On the Images of Braid Group Representations Coming from Braided Fusion Categories

Braided fusion categories are algebraic structures with strong ties to the representation theory of finite groups, Hopf algebras, and quantum groups. These structures also have strong connections with braid groups and low-dimensional topology. Recently, braid group representations coming from braided fusion categories have become a topic of interest in areas of condensed matter physics and topological quantum computation. Particularly interesting are the properties of the images of these representations. Calculations to determine the finiteness of these images have been performed for a few cases. A class of braided fusion categories coming from finite groups (group-theoretical) has been shown to yield finite images. We show that the images of braid group representations coming from the larger class of weakly group-theoretical braided fusion categories are also finite. We then compute the images of the pure braid groups for some specific representations