Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation

Bai, Chengming; Guo, Li; Sheng, Yunhe; Tang, Rong

doi:10.1007/s00208-023-02592-z

Cited by 11 publications

(3 citation statements)

References 46 publications

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“…In 2017, Guarnieri and Vendramin introduced the concept of skew left braces [8] and proved that every skew left brace provides a non-involutive solution to the Yang-Baxter equation. Skew left braces have been widely applied in various branches of mathematics, including connections to regular subgroups [9], Hopf-Galois extensions [10], triply factorized groups [11], Garside theory [12,13], ring theory [14,15], flat manifolds [16], and pre-Lie algebras [17].…”

Section: Introductionmentioning

confidence: 99%

Rota–Baxter Operators on Skew Braces

Wang,

Zhang,

Zhang

2024

Mathematics

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In this paper, we introduce the concept of Rota–Baxter skew braces, and provide classifications of Rota–Baxter operators on various skew braces, such as (Z,+,∘) and (Z/(4),+,∘). We also present a necessary and sufficient condition for a skew brace to be a co-inverse skew brace. Additionally, we describe some constructions of Rota–Baxter quasiskew braces, and demonstrate that every Rota–Baxter skew brace can induce a quasigroup and a Rota–Baxter quasiskew brace.

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Section: Introductionmentioning

confidence: 99%

Rota–Baxter Operators on Skew Braces

Wang,

Zhang,

Zhang

2024

Mathematics

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“…A braided vector space is said to be of set-theoretical type if it is associated to a set-theoretical solution of the Yang-Baxter equation. The Yang-Baxter equation was first introduced in physics [42,10] and set-theoretical solutions are closely related to many algebraic structures such as Hopf algebras, Nichols algebras, racks, 2 SHI (skew) braces, (relative) Rota-Baxter groups and so on; see for example [3,32,17,18,12,8,9].…”

Section: Introductionmentioning

confidence: 99%

Multinomial expansion and Nichols algebras associated to non-degenerate involutive solutions of the Yang-Baxter equation

Shi

2023

Communications in Algebra

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Let (X, r) be any set-theoretical non-degenerate solution of the Yang-Baxter equation and (X, r) be the derived solution of (X, r). As for any braided vector space (W X,r , c) associated to (X, r), is it possible to find some braided vector space (W X,r , c) which is t-equivalent to (W X,r , c)? In case that (X, r) is a near-rack solution, we give a sufficient condition to make an affirmative answer to the question. Examples of t-equivalence are constructed, hence finite dimensional Nichols algebras are obtained. In particular, all finite dimensional Nichols algebras associated to involutive near-rack solutions are classified.

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“…Recently, these tools have been extended, by Guarnieri and Vendramin [37] and Rump [62], in order to tackle general bijective non-degenerate solutions. These algebraic structures have since been widely studied, see for example [3,4,8,10,28,46,48,[60][61][62] and there are strong and fruitful connections with other areas, such as for example Hopf algebras [65], knot theory [47], group rings [1], nil and nilpotent rings [63], and pre-Lie algebras [5,64].…”

Section: Introductionmentioning

confidence: 99%