Category $\mathcal{O}$ for Lie superalgebras

TL;DR

This work defines Category $\mathcal{O}^{\mathfrak{p}}$ for quasi-reductive Lie superalgebras with a triangular decomposition, unifying parabolic and Type I variants of Category $\mathcal{O}$. It proves that for principal parabolic subalgebras, $\operatorname{Ext}^{i}_{\mathcal{O}^{\mathfrak{p}}}(M,N)$ coincides with relative Lie superalgebra cohomology $\operatorname{Ext}^{i}_{(\mathfrak{g},\mathfrak{l}_{\bar{0}})}(M,N)$, and that the cohomology ring $R=\operatorname{Ext}^{\bullet}_{\mathcal{O}^{\mathfrak{p}}}(\mathbb{C},\mathbb{C})$ is finitely generated and decomposes as $R\cong S^{\bullet}(\mathfrak{g}_{\bar{1}}^{*})^{G_{\bar{0}}}\otimes \operatorname{H}^{\bullet}(\mathfrak{g}_{\bar{0}},\mathfrak{l}_{\bar{0}},\mathbb{C}) \cong S^{\bullet}(\mathfrak{g}_{\bar{1}}^{*})^{G_{\bar{0}}}\otimes \operatorname{Ext}^{\bullet}_{\mathcal{O}^{\mathfrak{p}_{\bar{0}}}}(\mathbb{C},\mathbb{C})$. Finite-dimensional modules yield finitely generated Ext-modules over $R$, and if $\mathfrak{g}$ is almost simple, all Ext groups are still finitely generated over $R$. The authors introduce a generalized standardly stratified framework accommodating infinitely many simples and prove a universal bound $c_{\mathfrak{p}}(M) \le \dim \mathfrak{g}_{\bar{1}}$, linking homological properties to relative cohomology and laying groundwork for a geometric interpretation via support varieties in this setting. These results extend known parabolic and cohomological theories from classical Lie algebras to a broad class of Lie superalgebras, including Brundan-type and BBW-parabolic examples.

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 $\mathfrak{g}_{\bar 0}$.

Theorems & Definitions (3)

• Theorem A
• Theorem B
• Theorem C