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Factorizable Lie Bialgebras, Quadratic Rota–Baxter Lie Algebras and Rota–Baxter Lie Bialgebras

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In this paper, first we introduce the notion of quadratic Rota–Baxter Lie algebras of arbitrary weight, and show that there is a one-to-one correspondence between factorizable Lie bialgebras and quadratic Rota–Baxter Lie algebras of nonzero weight. Then we introduce the notions of matched pairs, bialgebras and Manin triples of Rota–Baxter Lie algebras of arbitrary weight, and show that Rota–Baxter Lie bialgebras, Manin triples of Rota–Baxter Lie algebras and certain matched pairs of Rota–Baxter Lie algebras are equivalent. The coadjoint representations and quadratic Rota–Baxter Lie algebras play important roles in the whole study. Finally we generalize some results to the Lie group context. In particular, we show that there is a one-to-one correspondence between factorizable Poisson Lie groups and quadratic Rota–Baxter Lie groups.

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Acknowledgements

This research is supported by NSFC (11922110, 12171006), the National Key Research and Development Program of China (No. 2021YFA1002000) and the Sino-Russian Mathematics Center. We give our warmest thanks to Maxim Goncharov for discussions. We also thank the referee for helpful comments.

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Authors and Affiliations

  1. College of Science, China Agricultural University, Beijing, 100083, China

    Honglei Lang

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

    Yunhe Sheng

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  1. Honglei Lang
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Correspondence to Yunhe Sheng.

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Communicated by C. Schweigert.

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Lang, H., Sheng, Y. Factorizable Lie Bialgebras, Quadratic Rota–Baxter Lie Algebras and Rota–Baxter Lie Bialgebras. Commun. Math. Phys. 397, 763–791 (2023). https://doi.org/10.1007/s00220-022-04501-y

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