On the structure of simple bounded weight modules of sl(∞),o(∞),sp(∞)

Abstract

We study the structure of bounded simple weight $\mathfrak{s}\mathfrak{l}(\infty )$-, $\mathfrak{o}(\infty )$-, $\mathfrak{s}\mathfrak{p}(\infty )$-modules, which have been recently classified by D. Grantcharov and I. Penkov. Given a splitting parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{s}\mathfrak{l}(\infty ),\mathfrak{o}(\infty ),\mathfrak{s}\mathfrak{p}(\infty ),$ we introduce the concepts of $\mathfrak{p}$-aligned and pseudo $\mathfrak{p}$-aligned $\mathfrak{s}\mathfrak{l}(\infty )$-,$\mathfrak{o}(\infty )$-, $\mathfrak{s}\mathfrak{p}(\infty )$-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.

Keywords:

• Weight module
• induced module
• direct limit Lie algebra

2010 MSC:

• 17B65
• 17B10

Acknowledgements

This paper has been written during a post-doctoral period at the Jacobs University, Bremen, under supervision of Ivan Penkov. I am grateful to Ivan Penkov for proposing the problem, for all stimulating discussions, and for valuable suggestions. I also thank Jacobs University for its hospitality.

Additional information

Funding

L. C. was supported by the Capes grant (88881.119190/2016-01) and by the PRPq grant ADRC-05/2016.