Factorizations of skew braces

Abstract

We introduce strong left ideals of skew braces and prove that they produce non-trivial decomposition of set-theoretic solutions of the Yang-Baxter equation. We study factorization of skew left braces through strong left ideals and we prove analogs of Itô's theorem in the context of skew left braces. As a corollary, we obtain applications to the retractability problem of involutive non-degenerate solutions of the Yang-Baxter equation. Finally, we classify skew braces that contain no non-trivial proper ideals.2010 Mathematics Subject Classification. Primary:16T25; Secondary: 81R50.

From Braces to Hecke algebras & Quantum Groups

“…The theory of braces and skew braces has connections with numerous research areas, for example with group theory (Garside groups, regular subgroups, factorised groups-see for example [3,30,31,49]), algebraic number theory, Hopf-Galois extensions [2,47], non-commutative ring theory [46,40,41], Knot theory [39,42], Hopf algebras, quantum groups [16], universal algebra, groupoids [29], semi-braces [8], trusses [7] and Yang-Baxter maps. Moreover, skew braces are related to non-commutative physics, Yetter-Drinfield modules and Nichols algebras.…”

Radical and weight of skew braces and their applications to structure groups of solutions of the Yang-Baxter equation

“…We denote the commutator subgroup of the group (A, +) as [A, A] + . Note that if (A, +, •) is a skew left brace, then [A, A] + is a strong left ideal of A (see [19] for the definition of strong left ideals in skew braces).…”

“…For a skew left brace A, [19]). The following result can be understood as a bracetheoretic analog of Schur's theorem, which says that the derived subgroup [G, G] of a group G is finite provided Z(G) is of finite index in G; the converse is true if G is finitely generated.…”

Set-theoretic solutions of the Yang–Baxter equation associated to weak braces

“…Moreover, r is involutive if and only if S is a brace. In the last years, several results about skew braces have been provided in [6,11,12,20,23,25,28], just to recall a few.…”

“…) =(23) and (G, •) = (13) =(23) . In addition, the solutions r • and r • are described byλ • id 3 = λ • (12) = id Sym 3 λ • (23) = ((12) (13) (23)) λ • (123) = λ • (132) = λ • (13) = ((12) (23) (13)) ρ • id 3 = ρ • (123) = ρ • (132) = id Sym 3 ρ • (12) = ρ • (13) = ρ • (23) = ((13) (23)) ((123) (132))andλ • id 3 = λ • (123) = λ • (132) = id Sym 3 λ • (12) = λ • (13) = λ • (23) = ((13) (23)) ((123) (132)) ρ • id 3 = ρ • (12) = id Sym 3 ρ • (23) = ρ •(123) = ((12) (13) (23)) ρ • (13) = ρ • (132) = ((12) (23) (…”