The Structure of Multiplicative Preprojective Algebras
Crawley-Boevey defined deformed preprojective algebras as a means of viewing solutions to the additive Deligne-Simpson problem as irreducible representations of these algebras. In the same vein, Crawley-Boevey--Shaw defined multiplicative preprojective algebras (MPAs) to encode solutions of the multiplicative Deligne-Simpson problem. For historical motivation, I'll present the ideas behind these two constructions. More recently, MPAs have gained popularity in geometry thanks to work of Van den Bergh, Yamakawa, Etgu--Lekili, and Bezrukavnikov--Kapranov, among others. In joint work with Travis Schedler, we ask which algebraic properties of preprojective algebras still hold for MPAs. We prove that the behavior is similar for quivers containing a cycle, as the MPA is 2-Calabi-Yau and prime, with trivial center when the containment is proper. Moreover, the MPA for a cycle is a non-commutative resolution of its center. I'll summarize the key ideas behind this work without assuming intimate knowledge of preprojective algebras.