Generalized Kac-Moody Lie algebras, free Lie algebras and the structure of the Monster Lie algebra

Abstract

It is shown that any generalized Kac-Moody Lie algebra g that has no mutually orthogonal imaginary simple roots can be written as g = u + ⊕ ( g J + h ) ⊕ u −, where g J is a Kac-Moody algebra defined from a symmetrizable Cartan matrix, and u + and u − are subalgebras isomorphic to free Lie algebras over certain g J -modules. The denominator identity for such an algebra g is obtained by using a generalization of Wilt's formula that computes the graded dimension of the free Lie algebra u − and the denominator identity known for the Kac-Moody subalgebra g J . The main result and consequent proof of the denominator identity give a new proof that the radical of a generalized Kac-Moody algebra of the above type is zero. The main result is applied to the Monster Lie algebra m to obtain an elegant decomposition m = u + ⊕ g I 2 ⊕ u −. Also included is a detailed discussion of Borcherds' construction of the Monster Lie algebra from a vertex algebra and an elementary proof of Borcherds' theorem relating Lie algebras with “an almost positive definite bilinear form” to generalized Kac-Moody algebras.