My research is on the structure, representations, and cohomology of
various types of (noncommutative) rings,
including Hopf algebras,
and group-graded rings.
A Hopf algebra is a ring with some additional structure.
Many important examples arising in various fields in mathematics
include group algebras, universal enveloping algebras of Lie algebras,
and quantum groups. Quantum groups
are not groups, but instead are
Hopf algebras that share some essential properties with groups.
Groups arise in the study of symmetry, such as that of
atoms and molecules, or of geometric spaces or of solutions of equations.
Quantum groups have their origins in statistical mechanics and knot theory,
and they as well as the more general Hopf algebras also can be regarded
as exposing symmetry.
One important property of modules (or representations) for a Hopf algebra is that the
tensor product of two modules, over a commutative ground ring, is again
a module. This property gives its category of modules the structure
of a tensor category, which is very beneficial for understanding
its modules and their cohomology.
A group-graded ring is a ring that decomposes into a direct sum
of components indexed by elements from a group in such a way
that multiplication follows the group multiplication.
Some important examples of
such algebras arise in geometry: If a group acts on a space
(such as a manifold or variety), one may form an
algebra from the group and an algebra of functions on the space. This algebra is
graded by the group and it encodes information about the space.
One is interested in cohomology and deformations of such
algebras in relation to cohomology and deformations of the original spaces.
Information about deformations of the algebras is contained in
their Hochschild cohomology. Some important examples of their deformations
are rings that were discovered independently in several different contexts,
termed graded (Drinfeld) Hecke algebras, symplectic
reflection algebras, and rational Cherednik algebras.