Graded Lie superalgebras and the superdimension formula

Abstract

In this paper, we investigate the structure of graded Lie superalgebras = (, a) × (, a), where is a countable abelian semigroup and is a countable abelian group with a coloring map satisfying a certain finiteness condition. Given a denominator identity for the graded Lie superalgebra , we derive a superdimension formula for the homogeneous subspaces (, a) ( , a ), which enables us to study the structure of graded Lie superalgebras in a unified way. We discuss the applications of our superdimension formula to free Lie superalgebras, generalized Kac–Moody superalgebras, and Monstrous Lie superalgebras. In particular, the product identities for normalized formal power series are interpreted as the denominator identities for free Lie superalgebras. We also give a characterization of replicable functions in terms of product identities and determine the root multiplicities of Monstrous Lie superalgebras.