Representation theory of a semisimple extension of the Takiff superalgebra

TL;DR

This paper explores the representation theory of a semisimple extension of the Takiff superalgebra, revealing its block structure, Borel subalgebras, and Koszul properties of its modules.

Contribution

It provides a detailed classification of blocks, Borel subalgebras, and extension groups, and establishes Koszulity for non-principal blocks in the finite-dimensional category.

Findings

Classification of blocks in module categories

Identification of Borel subalgebras

Proof of Koszulity for non-principal blocks

Abstract

We study a semisimple extension of a Takiff superalgebra which turns out to have a remarkably rich representation theory. We determine the blocks in both the finite-dimensional and BGG module categories and also classify the Borel subalgebras. We further compute all extension groups between two finite-dimensional simple objects and prove that all non-principal blocks in the finite-dimensional module category are Koszul.