PhD Comprehensive Exam | Abdulmajid Osumanu, Bicrossproduct Quantum Group Model for 3d Topological Models

MC 6460

Candidate

Abdulmajid Osumanu
Applied Mathematics, University of Waterloo

Title

Bicrossproduct Quantum Group Model for 3d Topological Models

Abstract

Topological quantum field theories (TQFT) despite being simpler than local quantum field theories contain rich structures interesting both for the mathematical and physical standpoints. Defined in terms of (quasi) Hopf algebras, aka quantum groups, TQFT's in 3d provide models for different frameworks such as  quantum gravity (QG) and topological quantum information (TQI). In the TQI case, we deal with the Drinfeld quantum double of finite dimensional Hopf algebras which is also relevant to 3d QG, either in the BF formulation or the Chern-Simons formulation.

Recent works have shown how semi-dualization can give rise to the bicrossproduct quantum group from the quantum double. This was specifically used in the Chern-Simons formulation and the semi-dualization is associated to passing to different regimes. We intend to discuss such semi-dualization procedure in the BF formulation. Interestingly this could be relevant to quantum information models such as Kitaev's model defined in terms of Drinfeld's quantum double.