nLab classical r-matrix

Idea

An $R$-matrix is a solution to a Yang-Baxter equation:

A classical r-matrix is a solution of classical Yang-Baxter equation.

(A quantum R-matrix is a solution to the quantum Yang-Baxter equation.)

Related entries

• Lie bialgebra

• Poisson Lie group

Yang-Baxter equations

• classical Yang-Baxter equation

• quantum Yang-Baxter equation

• dynamical quantum Yang-Baxter equation

• associative Yang-Baxter equation

• infinitesimal braid relation

• set theoretic Yang-Baxter equation

• quasitriangular bialgebra

References

• M. A. Semenov-Tyan-Shanskii, Classical r-matrices and quantization, J. Math. Sciences 31 6 (1985) 3411-3416, [doi; Classical r-matrices and the method of orbits, J. Math. Sciences 28 4 (1985) 513-524 [doi:10.1007/BF02104981]

• 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]

• Pavel Etingof, Alexander Varchenko: Geometry and classification of solutions of the Classical Dynamical Yang-Baxter Equation, Comm Math Phys 192 (1998) 77–120 [arXiv:q-alg/9703040, doi;10.1007/s002200050292]

In relation to infinitesimal braid relations and braid representations:

• Toshitake Kohno: Linear representations of braid groups and classical Yang-Baxter equations, in: Joan S. Birman, Anatoly Libgober (eds.) Braids, Cont. Math. 78 (1988) 339-363 [doi:10.1090/conm/078]

  (in relation to the classical Yang-Baxter equation)

See also:

• Victor Mouquin, The Fock-Rosly Poisson structure as defined by a quasitriangular r-matrix, arxiv/1610.01782