Quantum Spaces Associated to Multipermutation Solutions of Level Two

We study finite set-theoretic solutions (X, r) of the Yang-Baxter equation of square-free multipermutation type. We show that each such solution over C with multipermutation level two can be put in diagonal form with the associated Yang-Baxter algebra A(C, X, r) having a q-commutation form of relations determined by complex phase factors. These complex factors are roots of unity and all roots of a prescribed form appear as determined by the representation theory of finite abelian group G of left actions on X. We study the structure of A(C, X, r) and show that they have a •-product form ‘quantizing’ the commutative algebra of polynomials in |X| variables. We obtain the •-product both as a Drinfeld cotwist for a certain canonical 2-cocycle and as a braided-opposite product for a certain crossed G-module (over any field k). We provide first steps in the noncommutative differential geometry of A(k,X, r) arising from these results. As a byproduct of our work we find that every such level 2 solution (X, r) factorises as r = f ◦ τ ◦ f where τ is the flip map and (X, f) is another solution coming from X as a crossed G-set.