Dan Rogalski, UC San Diego

Title: Homological integrals for weak Hopf algebras

Abstract: The integral is an important structure in a finite-dimensional Hopf algebra. Lu, Wu, and Zhang generalized this to define a homological integral for any Artin-Schelter Gorenstein Hopf algebra. This homological integral has many applications in the study of Hopf algebras of small GK-dimension. A weak Hopf algebra is a generalization of a Hopf algebra in which the comultiplication does not necessarily preserve the unit. Weak Hopf algebras arise naturally in the study of tensor categories, for example. In this work we show how to define a homological integral for an AS Gorenstein weak Hopf algebra, and that it has good properties. This is joint work in progress with Rob Won and James Zhang.