In mathematics, a Kac–Moody algebra (named for Victor Kac and Robert Moody, who independently and simultaneously discovered them in 1968) is a Lie algebra, usually infinite-dimensional, that can be defined by generators and relations through a generalized Cartan matrix. These algebras form a generalization of finite-dimensional semisimple Lie algebras, and many properties related to the structure of a Lie algebra such as its root system, irreducible representations, and connection to flag manifolds have natural analogues in the Kac–Moody setting. A class of Kac–Moody algebras called affine Lie algebras is of particular importance in mathematics and theoretical physics, especially two-dimensional conformal field theory and the theory of exactly solvable models. Kac discovered an elegant proof of certain combinatorial identities, the Macdonald identities, which is based on the representation theory of affine Kac–Moody algebras. Howard Garland and James Lepowsky demonstrated that Rogers–Ramanujan identities can be derived in a similar fashion.
History of Kac–Moody algebras The initial construction by Élie Cartan and Wilhelm Killing of finite dimensional simple Lie algebras from the Cartan integers was type dependent. In 1966 Jean-Pierre Serre showed that relations of Claude Chevalley and Harish-Chandra, with simplifications by Nathan Jacobson, give a defining presentation for the Lie algebra. One could thus describe a simple Lie algebra in terms of generators and relations using data from the matrix of Cartan integers, which is naturally positive definite. "Almost simultaneously in 1967, Victor Kac in the USSR and Robert Moody in Canada developed what was to become Kac–Moody algebra. Kac and Moody noticed that if Wilhelm Killing's conditions were relaxed, it was still possible to associate to the Cartan matrix a Lie algebra which, necessarily, would be infinite dimensional." – A. J. Coleman In his 1967 thesis, Robert Moody considered Lie algebras whose Cartan matrix is no longer positive definite. This still gave rise to a Lie algebra, but one which is now infinite dimensional. Simultaneously, Z-graded Lie algebras were being studied in Moscow where I. L. Kantor introduced and studied a general class of Lie algebras including what eventually became known as Kac–Moody algebras. Victor Kac was also studying simple or nearly simple Lie algebras with polynomial growth. A rich mathematical theory of infinite dimensional Lie algebras evolved. An account of the subject, which also includes works of many others is given in (Kac 1990). See also (Seligman 1987).
Introduction Given an n×n generalized Cartan matrix C = ( c i j ) {\displaystyle C={\begin{pmatrix}c_{ij}\end{pmatrix}}} , one can construct a Lie algebra g ′ ( C ) {\displaystyle {\mathfrak {g}}'(C)} defined by generators e i {\displaystyle e_{i}} , h i {\displaystyle h_{i}} , and f i ( i ∈ { 1 , … , n } ) {\displaystyle f_{i}\left(i\in \{1,\ldots ,n\}\right)} and relations given by:
[ h i , h j ] = 0 {\displaystyle \left[h_{i},h_{j}\right]=0\ } for all i , j ∈ { 1 , … , n } {\displaystyle i,j\in \{1,\ldots ,n\}} ;
[ h i , e j ] = c i j e j {\displaystyle \left[h_{i},e_{j}\right]=c_{ij}e_{j}} ;
[ h i , f j ] = − c i j f j {\displaystyle \left[h_{i},f_{j}\right]=-c_{ij}f_{j}} ;
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