indecomposable generalized Cartan matrices of finite and affine type have been completely classified, corresponding to Dynkin diagrams and affine Dynkin diagrams.

## Representation theory

The basic result of the theory of integrable highest-weight representations is the Weyl–Kac character formula, which gives an explicit expression for the character of such representations.<sup>[1](https://encyclopediaofmath.org/wiki/Kac-Moody_algebra)</sup> The main technical tool for this part of the theory, the generalized Casimir operator, requires the Cartan matrix to be symmetrizable.<sup>[1](https://encyclopediaofmath.org/wiki/Kac-Moody_algebra)</sup>

For affine Kac–Moody algebras, the associated structures include a normalized invariant bilinear form, a root system and a [Weyl group](https://www.edgepedia.org/weyl-group), treated alongside integrable representations in the standard monograph on the subject.<sup>[4](https://link.springer.com/book/10.1007/978-1-4757-1382-4)</sup>

## Applications

Their representation theory also became an important ingredient of string theory.<sup>[1](https://encyclopediaofmath.org/wiki/Kac-Moody_algebra)</sup>

## History

The initial construction by [Élie Cartan](https://www.edgepedia.org/elie-cartan) and [Wilhelm Killing](https://www.edgepedia.org/wilhelm-killing) of finite-dimensional simple Lie algebras from the Cartan integers was type dependent.

A systematic study of Kac–Moody algebras was then started independently by [Victor Kac](https://www.edgepedia.org/victor-kac) and Robert Moody.<sup>[1](https://encyclopediaofmath.org/wiki/Kac-Moody_algebra)</sup> In his 1967 thesis, Moody considered Lie algebras whose Cartan matrix is no longer positive definite, which still gave rise to a Lie algebra, now infinite dimensional. Simultaneously, Z-graded Lie algebras were studied in Moscow, where I. L. Kantor introduced a general class of Lie algebras including what became known as Kac–Moody algebras, while Victor Kac studied simple or nearly simple Lie algebras with polynomial growth. A rich mathematical theory of infinite-dimensional Lie algebras evolved.<sup>[3](https://math.berkeley.edu/~barrett/resources/km.pdf)</sup>

The standard reference for the field is Victor Kac's monograph *Infinite Dimensional Lie Algebras*, whose third edition, published by [Cambridge University Press](https://www.edgepedia.org/cambridge-university-press) in 1990, is a substantially revised treatment of Kac–Moody algebras and their representations.<sup>[5](https://www.cambridge.org/core/books/infinitedimensional-lie-algebras/053FE77E6E9B35C56B5AEF7336FE7306)</sup>

## References

1. [Kac-Moody algebra - Encyclopedia of Mathematics](https://encyclopediaofmath.org/wiki/Kac-Moody_algebra)
2. [Kac-Moody algebra in nLab](https://ncatlab.org/nlab/show/Kac-Moody%20algebra)
3. [Kac-Moody Algebras and Applications (lecture notes, UC Berkeley)](https://math.berkeley.edu/~barrett/resources/km.pdf)
4. [Infinite Dimensional Lie Algebras: An Introduction (Springer)](https://link.springer.com/book/10.1007/978-1-4757-1382-4)
5. [Infinite-Dimensional Lie Algebras, 3rd edition (Cambridge University Press)](https://www.cambridge.org/core/books/infinitedimensional-lie-algebras/053FE77E6E9B35C56B5AEF7336FE7306)
6. [Minicourse on Kac–Moody Groups, Timothée Marquis, UCLouvain](https://perso.uclouvain.be/timothee.marquis/papers/OverviewKM.pdf)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Advanced algebraic structures › Lie theory › Lie representations and modules › Representations of Kac–Moody and affine Lie algebras*

*Initially written Sep 17, 2026 · Reviewed: Sep 30, 2026 · Edited: Sep 30, 2026; Oct 15, 2026; Oct 25, 2026 · Last review: Sep 30, 2026*

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