Abstract. We study set-theoretic solutions (X, r) of the Yang-Baxter equations on a set X in terms of the induced left and right actions of X on itself. We give a characterization of involutive square-free solutions in terms of cyclicity conditions. We characterise general solutions in terms of an induced matched pair of unital semigroups S(X, r) and construct (S, rS) from the matched pair. Finally, we study extensions of solutions in terms of matched pairs of their associated semigroups. We also prove several general results about matched pairs of unital semigroups of the required type, including iterated products S ⊲ ⊳ S ⊲ ⊳ S underlying the proof that rS is a solution, and extensions (S ⊲ ⊳ T, rS⊲⊳T). Examples include a general ‘double ’ construction (S ⊲ ⊳ S, rS⊲⊳S) and some concrete extensions, their actions and graphs based on small sets. 1.

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 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 the 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 −1 where τ is the flip map and (X, f) is another solution coming from X as a crossed G-set. 1.

of height one and a class of Noetherian

Abstract A review of some recent results on the dynamical theory of the Yang-Baxter maps (also known as set-theoretical solutions to the quantum Yang-Baxter equation) is given. The central question is the integrability of the transfer dynamics. The relations with matrix factorisations, matrix KdV solitons, Poisson Lie groups, geometric crystals and tropical combinatorics are discussed and demonstrated on several concrete examples.

presented algebras and groups defined by permutation relations ∗.

Abstract. We extend our recent work on set-theoretic solutions of the Yang-Baxter or braid relations with new results about their automorphism groups, strong twisted unions of solutions and multipermutation solutions. We introduce and study graphs of solutions and use our graphical methods for the computation of solutions of finite order and their automorphisms. Results include a detailed study of solutions of multipermutation level 2. 1.

Abstract. Let k be a field and X be a set of n elements. We introduce and study a class of quadratic k-algebras called quantum binomial algebras. Our main result shows that such an algebra A defines a solution of the classical Yang-Baxter equation (YBE), if and only if its Koszul dual A! is Frobenius of dimension n, with a regular socle and for each x, y ∈ X an equality of the type xyy = αzzt, where α ∈ k \ {0}, and z, t ∈ X is satisfied in A. We prove the equivalence of the notions a binomial skew polynomial ring and a binomial solution of YBE. This implies that the Yang-Baxter algebra of such a solution is of Poincaré-Birkhoff-Witt type, and possesses a number of other nice properties such as being Koszul, Noetherian, and an Artin-Schelter regular domain.

Abstract. We show the intimate connection between various mathematical notions that are currently under active investigation: a class of Garside monoids, with a “nice ” Garside element, certain monoids S with quadratic relations, whose monoidal algebra A = kS has a Frobenius Koszul dual A! with regular socle, the monoids of skew-polynomial type (or equivalently, binomial skew-polynomial rings) which were introduced and studied by the author and in 1995 provided a new class of Noetherian Artin-Schelter regular domains, and the square-free set-theoretic solutions of the Yang-Baxter equation. There is a beautiful symmetry in these objects due to their nice combinatorial and algebraic properties. 1.