Work in Progress

In my ongoing thesis work, I am characterizing tensor algebras in pointed fusion categories. So far, I have proven a result giving the decomposition of the tensor product of two bimodules in pointed fusion categories based on the classification of these bimodules by Ostrik and Natale. As a corollary, the fusion rules of a group-theoretical fusion category can be determined up to the decomposition of induced representations of twisted group algebras.


Actions of quantum groups on path algebras

Quantum groups are a natural extension of the notion of a group. Just as group actions describe symmetries of objects in classical geometry, Hopf actions of quantum groups give quantum symmetries of objects in quantum geometry. We investigate examples of quantum symmetry by parameterizing Hopf actions of U_q(\mathfrak{b}), U_q(\mathfrak{sl}_2), generalized Taft algebras, and the small quantum group on path algebras. Following work of Etingof, Kinser, Walton, we attempt to classify the “building blocks” of these actions by viewing path algebras as tensor algebras in the tensor category \mathsf{rep}(H) for the appropriate quantum group H and classifying the minimal, faithful tensor algebras.

Ryan Kinser and Amrei Oswald. Hopf actions of some quantum groups on path algebras. Journal of Algebra, 587:85-117, 2021.

Hopf actions of some quantum groups on path algebras – recorded short talk at the Special Session on Quantum Symmetries at the Mathematical Congress of the Americas 2021