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

TL;DR

This paper explores quantum algebras derived from set-theoretic solutions of the Yang-Baxter equation, establishing their structure as quasi-triangular quasi-bialgebras and extending recent results with explicit examples and q-deformations.

Contribution

It provides universal results on quasi-bialgebras from set-theoretic solutions and constructs admissible Drinfeld twists in the q-deformed setting, generalizing prior work.

Findings

Quantum algebras from set-theoretic solutions are quasi-triangular quasi-bialgebras.

Explicit examples illustrating the theoretical framework are provided.

Admissible Drinfeld twists are constructed for q-deformed solutions under additional constraints.

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 generalise recent relevant results on set theoretic solutions and their q-deformed analogues.