Bounded weight modules for basic classical Lie superalgebras at infinity

We classify simple bounded weight modules over the complex simple Lie superalgebras \(\mathfrak {sl}\hspace{0.55542pt}(\infty \,{|}\,\infty )\) $\mathfrak{sl}(\infty |\infty )$ and \(\mathfrak {osp}\hspace{0.55542pt}(m\,{|}\,2n)\) $\mathfrak{osp}(m|2n)$ , when at least one of m and n equals \(\infty \) $\infty$ . For \(\mathfrak {osp}\hspace{0.55542pt}(m\,{|}\,2n)\) $\mathfrak{osp}(m|2n)$ such modules are of spinor-oscillator type, i.e., they combine into one of the known classes of spinor \(\mathfrak {o}(m)\) $\mathfrak{o}(m)$ -modules and oscillator-type \(\mathfrak {sp}\hspace{0.55542pt}(2n)\) $\mathfrak{sp}(2n)$ -modules. In addition, we characterize the category of bounded weight modules over \(\mathfrak {osp}\hspace{0.55542pt}(m\,{|}\,2n)\) $\mathfrak{osp}(m|2n)$ (under the assumption \(\dim \mathfrak {osp}\hspace{0.55542pt}(m\,{|}\, 2n) = \infty \) $dim\mathfrak{osp}(m|2n)=\infty$ ) by reducing its study to already known categories of representations of \(\mathfrak {sp}\hspace{0.55542pt}(2n)\) $\mathfrak{sp}(2n)$ , where n possibly equals \(\infty \) $\infty$ . When classifying simple bounded weight \(\mathfrak {sl}\hspace{0.55542pt}(\infty \,{|}\,\infty )\) $\mathfrak{sl}(\infty |\infty )$ -modules, we prove that every such module is integrable over one of the two infinite-dimensional ideals of the Lie algebra \(\mathfrak {sl}\hspace{0.55542pt}(\infty \,{|}\,\infty )_{\bar{0}}\) $\mathfrak{sl}{(\infty |\infty )}_{\overline{0}}$ . We finish the paper by establishing some first facts about the category of bounded weight \(\mathfrak {sl}\hspace{0.55542pt}(\infty \,{|}\,\infty )\) $\mathfrak{sl}(\infty |\infty )$ -modules.