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

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 ${\mathrm{Ext}}_{\mathcal{O}}^1(L(\mu ),\Delta (\lambda ))$ 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 $\mathfrak{osp}(2|2n)$.

25 pages

References in corpus (3)

• On associated variety for Lie superalgebras
• On representations of the Lie superalgebra p(n)
• Gorenstein homological algebra for rngs and Lie superalgebras