Construction of free quasi-idempotent differential Rota-Baxter algebras by Gröbner-Shirshov bases

References

• Aguiar, M., Moreira, W. (2006). Combinatorics of the free Baxter algebra. Electron. J. Combin. 13:#R17.
   Web of Science ®Google Scholar
• Atkinson, F. V. (1963). Some aspects of Baxter’s functional equation. J. Math. Anal. Appl. 7:1–30. DOI: 10.1016/0022-247X(63)90075-1.
   Google Scholar
• Bai, C. (2007). A unified algebraic approach to the classical Yang-Baxter equations. J. Phys. A: Math. Theor. 40:11073–11082. DOI: 10.1088/1751-8113/40/36/007.
   Web of Science ®Google Scholar
• Bai, C., Bellier, O., Guo, L., Ni, X. (2013). Splitting of operations, Manin products and Rota-Baxter operators. Int. Math. Res. Not. 2013:485–524. DOI: 10.1093/imrn/rnr266.
   Web of Science ®Google Scholar
• Bavula, V. V. (2011). The algebra of integro-differential operators on a polynomial algebra. J. London Math. Soc. 83:517–543. DOI: 10.1112/jlms/jdq081.
   Web of Science ®Google Scholar
• Baxter, G. (1960). An analytic problem whose solution follows from a simple algebraic identity. Pac. J. Math 10: 731–742. DOI: 10.2140/pjm.1960.10.731.
   Google Scholar
• Belavin, A. A., Drinfeld, V. G. (1982). Solutions of the classical Yang-Baxter equation for simple Lie algebras. Funct. Anal. Appl. 16:159–180. DOI: 10.1007/BF01081585.
   Web of Science ®Google Scholar
• Bokut, L. A., Chen, Y. (2014). Gröbner-Shirshov bases and their calculation. Bull. Math. Sci. 4:325–395. DOI: 10.1007/s13373-014-0054-6.
   Web of Science ®Google Scholar
• Bokut, L. A., Chen, Y., Deng, X. (2010). Gröbner-Shirshov bases for Rota-Baxter algebras. Siberian Math. J. 51: 978–988. DOI: 10.1007/s11202-010-0097-1.
   Web of Science ®Google Scholar
• Bokut, L. A., Chen, Y., Qiu, J. (2010). Gröbner-Shirshov bases for associative algebras with multiple operators and free Rota-Baxter algebras. J. Pure Appl. Algebra 214:89–100. DOI: 10.1016/j.jpaa.2009.05.005.
   Web of Science ®Google Scholar
• Cartier, P. (1972). On the structure of free Baxter algebras. Adv. Math. 9:253–265. DOI: 10.1016/0001-8708(72)90018-7.
   Web of Science ®Google Scholar
• Connes, A., Kreimer, D. (2000). 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. DOI: 10.1007/s002200050779.
   Web of Science ®Google Scholar
• Ebrahimi-Fard, K. (2004). On the associative Nijenhuis relation. Electron. J. Combin. 11:#R38.
   Web of Science ®Google Scholar
• Ebrahimi-Fard, K., Guo, L. (2008). Free Rota-Baxter algebras and dendriform algebras. J. Pure Appl. Algebra 212:320–339. DOI: 10.1016/j.jpaa.2007.05.025.
   Web of Science ®Google Scholar
• Ebrahimi-Fard, K., Guo, L. (2008). Free Rota-Baxter algebras and rooted trees. J. Algebra Appl. 7:167–194. DOI: 10.1142/S0219498808002746.
   Web of Science ®Google Scholar
• Ebrahimi-Fard, K., Guo, L. (2006). Quasi-shuffles, mixable shuffles and Hopf algebras. J. Algebraic Combin. 24: 83–101. DOI: 10.1007/s10801-006-9103-x.
   Web of Science ®Google Scholar
• Gao, X., Guo, L., Zheng, S. (2014). Construction of free commutative integro-differential algebras by the method of Gröbner-Shirshov bases. J. Algebra Appl. 13:1350160. DOI: 10.1142/S0219498813501600.
   Web of Science ®Google Scholar
• Gao, X., Guo, L., Rosenkranz, M. (2015). Free integro-differential algebras and Gröbner-Shirshov bases. J. Algebra 442:354–396. DOI: 10.1016/j.jalgebra.2014.10.016.
   Web of Science ®Google Scholar
• Gao, X., Guo, L., Rosenkranz, M. (2018). On rings of differential Rota-Baxter operators. Int. J. Algebra Comput. 28:1–36. DOI: 10.1142/S0218196718500017.
   Web of Science ®Google Scholar
• Guo, L. (2012). An Introduction to Rota-Baxter Algebra. Somerville, MA: International Press.
   Google Scholar
• Guo, L., Keigher, W. (2000). Baxter algebras and shuffle products. Adv. Math. 150:117–149. DOI: 10.1006/aima.1999.1858.
   Web of Science ®Google Scholar
• Guo, L., Keigher, W. (2008). On differential Rota-Baxter algebras. J. Pure Appl. Algebra 212:522–540. DOI: 10.1016/j.jpaa.2007.06.008.
   Web of Science ®Google Scholar
• Guo, L., Regensburger, G., Rosenkranz, M. (2014). On integro-differential algebras. J. Pure Appl. Algebra 218: 456–473. DOI: 10.1016/j.jpaa.2013.06.015.
