Bounded weight modules for basic classical Lie superalgebras at infinity

TL;DR

This paper classifies simple bounded weight modules over certain infinite-dimensional Lie superalgebras, revealing their structure and categorization, and connecting them to known representation theories of classical Lie algebras.

Contribution

It provides the first classification of simple bounded weight modules for infinite-dimensional Lie superalgebras like rak{sl}(\u221e|) and rak{osp}(m|), linking them to classical representation categories.

Findings

Modules over rak{osp}(m|) are of spinor-oscillator type.

Every simple bounded rak{sl}(|)-module is integrable over certain ideals.

The category of bounded modules over rak{osp}(m|) reduces to known symplectic representations.

Abstract

We classify simple bounded weight modules over the complex simple Lie superalgebras $\mathfrak{sl}(\infty |\infty)$ and $\mathfrak{osp} (m | 2n)$, when at least one of $m$ and $n$ equals $\infty$. For $\mathfrak{osp} (m | 2n)$ such modules are of spinor-oscillator type, i.e., they combine into one the known classes of spinor $\mathfrak{o} (m)$-modules and oscillator-type $\mathfrak{sp} (2n)$-modules. In addition, we characterize the category of bounded weight modules over $\mathfrak{osp} (m | 2n)$ (under the assumption $\dim \, \mathfrak{osp} (m | 2n) = \infty$) by reducing its study to already known categories of representations of $\mathfrak{sp} (2n)$, where $n$ possibly equals $\infty$. When classifying simple bounded weight $\mathfrak{sl}(\infty |\infty)$-modules, we prove that every such module is integrable over one of the two infinite-dimensional ideals of the Lie algebra $\mathfrak{sl}(\infty |\infty)_{\bar{0}}$. We finish the paper by establishing some first facts about the category of bounded weight $\mathfrak{sl} (\infty |\infty)$-modules.