Weight modules of D(2,1,a)

TL;DR

This paper classifies simple weight modules of the Lie superalgebra D(2,1,a), showing they are all bounded with degrees up to 8, and characterizing typical and atypical modules based on their degree.

Contribution

It proves that all simple weight modules of D(2,1,a) are bounded and establishes precise degree bounds, including the distinction between typical and atypical modules.

Findings

All simple weight modules are bounded.

Degree of modules is at most 8.

Typical modules attain degree 8, atypical modules have degree between 2 and 6.

Abstract

A weight module of a basic Lie superalgebra is called finite if all of its weight spaces are finite dimensional, and it is called bounded if there is a uniform bound on the dimension of a weight space. The minimum bound is called the degree of the module. For the basic Lie superalgebra D(2,1,a), we prove that every simple weight module is bounded and has degree less than or equal to 8. This bound is attained by a cuspidal module if and only if it is "typical". Atypical cuspidal modules have degree less than or equal to 6 and greater than or equal to 2.