The Yang–Baxter equation and Thompson’s group F

Abstract

We define non-degenerate involutive partial solutions as a generalization of non-degenerate involutive set-theoretical solutions of the quantum Yang–Baxter equation (QYBE). The induced operator is not a classical solution of the QYBE, but a braiding operator as in conformal field theory. We define the structure inverse monoid of a non-degenerate involutive partial solution and prove that if the partial solution is square-free, then it embeds into the restricted product of a commutative inverse monoid and an inverse symmetric monoid. Furthermore, we show that there is a connection between partial solutions and Thompson’s group F. This raises the question of whether there are further connections between partial solutions and Thompson’s groups in general.