5 - Borcherds Kac-Moody Lie superalgebras

Abstract

Wakimoto [142] introduced the definition of BKM supermatrix and BKM Lie superalgebras. A BKM supermatrix can be defined from a GKM matrix by introducing an additional subset corresponding to odd roots on the index set and hence we get a superalgebra structure. Hence BKM superalgebras can be considered as the Lie superalgebras associated to BKM supermatrices. In Kac-Moody algebras, all simple roots are real whereas just like GKM algebras, BKM Lie superalgebras can have imaginary simple roots. Moreover in the case of BKM superalgebras, simple roots (both real and imaginary) are basically of two types, namely, odd and even simple roots (both real and imaginary). Since an additional structure on the index set is introduced on a BKM supermatrix, the properties of imaginary roots in BKM Lie superalgebras will be different from those of BKM algebras. In BKM superalgebras, in addition to these classes of fundamental root systems of imaginary roots, there are alien and domestic type imaginary roots. The notion of Kac-Moody superalgebras was introduced by Kac [34] and therein the Weyl-Kac character formula for the irreducible highest weight modules with dominant integral highest weight which yields a denominator identity when applied to 1-dimensional representation was also derived. For canonical q-deformation of noncompact Lie (super)algebras, one can refer [143].