On the structure of simple bounded weight modules of $\mathfrak{sl}(\infty)$, $\mathfrak{o}(\infty)$, $\mathfrak{sp}(\infty)$

TL;DR

This paper analyzes the structure of bounded simple weight modules over infinite-dimensional Lie algebras like rak{sl}(infty), rak{o}(infty), and rak{sp}(infty), introducing new concepts of rak{p}-alignment and exploring their conditions.

Contribution

It introduces the notions of rak{p}-aligned and pseudo rak{p}-aligned modules for these algebras and characterizes their existence and properties.

Findings

Necessary and sufficient conditions for rak{p}-aligned modules.

Existence of pseudo rak{p}-aligned modules due to infinite rank.

Classification of bounded simple weight modules.

Abstract

We study the structure of bounded simple weight $\mathfrak{sl}(\infty)$-, $\mathfrak{o}(\infty)$-, $\mathfrak{sp}(\infty)$-modules, which have been recently classified in [6]. Given a splitting parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{sl}(\infty)$, $\mathfrak{o}(\infty)$, $\mathfrak{sp}(\infty)$, we introduce the concepts of $\mathfrak{p}$-aligned and pseudo $\mathfrak{p}$-aligned $\mathfrak{sl}(\infty)$-, $\mathfrak{o}(\infty)$-, $\mathfrak{sp}(\infty)$-modules, and give necessary and sufficient conditions for bounded simple weight modules to be $\mathfrak{p}$-aligned or pseudo $\mathfrak{p}$-aligned. The existence of pseudo $\mathfrak{p}$-aligned modules is a consequence of the fact that the Lie algebras considered have infinite rank.