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

It is shown that any generalized Kac-Moody Lie algebra image that has no mutually orthogonal imaginary simple roots can be written as image = image+ circled plus (imageJ + image) circled plus image−, where imageJ is a Kac-Moody algebra defined from a symmetrizable Cartan matrix, and image+ and image− are subalgebras isomorphic to free Lie algebras over certain imageJ-modules. The denominator identity for such an algebra image is obtained by using a generalization of Wiltʹs formula that computes the graded dimension of the free Lie algebra image− and the denominator identity known for the Kac-Moody subalgebra image. 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 image to obtain an elegant decomposition image = image+ circled plus imageI2 circled plus image−.