Properties of the fixed ring of a preprojective algebra

TL;DR

This paper investigates the structure of fixed rings in preprojective algebras associated with extended Dynkin diagrams, extending classical invariant theory to a noncommutative, non-connected setting.

Contribution

It identifies conditions under which fixed rings of preprojective algebras exhibit desirable properties, expanding the understanding of invariants in noncommutative algebra.

Findings

Conditions for fixed rings to have rich algebraic structures

Differences between preprojective and regular algebras in quasi-reflection theory

Identification of obstacles in extending classical invariant results

Abstract

For a finite group acting on a polynomial ring, the Chevalley-Shephard-Todd Theorem proves that the fixed subring is isomorphic to a polynomial ring if and only if the group is generated by pseudo-reflections. In recent years, progress was made in work of Kirkman, Kuzmanovich, Zhang, and others to extend this result to regular algebras by expanding pseudo-reflections to quasi-reflections. Naturally, the question arises if the theory generalizes further to non-connected noncommutative algebras. Our objects of study will be preprojective algebras which are certain factor algebras of path algebras corresponding to extended Dynkin diagrams of type $A$, $D$ or $E$. This work answers the question what conditions need to be satisfied by the fixed ring in order to make a rich theory possible. On our way, we will point out additional difficulties in establishing quasi-reflections using the trace and reveal situations which do not occur for regular algebras.