Root multiplicities of Kac-Moody algebras (Q1331759)

The Kac-Moody Lie algebras admit a decomposition into a direct sum of finite-dimensional subspaces called ``root spaces''. The dimension of a root space is called the multiplicity of the corresponding root. The determination of the root multiplicities for non-finite and non-affine Kac-Moody Lie algebras is still an open problem. In this paper, the author has obtained two new root multiplicity formulas for symmetrizable Kac-Moody Lie algebras. One of these formulas is recursive and the other is closed form. He has also used these formulas to calculate root multiplicities for the hyperbolic Kac-Moody Lie algebra \(HA^{(1)}_ 1\) and for rank two symmetric Kac-Moody Lie algebras.