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Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation. (English) Zbl 1517.16028

There is an interest in studying classes of quantum algebras arising from set-theoretic solutions of the Yang-Baxter equation. The motivations for deepening this investigation lie in the paper [A. Doikou and A. Smoktunowicz, Lett. Math. Phys. 111, No. 4, Paper No. 105, 40 p. (2021; Zbl 1486.16039)].
In the paper under review, the authors investigate algebras coming from involutive and non-degenerate solutions and their \(q\)-analogues. It is shown that they are quasi-triangular quasi-bialgebras. To get this result, they provide some universal results on quasi-bialgebras and admissible Drinfeld twists. In fact, the property of being quasi-triangular (quasi-)bialgebra is preserved by twisting (see [V. G. Drinfel’d, Sov. Math., Dokl. 32, 256–258 (1985; Zbl 0588.17015); translation from Dokl. Akad. Nauk SSSR 283, 1060–1064 (1985)]). Moreover, they make use of some first results on the admissible Drinfeld twist for involutive set-theoretic solution already derived in [A. Doikou, J. Phys. A, Math. Theor. 54, No. 41, Article ID 415201, 21 p. (2021; Zbl 1519.16027)].

MSC:

16T20 Ring-theoretic aspects of quantum groups
16T25 Yang-Baxter equations
17B37 Quantum groups (quantized enveloping algebras) and related deformations
20G42 Quantum groups (quantized function algebras) and their representations
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