Skip to main content
SpringerLink
Log in

Rota-Baxter Modules Toward Derived Functors

  • Published:
Algebras and Representation Theory Aims and scope Submit manuscript

Abstract

In this paper we study Rota-Baxter modules with emphasis on the role played by the Rota-Baxter operators and the resulting difference between Rota-Baxter modules and the usual modules over an algebra. We introduce the concepts of free, projective, injective and flat Rota-Baxter modules. We give the construction of free modules and show that there are enough projective, injective and flat Rota-Baxter modules to provide the corresponding resolutions for derived functors.

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

Similar content being viewed by others

References

  1. Aguiar, M.: On the associative analog of Lie bialgebras. J. Algebra. 244, 492–532 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  2. Atkinson, F.V.: Some aspects of Baxter’s functional equation. J. Math. Anal. Appl. 7, 1–30 (1963)

    Article  MathSciNet  MATH  Google Scholar 

  3. 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 

  4. Bakayoko, I., Banagoura, M.: Bimodules and Rota-Baxter relations. J. Appl. Mech. Eng. 4(178), 8 (2015). https://doi.org/10.4172/2168-9873.1000178

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  6. Bokut, L.A., Chen, Y., Qiu, J.: Greobner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras. J. Pure Appl. Algebra 214, 89–110 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  8. 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. 210, 249–273 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  9. Ebrahimi-Fard, K., Guo, L., Manchon, D.: Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion. Comm. Math. Phys. 267, 821–845 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  10. Guo, L.: Operated semigroups, Motzkin paths and rooted trees. J. Algeb. Combinatorics 29, 35–62 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  11. Guo, L.: WHAT IS a Rota-Baxter algebra. Notice Amer: Math. Soc. 56, 1436–1437 (2009)

    MathSciNet  MATH  Google Scholar 

  12. Guo, L.: An Introduction to Rota-Baxter Algebra. International Press (US) and Higher Education Press (China) (2012)

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

    Article  MathSciNet  MATH  Google Scholar 

  14. Guo, L., Lin, Z.: Representations and modules of Rota-Baxter algebras, preprint

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

    Article  MathSciNet  MATH  Google Scholar 

  16. Lin, Z., Qiao, L.: Representations of Rota-Baxter algebras and regular singular decompositions. J. Pure Apply Algebra, accepted

  17. Qiao, L., Pei, J.: Representations of polynomial Rota-Baxter algebras. J. Pure Appl. Algebra. https://doi.org/10.1016/j.jpaa.2017.08.003

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Rota, G.-C.: Baxter operators, an introduction. In: Kung, J. P.S. (ed.) Gian-Carlo Rota on Combinatorics, Introductory Papers and Commentaries. Birkhäuser, Boston (1995)

  20. Rotman, J.J.: An Introduction to Homological Algebra. Springer Science+Business Media LLC (2009)

  21. Semenov-Tian-Shansky, M.A.: What is a classical r-matrix? Funct. Ana. Appl. 17, 259–272 (1983)

    Article  MATH  Google Scholar 

  22. Weibel, C.: An Introduction to Homological Algebra. Cambridge University Press (1994)

  23. Zheng, H., Guo, L., Zhang, L.: Rota-Baxter paired modules and their constructions from Hopf algebras, arXiv:http://arXiv.org/abs/1710.03880

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grant No. 11771190 and 11501466), Fundamental Research Funds for the Central Universities (Grant No. lzujbky-2017-162), the Natural Science Foundation of Gansu Province (Grant No. 17JR5RA175). X. Gao thanks Rutgers University at Newark for its hospitality during his visit in 2015-2016.

Author information

Authors and Affiliations

  1. School of Mathematics and Statistics, Southwest University, Chongqing, 400715, People’s Republic of China

    Li Qiao

  2. School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu, 730000, People’s Republic of China

    Li Qiao & Xing Gao

  3. Department of Mathematics, Jiangxi Normal University, Nanchang, Jiangxi, 330022, China

    Li Guo

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

    Li Guo

Authors
  1. Li Qiao
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Xing Gao
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Li Guo
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Li Guo.

Additional information

Presented by Jon F. Carlson.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Qiao, L., Gao, X. & Guo, L. Rota-Baxter Modules Toward Derived Functors. Algebr Represent Theor 22, 321–343 (2019). https://doi.org/10.1007/s10468-018-9769-5

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s10468-018-9769-5

Keywords

Mathematics Subject Classification (2010)

Search

Navigation