Category $\mathcal{O}$ for Lie superalgebras

Abstract

The authors define a Category $\mathcal{O}$ for any quasi-reductive Lie superalgebra $\mathfrak{g}$ with respect to a triangular decomposition. This much needed approach unifies many important constructions in the existing literature in a rigorous fashion. Our Category $\mathcal{O}$ encompasses all highest weight categories for Lie (super)algebras as well as specific examples which may not be highest weight categories. When the decomposition arises from a principal parabolic subalgebra $\mathfrak{p}$ of $\mathfrak{g}$, the Category $\mathcal{O}$ exhibits rich homological properties. For one, the authors show that in contrast to the case of a semisimple Lie algebra, the Category $\mathcal{O}$ is standardly stratified. Furthermore, the categorical cohomology of $\mathcal{O}$ is a finitely generated ring. This provides a first step towards developing a support variety theory for Category $\mathcal{O}$. 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 $\mathfrak{g}$. This upgrades results known for $\mathfrak{gl}(m|n)$ to the more general setting. Our arguments are based on foundational connections between the categorical cohomology and the relative Lie superalgebra cohomology as well as the interplay between Category $\mathcal{O}$ for $\mathfrak{g}$ and the Category $\mathcal{O}$ for its corresponding Lie algebra $\mathfrak{g}_{\bar 0}$.