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.
Similar content being viewed by others
References
Aguiar, M.: On the associative analog of Lie bialgebras. J. Algebra. 244, 492–532 (2001)
Atkinson, F.V.: Some aspects of Baxter’s functional equation. J. Math. Anal. Appl. 7, 1–30 (1963)
Bai, C.: A unified algebraic approach to the classical Yang-Baxter equations. J. Phys. A: Math. Theor. 40, 11073–11082 (2007)
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
Baxter, G.: An analytic problem whose solution follows from a simple algebraic identity. Pacific J. Math. 10, 731–742 (1960)
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)
Cartier, P.: On the structure of free Baxter algebras. Adv. Math. 9, 253–265 (1972)
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)
Ebrahimi-Fard, K., Guo, L., Manchon, D.: Birkhoff type decompositions and the Baker-Campbell-Hausdorff recursion. Comm. Math. Phys. 267, 821–845 (2006)
Guo, L.: Operated semigroups, Motzkin paths and rooted trees. J. Algeb. Combinatorics 29, 35–62 (2009)
Guo, L.: WHAT IS a Rota-Baxter algebra. Notice Amer: Math. Soc. 56, 1436–1437 (2009)
Guo, L.: An Introduction to Rota-Baxter Algebra. International Press (US) and Higher Education Press (China) (2012)
Guo, L., Keigher, W.: Baxter algebras and shuffle products. Adv. Math. 150, 117–149 (2000)
Guo, L., Lin, Z.: Representations and modules of Rota-Baxter algebras, preprint
Guo, L., Zhang, B.: Renormalization of multiple zeta values. J. Algebra. 319, 3770–3809 (2008)
Lin, Z., Qiao, L.: Representations of Rota-Baxter algebras and regular singular decompositions. J. Pure Apply Algebra, accepted
Qiao, L., Pei, J.: Representations of polynomial Rota-Baxter algebras. J. Pure Appl. Algebra. https://doi.org/10.1016/j.jpaa.2017.08.003
Rota, G.-C.: Baxter algebras and combinatorial identities I, II. Bull. Amer. Math. Soc. 75, 325–329, 330–334 (1969)
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)
Rotman, J.J.: An Introduction to Homological Algebra. Springer Science+Business Media LLC (2009)
Semenov-Tian-Shansky, M.A.: What is a classical r-matrix? Funct. Ana. Appl. 17, 259–272 (1983)
Weibel, C.: An Introduction to Homological Algebra. Cambridge University Press (1994)
Zheng, H., Guo, L., Zhang, L.: Rota-Baxter paired modules and their constructions from Hopf algebras, arXiv:http://arXiv.org/abs/1710.03880
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
Corresponding author
Additional information
Presented by Jon F. Carlson.
Rights and permissions
About this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10468-018-9769-5
Keywords
- Rota-Baxter algebra
- Rota-Baxter module
- Free module
- Projective module
- Injective module
- Flat module
- Ring of Rota-Baxter operators