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

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.

minor corrections, published version

References in corpus (1)

• Simple bounded weight modules of $\mathfrak{sl}(\infty )$, $\mathfrak{o}(\infty )$, $\mathfrak{sp}(\infty )$