Quasi-bialgebras from set-theoretic type solutions of the Yang-Baxter equation
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 [\textit{A. Doikou} and \textit{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 [\textit{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 [\textit{A. Doikou}, J. Phys. A, Math. Theor. 54, No. 41, Article ID 415201, 21 p. (2021; Zbl 07654521)].
scientific article; zbMATH DE number 7583939
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