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The classical Yang-Baxter equation on alternative algebras: The alternative D-bialgebra structure on Cayley-Dickson matrix algebras

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Abstract

We consider the Yang-Baxter equations on alternative algebras and prove that the bialgebras induced by the solutions to these equations are alternative D-bialgebras. We describe the alternative D-bialgebra structure on Cayley-Dickson matrix algebras.

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References

  1. Drinfeld V. G., “Hamiltonian structures on Lie groups, Lie bialgebras and the geometric meaning of classical Yang-Baxter equations,” Soviet Math. Dokl., 268, 68–71 (1983).

    Google Scholar 

  2. Zhelyabin V. N., “Jordan bialgebras and their connection with Lie bialgebras,” Algebra and Logic, 36, No. 1, 1–15 (1997).

    Article  MathSciNet  Google Scholar 

  3. Zhelyabin V. N., “Jordan bialgebras of symmetric elements and Lie bialgebras,” Siberian Math. J., 39, No. 2, 261–276 (1998).

    Article  MathSciNet  Google Scholar 

  4. Joni S. A. and Rota G. C., “Coalgebras and bialgebras in combinatorics,” Stud. Appl. Math., 61, No. 2, 93–139 (1979).

    MATH  MathSciNet  Google Scholar 

  5. Aguiar M., “On the associative analog of Lie bialgebras,” J. of Algebra, 244, No. 2, 492–532 (2001).

    Article  MATH  MathSciNet  Google Scholar 

  6. Polishchuk A., “Classical Yang-Baxter equation and the A ∞-constraint,” Adv. Math., 168, No. 1, 56–95 (2002).

    Article  MATH  MathSciNet  Google Scholar 

  7. Zhelyabin V. N., “On a class of Jordan D-bialgebras,” St. Petersburg Math. J., 11, No. 4, 589–609 (2000).

    MathSciNet  Google Scholar 

  8. Mudrov A. I., “Associative triples and the Yang-Baxter equation,” Israel J. Math., 139, 11–28 (2004).

    MATH  MathSciNet  Google Scholar 

  9. Anquela J. A., Cortes T., and Montaner F., “Nonassociative coalgebras,” Comm. Algebra, 22, No. 12, 4693–4716 (1994).

    Article  MATH  MathSciNet  Google Scholar 

  10. Drinfeld V. G. “Quantum groups,” in: Proc. Intern. Congr. Math. Berkeley, 1986; Amer. Math. Soc., Providence, RI, 1987, pp. 798–820.

    Google Scholar 

  11. Zhelyabin V. N., “Jordan D-bialgebras and symplectic forms on Jordan algebras,” Siberian Adv. in Math., 10,No. 2, 134–142 (2000).

    MathSciNet  Google Scholar 

  12. Zhevlakov K. A., Slinko A. M., Shestakov I. P., and Shirshov A. I., Rings That Are Nearly Associative, Academic Press, New York; London (1982) (Pure and Applied Mathematics; 104).

    MATH  Google Scholar 

  13. Jacobson N. R., Structure and Representations of Jordan Algebras, Amer. Math. Soc., Providence, RI (1968) (Amer. Math. Soc. Colloq. Publ.; 39).

    MATH  Google Scholar 

  14. Schafer R. D., An Introduction to Nonassociative Algebras, Academic Press, New York (1966).

    MATH  Google Scholar 

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Authors and Affiliations

  1. Novosibirsk State University, Novosibirsk, Russia

    M. E. Goncharov

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  1. M. E. Goncharov
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Correspondence to M. E. Goncharov.

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__________

Translated from Sibirskiĭ Matematicheskiĭ Zhurnal, Vol. 48, No. 5, pp. 1008–1024, September–October, 2007.

Original Russian Text Copyright © 2007 Goncharov M. E.

The author was supported by the Russian Foundation for Basic Research (Grant 05-01-00230) and the Ministry for Education (Grant No. 11617).

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Goncharov, M.E. The classical Yang-Baxter equation on alternative algebras: The alternative D-bialgebra structure on Cayley-Dickson matrix algebras. Sib Math J 48, 809–823 (2007). https://doi.org/10.1007/s11202-007-0083-4

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  • DOI: https://doi.org/10.1007/s11202-007-0083-4

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