Current Research: Noncommutative Representation Theory

Group actions are ubiquitous in mathematics. To understand a mathematical object, it is often helpful to understand its symmetries as expressed by a group. For example, a group acts on a ring by automorphisms (preserving its structure). Analogously, a Lie algebra acts on a ring by derivations. Unifying these two types of actions are Hopf algebras acting on rings. A Hopf algebra is not only an algebra, but also a coalgebra, and the notion of an action preserving the structure of a ring uses this property. The category of representations of a Hopf algebra is rich due to this extra structure: It is a tensor category. Those who study tensor categories and their many applications are led to study (quasi) Hopf algebras, and vice versa. Hopf algebras can behave quite differently from groups and Lie algebras. Their representations can have a noncommutative flavor (due to asymmetry in the tensor product), making them more challenging to study. I take a homological and geometric approach to the representation theory of Hopf algebras, expanding what is known for finite groups and more generally finite group schemes (equivalently, cocommutative Hopf algebras) to accommodate noncommutativity. My research program, which involves collaborations with many mathematicians including postdocs and graduate students, targets understanding of Hopf algebras, their categories of representations, algebras on which they act, and related structures such as smash or crossed products (i.e. rings built out of Hopf algebras and rings on which they act) and their deformations. My recent and current projects are aimed at such understanding and can be grouped into three areas as described below: support variety theory, cohomology of Hopf algebras, and algebraic deformations, Hopf actions, and Hochschild cohomology.