Quiver-theoretical approach to dynamical Yang–Baxter maps

Matsumoto, Diogo Kendy; Shimizu, Kenichi

doi:10.1016/j.jalgebra.2018.04.003

Cited by 11 publications

(13 citation statements)

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“…In this regard, we highlight that, recently, in [39] it has been provided an algorithm for the enumeration of the inverse semigroups of order n (up to isomorphism). Furthermore, these maps are added to the class of idempotent solutions which are known until now [37,40,26,15,17].…”

Section: A Preprintmentioning

confidence: 99%

Semi-braces and the Yang–Baxter equation

2017

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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,

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Section: A Preprintmentioning

confidence: 99%

Semi-braces and the Yang–Baxter equation

2017

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“…In this section, we induce a left bialgebroid A(w σ ) in Subsection 3.2 from the settings of the left bialgebroid A σ in Subsection 3.1, and construct a left bialgebroid homomorphism Φ from A(w σ ) to A σ . This is a generalization of [12].…”

Section: Left Bialgebroid Homomorphism φmentioning

confidence: 92%

“…There is an interesting relation between Hayashi's generalization and Shibukawa-Takeuchi's generalization of the FRT construction. If the parameter set Λ of a DYBM σ is finite, the left bialgebroid A σ becomes a weak bialgebra since the base algebra M Λ (K) consisting of maps from Λ to a field K is a Frobeniusseparable algebra over K. Matsumoto and Shimizu [12] showed that a DYBM σ gives birth to a solution w σ to the quiver-theoretical YBE and gave a weak bialgebra homomorphism Φ from A(w σ ) to A σ .…”

Section: Introductionmentioning

confidence: 99%

Two FRT bialgebroids and their relations

Otsuto

2020

Preprint

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“…Selain itu, Agata Smoktunowicz dalam [10] dan [11] juga memberikan solusi alternatif dari sudut pandang teori himpunan. Selanjutnya, solusi dengan pendekatan teori himpunan juga diberikan oleh Castelli, et.al dalam papernya [12] dan Matsumoto & Shimizu dalam [13]. Solusi yang diperumum diberikan oleh [14] dengan menggunakan struktur aljabar baru yang disebut dengan brace.…”

Section: Gambarunclassified

Konstruksi Brace Dua Sisi Dengan Menggunakan Ring Jacobson

Prasetyo

Melati

2020

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Quiver-theoretical approach to dynamical Yang–Baxter maps

Cited by 11 publications

References 20 publications

Semi-braces and the Yang–Baxter equation

Semi-braces and the Yang–Baxter equation

Two FRT bialgebroids and their relations

Konstruksi Brace Dua Sisi Dengan Menggunakan Ring Jacobson

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