Koszulity in the category $\mathcal{O}$ of the periplectic Lie Superalgebra $\mathfrak{pe}(2)$

TL;DR

This paper determines which blocks in the category of the periplectic Lie superalgebra (2) are Koszul, showing the remaining block is Koszul through explicit algebra computations aided by computer algebra tools.

Contribution

It explicitly computes the endomorphism algebra of projective modules in the remaining block and proves its Koszulity, completing the classification for (2).

Findings

The generic block of (2) is Koszul.

The principal integral block is not Koszul.

The remaining block is proven to be Koszul using explicit algebra computations.

Abstract

The main result of this paper is to establish precisely which blocks in the Category $\mathcal{O}$ of the periplectic Lie superalgebra $\mathfrak{pe}(2)$ are Koszul. It is known that $\mathcal{O}(\mathfrak{pe}(2))$ has three blocks up to equivalence; one generic block and two integral blocks. The generic block is known to be Koszul, and the principle integral block is verifiably not Koszul. In this paper, we prove that the remaining of the three blocks of $\mathcal{O}(\mathfrak{pe}(2))$ is Koszul. This is done by explicitly computing the endomorphism algebra of projective modules in this block and then proving that it is Koszul inductively. Along the way, we compute all $\mathrm{Ext}$ groups between simples in this block. To compute the endomorphism algebra we are aided by a computer algebra tool developed in Mathematica, inspired by a post on Stack Exchange.