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  Solutions of the Yang-Baxter equation and the Tetrahedron equation

+ 5 like - 0 dislike

Is there a systematic way to find solutions of the Yang-Baxter equation independent of the context (statistical physics in 2d or braids etc) ? And if yes can someone please a) sketch the procedure, b) provide some reference, c) comment on possible generalization to the tetrahedron equations?

As far as b) is concerned I have only seen some solutions in terms of statistical physics (e.g. Van Nieuwenhuizen notes on QFT). So, I would really find extremely useful some reference on the subject.

Thanks a lot in advance.

asked Oct 5, 2014 in Theoretical Physics by conformal_gk (3,625 points) [ revision history ]
edited Oct 6, 2014 by suresh

A good place to start would be Baxter's book "Exactly Solvable Models in Statistical Mechanics". See also the books by Michio Jimbo: Book1 and Book2 (this is a collection edited by Jimbo). Both approach it from the viewpoint of statistical mechanics and/or representation of affine algebras.

Highly recommend Baxter's book as suggested by Suresh. For methods of solution of the YB equation a good place to get started is "Braiding Operators are Universal Quantum Gates" by Lomonaco & Kauffman and references therein.

Another very useful reference is Zamolodchikov's classic work which you can find at http://projecteuclid.org/euclid.cmp/1103909139, where the physical origin of the tetrahedron equation is explained.

A systematic way to get solutions of Yang-Baxter equations is through braids, using ribbon fusion categories, which are the essential algebraic data contained in many of the knot/link/statistical mechanics approaches to the problem. See e.g. http://arxiv.org/pdf/1203.1063.pdf

1 Answer

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A method outlined by Grime entitled the hook fusion procedure provides a way to construct new solutions to the Young-Baxter equation. The construction differs from the standard approach of decomposing the Young diagram into its rows and columns by instead using 'hooks.' The thesis is here: http://arxiv.org/abs/0708.1109.

answered Jan 15, 2015 by anonymous [ revision history ]
edited Jan 15, 2015 by Dilaton

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