Generalizations of the classical Yang-Baxter equation and ${\mathcal O}$O-operators

Abstract

Tensor solutions (r-matrices) of the classical Yang-Baxter equation (CYBE) in a Lie algebra, obtained as the classical limit of the R-matrix solution of the quantum Yang-Baxter equation, is an important structure appearing in different areas such as integrable systems, symplectic geometry, quantum groups, and quantum field theory. Further study of CYBE led to its interpretation as certain operators, giving rise to the concept of \documentclass[12pt]{minimal}\begin{document}${{\mathcal O}}$\end{document}O-operators. The \documentclass[12pt]{minimal}\begin{document}${\mathcal O}$\end{document}O-operators were in turn interpreted as tensor solutions of CYBE by enlarging the Lie algebra [Bai, C., “A unified algebraic approach to the classical Yang-Baxter equation,” J. Phys. A: Math. Theor. 40, 11073 (2007)]10.1088/1751-8113/40/36/007. The purpose of this paper is to extend this study to a more general class of operators that were recently introduced [Bai, C., Guo, L., and Ni, X., “Nonabelian generalized Lax pairs, the classical Yang-Baxter equation and PostLie algebras,” Commun. Math. Phys. 297, 553 (2010)]10.1007/s00220-010-0998-7 in the study of Lax pairs in integrable systems. Relations between \documentclass[12pt]{minimal}\begin{document}${\mathcal O}$\end{document}O-operators, relative differential operators, and Rota-Baxter operators are also discussed.