UC San Diego

Studying invariant theory of commutative polynomial rings has motivated many developments in commutative algebra and algebraic geometry. For a finite group acting on a polynomial ring, the remarkable 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. Another interesting question is to find properties of a fixed ring for a group action satisfying certain attributes. In recent years, progress was made in work of Jing, J{\o}rgensen, Kirkman, Kuzmanovich, Walton, Zhang, and others to extend the theory to regular algebras which are a noncommutative generalization of polynomial rings.

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 dissertation answers the question what conditions need to be satisfied by the fixed ring in order to make a rich theory possible. Moreover, we give a sufficient condition on the finite group acting on a preprojective algebra to guarantee that the fixed ring has finite injective dimension and satisfies a generalized Gorenstein condition. Part of this result is the construction of a homological determinant of a non-connected algebra which turns out to be particularly nice for the examined preprojective algebras.