Skip to main content
SpringerLink
Log in

Embedding of dendriform algebras into Rota-Baxter algebras

  • Research Article
  • Published:
Central European Journal of Mathematics

Abstract

Following a recent work [Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266] we define what is a dendriform dior trialgebra corresponding to an arbitrary variety Var of binary algebras (associative, commutative, Poisson, etc.). We call such algebras di- or tri-Var-dendriform algebras, respectively. We prove in general that the operad governing the variety of di- or tri-Var-dendriform algebras is Koszul dual to the operad governing di- or trialgebras corresponding to Var!. We also prove that every di-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of weight zero in the variety Var, and every tri-Var-dendriform algebra can be embedded into a Rota-Baxter algebra of nonzero weight in Var.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Aguiar M., Pre-Poisson algebras, Lett. Math. Phys., 2000, 54(4), 263–277

    Article  MathSciNet  MATH  Google Scholar 

  2. Andrews G.E., Guo L., Keigher W., Ono K., Baxter algebras and Hopf algebras, Trans. Amer. Math. Soc., 2003, 355(11), 4639–4656

    Article  MathSciNet  MATH  Google Scholar 

  3. Bai C., Bellier O., Guo L., Ni X., Splitting of operations, Manin products, and Rota-Baxter operators, Int. Math. Res. Not. IMRN (in press), DOI: 10.1093/imrn/rnr266

  4. Bai C., Guo L., Ni X., O-operators on associative algebras and dendriform algebras, preprint available at http://arxiv.org/abs/1003.2432

  5. Bai C., Liu L., Ni X., Some results on L-dendriform algebras, J. Geom. Phys., 2010, 60(6–8), 940–950

    Article  MathSciNet  MATH  Google Scholar 

  6. Bakalov B., D’Andrea A., Kac V.G., Theory of finite pseudoalgebras, Adv. Math., 2001, 162(1), 1–140

    Article  MathSciNet  MATH  Google Scholar 

  7. Baxter G., An analytic problem whose solution follows from a simple algebraic identity, Pacific J. Math., 1960, 10, 731–742

    MathSciNet  MATH  Google Scholar 

  8. Belavin A.A., Drinfel’d V.G., Solutions of the classical Yang-Baxter equation for simple Lie algebras, Funct. Anal. Appl., 1982, 16(3), 159–180

    Article  MathSciNet  Google Scholar 

  9. Bokut L.A., Chen Yu., Deng X., Gröbner-Shirshov bases for Rota-Baxter algebras, Sib. Math. J., 2010, 51(6), 978–988

    Article  MathSciNet  MATH  Google Scholar 

  10. Cartier P., On the structure of free Baxter algebras, Adv. in Math., 1972, 9, 253–265

    Article  MathSciNet  MATH  Google Scholar 

  11. Chapoton F., Un endofoncteur de la catégorie des opérades, In: Dialgebras and Related Operads, Lecture Notes in Math., 1763, Springer, Berlin, 2001, 105–110

    Chapter  Google Scholar 

  12. Chen Y., Mo Q., Embedding dendriform algebra into its universal enveloping Rota-Baxter algebra, Proc. Amer. Math. Soc., 2011, 139(12), 4207–4216

    Article  MathSciNet  MATH  Google Scholar 

  13. Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. I: The Hopf algebra structure of graphs and the main theorem, Comm. Math. Phys., 2000, 210(1), 249–273

    Article  MathSciNet  MATH  Google Scholar 

  14. Connes A., Kreimer D., Renormalization in quantum field theory and the Riemann-Hilbert problem. II: The β-function, diffeomorphisms and the renormalization group, Comm. Math. Phys., 2001, 216(1), 215–241

    Article  MathSciNet  MATH  Google Scholar 

  15. Ebrahimi-Fard K., Loday-type algebras and the Rota-Baxter relation, Lett. Math. Phys., 2002, 61(2), 139–147

    Article  MathSciNet  MATH  Google Scholar 

  16. Ebrahimi-Fard K., Guo L., On products and duality of binary, quadratic, regular operads, J. Pure Appl. Algebra, 2005, 200(3), 293–317

    Article  MathSciNet  MATH  Google Scholar 

  17. Ebrahimi-Fard K., Guo L., Mixable shuffles, quasi-shuffles and Hopf algebras, J. Algebraic Combin., 2006, 24(1), 83–101

    Article  MathSciNet  MATH  Google Scholar 

  18. Ebrahimi-Fard K., Guo L., Rota-Baxter algebras and dendriform algebras, J. Pure Appl. Algebra, 2008, 212(2), 320–339

    Article  MathSciNet  MATH  Google Scholar 

  19. Ginzburg V., Kapranov M., Koszul duality for operads, Duke Math. J., 1994, 76(1), 203–272

    Article  MathSciNet  MATH  Google Scholar 

  20. Golenishcheva-Kutuzova M.I., Kac V.G., Γ-conformal algebras, J. Math. Phys., 1998, 39(4), 2290–2305

    Article  MathSciNet  MATH  Google Scholar 

  21. Gubarev V.Yu., Kolesnikov P.S., The Tits-Kantor-Koecher construction for Jordan dialgebras, Comm. Algebra, 2011, 39(2), 497–520

