Singular blocks of restricted sl3

TL;DR

This paper explicitly computes the structure, center, and Hochschild cohomology of a specific singular block of the restricted enveloping algebra of sl3, revealing its Koszul property.

Contribution

It provides the first detailed algebraic description of this singular block, including generators, relations, and homological properties, advancing understanding of modular Lie algebra representations.

Findings

Computed generators and relations for the basic algebra

Determined the algebra's center and Hochschild cohomology

Proved the Verma modules are Koszul

Abstract

We compute generators and relations for the basic algebra of a non-semisimple singular block of the restricted enveloping algebra of $\mathfrak{sl}_3$ over an algebraically closed field of characteristic $p>3$. Working directly with the basic algebra we compute its centre and the internal degree zero part of its first Hochschild cohomology, and show its Verma modules are Koszul.