Dimension formula for graded Lie algebras and its applications

Abstract

In this paper, we investigate the structure of in nite dimensional Lie algebras L = L 2􀀀 L graded by a countable abelian semigroup 􀀀 sat- isfying a certain niteness condition. The Euler-Poincar e principle yields the denominator identities for the 􀀀-graded Lie algebras, from which we derive a dimension formula for the homogeneous subspaces L ( 2 􀀀). Our dimen- sion formula enables us to study the structure of the 􀀀-graded Lie algebras in a uni ed way. We will discuss some interesting applications of our dimension formula to the various classes of graded Lie algebras such as free Lie algebras, Kac-Moody algebras, and generalized Kac-Moody algebras. We will also dis- cuss the relation of graded Lie algebras and the product identities for formal power series.