arXiv subjects
Publications and source records attributed to Andrew Kimball.
We unify and generalize several approaches to constructing braid group representations from finite groups, using iterated twisted tensor products. Our results hint at a relationship between the braidings on the $G$-gaugings of a pointed modular category $\mathcal{C}(A,Q)$ and that of $\mathcal{C}(A,Q)$ itself.
The necklace braid group $\mathcal{NB}_n$ is the motion group of the $n+1$ component necklace link $\mathcal{L}_n$ in Euclidean $\mathbb{R}^3$. Here $\mathcal{L}_n$ consists of $n$ pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group $\mathcal{NB}_n$, especially those obtained as extensions of representations of the braid group $\mathcal{B}_n$ and the loop braid group $\mathcal{LB}_n$. We show that any irreducible $\mathcal{B}_n$ representation extends to $\mathcal{NB}_n$ in a standard way. We also find some non-standard extensions of several well-known $\mathcal{B}_n$-representations such as the Burau and LKB representations. Moreover, we prove that any local representation of $\mathcal{B}_n$ (i.e. coming from a braided vector space) can be extended to $\mathcal{NB}_n$, in contrast to the situation with $\mathcal{LB}_n$. We also discuss some directions for future study from categorical and physical perspectives.