A result on exponents of finite-dimensional simple lie algebras and its application to Kac-Moody algebras

Abstract

The centralizer of the principal nilpotent element of a finite-dimensional simple Lie algebra is a naturally graded Cartan subalgebra, the degrees of the non-trivial components being the so-called exponents. If every nonzero element of a homogeneous subspace is a linear combination of all root spaces of the corresponding height, the exponent is called complete. We will show that there are precisely three non-complete exponents and apply this to affine Kac-Moody algebras.