nLab quantum Yang-Baxter equation

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Context

Algebra

Quantum systems

quantum logic


quantum physics


quantum probability theoryobservables and states


quantum information


quantum computation

quantum algorithms:

Contents

Idea

Given a monoidal category CC with tensor product \otimes, and an object VV in CC, a quantum Yang-Baxter operator is a morphism of the form

R:VVVV R \,\colon\, V\otimes V\to V\otimes V

which satisfies the following quantum Yang-Baxter equation in VVVV\otimes V\otimes V

R 12R 13R 23=R 23R 13R 12, R_{12} \, R_{13} \, R_{23} \;=\; R_{23} \, R_{13} \, R_{12} \,,

where the subscripts indicate which tensor factors are being utilized, for instance R 12=Rid VVVVR_{12} = R\otimes id_V\in V\otimes V\otimes V.

This equation is in particular satisfied by the component VV\mathcal{R}_{V V} at VV of any braiding \mathcal{R} on CC.

Typical categories where the equation is considered are

  1. the category of vector spaces when the solutions are called RR-matrices (or quantum Yang-Baxter matrices),

  2. categories of representations of quantum groups; often (a completion of) a quantum group GG itself has a particular element \mathcal{R}, called a universal \mathcal{R}-element, satisfying axioms (quasitriangularity) ensuring that its image in every representation satisfies qYBE

  3. the category of sets where one spekas about set theoretic solutions of Yang-Baxter equation.

Remark

(Historical motivation)

The quantum Yang-Baxter equation has been proposed by Baxter in the context of a particular model of statistical mechanics (the 8-vertex model) and called star-triangle relation [Baxter (1978), Baxter (1982)].

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. 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 spectral parameter

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} (u v) \, R_{23} (v) \;=\; R_{23}(v) \, R_{13}(u v) \, R_{12}(u)

where the subscripts indicate which tensor factors are being utilized.

Yang-Baxter equations

References

Original texts:

Introductions:

  • Michio Jimbo, Introduction to the Yang-Baxter Equation, Int. J. Modern Physics A 4 15 (1989) 3759-3777 [doi:10.1142/S0217751X89001503]

    reprinted in: Braid Group, Knot Theory and Statistical Mechanics, Advanced Series in Mathematical Physics 9, World Scientific (1991) [doi:10.1142/0796]

See also:

Further discussion in the context of quantum groups:

  • 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, Nicolai Reshetikhin, E. K. Sklyanin, Yang-Baxter equation and representation theory: I, Lett. Math. Phys. 5:5 (1981), 393-403, doi

Discussion in the context of braid group representations:

Last revised on December 29, 2022 at 12:43:42. See the history of this page for a list of all contributions to it.