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Infinite-Dimensional Lie Bialgebras via Affinization of Novikov Bialgebras and Koszul Duality

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Abstract

Balinsky and Novikov showed that the affinization of a Novikov algebra naturally defines a Lie algebra, a property that in fact characterizes the Novikov algebra. It is also an instance of the operadic Koszul duality. In this paper, we develop a bialgebra theory for the Novikov algebra, namely the Novikov bialgebra, which is characterized by the fact that its affinization (by a quadratic right Novikov algebra) gives an infinite-dimensional Lie bialgebra, suggesting another instance of Koszul duality for properads. A Novikov bialgebra is also characterized as a Manin triple of Novikov algebras. The notion of Novikov Yang–Baxter equation is introduced, whose skewsymmetric solutions can be used to produce Novikov bialgebras and hence Lie bialgebras. Moreover, these solutions also give rise to skewsymmetric solutions of the classical Yang–Baxter equation in the infinite-dimensional Lie algebras from the Novikov algebras.

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Notes

  1. See §19 where he wrote “The moral is that Novikov algebras are not proper”.

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Acknowledgements

This research is supported by NSFC (12171129, 11871421, 11931009, 12271265, 12261131498), the Zhejiang Provincial Natural Science Foundation of China (LY20A010022) and the Scientific Research Foundation of Hangzhou Normal University (2019QDL012), the Fundamental Research Funds for the Central Universities and Nankai Zhide Foundation. We thank Jun Pei for helpful discussions. We give our appreciation to the referees for their suggestions that have improved the paper.

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Authors and Affiliations

  1. School of Mathematics, Hangzhou Normal University, Hangzhou, 311121, People’s Republic of China

    Yanyong Hong

  2. Chern Institute of Mathematics & LPMC, Nankai University, Tianjin, 300071, People’s Republic of China

    Chengming Bai

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

    Li Guo

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  1. Yanyong Hong
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  2. Chengming Bai
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Correspondence to Li Guo.

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Communicated by Y. Kawahigashi.

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Hong, Y., Bai, C. & Guo, L. Infinite-Dimensional Lie Bialgebras via Affinization of Novikov Bialgebras and Koszul Duality. Commun. Math. Phys. 401, 2011–2049 (2023). https://doi.org/10.1007/s00220-023-04684-y

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