Celeste Damiani (London)
Abstract
This work takes inspiration from the braid group revolution ignited by Jones in the early 80s, to study representations of the motion group of the free unlinked circles in the $3$ dimensional space, the loop braid group~$LB_n$. Since $LB_n$ contains a copy of the braid group $B_n$ as a subgroup, a natural approach to look for linear representations is to extend known representations of the braid group $B_n$. 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 $LB_n$ we quotient its group algebra by the ideal generated by $(\sigma_i + 1)(\sigma_i-1)$ 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~$LH_n$. We proceed then to analyse this structure. Our hope is that this work could be one of the first steps to find invariants Ă la Jones for knotted objects related to loop braid groups.