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Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation

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Abstract

The notions of a post-group and a pre-group are introduced as a unification and enrichment of several group structures appearing in diverse areas from numerical integration to the Yang–Baxter equation. First the Butcher group from numerical integration on Euclidean spaces and the -group of an operad naturally admit a pre-group structure. Next a relative Rota–Baxter operator on a group naturally splits the group structure to a post-group structure. Conversely, a post-group gives rise to a relative Rota–Baxter operator on the sub-adjacent group. Further a post-group gives a braided group and a solution of the Yang–Baxter equation. Indeed the category of post-groups is isomorphic to the category of braided groups and the category of skew-left braces. Moreover a post-Lie group differentiates to a post-Lie algebra structure on the vector space of left invariant vector fields, showing that post-Lie groups are the integral objects of post-Lie algebras. Finally, post-Hopf algebras and post-Lie Magnus expansions are utilized to study the formal integration of post-Lie algebras. As a byproduct, a post-group structure is explicitly determined on the Lie–Butcher group from numerical integration on manifolds.

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Acknowledgements

This work is supported by NSFC (11931009,11922110,12001228,12271265), Sino-Russian Mathematical Center and the Fundamental Research Funds for the Central Universities and Nankai Zhide Foundation. We give our warmest thanks to the referee for the very helpful suggestions that improve the paper.

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Authors and Affiliations

  1. Chern Institute of Mathematics and LPMC, Nankai University, Tianjin, 300071, China

    Chengming Bai

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

    Li Guo

  3. Department of Mathematics, Jilin University, Changchun, 130012, Jilin, China

    Yunhe Sheng & Rong Tang

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  1. Chengming Bai
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  2. Li Guo
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  3. Yunhe Sheng
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Correspondence to Li Guo.

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Bai, C., Guo, L., Sheng, Y. et al. Post-groups, (Lie-)Butcher groups and the Yang–Baxter equation. Math. Ann. 388, 3127–3167 (2024). https://doi.org/10.1007/s00208-023-02592-z

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