Unitary Yang--Baxter Operators: Towards a Classification

Abstract

There is a well-known circle of conjectures relating unitary solutions of the Yang--Baxter equation, unitary braided fusion categories, topological quantum computation, and link invariants. Progress is limited by the lack of a classification of unitary Yang--Baxter operators. We propose a conjectural classification with three generating sources: monomial solutions, group-type solutions arising from Yetter--Drinfeld modules, and solutions from twisted group-algebra towers. We conjecture that, up to unit scalars and local unitary basis changes, all solutions are generated by these sources. We provide evidence for our classification conjecture by means of computational searches for unitary Yang--Baxter operators among Clifford groups and a polynomial-in-Pauli ansatz. This search yields only a few solutions that do not appear to be equivalent to monomial or group type solutions. Direct calculations identify their link invariants with specializations of the HOMFLYPT and BMW/Kauffman invariants; the latter is a squared Jones specialization at a primitive sixth root, and the corresponding qutrit operator strictly localizes the semisimple quotient of the BMW algebra specializations $\mathcal{C}_n(q^3,e^{\pi\ii/6})$ associated with the braided fusion category $SO(4)_2$. The quaternionic Family III specialization realizes a two-eigenvalue class and gives tensor-power realization of the semisimple Hecke algebra tower associated with \(SU(3)_3\). Significantly, this answers a recent existence question of Lechner, and verifies an open case of the Rowell-Wang localization conjecture from 2010.