Construction of quasi-linear left cycle sets

Abstract

In this paper, we produce a method to construct quasi-linear left cycle sets A with Rad (A) ⊆ Fix (A). Moreover, among these cycle sets, we give a complete description of those for which Fix (A) = Soc (A) and the underlying additive group is cyclic. Using such cycle sets, we obtain left non-degenerate involutive set-theoretic solutions of the Yang–Baxter equation which are different from those obtained in [P. Etingof, T. Schedler and A. Soloviev, Set-theoretical solutions to the quantum Yang–Baxter equation, Duke Math. J. 100 (1999) 169–209; P. Etingof, A. Soloviev and R. Guralnick, Indecomposable set-theoretical solutions to the quantum Yang–Baxter equation on a set with a prime number of elements, J. Algebra 242 (2001) 709–719].

Dynamical extensions of quasi-linear left cycle sets and the Yang–Baxter equation

“…Precisely, we prove that a quasi-linear left cycle set A has Rad(A) ⊆ Soc(A) if and only if its associated permutation is a unitary metahomomorphism. Moreover, among these left cycle sets, we give a complete description of those with Fix(A) = Soc(A), improving the results obtained in [3].…”

“…Another classical tool is the Fixator of A, denoted by Fix(A), that is always a subgroup of A contained in Soc(A), but in general is not an ideal. In [3], the second and the third author produced a method to construct quasi-linear left cycle sets with Rad(A) ⊆ Soc(A) and among these left cycle sets they give a complete description of those with Fix(A) = Soc(A) and cyclic underlying additive group.…”

On the indecomposable involutive set-theoretic solutions of the Yang-Baxter equation of prime-power size

“…Conversely, if (X, r) is a non-degenerate involutive solution and · the binary operation given by x · y := λ −1 x (y) for all x, y ∈ X, then (X, ·) is a non-degenerate cycle set. The existence of this bijective correspondence allows us to move the study of involutive non-degenerate solutions to non-degenerate cycle sets, as already made in [2,18,5,8,4,6,3,23,25,19].…”

“…In 1999 Etingof, Schedler and Soloviev [14] found, by computer calculations, all the indecomposable involutive non-degenerate solutions of cardinality at most eight; moreover, they showed that, up to isomorphism, the unique indecomposable involutive non-degenerate solution having a prime number p of elements is given by (Z/pZ, u) and u(x, y) := (y − 1, x + 1), for every x, y ∈ Z/pZ. In recent years, some authors used the links between involutive solutions and other algebraic structures (see for example [11,22,16,7,12]) to provide new descriptions of the indecomposable ones: in particular, in 2010 Chouraqui [11] give a characterization of indecomposable involutive non-degenerate solutions by Garside monoids, while A. Smoktunowicz and A.…”

About a question of Gateva-Ivanova and Cameron on square-free set-theoretic solutions of the Yang-Baxter equation

“…Conversly, if (X, r) is a non-degenerate solution, the binary operation · defined by x · y := λ −1 x (y) for all x, y ∈ X makes X into a left cycle set. The existence of this correspondence allows to move the study of involutive non degenerate solutions to left cycle sets, as recently done in [19,3,4,16,15,18,5,2], and clearly to translate in terms of left cycle set the classical concepts related to the non-degenerate involutive set-theoretic solutions. Therefore, a left cycle set is said to be square-free if the squaring-map q is the identity on X.…”