Abstract

Preprojective algebras are one of the central objects in representation theory. The preprojective algebra of a quiver Q is a graded algebra whose degree zero part is the path algebra kQ of Q, and each degree i part gives a distinguished class of representations of Q, called the preprojective modules. It categorifies the Coxeter groups as reflection functors, and their structure depends on the trichotomy of quivers: Dynkin, extended Dynkin, and wild. From homological algebra point of view, the algebra kQ is hereditary (i.e. global dimension at most one), and its preprojective algebra is 2-Calabi-Yau.
In this talk, I will discuss the higher preprojective algebras P of algebras A of finite global dimension d. When d=2, then P is the Jacobi algebra of a certain quiver with potential. When A is a d-hereditary algebra, a certain distinguished class of algebras of global dimension d, then its higher preprojective algebra is (d+1)-Calabi-Yau. I will explain results and examples of higher preprojective algebras based on joint works with Herschend and Oppermann. If time permits, I will explain a joint work with Amiot and Reiten on algebraic McKay correspondence for higher preprojective algebras.