Integrable highest weight modules over affine superalgebras and number theory

Abstract

In the first part of the paper we give the denominator identity for all simple finite-dimensional Lie super algebras $\frak g\/$ with a non-degenerate invariant bilinear form. We give also a character and (super) dimension formulas for all finite-dimensional irreducible $\frak g\/$-modules of atypicality $\leq 1\/$ . In the second part of the paper we give the denominator identity for the affine superalgebras $\hat{\frak g}\/$ associated to $\frak g\/$. Specializations of this identity give almost all old and many new formulas for the number of representations of an integer as sums of squares and sums of triangular numbers. At the end, we introduce the notion of an integrable $\hat{\frak g}\/$-module and give a classification of irreducible integrable highest weight $\hat{\frak g}\/$-modules.