Loop braid groups and a generalisation of Hecke algebras

Celeste Damiani (Queen Mary University of London)

Monday 31st January 16:00-17:00 Online + Maths 110B

Abstract

The study of loop braid groups LBn has been widely developed during the last twenty years, in different domains of mathematics and mathematical physics. This work takes inspiration by from the braid group revolution ignited by Jones in the early 80's, and has the aim to understand representations of LBn seen as the motion group of the free unlinked circles in the 3-dimensional space. Since LBn contains a copy of the braid group Bn as a subgroup, a natural approach to look for linear representations is to extend known representations of the braid group Bn. Another possible strategy is to look for finite dimensional quotients of the group algebra, mimicking the braid group / Iwahori-Hecke algebra / Temperlely-Lieb algebra paradigm. Here we combine the two in a hybrid approach: starting from the loop braid group LBn we quotient its group algebra by the ideal generated by (σi + 1)(σi − 1) as in classical Iwahori-Hecke algebras. We then add certain quadratic relations, satisfied by the extended Burau representation, to obtain a finite dimensional quotient that we denote by LHn. We proceed then to analyse this structure. Our hope is that this work could be one of the first steps to find invariants à la Jones for knotted objects related to loop braid groups.

The talk will be preceded by a tea time at 3:45pm. The Zoom link for the seminar is https://uofglasgow.zoom.us/j/98078798957 and the passcode is the genus of the two-dimensional sphere (4 letters, all lowercase).

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