The Loop Braid Group and Charge Conserving Yang–Baxter Operators
Moduli and Friends seminar
Monday 22 May 2023
Eric Rowell (Texas A&M University) — The Loop Braid Group and Charge Conserving Yang-Baxter Operators
The Loop Braid Group LBn is the motion group of n free loops in S3, with generators the "leapfrog" motions and the symmetric exchanges. This can be viewed as a 3-dimensional analogue of the braid group, which is the motion group of points in a disk. The representation theory of LBn is largely unexplored, but is of interest in the study of 3D topological phases as well as in algebra and topology. In recent work with Celeste Damiani and Paul Martin we defined finite dimensional quotients of the loop braid group algebras, analogous to the Hecke algebra construction from braid group algebras. We also found (conjecturally faithful) local representations via a "charge conserving" Yang-Baxter operator. In a follow up paper with Martin we classified all charge conserving Yang-Baxter operators, and determined, in a 3rd paper with Fiona Torzewska, when these give rise to representations of the Loop braid group. I will summarize this trilogy, emphasizing the surprising combinatorics we encountered along the way.
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