Hopf algebras arise naturally in many different areas of mathematics, such as combinatorics, topology, and mathematical physics. Many commonly studied algebraic structures are either Hopf algebras themselves or are directly related to them: for example, groups, Lie algebras, and quantum groups. Hopf algebras act on rings, generalizing the notion of a group of automorphisms. Modules for a Hopf algebra can be added (direct sum) and multiplied (tensor product), giving their categories of modules the structure of tensor categories. In this first talk, we will define Hopf algebras and their actions on rings, give examples, and explain how their modules fit into this larger picture of tensor categories.
Current research on Hopf algebras includes programs to classify some types and to understand their categories of modules. In this second talk, we will consider this second problem for nonsemisimple Hopf algebras, and homological techniques for approaching it. We will define the cohomology ring and state a finite generation conjecture for finite dimensional Hopf algebras and more generally for finite tensor categories. We will survey what is known and touch on some recent research.
From groups to Hopf algebras: Cohomology and varieties for modules
Sarah Witherspoon (Texas A & M University)
MSRI: Simons Auditorium
Group cohomology is a powerful tool in group representation theory.
To a group action on a vector space, one associates a geometric object called its support variety that is defined using group cohomology. Hopf algebras generalize groups and include many important classes of algebras such as Lie algebras and quantum groups. The theory of varieties for modules generalizes to Hopf algebras to some extent, but there are many open questions.
In this introductory talk, we will define Hopf algebras, their cohomology, and the corresponding varieties for modules. We will discuss known and unknown properties and recent and current research on open problems