    Article  MathSciNet  MATH  Google Scholar 

  22. Guo L., An Introduction to Rota-Baxter Algebra, available at http://math.newark.rutgers.edu/_liguo/rbabook.pdf

  23. Kolesnikov P., Identities of conformal algebras and pseudoalgebras, Comm. Algebra, 2006, 34(6), 1965–1979

    Article  MathSciNet  MATH  Google Scholar 

  24. Kolesnikov P.S., Varieties of dialgebras and conformal algebras, Sib. Math. J., 2008, 49(2), 257–272

    Article  MathSciNet  Google Scholar 

  25. Leinster T., Higher Operads, Higher Categories, London Math. Soc. Lecture Note Ser., 298, Cambridge University Press, Cambridge, 2004

    Google Scholar 

  26. Leroux P., Construction of Nijenhuis operators and dendriform trialgebras, Int. J. Math. Math. Sci., 2004, 49–52, 2595–2615

    Article  MathSciNet  Google Scholar 

  27. Loday J.-L., Une version non commutative des algèbres de Lie: les algèbres de Leibniz, Enseign. Math., 1993, 39(3–4), 269–293

    MathSciNet  MATH  Google Scholar 

  28. Loday J.-L., Dialgebras, In: Dialgebras and Related Operads, Lecture Notes in Math., 1763, Springer, Berlin, 2001, 7–66

    Google Scholar 

  29. Loday J.-L., Pirashvili T., Universal enveloping algebras of Leibniz algebras and (co)homology, Math. Ann., 1993, 296(1), 139–158

    Article  MathSciNet  MATH  Google Scholar 

  30. Loday J.-L., Ronco M., Trialgebras and families of polytopes, In: Homotopy Theory: Relations with Algebraic Geometry, Group Cohomology, and Algebraic K-Theory, Contemp. Math., 346, American Mathematical Society, Providence, 2004, 369–398

    Chapter  Google Scholar 

  31. Loday J.-L., Vallette B., Algebraic Operads, Grundlehren Math. Wiss., 346, Springer, Heidelberg, 2012

    Google Scholar 

  32. May J.P., Geometry of Iterated Loop Spaces, Lecture Notes in Math., 271, Springer, Berlin-New York, 1972

    Google Scholar 

  33. Pozhidaev A.P., 0-dialgebras with bar-unity, Rota-Baxter and 3-Leibniz algebras, In: Groups, Rings and Group Rings, Contemp. Math., 499, American Mathematical Society, Providence, 2009, 245–256

    Google Scholar 

  34. Rota G.-C., Baxter algebras and combinatorial identities I, II, Bull. Amer. Math. Soc., 1969, 75(2), 325–329, 330–334

    Article  MathSciNet  MATH  Google Scholar 

  35. Semenov-Tyan-Shanskii M.A., What is a classical r-matrix?, Funct. Anal. Appl., 1983, 17(4), 259–272

    Article  MathSciNet  Google Scholar 

  36. Spitzer F., A combinatorial lemma and its application to probability theory, Trans. Amer. Math. Soc., 1956, 82, 323–339

    Article  MathSciNet  MATH  Google Scholar 

  37. Stasheff J., What is … an operad? Notices Amer. Math. Soc., 2004, 51(6), 630–631

    MathSciNet  MATH  Google Scholar 

  38. Uchino K., Derived bracket construction and Manin products, Lett. Math. Phys., 2010, 90(1), 37–53

    Article  MathSciNet  Google Scholar 

  39. Uchino K., On distributive laws in derived bracket construction, preprint available at http://arxiv.org/abs/1110.4188v1

  40. Vallette B., Homology of generalized partition posets, J. Pure Appl. Algebra, 2007, 208(2), 699–725

    Article  MathSciNet  MATH  Google Scholar 

  41. Vallette B., Manin products, Koszul duality, Loday algebras and Deligne conjecture, J. Reine Angew. Math., 2008, 620, 105–164

    MathSciNet  MATH  Google Scholar 

  42. Voronin V., Special and exceptional Jordan dialgebras, J. Algebra Appl., 2012, 11(2), #1250029

  43. Zinbiel G.W., Encyclopedia of types of algebras 2010, preprint available at http://arxiv.org/abs/1101.0267

Download references

Author information

Authors and Affiliations

  1. Novosibirsk State University, Pirogova Str. 2, Novosibirsk, 630090, Russia

    Vsevolod Yu. Gubarev

  2. Sobolev Institute of Mathematics, Akad. Koptyug Ave. 4, Novosibirsk, 630090, Russia

    Pavel S. Kolesnikov

Authors
  1. Vsevolod Yu. Gubarev
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Pavel S. Kolesnikov
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Vsevolod Yu. Gubarev.

About this article

Cite this article

Gubarev, V.Y., Kolesnikov, P.S. Embedding of dendriform algebras into Rota-Baxter algebras. centr.eur.j.math. 11, 226–245 (2013). https://doi.org/10.2478/s11533-012-0138-z

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.2478/s11533-012-0138-z

MSC

Keywords

Search

Navigation