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

Anastasia Doikou; Alexandros Ghionis; Bart Vlaar

doi:10.1007/s11005-022-01572-9

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

Anastasia Doikou, Alexandros Ghionis, Bart Vlaar

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Abstract

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.

Original language	English
Article number	78
Journal	Letters in Mathematical Physics
Volume	112
Issue number	4
DOIs	https://doi.org/10.1007/s11005-022-01572-9
Publication status	Published - 8 Aug 2022

Keywords

Drinfeld twists
Quasi-bialgebras
Set-theoretic Yang-Baxter equation

ASJC Scopus subject areas

Statistical and Nonlinear Physics
Mathematical Physics

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10.1007/s11005-022-01572-9Licence: CC BY

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title = "Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation",

abstract = "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.",

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author = "Anastasia Doikou and Alexandros Ghionis and Bart Vlaar",

note = "Funding Information: We are grateful to A. Smoktunowicz and B. Rybolowicz for useful discussions. Support from the EPSRC research grants EP/V008129/1 and EP/R009465/1 is acknowledged. A.G. acknowledges support from Heriot-Watt University via a James Watt scholarship. Publisher Copyright: {\textcopyright} 2022, The Author(s).",

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AU - Doikou, Anastasia

AU - Ghionis, Alexandros

AU - Vlaar, Bart

N1 - Funding Information: We are grateful to A. Smoktunowicz and B. Rybolowicz for useful discussions. Support from the EPSRC research grants EP/V008129/1 and EP/R009465/1 is acknowledged. A.G. acknowledges support from Heriot-Watt University via a James Watt scholarship. Publisher Copyright: © 2022, The Author(s).

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N2 - 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.

AB - 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.

KW - Drinfeld twists

KW - Quasi-bialgebras

KW - Set-theoretic Yang-Baxter equation

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DO - 10.1007/s11005-022-01572-9

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JO - Letters in Mathematical Physics

JF - Letters in Mathematical Physics

IS - 4

M1 - 78

ER -

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

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Keywords

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