Quantum Spaces Associated to Multipermutation Solutions of Level Two

Abstract

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 $\mathcal{A}(\mathbb{C},X,r)$ 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 the finite abelian group $\mathcal{G}$ of left actions on X. We study the structure of $\mathcal{A}(\mathbb{C},X,r)$ 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 $\mathcal{G}$-module (over any field k). We provide first steps in the noncommutative differential geometry of $\mathcal{A}(k,X,r)$ 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 ^− 1 where τ is the flip map and (X,f) is another solution coming from X as a crossed $\mathcal{G}$-set.