Unitary braidings
Gandalf Lechner
Abstract: The Yang-Baxter equation is a cubic equation for an endomorphism R („R-matrix“) of the tensor square of a vector space V which naturally induces representations of braid groups on higher tensor powers of V. In applications in quantum physics, one is often interested in the situation where V is a finite-dimensional Hilbert space and R is unitary. This talk will review research aiming at understanding the set of all unitary R-matrices up to a natural equivalence given by characters of the infinite braid group. The case of involutive unitary R-matrices, connected to extremal characters of the infinite symmetric group, will be described in some detail. Although the question asked are purely algebraic, our main tools are taken from operator algebras (von Neumann algebras, subfactors) and quantum field theory.