Bounded weight modules for basic classical Lie superalgebras at infinity

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.

37 pages