Extended Affine Root Supersystems

TL;DR

This paper introduces extended affine root supersystems, generalizing affine reflection systems and root supersystems, to facilitate the structural study of extended affine Lie superalgebras.

Contribution

It develops a new framework for understanding root systems of extended affine Lie superalgebras, expanding the theory beyond Lie algebras to superalgebras.

Findings

Definition and properties of extended affine root supersystems

Connection to root systems of extended affine Lie superalgebras

Framework for structural analysis of extended affine Lie superalgebras

Abstract

The interaction of a Lie algebra $\LL,$ having a weight space decomposition with respect to a nonzero toral subalgebra, with its corresponding root system forms a powerful tool in the study of the structure of $\LL.$ This, in particular, suggests a systematic study of the root system apart from its connection with the Lie algebra. Although there have been a lot of researches in this regard on Lie algebra level, such an approach has not been considered on Lie superalgebra level. In this work, we introduce and study extended affine root supersystems which are a generalization of both affine reflection systems and locally finite root supersystems. Extended affine root supersystems appear as the root systems of the super version of extended affine Lie algebras and invariant affine reflection algebras including affine Lie superalgebras. This work provides a framework to study the structure of this kind of Lie superalgebras refereed to as extended affine Lie superalgebras.