Semi-braces and the Yang–Baxter equation

Abstract

The main aim of this paper is to provide set-theoretical solutions of the Yang-Baxter equation that are not necessarily bijective, among these new idempotent ones. In the specific, we draw on both to the classical theory of inverse semigroups and to that of the most recently studied braces, to give a new research perspective to the open problem of finding solutions. Namely, we have recourse to a new structure, the inverse semi-brace, that is a triple (S, +, •) with (S, +) a semigroup and (S, •) an inverse semigroup satisfying the relation a (b + c) = ab + a a −1 + c , for all a, b, c ∈ S, where a −1 is the inverse of a in (S, •). In particular, we give several constructions of inverse semi-braces which allow for obtaining solutions that are different from those until known. Keywords Quantum Yang-Baxter equation • set-theoretical solution • inverse semigroups • brace • semi-brace • skew brace • asymmetric product MSC 2020 16T25 81R50 16Y99 16N20 20M18 r : S × S −→ S × S satisfying the relation (r × id S) (id S ×r) (r × id S) = (id S ×r) (r × id S) (id S ×r) A PREPRINT is said to be a set-theoretical solution of the Yang-Baxter equation, or briefly a solution. The map r is usually written as r(x, y) = (λ x (y), ρ y (x)), with λ x and ρ y maps from S into itself, for all x, y ∈ S. One says that a solution r is left non-degenerate if λ x is bijective, for every x ∈ S, right non-degenerate if ρ y is bijective, for every y ∈ S, and non-degenerate if r is both left and right non-degenerate. If r is neither left nor right non-degenerate, then it is called degenerate. Determining all the solutions is still an open problem and it has drawn the attention of several mathematicians. A large number of works related to this topic has been produced in recent years, actually. The milestones are the papers by Etingof, Schedler and Soloviev [28], Gateva-Ivanova and Van den Bergh [30], Lu, Yan, and Zhu [38], and Soloviev [53], where a greater attention has been posed on non-degenerate bijective solutions. Subsequently, involutive solutions have been profusely investigated by many authors, as mostly illustrated into details in the introduction of the paper by Cedó, Jespers, and Okniński [21]. The most used approach is based on left braces, algebraic structures introduced by Rump [47] that include the Jacobson radical rings. In particular, such structures are involved for obtaining non-degenerate solutions which are also involutive, i.e., r 2 = id. In this way, Rump traced a novel research direction and later fruitful results on these kind of solutions appeared, as one can see in the survey by Cedó [19], along the references therein. To classify involutive solutions, Rump [46] also involved another algebraic structure, that is the left cycle set. Interesting contributions in this framework have been obtained, for example, see [48,

Skew lattices and set-theoretic solutions of the Yang-Baxter equation

“…Using absorption and regularity (2) Thus, the class of weak distributive solutions is defined by the identity Proof. Denote by S L and S R the left and the right factor of S, respectively.…”

“…Later braces were generalized by Guarnieri and Vendramin [12] to study non-degenerate solutions that are not necessarily involutive. Catino, Colazzo, and Stefanelli [2], and Jespers and Van Antwerpen [14] introduced (left cancellative) left semi-braces to deal with solutions that are not necessarily non-degenerate or solutions that are idempotent or cubic. Some of these solutions are degenerate.…”

The matched product of set-theoretical solutions of the Yang-Baxter equation

“…Lebed in [23] drew attention on idempotent solutions that, although of little interest in physics, provide a tool for dealing with very different algebraic structures ranging from free (commutative) monoids to factorizable monoids, and from distributive lattices to Young tableaux and plactic monoids. Catino, Colazzo, and Stefanelli [6], and Jespers and Van Antwerpen [34] introduced the algebraic structure called semi-brace to deal with solutions that are not necessarily non-degenerate or that are idempotent. Given a set X, a solution r is said to be idempotent if r 2 = r. In [8] the authors proved that the matched product of solutions, a novel construction technique for solutions of the Yang-Baxter equation introduced in [7], is a unifying tool for treating solutions with finite order, that includes involutive and idempotent solutions as particular cases.…”

The Matched Product of the Solutions to the Yang–Baxter Equation of Finite Order

“…Namely, given a set X, a solution r is said to be idempotent if r 2 = r. The question arises whether there is an algebraic structure similar to the brace structure useful for studying solutions not necessarily bijective. In [9], we gave an initial answer to the question by introducing semi-braces. A left (cancellative) semi-brace is a set B with two operations + and • such that (B, +) is a left cancellative semigroup, (B, •) is a group and a • (b + c) = a • b + a • (a − + c) holds for all a, b, c ∈ B, where a − denotes the inverse of a in (B, •).…”