Low-dimensional representations of the three component loop braid group

Abstract

Motivated by physical and topological applications, we study representations of the group ${\mathcal{LB}}_3$ of motions of $3$ unlinked oriented circles in ${\mathbb{R}}^3$. Our point of view is to regard the three strand braid group ${\mathcal{B}}_3$ as a subgroup of ${\mathcal{LB}}_3$ and study the problem of extending ${\mathcal{B}}_3$ representations. We introduce the notion of a \emph{standard extension} and characterize ${\mathcal{B}}_3$ representations admiting such an extension. In particular we show, using a classification result of Tuba and Wenzl, that every irreducible ${\mathcal{B}}_3$ representation of dimension at most $5$ has a (standard) extension. We show that this result is sharp by exhibiting an irreducible $6$-dimensional ${\mathcal{B}}_3$ representation that has no extensions (standard or otherwise). We obtain complete classifications of (1) irreducible $2$-dimensional ${\mathcal{LB}}_3$ representations (2) extensions of irreducible $3$-dimensional ${\mathcal{B}}_3$ representations and (3) irreducible ${\mathcal{LB}}_3$ representations whose restriction to ${\mathcal{B}}_3$ has abelian image.

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Cited by in corpus (5)

• Representations of the Necklace Braid Group: Topological and Combinatorial Approaches
• An Invitation to the Mathematics of Topological Quantum Computation
• High-dimensional representations of the 3-component loop braid group
• Representations of the loop braid groups from braided tensor categories
• Representations of the necklace braid group {NB}}_n of dimension (n=2,3,4)