Solutions to Generalized Yang-baxter Equations via Ribbon Fusion Categories

Abstract. Inspired by quantum information theory, we look for representations of the braid groups Bn on V ⊗(n+m−2) for some fixed vector space V such that each braid generator σi, i = 1, ..., n−1, acts on m consecutive tensor factors from i through i+m− 1. The braid relation for m = 2 is essentially the YangBaxter equation, and the cases for m > 2 are called generalized Yang-Baxter equations. We observe that certain objects in ribbon fusion categories naturally give rise to such representations for the case m = 3. Examples are given from the Ising theory (or the closely related SU(2)2), SO(N)2 for N odd, and SU(3)3. The solution from the Jones-Kauffman theory at a 6 th root of unity, which is closely related to SO(3)2 or SU(2)4, is explicitly described in the end.