On the correspondence of Affine Generalized Root Systems and symmetrizable affine Kac-Moody superalgebras

TL;DR

This paper classifies affine generalized root systems (AGRS) and establishes their correspondence with real roots of symmetrizable affine indecomposable Kac-Moody superalgebras, extending the understanding of root systems in superalgebra theory.

Contribution

It introduces AGRS as a new generalization of affine root systems and proves their classification and relation to Kac-Moody superalgebras.

Findings

All AGRSs are classified.

Almost every irreducible AGRS corresponds to a Kac-Moody superalgebra.

The set of real roots of these superalgebras forms an AGRS.

Abstract

Generalized root systems (GRS), that were introduced by V. Serganova, are a generalization of finite root systems (RS). We define a generalization of affine root systems (ARS), which we call $\textit{affine generalized root systems}$ (AGRS). The set of real roots of almost every symmetrizable affine indecomposable Kac-Moody superalgebra is an irreducible AGRS. In this paper we classify all AGRSs and show that almost every irreducible AGRS is the set of real roots of a symmetrizable affine indecomposable Kac-Moody superalgebra.