In this paper, we investigate the structure of graded Lie superalgebras L = +((alpha,a)is an element of Gamma XA) L-(alpha,L-a), where Gamma is a countable abelian semigroup and A is a countable abelian group with a coloring map satisfying a certain finiteness condition. Given a denominator identity for the graded Lie superalgebra L, we derive a superdimension formula for the homogeneous subspaces L-(alpha,L-a) (alpha is an element of Gamma, a is an element of 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. (C) 1998 Academic Press. [References: 60]
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