nLab quantum Yang-Baxter matrix




The 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.


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) 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.

Yang-Baxter equations


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