Some homological properties of category $\mathcal O$ for Lie superalgebras

TL;DR

This paper investigates homological properties of category O for classical Lie superalgebras, providing criteria for Verma supermodules, analyzing extensions, and determining projective and injective dimensions for key superalgebras.

Contribution

It offers necessary and sufficient conditions for homomorphisms between Verma supermodules and computes projective/injective dimensions for important superalgebras.

Findings

Criteria for homomorphisms between Verma supermodules.

Description of the socle of cokernels in periplectic Lie superalgebras.

Complete determination of projective and injective dimensions for certain supermodules.

Abstract

For classical Lie superalgebras of type I, we provide necessary and sufficient conditions for a Verma supermodule $\Delta(\lambda)$ to be such that every non-zero homomorphism from another Verma supermodule to $\Delta(\lambda)$ is injective. This is applied to describe the socle of the cokernel of an inclusion of Verma supermodules over the periplectic Lie superalgebras $\mathfrak{pe}(n)$ and, furthermore, to reduce the problem of description of $\mathrm{Ext}^1_{\mathcal O}(L(\mu),\Delta(\lambda))$ for $\mathfrak{pe}(n)$ to the similar problem for the Lie algebra $\mathfrak{gl}(n)$. Additionally, we study the projective and injective dimensions of structural supermodules in parabolic category $\mathcal O^{\mathfrak p}$ for classical Lie superalgebras. In particular, we completely determine these dimensions for structural supermodules over the periplectic Lie superalgebra $\mathfrak{pe}(n)$ and the ortho-symplectic Lie superalgebra $\mathfrak{osp}(2|2n)$.