In this talk, the speaker will define a general Category $\mathcal{O}$ for any quasi-simple Lie superalgebra. Our 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, the speaker will develop a parabolic Category ${\mathcal O}$ for classical Lie superalgebras.

Connections between the categorical cohomology and the relative Lie superalgebra cohomology will be firmly established. These results will be used to show that the Category ${\mathcal O}$ is standardly co-stratified. The definition of standardly co-stratified used in this context is generalized from the original definition of Cline, Parshall, and Scott. An explicit description of the categorical cohomology ring will be given, and finite generation results will be presented.

Furthermore, it will be 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. This result provides a generalization of prior work for $\mathfrak{gl}(m|n)$ due to Coulembier and Serganova.

This talk represents joint work with Chun-Ju Lai and Arik Wilbert.