   Web of Science ®Google Scholar
• Guo, L., Lang, H., Sheng, Y. (2021). Integration and geometrization of Rota-Baxter Lie algebras. Adv. Math. 387:107834. DOI: 10.1016/j.aim.2021.107834.
   Web of Science ®Google Scholar
• Hoffman, M. (2000). Quasi-shuffle products. J. Algebraic Combin. 11:49–68. DOI: 10.1023/A:1008791603281.
   Web of Science ®Google Scholar
• Jian, R. (2017). Quasi-idempotent Rota-Baxter operators arising from quasi-idempotent elements. Lett. Math. Phys. 107:367–374. DOI: 10.1007/s11005-016-0905-z.
   Web of Science ®Google Scholar
• Kolchin, E. R. (1973). Differential Algebras and Algebraic Groups. New York: Academic Press.
   Google Scholar
• Lang, H., Sheng, Y. (2023). Factorizable Lie bialgebras, quadratic Rota-Baxter Lie algebras and Rota-Baxter Lie bialgebras. Commun. Math. Phys. 397:763–791. DOI: 10.1007/s00220-022-04501-y.
   Web of Science ®Google Scholar
• Li, Y., Guo, L. (2021). Construction of free differential algebras by extending Gröbner-Shirshov bases. J. Symb. Comput. 107:167–189. DOI: 10.1016/j.jsc.2021.03.002.
   Web of Science ®Google Scholar
• Ma, T., Li, J., Yang, H. (2021). Rota-Baxter bialgebra structures arising from (co-)quasi-idempotent elements. Hacet. J. Math. Stat. 50:216–223. DOI: 10.15672/hujms.685742.
   Web of Science ®Google Scholar
• Magid, A. R. (1994). Lectures on Differential Galois Theory, University Lecture Series, Vol. 7. Providence, RI: American Mathematical Society.
   Google Scholar
• Qi, Z., Qin, Y., Wang, K., Zhou, G. (2021). Free objects and Gröbner-Shirshov bases in operated contexts. J. Algebra 584:89–124. DOI: 10.1016/j.jalgebra.2021.04.042.
   Web of Science ®Google Scholar
• Qiu, J., Chen, Y. (2010). Composition-Diamond lemma for λ-differential associative algebras with multiple operators. J. Algebra Appl. 9:223–239. DOI: 10.1142/S0219498810003859.
   Web of Science ®Google Scholar
• Qiu, J., Chen, Y. (2017). Free Lie differential Rota-Baxter algebras and Gröbner-Shirshov bases. Int. J. Algebra Comput. 27:1041–1060. DOI: 10.1142/S0218196717500485.
   Web of Science ®Google Scholar
• Qiu, J. (2014). Gröbner-Shirshov bases for commutative algebras with multiple operators and free commutative Rota-Baxter algebras. Asian-Eur. J. Math. 7:1450033. DOI: 10.1142/S1793557114500338.
   Google Scholar
• Ritt, J. F. (1950). Differential Algebra. American Mathematical Society Colloquium Publications, Vol. 33. New York: American Mathematical Society.
   Google Scholar
• Rota, G.-C. (1969). Baxter algebras and combinatorial identities I, II. Bull. Amer. Math. Soc. 75:325–329, 330–334. DOI: 10.1090/S0002-9904-1969-12158-0.
   Google Scholar
• Rota, G.-C. (1995). Baxter operators, an introduction. In: Kung, J. P. S., ed. Gian-Carlo Rota on Combinatorics, Introductory Papers and Commentaries. Boston: Birkhäuser, 504–512.
   Google Scholar
• Semenov-Tian-Shansky, M. A. (1983). What is a classical r-matrix? Funct. Anal. Appl. 17:254–272.
   Google Scholar
• Shirshov, A. I. (1962). Some algorithmic problem for Lie algebras. Sibirsk. Mat. Zh. 3:292–296 (in Russian), English translation in SIGSAM Bull. 33(2):3–6.
   Google Scholar
• Zhang, S., Guo, L., Keigher, W. (2016). Monads and distributive laws for Rota-Baxter and differential algebras. Adv. Appl. Math. 72:139–165. DOI: 10.1016/j.aam.2015.09.014.
   Web of Science ®Google Scholar
• Yu, H., Guo, L., Thibon, J.-Y. (2019). Weak quasi-symmetric functions, Rota-Baxter algebras and Hopf algebras. Adv. Math. 344:1–34. DOI: 10.1016/j.aim.2018.12.001.
   Web of Science ®Google Scholar
• Zhang, T., Gao, X., Guo, L. (2023). Construction of free commutative Reynolds algebras by Gröbner-Shirshov bases. J. Symb. Comput. 119:64–80. DOI: 10.1016/j.jsc.2023.02.008.
   Web of Science ®Google Scholar
• Zheng, S., Gao, X., Guo, L., Sit, W. Rota-Baxter type operators, rewritting systems and Gröbner-Shirshov bases. J. Symb. Comput., accepted, arXiv:1412.8055.
   Google Scholar
• Zheng, H., Guo, L., Zhang, L. (2020). Rota-Baxter paired modules and their constructions from Hopf algebras. J. Algebra 559:601–624. DOI: 10.1016/j.jalgebra.2020.04.023.
   Web of Science ®Google Scholar