Low-dimensional representations of the three component loop braid group

• Autores: Bruillard, Paul; Chang, Liang; Hong, Seung-Moon; Plavnik, Julia Yael; Rowell, Eric C.; Sun, Michael Yuan
• Año de publicación: 2015
• Idioma: inglés
• Tipo de recurso: artículo
• Materia: REPRESENTATIONS; LOOP BRAID GROUP; BRAID GROUP; STANDARD EXTENSION; LOW DIMENSIONAL; IRREDUCIBLE REPRESENTATIONS
• Institución: Consejo Nacional de Investigaciones Científicas y Técnicas

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.
Fil: Bruillard, Paul. Pacific Northwest National Laboratory; Estados Unidos
Fil: Chang, Liang. Texas A&M University; Estados Unidos
Fil: Hong, Seung-Moon. University of Toledo; Estados Unidos
Fil: Plavnik, Julia Yael. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Texas A&M University; Estados Unidos
Fil: Rowell, Eric C.. Texas A&M University; Estados Unidos
Fil: Sun, Michael Yuan. Mathematisches Institut der Universität Münster; Alemania