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.

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., 2024) in Introduction (first paragraphs)

We conjecture that, up to unit scalars and local unitary basis changes, all solutions are generated by these sources.

— Unitary Yang--Baxter Operators: Towards a Classification  (2608.16865 - Galindo et al., 17 Aug 2026) in Abstract; Conjecture 1.1, Section 1; Section 7, Definition 7.2

The following conjecture was formulated in Conjecture~1.1 as a refinement of one found in ; we restate it in the present terminology. For every unitary Yang--Baxter operator R and every n\geq2, the image of the associated braid group representation \rho_{R,n}:B_n\longrightarrow\mathrm U(V{ n}) is virtually abelian.

— Unitary Yang--Baxter Operators: Towards a Classification  (2608.16865 - Galindo et al., 17 Aug 2026) in Conjecture 1.3, Section 1

A conjecture for fixed Pauli directions

— Unitary Yang--Baxter Operators: Towards a Classification  (2608.16865 - Galindo et al., 17 Aug 2026) in Conjecture 9.1, Section 9.4.4

Even though the $FK_3$ Nichols algebra has no deformations (within the category of Nichols algebras), does its $R$-matrix have any non-trivial deformations?

— The $\mathrm{FK}_3$ knot polynomial  (2609.01506 - Garoufalidis et al., 1 Sep 2026) in Question 1, Section 6.1 (“Summary”)

They conjecture that, up to an overall phase and a homogeneous local change of basis, every such operator is generated from three sources: monomial solutions, group-type solutions, and solutions from twisted group-algebra towers.

— On the growth of operator entanglement in brickwork circuits with Yang--Baxter gates  (2609.05121 - Pozsgay, 4 Sep 2026) in Section 2, Subsection 2.7