Graded Lie superalgebras and the superdimension formula (Q1270970)

The author studies the structure of graded Lie superalgebras \(\mathcal L=\oplus_{(\alpha, a)\in\Gamma\times\mathcal A}\mathcal L_{(\alpha,a)}\), where \(\Gamma\) is a countable abelian semigroup and \(\mathcal A\) is a countable abelian group with a coloring map satisfying a certain finiteness condition. Given a denominator identity for the graded Lie \(\mathcal L\) superalgebra, he derives a superdimension formula for the homogeneous subspaces \(\mathcal L_{(\alpha,a)}(\alpha\in\Gamma,\;a\in\mathcal)\), which enables him to study the structure of graded Lie superalgebras in a unified way. He discusses the applications of his 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. He also gives a characterization of replicable functions in terms of product identities and determine the root multiplicities of Monstrous Lie superalgebras.