# nLab quantum Yang-Baxter matrix

Equation

Quantum Yang-Baxter equation has been proposed by Baxter in the context of a particular model of statistical mechanics (6-vertex model ??) and called star-triangle relation. Later it has been generalized and axiomatized to a number of contexts: it is most notably satisfied by the universal R-element in a quasitriangular Hopf algebra. In some context it is equivalent to a braid relation for certain transposed matrix. The solution to a quantum Yang-Baxter equation for matrices is called the quantum Yang-Baxter matrix or quantum R-matrix; some solutions to quantum Yang-Baxter equation have good limits in classical mechanics which are classical r-matrices, and the latter satisfy the classical Yang-Baxter equation.

## Equation

With multiplicative spectral parameter, the equation reads

$R_{12} (u) R_{13} (uv) R_{23} (v) = R_{23}(v) R_{13}(uv) R_{12}(u)$

where the subscripts indicate which tensor factors are being utilized.

• A. U. Klymik, K. Schmuedgen, Quantum groups and their representations, Springer 1997.
• V. Chari, A. Pressley, A guide to quantum groups, Cambridge Univ. Press 1994
• V. G. Drinfel'd, Quantum groups, Proceedings of the International Congress of Mathematicians 1986, Vol. 1, 798–820, AMS 1987, djvu:1.3M, pdf:2.5M
• D. Gurevich, V. Rubtsov, Yang-Baxter equation and deformation of associative and Lie algebras, in: Quantum Groups, Springer Lecture Notes in Math. 1510 (1992) 47-55,doi
• P. P. Kulish, N. Yu. Reshetikhin, E. K. Sklyanin, Yang-Baxter equation and representation theory: I, Lett. Math. Phys. 5:5 (1981), 393-403, doi

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