Gavrilo Sipka is currently a mathematics student at the University of Queensland. He will be starting honours in the coming year. His interests lie in pure mathematics particularly abstract algebra, representation theory and number theory.
Fusion Categories from Representations of Quantum sl(2) at Roots of Unity
The project will be based around the study of quantum deformations of the universal enveloping algebra of a semi-simple Lie algebra depending on a parameter q. The project will focus on the quantum group of the special linear algebra sl(2). The project will go through understanding examples of fusion categories the come from the theory of quantum groups. We will study the representation theory of the quantum group associated to sl(2) in both the cases where the parameter q is not a root of unity and when it is a root of unity. We hope to understand how one can build interesting structures from the representation theory at roots of unity by using the theory of tilting modules. These ideas will be applied to construct semi-simple fusion categories from the representation theory when the parameter q is an odd root of unity. Lastly, the project will aim to look at the Lie super algebra osp(1|2) and investigate fusion categories that come from it’s quantum group. The key focus is to understand if one can obtain different fusion categories from the quantum groups of osp(1|2) and sl(2).