[go: up one dir, main page]
More Web Proxy on the site http://driver.im/ Skip to main content
Springer Nature Link
Log in

Extensions of Operators, Liftings of Monads, and Distributive Laws

  • Published:
Applied Categorical Structures Aims and scope Submit manuscript

Abstract

In a previous study, the algebraic formulation of the First Fundamental Theorem of Calculus (FFTC) is shown to allow extensions of differential and Rota–Baxter operators on the one hand, and to give rise to categorical explanations using the ideas of liftings of monads and comonads, and mixed distributive laws on the other. Generalizing the FFTC, we consider in this paper a class of constraints between a differential operator and a Rota–Baxter operator. For a given constraint, we show that the existences of extensions of differential and Rota–Baxter operators, of liftings of monads and comonads, and of mixed distributive laws are equivalent.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
£29.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

Price includes VAT (United Kingdom)

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Bai, C.: A unified algebraic approach to the classical Yang–Baxter equations. J. Phys. A: Math. Theor. 40, 11073–11082 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  2. Baxter, G.: An analytic problem whose solution follows from a simple algebraic identity. Pac. J. Math. 10, 731–742 (1960)

    Article  MathSciNet  MATH  Google Scholar 

  3. Beck, J.: Distributive Laws, Lecture Notes in Mathematics, vol. 80, pp. 119–140. Springer, New York (1969)

  4. Cartier, P.: On the structure of free Baxter algebras. Adv. Math. 9, 253–265 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  5. 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. Commun. Math. Phys. 210, 249–273 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  6. Gao, X., Guo, L., Rosenkranz, M.: Free integro-differential algebras and Gröbner–Shirshov bases. J. Algebra 442, 354–396 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  7. Guo, L.: An Introduction to Rota–Baxter Algebra. International Press and Higher Education Press, Beijing (2012)

    MATH  Google Scholar 

  8. Guo, L.: Operated semigroups, Motzkin paths and rooted trees. J. Algebr. Comb. 29, 35–62 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  9. Guo, L., Keigher, W.: Baxter algebras and shuffle products. Adv. Math. 150, 117–149 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guo, L., Keigher, W.: On differential Rota–Baxter algebras. J. Pure Appl. Algebra 212, 522–540 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guo, L., Regensburger, G., Rosenkranz, M.: On integro-differential algebras. J. Pure Appl. Algebra 218, 456–471 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  12. Guo, L., Zhang, B.: Renormalization of multiple zeta values. J. Algebra 319, 3770–3809 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  13. Hoffman, M.: Quasi-shuffle products. J. Algebra. Combin. 11, 49–68 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  14. Keigher, W.: On the ring of Hurwitz series. Commun. Algebra 25, 1845–1859 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  15. Kolchin, E.: Differential Algebra and Algebraic Groups. Academic Press, New York (1973)

    MATH  Google Scholar 

  16. Kurosh, A.G.: Free sums of multiple operator algebras. Sib. Math. J. 1, 62–70 (1960)

    MathSciNet  MATH  Google Scholar 

  17. Mac Lane, S.: Categories for the Working Mathematician. Springer, New York (1971)

    MATH  Google Scholar 

  18. Ritt, J .F.: Differential Equations from the Algebraic Standpoint. AMS, Providence (1932)

    Book  MATH  Google Scholar 

  19. Rosenkranz, M., Regensburger, G.: Solving and factoring boundary problems for linear ordinary differential equations in differential algebra. J. Symb. Comput. 43, 515–544 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  20. Rota, G.: Baxter algebras and combinatorial identities I. Bull. AMS 5, 325–329 (1969)

    Article  MATH  Google Scholar 

  21. Rotman, J .J.: Introduction to Homological Algebra, 2nd edn. Springer, New York (2008)

    MATH  Google Scholar 

  22. Wolff, H.: \(\cal{V}\)-Localizations and \(\cal{V}\)-Monads. J. Algebra 24, 405–438 (1973)

    Article  MathSciNet  Google Scholar 

  23. Wolff, H.: Distributive laws and lifting of triples. Commun. Algebra 5, 981–996 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  24. Wen-Tsun, W.: A constructive theory of differential algebraic geometry based on works of J. F. Ritt with particular applications to mechanical theorem-proving of differential geometries. Differential Geometry and Differential Equations (Shanghai, 1985), Lecture Notes in Mathematics, vol. 1255, pp. 173–189. Springer, New York (1987)

  25. Zhang, S., Guo, L., Keigher, W.: Monads and distributive laws for Rota–Baxter and differential algebras. Adv. Appl. Math. 72, 139–165 (2016)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (Grant No. 11771190) and the China Scholarship Council (Grant No. 201606180084). Shilong Zhang thanks Rutgers University, Newark for its hospitality during his visit August 2016–August 2017. The authors thank the referee for helpful suggestions.

Author information

Authors and Affiliations

  1. Department of Mathematics, Lanzhou University, Lanzhou, 730000, Gansu, China

    Shilong Zhang

  2. Department of Mathematics and Computer Science, Rutgers University, Newark, NJ, 07102, USA

    Li Guo & William Keigher

Authors
  1. Shilong Zhang
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Li Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. William Keigher
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Guo.

Additional information

Communicated by Richard Garner.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Zhang, S., Guo, L. & Keigher, W. Extensions of Operators, Liftings of Monads, and Distributive Laws. Appl Categor Struct 26, 747–765 (2018). https://doi.org/10.1007/s10485-018-9517-3

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10485-018-9517-3

Keywords

Mathematics Subject Classification

Search

Navigation