Irreducible components of varieties of representations of a class of finite dimensional algebras

TL;DR

This paper investigates the structure of representation varieties of a specific class of finite-dimensional algebras, identifying their irreducible components through radical and socle layering techniques.

Contribution

It provides a detailed description of the irreducible components of representation varieties for a class of truncated non-commutative polynomial algebras with an added relation.

Findings

Subvarieties with fixed radical layering are irreducible.

Subvarieties with fixed socle layering are irreducible.

Comparison of radical and socle layerings helps determine the irreducible components.

Abstract

We study the representations of a class of non-commutative polynomial algebras truncated at degree 3, with one additional relation. We determine the irreducible components of their varieties of representations. We do this by showing that the subvarieties with fixed radical or socle layering are irreducible. We then compare the coverings we get using radical and socle layerings to determine the components.