Classification of irreducible real modules of real Lie superalgebras

TL;DR

This paper classifies all irreducible finite-dimensional modules of various real Lie superalgebras, including simple, classical, and complex types, by analyzing orbits of certain functors on complex modules.

Contribution

It introduces a unified approach to classify irreducible modules of real Lie superalgebras via orbit analysis of complexified modules, extending previous classifications.

Findings

Explicit computation of orbits for parity and conjugation functors.

Classification applies to basic type and Q(n) Lie superalgebras with arbitrary Borel subalgebras.

Provides a new perspective on classifying real simple Lie algebra modules.

Abstract

We classify irreducible finite-dimensional modules of a collection of real Lie superalgebras that includes the simple ones, their classical variants, complex Lie superalgebras after restriction of scalars, and all real Lie algebras. Our strategy is to reduce this classification to determining the orbits of the parity and conjugation functors on irreducible modules of the complexifications of the aforementioned algebras. Then we provide explicit results for the computation of these orbits. For Lie superalgebras of basic type or of type $\mathbf Q(n)$, our classification applies to any highest-weight parametrization of irreducible complex modules with respect to an arbitrary Borel subalgebra. As a consequence, in the special case of real simple Lie algebras we obtain a new perspective on the classification of real simple modules and establish a conceptual connection with Kostant's cascade of strongly orthogonal roots.