Theoretical and Mathematical Physics

, Volume 176, Issue 3, pp 1156–1162 | Cite as

Classification of constant solutions of the associative Yang-Baxter equation on Mat3



We find all nonequivalent constant solutions of the classical associative Yang-Baxter equation for Mat 3 . New examples found in the classification yield the corresponding Poisson brackets on traces, double Poisson brackets on a free associative algebra with three generators, and anti-Frobenius associative algebras.


associative Yang-Baxter equation constant solution classification 


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  1. 1.
    G.-C. Rota, “Baxter operators, an introduction,” in: Gian-Carlo Rota on Combinatorics (J. P. S. Kung, eds.) (Contemp. Mathematicians, Vol. 57, No. 6), Birkhäuser, Boston, Mass. (1995), pp. 504–512.Google Scholar
  2. 2.
    T. Schedler, “Poisson algebras and Yang-Baxter equations,” in: Advances in Quantum Computation (Contemp. Math., Vol. 482), Amer. Math. Soc., Providence, R. I. (2009), pp. 91–106.CrossRefGoogle Scholar
  3. 3.
    M. Van den Bergh, Trans. Amer. Math. Soc., 360, 5711–5769 (2008).MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    A. V. Odesskii, V. N. Rubtsov, and V. V. Sokolov, “Double Poisson brackets on free associative algebras,” Preprint No. 52, Max-Planck-Institut für Mathematik, Bonn (2012); arXiv:1208.2935v1 [nlin.SI] (2012).Google Scholar
  5. 5.
    M. Aguiar, J. Algebra, 244, 492–532 (2001).MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    A. V. Odesskii, V. N. Rubtsov, and V. V. Sokolov, Theor. Math. Phys., 171, 442–447 (2012).CrossRefMATHGoogle Scholar
  7. 7.
    A. V. Mikhailov and V. V. Sokolov, Commun. Math. Phys., 211, 231–251 (2000).MathSciNetADSCrossRefMATHGoogle Scholar
  8. 8.
    C. Procesi, Adv. Math., 19, 306–381 (1976).MathSciNetCrossRefMATHGoogle Scholar
  9. 9.
    A. G. Élashvili,, Funct. Anal. Appl., 16, 326–328 (1982).CrossRefGoogle Scholar

Copyright information

© Pleiades Publishing, Ltd. 2013

Authors and Affiliations

  1. 1.Landau Institute for Theoretical PhysicsMoscowRussia

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