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, 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 Witt's formula that computes the graded dimension of the free Lie algebra u(-) and the denominator identity known for the Kac-Moody subalgebra g,. 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(+) + gI(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. (C) 1998 Elsevier Science B.V. [References: 25]
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