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TMF, 2006, Volume 146, Number 2, Pages 195–207 (Mi tmf2031)  

This article is cited in 12 scientific papers (total in 12 papers)

Compatible Lie Brackets and the Yang–Baxter Equation

I. Z. Golubchika, V. V. Sokolovb

a Bashkir State Pedagogical University
b L. D. Landau Institute for Theoretical Physics, Russian Academy of Sciences

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

DOI: https://doi.org/10.4213/tmf2031

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English version:
Theoretical and Mathematical Physics, 2006, 146:2, 159–169

Bibliographic databases:

Received: 18.03.2005

Citation: I. Z. Golubchik, V. V. Sokolov, “Compatible Lie Brackets and the Yang–Baxter Equation”, TMF, 146:2 (2006), 195–207; Theoret. and Math. Phys., 146:2 (2006), 159–169

Citation in format AMSBIB
\by I.~Z.~Golubchik, V.~V.~Sokolov
\paper Compatible Lie Brackets and the Yang--Baxter Equation
\jour TMF
\yr 2006
\vol 146
\issue 2
\pages 195--207
\jour Theoret. and Math. Phys.
\yr 2006
\vol 146
\issue 2
\pages 159--169

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    This publication is cited in the following articles:
    1. Odesskii AV, Sokolov VV, “Integrable matrix equations related to pairs of compatible associative algebras”, Journal of Physics A-Mathematical and General, 39:40 (2006), 12447–12456  crossref  mathscinet  zmath  isi  scopus  scopus
    2. Odesskii, AV, “Compatible Lie brackets related to elliptic curve”, Journal of Mathematical Physics, 47:1 (2006), 013506  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    3. Odesskii A., Sokolov V., “Algebraic structures connected with pairs of compatible associative algebras”, International Mathematics Research Notices, 2006, 43734  mathscinet  zmath  isi  elib
    4. Odesskii A, Sokolov V, “Pairs of compatible associative algebras, classical Yang–Baxter equation and quiver representations”, Communications in Mathematical Physics, 278:1 (2008), 83–99  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    5. Bolsinov, AV, “Bi-Hamiltonian structures and singularities of integrable systems”, Regular & Chaotic Dynamics, 14:4–5 (2009), 431  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    6. Zhang Y., “Homotopy Transfer Theorem for Linearly Compatible Di-Algebras”, J. Homotopy Relat. Struct., 8:1 (2013), 141–150  crossref  mathscinet  zmath  isi  elib  scopus  scopus
    7. Zhang Y., Bai Ch., Guo L., “The Category and Operad of Matching Dialgebras”, Appl. Categ. Struct., 21:6 (2013), 851–865  crossref  mathscinet  zmath  isi  scopus  scopus
    8. Zhang Yong, Bai ChengMing, Guo Li, “Totally Compatible Associative and Lie Dialgebras, Tridendriform Algebras and Postlie Algebras”, Sci. China-Math., 57:2 (2014), 259–273  crossref  mathscinet  zmath  isi  scopus  scopus
    9. Pumei Zhang, “Algebraic Properties of Compatible Poisson Brackets”, Regul. Chaotic Dyn., 19:3 (2014), 267–288  mathnet  crossref  mathscinet
    10. Dobrogowska A., “R-Matrix, Lax pair, and Multiparameter Decompositions of Lie Algebras”, J. Math. Phys., 56:11 (2015), 113508  crossref  mathscinet  zmath  adsnasa  isi  scopus  scopus
    11. Wu M., “Double Constructions of Compatible Associative Algebras”, Algebr. Colloq., 26:3 (2019), 479–494  crossref  mathscinet  isi
    12. Liu J., Bai Ch., Sheng Yu., “Noncommutative Poisson Bialgebras”, J. Algebra, 556 (2020), 35–66  crossref  mathscinet  isi
  • Теоретическая и математическая физика Theoretical and Mathematical Physics
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