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Teoreticheskaya i Matematicheskaya Fizika, 2006, Volume 146, Number 2, Pages 195–207
DOI: https://doi.org/10.4213/tmf2031 (Mi tmf2031) 		 
This article is cited in 28 scientific papers (total in 28 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
Full-text PDF (188 kB) Citations (28)
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DOI: https://doi.org/10.4213/tmf2031
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.
Received: 18.03.2005
English version:
Theoretical and Mathematical Physics, 2006, Volume 146, Issue 2, Pages 159–169
DOI: https://doi.org/10.1007/s11232-006-0016-6
Bibliographic databases:
Language: Russian
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
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    • This publication is cited in the following 28 articles:
    1. Taoufik Chtioui, Ripan Saha, “On deformation cohomology of compatible Hom-associative algebras”, Afr. Mat., 36:1 (2025)  crossref
    2. Manuel Ladra, Bernardo Leite da Cunha, Samuel A. Lopes, “A classification of nilpotent compatible Lie algebras”, Rend. Circ. Mat. Palermo, II. Ser, 74:1 (2025)  crossref
    3. Abdenacer Makhlouf, Ripan Saha, “On compatible Leibniz algebras”, J. Algebra Appl., 24:04 (2025)  crossref
    4. 芸文 宋, “Compatiable Rota-Baxter Jordan Algebra”, PM, 15:01 (2025), 282  crossref
    5. Taoufik Chtioui, Apurba Das, Sami Mabrouk, “(Co)homology of compatible associative algebras”, Communications in Algebra, 52:2 (2024), 582  crossref
    6. Xinyue Wang, Yao Ma, Liangyun Chen, “Cohomology and Deformations of Compatible Lie Triple Systems”, Mediterr. J. Math., 21:2 (2024)  crossref
    7. Shuai Hou, Yunhe Sheng, Yanqiu Zhou, “Deformations, cohomologies and abelian extensions of compatible 3-Lie algebras”, Journal of Geometry and Physics, 202 (2024), 105218  crossref
    8. Shanshan Liu, Liangyun Chen, “Deformations and abelian extensions of compatible pre-Lie algebras”, Journal of Geometry and Physics, 2024, 105335  crossref
    9. Hani Abdelwahab, Ivan Kaygorodov, Abdenacer Makhlouf, “The Algebraic and Geometric Classification of Compatible Pre-Lie Algebras”, SIGMA, 20 (2024), 107, 20 pp.  mathnet  crossref
    10. Apurba Das, “Cohomology and deformations of compatible Hom-Lie algebras”, Journal of Geometry and Physics, 192 (2023), 104951  crossref
    11. Jiefeng Liu, Yunhe Sheng, Chengming Bai, “Maurer-Cartan characterizations and cohomologies of compatible Lie algebras”, Sci. China Math., 66:6 (2023), 1177  crossref
    12. Vsevolod Gubarev, “Universal enveloping algebra of a pair of compatible Lie brackets”, Int. J. Algebra Comput., 32:07 (2022), 1335  crossref
    13. Apurba Das, “Compatible L∞-algebras”, Journal of Algebra, 610 (2022), 241  crossref
    14. Dobrogowska A., Jakimowicz G., “Generalization of the Concept of Classical R-Matrix to Lie Algebroids”, J. Geom. Phys., 165 (2021), 104227  crossref  mathscinet  isi
    15. Liu J., Bai Ch., Sheng Yu., “Noncommutative Poisson Bialgebras”, J. Algebra, 556 (2020), 35–66  crossref  mathscinet  isi
    16. Wu M., “Double Constructions of Compatible Associative Algebras”, Algebr. Colloq., 26:3 (2019), 479–494  crossref  mathscinet  isi
    17. 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
    18. Ming-Zhong Wu, Cheng-Ming Bai, “Compatible Lie Bialgebras*”, Commun. Theor. Phys., 63:6 (2015), 653  crossref
    19. 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
    20. Pumei Zhang, “Algebraic Properties of Compatible Poisson Brackets”, Regul. Chaotic Dyn., 19:3 (2014), 267–288  mathnet  crossref  mathscinet
    Citing articles in Google Scholar: Russian citations, English citations
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