Indecomposable representations of Lie superalgebras

TL;DR

This paper classifies finite-dimensional indecomposable representations of vect(0|2) and related Lie superalgebras, extending previous work and identifying key differences in representation structures across various superalgebras.

Contribution

It provides a detailed description of indecomposable representations for vect(0|2), sl(1|n), and osp(2|2n), and reviews partial results for other superalgebras, highlighting new classifications.

Findings

Finite-dimensional indecomposable representations of vect(0|2) are described.

Representations of sl(1|2) and osp(2|2) are also characterized.

For most simple Lie superalgebras without Cartan matrix, indecomposable modules with arbitrary composition factors exist.

Abstract

In 1960's I. Gelfand posed a problem: describe indecomposable representations of any simple infinite dimensional Lie algebra of polynomial vector fields. Here, by applying the elementary technique of Gelfand and Ponomarev, a toy model of the problem is solved: finite dimensional indecomposable representations of vect(0|2), the Lie superalgebra of vector fields on the (0|2)-dimensional superspace, are described. Since vect (0|2) is isomorphic to sl(1|2) and osp(2|2), their representations are also described. The result is generalized in two directions: for sl(1|n) and osp(2|2n). Independently and differently J. Germoni described indecomposable representation of the series sl(1|n) and several individual Lie superalgebras. Partial results for other simple Lie superalgebras without Cartan matrix are reviewed. In particular, it is only for vect(0|2) and sh(0|4) that the typical irreducible representations can not participate in indecomposable modules; for other simple Lie superalgebras without Cartan matrix (of series vect(0|n), svect(0|n)$, svect(0|n)', spe(n) for n>2 and sh(0|m) for m>4) one can construct indecomposable representations with arbitrary composition factors. Several tame open problems are listed, among them a description of odd parameters