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General solutions to the quantum Yang–Baxter equation

Determine general methods to construct or classify solutions of the quantum Yang–Baxter equation (QYBE), for which the overall search for solutions remains unresolved.

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Background

The paper recalls that the Yang–Baxter equation was introduced by Yang and Baxter, and that substantial attention has been devoted to the quantum Yang–Baxter equation because of its central role in quantum integrable systems.

Due to the complexity of the quantum case, Drinfel'd proposed studying simplified set-theoretic solutions; this paper focuses on reflections for set-theoretic solutions. However, the broader problem of finding solutions to the quantum Yang–Baxter equation itself is acknowledged as open.

References

Much of the attention has been directed to the so-called quantum Yang-Baxter equation, which is fundamental in mathematical physics since it occupies a central role in the theory of quantum integrable systems . The search for solutions to this equation has attracted numerous authors, and it is still an open problem.

— Reflections to set-theoretic solutions of the Yang-Baxter equation (2405.19105 - Albano et al., 29 May 2024) in Introduction (first paragraphs)