Low-dimensional representations of the three component loop braid group Academic Article uri icon

abstract

  • © 2015 AIP Publishing LLC. Motivated by physical and topological applications, we study representations of the group LB3 of motions of 3 unlinked oriented circles in R3. Our point of view is to regard the three strand braid group B3 as a subgroup of LB3 and study the problem of extending B3 representations. We introduce the notion of a standard extension and characterize B3 representations admitting such an extension. In particular we show, using a classification result of Tuba and Wenzl [Pacific J. Math. 197, 491-510 (2001)], that every irreducible B3 representation of dimension at most 5 has a (standard) extension. We show that this result is sharp by exhibiting an irreducible 6-dimensional B3 representation that has no extensions (standard or otherwise). We obtain complete classifications of (1) irreducible 2-dimensional LB3 representations, (2) extensions of irreducible 3-dimensional B3 representations, and (3) irreducible LB3 representations whose restriction to B3 has abelian image.

altmetric score

  • 0.25

author list (cited authors)

  • Bruillard, P., Chang, L., Hong, S., Plavnik, J. Y., Rowell, E. C., & Sun, M. Y.

citation count

  • 6

publication date

  • November 2015