Kac–Moody algebra
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). Definition. Given an "n"×"n" generalized Cartan matrix , the "Kac-Moody algebra" of C is the Lie algebra formula_1 defined by generators formula_2, formula_3, and formula_4 and relations given by: A real (possibly infinite-dimensional) Lie algebra is also considered a Kac–Moody algebra if its complexification is a Kac–Moody algebra. Generalization. One may also define a modified form of a Kac-Moody algebra given a choice of roots in a given vector space and given coroots in the dual space. Specifically, assume we are given the following data: Then we can consider the Lie  algebra formula_1 defined by generators formula_2 and formula_4 and the elements of formula_17 and relations Root-space decomposition of a Kac–Moody algebra. formula_17 is the analogue of a Cartan subalgebra for the Kac–Moody algebra formula_1. If formula_45 is an element of formula_1 such that for some formula_48, then formula_49 is called a root vector and formula_50 is a root of formula_1. (The zero functional is not considered a root by convention.) The set of all roots of formula_1 is often denoted by formula_53 and sometimes by formula_54. For a given root formula_50, one denotes by formula_56 the root  space of formula_50; that is, It follows from the defining relations of formula_1 that formula_60 and formula_61. Also, if formula_62 and formula_63, then formula_64 by the Jacobi identity. A fundamental result of the theory is that any Kac–Moody algebra can be decomposed into the direct sum of formula_17 and its root spaces, that is and that every root formula_50 can be written as formula_68 with all the formula_69 being integers of the same sign. Types of Kac–Moody algebras. Properties of a Kac–Moody algebra are controlled by the algebraic properties of its generalized Cartan matrix "C". In order  to classify Kac–Moody algebras, it is enough to consider the case of an "indecomposable" matrix "C", that is, assume that there is no decomposition of the set of indices "I" into a disjoint union of non-empty subsets "I" and "I" such that "C" = 0 for all "i" in "I" and "j" in "I". Any decomposition of the generalized Cartan matrix leads to the direct sum decomposition of the corresponding Kac–Moody algebra: where the two Kac–Moody algebras in the right hand side are associated with the submatrices of "C" corresponding to the index sets "I" and  I. An important subclass of Kac–Moody algebras corresponds to "symmetrizable" generalized Cartan matrices "C", which can be decomposed as "DS", where "D" is a diagonal matrix with positive integer entries and "S" is a symmetric matrix. Under the assumptions that "C" is symmetrizable and indecomposable, the Kac–Moody algebras are divided into three classes: Symmetrizable indecomposable generalized Cartan matrices of finite and affine type have been completely classified. They correspond to Dynkin diagrams and affine Dynkin diagrams. Little is known about the Kac–Moody algebras of indefinite type, although the groups corresponding to these Kac–Moody algebras were constructed  over arbitrary fields by Jacques Tits. Among the Kac–Moody algebras of indefinite type, most work has focused on those hyperbolic type, for which the matrix "S" is indefinite, but for each proper subset of "I", the corresponding submatrix is positive definite or positive semidefinite. Hyperbolic Kac–Moody algebras have rank at most 10, and they have been completely classified. There are infinitely many of rank 2, and 238 of ranks between 3 and 10.