On the braid group representations coming from weakly group-theoretical fusion categories

On weakly group-theoretical non-degenerate braided fusion categories

We show that the Witt class of a weakly group-theoretical non-degenerate braided fusion category belongs to the subgroup generated by classes of non-degenerate pointed braided fusion categories and Ising braided categories. This applies in particular to solvable nondegenerate braided fusion categories. We also give some sufficient conditions for a braided fusion category to be weakly group-theo...

New Permutation Representations of The Braid Group

We give a new infinite family of group homomorphisms from the braid group Bk to the symmetric group Smk for all k and m ≥ 2. Most known permutation representations of braids are included in this family. We prove that the homomorphisms in this family are non-cyclic and transitive. For any divisor l of m, 1 ≤ l < m, we prove in particular that if m l is odd then there are 1+ m l non-conjugate hom...

Braid Group Actions on Derived Categories of Coherent Sheaves

This paper gives a construction of braid group actions on the derived category of coherent sheaves on a variety X. The motivation for this is M. Kontsevich’s homological mirror conjecture, together with the occurrence of certain braid group actions in symplectic geometry. One of the main results is that when dimX ≥ 2, our braid group actions are always faithful. We describe conjectural mirror s...

Localization of Unitary Braid Group Representations

Governed by locality, we explore a connection between unitary braid group representations associated to a unitary R-matrix and to a simple object in a unitary braided fusion category. Unitary R-matrices, namely unitary solutions to the Yang-Baxter equation, afford explicitly local unitary representations of braid groups. Inspired by topological quantum computation, we study whether or not it is...

DIMENSION n REPRESENTATIONS OF THE BRAID GROUP ON n STRINGS