Classification of bases of twisted affine root supersystems

Abstract

Following the definition of a root basis of an affine root system, we define a base of the root system R of an affine Lie superalgebra to be a linearly independent subset B of the linear span of R such that \(B\subseteq R\) and each root can be written as a linear combination of elements of B with integral coefficients such that either all coefficients are nonnegative or all coefficients are non-positive. 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 the root systems of twisted affine Lie superalgerbas with nontrivial odd part. We precisely describe and classify them.