All involutive solutions of the Yang-Baxter equation

par Gandalf Lechner (Cardiff University)

vendredi 2 mars 2018, 14:00 → 15:00 Europe/Paris

Fokko du Cloux (Institut Camille Jordan)

The Yang-Baxter equation (YBE) is a nonlinear matrix equation that lies at the heart of many subjects, including quantum statistical mechanics, integrable quantum field theory, knot theory, braid groups, subfactors, quantum groups, quantum information ... . Due to the nonlinearity and noncommutativity of the YBE, its solutions are notoriously difficult to obtain. In this talk, I will consider mainly involutive solutions R of the YBE ("R-matrices", satisfying R^2=1) and describe a new way of completely classifying them in any dimension. The upshot of this classification is that any involutive R-matrix defines a representation and an extremal character of the infinite symmetric group as well as a corresponding tower of subfactors. Using these structures, I will describe how to find all R-matrices up to a natural notion of equivalence inherited from applications in QFT (given by the character and the dimension), how to completely parameterize the set of solutions, and how to decide efficiently whether two given R-matrices are equivalent. Time permitting, I will also indicate how these results carry over to R-matrices underlying knot polynomials which are no longer involutive.