$\widehat{sl(2)}$ decomposition of denominator formulae of some BKM Lie superalgebras

TL;DR

This paper explores the decomposition of denominator formulae of certain Borcherds-Kac-Moody Lie superalgebras using an $ ext{sl}(2)$ subalgebra, linking modular forms, Jacobi forms, and root multiplicities.

Contribution

It introduces a method to decompose denominator formulae via $ ext{sl}(2)$ characters, providing explicit formulas for associated vector-valued modular forms.

Findings

Decomposition of Siegel modular forms into $ ext{sl}(2)$ characters.

Closed-form expressions for vector-valued modular forms.

Connection between Fourier coefficients and root multiplicities.

Abstract

We study a family of Siegel modular forms that are constructed using Jacobi forms that arise in Umbral moonshine. All but one of them arise as the Weyl-Kac-Borcherds denominator formula of some Borcherds-Kac-Moody (BKM) Lie superalgebras. These Lie superalgebras have a $\widehat{sl(2)}$ subalgebra which we use to study the Siegel modular forms. We show that the expansion of the Umbral Jacobi forms in terms of $\widehat{sl(2)}$ characters leads to vector-valued modular forms. We obtain closed formulae for these vector-valued modular forms. In the Lie algebraic context, the Fourier coefficients of these vector-valued modular forms are related to multiplicities of roots appearing on the sum side of the Weyl-Kac-Borcherds denominator formulae.