Classification of bases of twisted affine root supersystems

TL;DR

This paper provides a complete characterization and classification of bases in twisted affine root supersystems, extending known results from affine Lie algebras to the more complex super case.

Contribution

It introduces a full classification of bases for twisted affine root supersystems, addressing complexities due to self-orthogonal roots in the super case.

Findings

Complete classification of bases in twisted affine root supersystems

Extension of affine Lie algebra results to superalgebras

Detailed description of the structure of these bases

Abstract

Following the definition of a root basis of an affine root system, we define a base of the root system of an affine Lie superalgebra to be a linearly independent subset $B$ of its root system such that each root can be written as a linear combination of elements of $B$ with integral coefficients such that all coefficients are nonnegative or all coefficients are nonpositive. Characterization and classification of bases of root systems of affine Lie algebras are known in the literature; in fact, up to $\pm 1$-multiple, each base of an affine root system is conjugate with the standard base under the Weyl group action. In the super case, the existence of those self-orthogonal roots which are not orthogonal to at least one other root, makes the situation more complicated. In this work, we give a complete characterization of bases of a twisted affine root supersystem. We precisely describe and classify them.