Low-dimensional representations of the three component loop braid group

PAUL BRUILLARD; LIANG CHANG; SEUNG-MOON HONG; JULIA YAEL PLAVNIK; ERIC C. ROWELL; MICHAEL SUN

JOURNAL OF MATHEMATICAL PHYSICS

AMER INST PHYSICS

Lugar: New York; Año: 2015 vol. 56

0022-2488

Motivated by physical and topological applications, we study representations of the group LB_3 of motions of 3 unlinked oriented circles in R^3. Our point of view is to regard the three strand braid group B_3 as a subgroup of LB_3 and study the problem of extending B_3 representations. We introduce the notion of a standard extension and characterize B_3 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 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 B_3 representation that has no extensions (standard or otherwise). We obtain complete classifications of (1) irreducible 2-dimensional LB_3 representations, (2) extensions of irreducible 3-dimensional B_3 representations, and (3) irreducible LB_3 representations whose restriction to B_3 has abelian image.