Bounded Weight Modules for Basic Classical Lie Superalgebras at Infinity

Abstract

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