Theoretical and Mathematical Physics

, Volume 146, Issue 2, pp 159–169 | Cite as

Compatible Lie Brackets and the Yang-Baxter Equation

  • I. Z. Golubchik
  • V. V. Sokolov
Article

Abstract

We show that any pair of compatible Lie brackets with a common invariant form produces a nonconstant solution of the classical Yang-Baxter equation. We describe the corresponding Poisson brackets, Manin triples, and Lie bialgebras. It turns out that all bialgebras associated with the solutions found by Belavin and Drinfeld are isomorphic to some bialgebras generated by our solutions. For any compatible pair, we construct a double with a common invariant form and find the corresponding solution of the quantum Yang-Baxter equation for this double.

Keywords

Yang-Baxter equation Lie bialgebra Manin triple 

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Copyright information

© Springer Science+Business Media, Inc. 2006

Authors and Affiliations

  • I. Z. Golubchik
    • 1
  • V. V. Sokolov
    • 2
  1. 1.Bashkir State Pedagogical UniversityUfaRussia
  2. 2.Landau Institute for Theoretical PhysicsRASMoscowRussia

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