Simple supermodules over Lie superalgebras

TL;DR

This paper establishes a bijection between simple supermodules over certain Lie superalgebras and their even parts, simplifying classification and revealing that supermodule structures depend only on specific annihilators, with applications to classical types.

Contribution

It introduces a bijection via Kac induction for simple supermodules over many Lie superalgebras, reducing classification to the even part and analyzing structure dependence.

Findings

Bijection between simple supermodules and even part supermodules for many Lie superalgebras.

Supermodule structures depend only on the annihilator of the input.

Applicable to all classical Lie superalgebras of type I, including (m|n).

Abstract

We show that, for many Lie superalgebras admitting a compatible $\mathbb{Z}$-grading, Kac induction functor gives rise to a bijection between simple supermodules over a Lie superalgebra and simple supermodules over the even part of this Lie superalgebra. This reduces the classification problem for the former to the one for the latter. Our result applies to all classical Lie superalgebra of type $I$, in particular, to the general linear Lie superalgebra $\gl(m|n)$. In the latter case we also show that the rough structure of simple $\gl(m|n)$-supermodules and also that of Kac supermodules depends only on the annihilator of the $\mf{gl}(m)\oplus \mf{gl}(n)$-input and hence can be computed using the combinatorics of BGG category $\mathcal{O}$.