Noncommutative Representation Theory
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Symmetry is ubiquitous in nature and in mathematics. Mathematicians understand and quantify symmetry via symmetry groups and their actions (or representations) on objects. For example, the bilateral symmetry of many animals is encoded by a reflection operation, while one needs both rotations and reflections to describe the more elaborate symmetry of a crystal. The noncommutativity inherent in some settings such as quantum physics, that nevertheless possess some symmetry, is sometimes viewed via actions of the Hopf algebras that generalize symmetry groups. This project investigates questions about representations of Hopf algebras and related algebraic structures. Hopf algebras arise in many fields of mathematics, such as combinatorics, number theory, topology, and mathematical physics, and results on their representations are of wide interest. The research project involves many collaborators and serves as a good training ground for the many graduate students, postdocs, and other young mathematicians with whom the PI works.Categories of representations of Hopf algebras are tensor categories, a rich structure arising from their coproducts. In comparison to groups, Hopf algebras can behave quite differently and surprisingly. Tensor categories of representations can be noncommutative, in contrast to those of groups, leading to some curious behavior. This project investigates support variety theory and the triangulated category structure of these noncommutative categories of representations, with the goal of developing it much more completely, as has been done in the (commutative) case of finite groups and finite group schemes. In order to make such progress, the project will also need to advance homological understanding of (finite dimensional) Hopf algebras, whose cohomology is conjectured to be finitely generated. In work with collaborators, the PI aims to prove the conjecture for some important classes of Hopf algebras, namely the pointed Hopf algebras of diagonal type that include as a special case the small quantum groups. In contrast to the better-understood small quantum groups, many in this larger class of Hopf algebras may have noncommutative categories of representations, and one must take more care in homological work as this leads to subtle differences. Some related homological projects involve an action of a Hopf algebra on an algebra, which may be studied by constructing a larger algebra built out of the two. The PI will study such smash product algebras and their Hochschild cohomology, in particular the Gerstenhaber algebra structure, and related deformations.