Category $\mathcal{O}$ for Lie superalgebras

In this paper, the authors define a general Category $\mathcal{O}$ for any quasi-reductive Lie superalgebra. The construction encompasses (i) the parabolic Category ${\mathcal O}$ for complex semisimple Lie algebras, and (ii) the known constructions of Category ${\mathcal O}$ for specific examples of classical Lie superalgebras. In particular, a parabolic version of Category ${\mathcal O}^{\mathfrak p}$ for quasi-reductive Lie superalgebras ${\mathfrak g}\cong {\mathfrak g}_{\overline{0}}\oplus {\mathfrak g}_{\overline{1}}$ is developed. In this paper, the authors demonstrate how the parabolic category ${\mathcal O}^{{\mathfrak p}_{\overline{0}}}$ for the reductive Lie algebra ${\mathfrak g}_{\overline{0}}$ plays an important role in the representation theory and cohomology for ${\mathcal O}^{\mathfrak p}$. Connections between the categorical cohomology and the relative Lie superalgebra cohomology are firmly established. These results are then used to show that the Category ${\mathcal O}$ is standardly stratified. The definition of standardly stratified used in this context is generalized from the original definition of Cline, Parshall, and Scott. Other applications involve showing that the categorical cohomology for ${\mathcal O}^{\mathfrak p}$ is a finitely generated ring. Furthermore, it is shown that the complexity of modules in Category ${\mathcal O}$ is finite with an explicit upper bound given by the dimension of the subspace of the odd degree elements in the given Lie superalgebra.