Algebras with finitely many conjugacy classes of left ideals versus algebras of finite representation type

TL;DR

This paper establishes a precise equivalence between having finitely many conjugacy classes of left ideals and being of finite representation type for certain finite-dimensional algebras, under specific module dimension conditions.

Contribution

It proves that for algebras with radical nilpotent of index 2 and simple modules of dimension at least 6, the finiteness of conjugacy classes of left ideals characterizes finite representation type.

Findings

Finitely many conjugacy classes of left ideals imply finite representation type under given conditions.

The equivalence holds specifically for algebras with radical nilpotent of index 2 and simple modules of dimension ≥6.

Provides a classification criterion linking algebraic structure and module theory.

Abstract

Let A be a finite dimensional algebra over an algebraically closed field with the radical nilpotent of index 2. It is shown that A has finitely many conjugacy classes of left ideals if and only if A is of finite representation type provided that all simple A-modules have dimension at least 6. This is a revised version.