# Yang–Baxter equation

Yang-Baxter equations (or star-triangle equations) are mathematical physics equations with importance in statistical mechanics and the theory of quantum groups.

## Quotes

* The origins of neutrino masses is one of the biggest mysteries in modern physics since they are beyond the realm of the Standard Model. As massive particles, neutrinos undergo flavor oscillations throughout their propagation. In this paper we show that when a neutrino oscillates from a flavor state α to a flavor state β, it follows three possible paths consistent with the Quantum Yang- Baxter Equations. These trajectories define the transition probabilities of the oscillations. Moreover, we define a probability matrix for flavor transitions consistent with the Quantum Yang-Baxter Equations, and estimate the values of the three neutrino mass eigenvalues within the framework of the triangular formulation.
** Ivan Arraut and Enrique Arrieta-Diaz. "Neutrino oscillations from the perspective of the quantum Yang-Baxter equations." arXiv preprint arXiv:2409.00560 (2024).https://arxiv.org/abs/2409.00560
* It has been known for some time that the Yang-Baxter equations can be solved using elliptic curves. More recently it was discovered … that the YBE for the N state chiral Potts model could be solved using special curves of genus (N – 1)$^2$.
** Michael Francis Atiyah and Michael Kevin Murray, (June 1990)"Monopoles and Yang-Baxter equations". Twistor Newsletter (30): 10–13. archive for Twistor Newsletters, issue No. 26 (March 1986) – No. 45 (August 2000)
* About 40 years ago, in the study of quantum integrable systems … , in particular in the framework of the quantum inverse scattering method … , new algebraic structures arose, the generalizations of which were later called quantum groups … The Yang-Baxter equations became a unifying basis of all these investigations. The most important nontrivial examples of quantum groups are quantizations (or deformations) of ordinary classical Lie groups and algebras (more precisely, one considers the deformations of the algebra of functions of a Lie group and the universal enveloping of a Lie algebra). The quantization is accompanied by the introduction of an additional parameter q (the deformation parameter), which plays a role analogous to the role of Planck’s constant in quantum mechanics. In the limit q → 1, the quantum groups and algebras go over into the classical ones.
** Alexey Petrovich Isaev, (2022). "Lectures on quantum groups and Yang-Baxter equations". arXiv preprint arXiv:2206.08902. (181 pp.; quote from p. 4)
* At an early stage the Yang-Baxter equation (YBE) appeared in several different guises in the literature, and sometimes its solutions have preceded the equation. One can trace basically three streams of ideas from which YBE has emerged: the Bethe Ansatz, commuting transfer matrices in statistical mechanics, and factorizable S matrices in field theory.
** Michio Jimbo, ed (1990). "Pioneering Works". Yang-Baxter Equation in Integrable Systems. Volume 10 of Advanced series in mathematical physics. World Scientific. pp. 5–6. ISBN 9810201206. (quote from p. 5)

## See also

* Yang Chen-Ning

## External links

Encyclopedic article on Yang–Baxter equation on Wikipedia