##### Department of Mathematics,

University of California San Diego

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### Final Defense

## Stephan Weispfenning

#### University of California San Diego

## Invariant Theory of Preprojective Algebras

##### Abstract:

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. Related questions try to find properties of the fixed ring under some special group actions. In recent years, progress was made in work of Jing, Jorgensen, 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. This talk answers the question what conditions need to be satisfied by the fixed ring in order to make a rich theory possible. Moreover, we construct a homological determinant for preprojective algebras and discuss how it being trivial for all elements of a finite group affects the related fixed ring.

Advisor: Daniel Rogalski

### October 3, 2018

### 9:00 AM

### AP&M 6402

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