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Nonabelian Generalized Lax Pairs, the Classical Yang-Baxter Equation and PostLie Algebras

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Abstract

We generalize the classical study of (generalized) Lax pairs, the related \({\mathcal O}\) -operators and the (modified) classical Yang-Baxter equation by introducing the concepts of nonabelian generalized Lax pairs, extended \({\mathcal O}\) -operators and the extended classical Yang-Baxter equation. We study in this context the nonabelian generalized r-matrix ansatz and the related double Lie algebra structures. Relationship between extended \({\mathcal O}\) -operators and the extended classical Yang-Baxter equation is established, especially for self-dual Lie algebras. This relationship allows us to obtain an explicit description of the Manin triples for a new class of Lie bialgebras. Furthermore, we show that a natural structure of PostLie algebra is behind \({\mathcal O}\) -operators and fits in a setup of the triple Lie algebra that produces self-dual nonabelian generalized Lax pairs.

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Authors and Affiliations

  1. Chern Institute of Mathematics& LPMC, Nankai University, Tianjin, 300071, P.R. China

    Chengming Bai & Xiang Ni

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

    Li Guo

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  1. Chengming Bai
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  2. Li Guo
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  3. Xiang Ni
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Correspondence to Xiang Ni.

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Communicated by A. Connes

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Bai, C., Guo, L. & Ni, X. Nonabelian Generalized Lax Pairs, the Classical Yang-Baxter Equation and PostLie Algebras. Commun. Math. Phys. 297, 553–596 (2010). https://doi.org/10.1007/s00220-010-0998-7

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