Graded Lie Superalgebras and the Superdimension Formula

✍ Scribed by Seok-Jin Kang

• Publisher: Elsevier Science
• Year: 1998
• Tongue: English
• Weight: 541 KB
• Volume: 204
• Category: Article
• ISSN: 0021-8693
• DOI: 10.1006/jabr.1997.7352

✦ Synopsis

In this paper, we investigate the structure of graded Lie superalgebras

, where ⌫ is a countable abelian semigroup and A A is a Ž ␣ , a. Ž ␣ , a.g ⌫=A A countable abelian group with a coloring map satisfying a certain finiteness condition. Given a denominator identity for the graded Lie superalgebra L L , we derive a Ž . superdimension formula for the homogeneous subspaces

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.