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

Jespers, Eric; Kubat, Łukasz; Antwerpen, Arne Van; Vendramin, L.

doi:10.1016/j.aim.2021.107767

Cited by 21 publications

(25 citation statements)

References 30 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Skew-braces on the other hand are used [23] to describe non-involutive set-theoretic solutions of the YBE, and they may be instrumental in identifying universal R-matrices. This rising research field has been particularly fertile and numerous relevant studies have been produced over the past several years (see for instance [2,3,6,7], [9]- [11], [18]- [20], [21,22], [23,25,26,30], [37]- [39]).…”

Section: Introductionmentioning

confidence: 99%

Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation

Doikou,

Ghionis,

Vlaar

2022

Preprint

View full text Add to dashboard Cite

We examine classes of quantum algebras emerging from involutive, non-degenerate set-theoretic solutions of the Yang-Baxter equation and their q-analogues. After providing some universal results on quasi-bialgebras and admissible Drinfeld twists we show that the quantum algebras produced from set-theoretic solutions and their q-analogues are in fact quasi-triangular quasi-bialgebras. Specific illustrative examples compatible with our generic findings are worked out. In the q-deformed case of set-theoretic solutions we also construct admissible Drinfeld twists similar to the set-theoretic ones, subject to certain extra constraints dictated by the q-deformation. These findings greatly generalize recent relevant results on set-theoretic solutions and their q-deformed analogues.

show abstract

Section: Introductionmentioning

confidence: 99%

Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation

Doikou,

Ghionis,

Vlaar

2022

Preprint

View full text Add to dashboard Cite

show abstract

“…In [1], Angiono, Galindo and Vendramin introduced the notion of Hopf braces, generalizing Rump's braces and their nonabelian generalizations, skew left braces [24,27,36,37], which are stemmed from group theory. All these algebraic objects have deep relations with the Yang-Baxter equation.…”

Section: 2mentioning

confidence: 99%

Post-Hopf algebras, relative Rota-Baxter operators and solutions of the Yang-Baxter equation

Li¹,

Sheng²,

Tang³

2022

Preprint

View full text Add to dashboard Cite

In this paper, first we introduce the notion of a post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman-Larsson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions of the Yang-Baxter equation. Then we introduce the notion of relative Rota-Baxter operators on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota-Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota-Baxter operator also induces a post-Hopf algebra. Then we show that relative Rota-Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota-Baxter operators give solutions of the Yang-Baxter equation in certain cocommutative Hopf algebras. Finally we characterize relative Rota-Baxter operators on Hopf algebras using relative Rota-Baxter operators on the Lie algebra of primitive elements, graphs and module bialgebra structures.

show abstract

“…We remark that, in fact, braces were considered by Kurosh in his lectures [26,Section 10] of 1969-1970 academic year. The subject received a tremendous attention of the mathematical community after the work of Rump; see [2,12,20,25,31,32] and the references therein. Interest in the study of set theoretic solutions of the quantum Yang-Baxter equations was intrigued by the paper [16] of Drinfeld, published in 1992.…”

Section: Introductionmentioning

confidence: 99%

Symmetric skew braces and brace systems

Bardakov¹,

Нещадим²,

Yadav³

2022

Preprint

View full text Add to dashboard Cite

For a skew left brace (G, •, •), the map λ : (G, •) → Aut (G, •), a → λa, where λa(b) = a −1 • (a • b) for all a, b ∈ G, is a group homomorphism. Then λ can also be viewed as a map from (G, •) to Aut (G, •), which, in general, may not be a homomorphism. A skew left brace will be called λ-anti-homomorphic (λ-homomorphic) if λ : (G, •) → Aut (G, •) is an antihomomorphism (a homomorphism). We mainly study such skew left braces. We device a method for constructing a class of binary operations on a given set so that the set with any two such operations constitute a λ-homomorphic symmetric skew brace. Most of the constructions of symmetric skew braces dealt with in the literature fall in the framework of our construction. We then carry out various such constructions on specific infinite sets.

show abstract

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

Cited by 21 publications

References 30 publications

Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation

Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation

Post-Hopf algebras, relative Rota-Baxter operators and solutions of the Yang-Baxter equation

Symmetric skew braces and brace systems

Contact Info

Product

Resources

About