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Conformal classical Yang–Baxter equation, S-equation and \({\mathcal {O}}\)-operators

  • Yanyong HongEmail author
  • Chengming Bai
Article
  • 15 Downloads

Abstract

Conformal classical Yang–Baxter equation and S-equation naturally appear in the study of Lie conformal bialgebras and left-symmetric conformal bialgebras. In this paper, they are interpreted in terms of a kind of operators, namely \(\mathcal O\)-operators in the conformal sense. Explicitly, the skew-symmetric part of a conformal linear map T where \(T_0=T_\lambda \mid _{\lambda =0}\) is an \({\mathcal {O}}\)-operator in the conformal sense is a skew-symmetric solution of conformal classical Yang–Baxter equation, whereas the symmetric part is a symmetric solution of conformal S-equation. One by-product is that a finite left-symmetric conformal algebra which is a free \({\mathbb {C}}[\partial ]\)-module gives a natural \({\mathcal {O}}\)-operator, and hence, there is a construction of solutions of conformal classical Yang–Baxter equation and conformal S-equation from the former. Another by-product is that the non-degenerate solutions of these two equations correspond to 2-cocycles of Lie conformal algebras and left-symmetric conformal algebras, respectively. We also give a further study on a special class of \({\mathcal {O}}\)-operators called Rota–Baxter operators on Lie conformal algebras, and some explicit examples are presented.

Keywords

Lie conformal algebra Left-symmetric conformal algebra Conformal CYBE Conformal S-equation \({\mathcal {O}}\)-operator Rota–Baxter operator 

Mathematics Subject Classification

17A30 17B62 17B65 17B69 

Notes

Acknowledgements

This work was supported by the National Natural Science Foundation of China (11425104, 11501515, 11931009), the Zhejiang Provincial Natural Science Foundation of China (LY20A010022) and the Scientific Research Foundation of Hangzhou Normal University (2019QDL012). C. Bai is also supported by the Fundamental Research Funds for the Central Universities and Nankai ZhiDe Foundation. This work was carried out during the first author’s stay at Chern Institute of Mathematics, Tianjin, China, from April 10 to April 24, 2016, and he would like to thank the CIM for its support and hospitality.

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Copyright information

© Springer Nature B.V. 2019

Authors and Affiliations

  1. 1.Department of MathematicsHangzhou Normal UniversityHangzhouPeople’s Republic of China
  2. 2.Chern Institute of Mathematics and LPMCNankai UniversityTianjinPeople’s Republic of China

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