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Free Rota-Baxter Family Algebras and Free (tri)dendriform Family Algebras

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Abstract

In this paper, we first construct the free Rota-Baxter family algebra generated by some set X in terms of typed angularly X-decorated planar rooted trees. As an application, we obtain a new construction of the free Rota-Baxter algebra only in terms of angularly decorated planar rooted trees (not forests), which is quite different from the known construction via angularly decorated planar rooted forests by K. Ebrahimi-Fard and L. Guo. We then embed the free dendriform (resp. tridendriform) family algebra into the free Rota-Baxter family algebra of weight zero (resp. one). Finally, we prove that the free Rota-Baxter family algebra is the universal enveloping algebra of the free (tri)dendriform family algebra.

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Acknowledgements

The first author is very grateful to the Laboratoire de Mathématiques Blaise Pascal of Université Clermont Auvergne (Clermont-Ferrand, France) for providing a position in France as a visiting PhD. student at the time of writing this paper. This work is supported in part by Natural Science Foundation of China (No. 12071191, 12101183), project funded by China Postdoctoral Science Foundation (No. 2021M690049) and the Natural Science Project of Shaanxi Province (No. 2022JQ-035).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Henan University, Kaifeng, 475004, People’s Republic of China

    Yuanyuan Zhang

  2. School of Mathematics and Statistics, Lanzhou University, Lanzhou, 730000, China

    Xing Gao

  3. School of Mathematics and Statistics, Qinghai Nationalities University, Xining, 810007, China

    Xing Gao

  4. Laboratoire de Mathématiques Blaise Pascal, CNRS–Université Clermont-Auvergne, 3 place Vasarély, CS 60026, F63178, Aubière, France

    Dominique Manchon

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  1. Yuanyuan Zhang
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  2. Xing Gao
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Correspondence to Xing Gao or Dominique Manchon.

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Zhang, Y., Gao, X. & Manchon, D. Free Rota-Baxter Family Algebras and Free (tri)dendriform Family Algebras. Algebr Represent Theor 26, 2967–3002 (2023). https://doi.org/10.1007/s10468-022-10198-3

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  • DOI: https://doi.org/10.1007/s10468-022-10198-3

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