Properties of the fixed ring of a preprojective algebra

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.