Advertisement

Rota–Baxter Operators on Quadratic Algebras

  • Pilar Benito
  • Vsevolod Gubarev
  • Alexander Pozhidaev
Open Access
Article
  • 86 Downloads

Abstract

We prove that all Rota–Baxter operators on a quadratic division algebra are trivial. For nonzero weight, we state that all Rota–Baxter operators on the simple odd-dimensional Jordan algebra of bilinear form are projections on a subalgebra along another one. For weight zero, we find a connection between the Rota–Baxter operators and the solutions to the alternative Yang–Baxter equation on the Cayley–Dickson algebra. We also investigate the Rota–Baxter operators on the matrix algebras of order two, the Grassmann algebra of plane, and the Kaplansky superalgebra.

Keywords

Rota–Baxter operator Yang–Baxter equation Quadratic algebra Grassmann algebra Jordan algebra of bilinear form Matrix algebra Kaplansky superalgebra 

Mathematics Subject Classification

Primary 17A45 Secondary 17C50 Thirdly 16T25 

Notes

Acknowledgements

Open access funding provided by Austrian Science Fund (FWF). Pilar Benito acknowledges financial support by Grant MTM2017-83506-C2-1-P (AEI/FEDER, UE). Vsevolod Gubarev is supported by the Austrian Science Foundation FWF, Grant P28079.

References

  1. 1.
    Aguiar, M.: Pre-Poisson algebras. Lett. Math. Phys. 54, 263–277 (2000)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aguiar, M.: On the associative analog of Lie bialgebras. J. Algebra 244, 492–532 (2001)MathSciNetCrossRefGoogle Scholar
  3. 3.
    An, H., Bai, C.: From Rota–Baxter algebras to pre-Lie algebras. J. Phys. A. Math. Theor. 1, 015201 (2008)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Bai, C., Bellier, O., Guo, L., Ni, X.: Splitting of operations, Manin products, and Rota–Baxter operators. Int. Math. Res. Not. 3, 485–524 (2013)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Baxter, G.: An analytic problem whose solution follows from a simple algebraic identity. Pac. J. Math. 10, 731–742 (1960)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Belavin, A.A., Drinfel’d, V.G.: Solutions of the classical Yang–Baxter equation for simple Lie algebras. Funct. Anal. Appl. 16, 159–180 (1982)MathSciNetCrossRefGoogle Scholar
  7. 7.
    de Bragança, S.L.: Finite dimensional Baxter algebras. Stud. Appl. Math. 54(1), 75–89 (1975)MathSciNetCrossRefGoogle Scholar
  8. 8.
    Ebrahimi-Fard, K.: Loday-type algebras and the Rota–Baxter relation. Lett. Math. Phys. 61(2), 139–147 (2002)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Flanders, H.: On spaces of linear transformations with bounded rank. J. Lond. Math. Soc. 37, 10–16 (1962)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Gantmacher, F.R.: The Theory of Matrices, vol. 2. Chelsea, New York (1959)zbMATHGoogle Scholar
  11. 11.
    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(5), 809–823 (2007)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Goncharov, M.E.: On Rota–Baxter operators of non-zero weight arisen from the solutions of the classical Yang–Baxter equation. Sib. Electron. Math. Rep. 14, 1533–1544 (2017)MathSciNetzbMATHGoogle Scholar
  13. 13.
    Gubarev, V.: Rota–Baxter operators of weight zero on simple Jordan algebra of Clifford type. Sib. Electron. Math. Rep. 14, 1524–1532 (2017)MathSciNetzbMATHGoogle Scholar
  14. 14.
    Gubarev, V., Kolesnikov, P.: Embedding of dendriform algebras into Rota–Baxter algebras. Cent. Eur. J. Math. 11(2), 226–245 (2013)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Guo, L.: An introduction to Rota—Baxter algebra. In: Surveys of Modern Mathematics. vol. 4. Beijing: Higher Education Press (2012)Google Scholar
  16. 16.
    Guo, L., Ebrahimi-Fard, K.: Rota-Baxter algebras and dendriform algebras. J. Pure Appl. Algebra 212, 320–339 (2008)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Guo, L., Keigher, W.: On differential Rota–Baxter algebras. J. Pure Appl. Algebra 212, 522–540 (2008)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Jian, R.-Q.: Quasi-idempotent Rota–Baxter operators arising from quasi-idempotent elements. Lett. Math. Phys. 107(2), 367–374 (2017)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Kolesnikov, P.S.: Homogeneous averaging operators on simple finite conformal Lie algebras. J. Math. Phys. 56, 071702 (2015)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Li, X.X., Hou, D.P., Bai, C.M.: Rota–Baxter operators on pre-Lie algebras. J. Nonlinear Math. Phys. 14(2), 269–289 (2007)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Meshulam, R.: On the maximal rank in a subspace of matrices. Q. J. Math. Oxf. 36(2), 225–229 (1985)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Pan, Yu., Liu, Q., Bai, C., Guo, L.: Post-Lie algebra structures on the Lie algebra \({\rm sl}(2,\mathbb{C})\). Electron. J. Linear Algebra 23, 180–197 (2012)MathSciNetGoogle Scholar
  23. 23.
    Pei, J., Bai, C., Guo, L.: Rota–Baxter operators on \({\rm sl} (2,\mathbb{C})\) and solutions of the classical Yang–Baxter equation. J. Math. Phys. 55, 021701 (2014)MathSciNetCrossRefGoogle Scholar
  24. 24.
    Pozhidaev, A.P.: 0-Dialgebras with bar-unity and nonassociative Rota–Baxter algebras. Sib. Math. J. 50(6), 1070–1080 (2009)MathSciNetCrossRefGoogle Scholar
  25. 25.
    Rota, G.-C.: Baxter algebras and combinatorial identities. I. Bull. Am. Math. Soc. 75, 325–329 (1969)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Rota, G.-C.: Gian-Carlo Rota on combinatorics, introductory papers and commentaries. Birkhäuser, Boston (1995)zbMATHGoogle Scholar
  27. 27.
    Semenov-Tyan-Shanskii, M.A.: What is a classical \(r\)-matrix? Funct. Anal. Appl. 17, 259–272 (1983)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Tang, X., Zhang, Y., Sun, Q.: Rota-Baxter operators on 4-dimensional complex simple associative algebras. Appl. Math. Comput. 229, 173–186 (2014)MathSciNetzbMATHGoogle Scholar
  29. 29.
    Zhelyabin, V.N.: Jordan bialgebras of symmetric elements and Lie bialgebras. Sib. Math. J. 39(2), 261–276 (1998)MathSciNetCrossRefGoogle Scholar
  30. 30.
    Zhevlakov, K.A., Slin’ko, A.M., Shestakov, I.P., Shirshov, A.I.: Rings that are Nearly Associative. Academic Press, New York (1982)zbMATHGoogle Scholar

Copyright information

© The Author(s) 2018

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  1. 1.Universidad de La RiojaLogroñoSpain
  2. 2.University of ViennaViennaAustria
  3. 3.Sobolev Institute of MathematicsNovosibirskRussia
  4. 4.Novosibirsk State UniversityNovosibirskRussia

Personalised recommendations