Tag - Other non-associative rings

We investigate a special class of solutions of the set-theoretic Yang-Baxter equation, called Frobenius-Separability (FS) type solutions. In particular, we show that the category of solutions of the set-theoretic Yang-Baxter equation of Frobenius-Separability (FS) type is equivalent to the category of pointed Kimura semigroups. As applications, all involutive, idempotent, non-degenerate, surjective, finite order, unitary, or indecomposable solutions of FS type are classified. For instance, if |X| = n, then the number of isomorphism classes of all such solutions on X that are (a) left non-degenerate, (b) bijective, (c) unitary or (d) indecomposable and left-non-degenerate is: (a) the Davis number d(n), (b) Σ_m|n p(m), where p(m) is the Euler partition number, (c) τ(n) + Σ_d|n[d/2], where τ is the number of divisors of n, or (d) the Harary number. The automorphism groups of such solutions can also be recovered as automorphism groups Aut(f) of sets X equipped with a single endo-function f : X → X. We describe all groups of the form Aut(f) as iterations of direct and (possibly infinite) wreath products of cyclic or full symmetric groups, characterize the abelian ones as products of cyclic groups, and produce examples of symmetry groups of FS solutions not of the form Aut(f).

Recall that a set-theoretic solution of the Yang-Baxter equation is a tuple (X,r), where X is a non-empty set and r: X × X → X × X a bijective map such that

(r × id_X) (id_X × r) (r × id_X) = (id_X × r) (r × id_X) (id_X × r),

where one denotes r(x,y)=(l_x(y), r_y(x)). Attention is often restricted to so-called non-degenerate solutions, i.e. l_x and r_y are bijective. We will call these solutions for short in the remainder of this abstract. To understand more general objects, it is an important technique to study 'minimal' objects and glue them together. For solutions both indecomposable and simple solutions fit the bill for being a minimal object. In this talk, we will report on recent work with I. Colazzo, E. Jespers and L. Kubat on simple solutions. In particular, we will discuss an extension of a result of M. Castelli that allows to identify whether a solution is simple, without having to know or calculate all smaller solutions. This method employs so-called skew braces, which were constructed to provide more examples of solutions, but also govern many properties of general solutions. In the latter part of the talk, we discuss the extension of a method to construct new indecomposable or simple solutions from old ones via cabling, originally introduced by V. Lebed, S. Ramirez, and L. Vendramin to unify the known results on indecomposability of solutions.

To each solution of the Yang-Baxter equation one may associate a quadratic algebra over a field, called the YB-algebra, encoding certain information about the solution. It is known that YB-algebras of finite non-degenerate solutions are (two-sided) Noetherian, PI and of finite Gelfand-Kirillov dimension. If the solution is additionally involutive then the corresponding YB-algebra shares many other properties with polynomial algebras in commuting variables (e.g., it is a Cohen-Macaulay domain of finite global dimension). The aim of this talk is to explain the intriguing relationship between ring-theoretical and homological properties of YB-algebras and properties of the corresponding solutions of the Yang-Baxter equation. The main focus is on when such algebras are Noetherian, (semi)prime and representable.