Advances in Theoretical and Mathematical Physics

Volume 26 (2022)

Number 7

Operator forms of the nonhomogeneous associative classical Yang–Baxter equation

Pages: 1965 – 2009



Chengming Bai (Chern Institute of Mathematics & LPMC, Nankai University, Tianjin, China)

Xing Gao (School of Mathematics and Statistics, Lanzhou University, Lanzhou, China; Gansu Provincial Research Center for Mathematics and Statistics, Lanzhou, China; and School of Mathematics and Statistics, Qinghai Nationalities University, Xining, China)

Li Guo (Department of Mathematics and Computer Science, Rutgers University, Newark, New Jersey, U.S.A.)

Yi Zhang (School of Mathematics and Statistics, Center for Applied Mathematics of Jiangsu Province, Nanjing University of Information Science & Technology, Nanjing, Jiangsu, China)


This paper studies operator forms of the nonhomogeneous associative classical Yang–Baxter equation (nhacYBe), extending and generalizing such studies for the classical Yang–Baxter equation and the associative Yang–Baxter equation that can be traced back to the works of Semenov–Tian–Shansky and Kupershmidt on Rota–Baxter Lie algebras and $\mathcal{O}$–operators. Solutions of the nhacYBe are characterized in terms of generalized $\mathcal{O}$–operators, and in terms of the classical $\mathcal{O}$–operators precisely when the solutions satisfy an invariant condition. When the invariant condition is compatible with a Frobenius algebra, such solutions have a close relationship with Rota–Baxter operators on the Frobenius algebra. In general, solutions of the nhacYBe can be produced from Rota–Baxter operators, and then from $\mathcal{O}$–operators when the solutions are taken in semi–direct product algebras. In the other direction, Rota–Baxter operators can be obtained from solutions of the nhacYBe in unitizations of algebras. Finally a classification is obtained for solutions of the nhacYBe satisfying the mentioned invariant condition in all unital complex algebras of dimensions two and three. All these solutions are shown to come from Rota–Baxter operators.

This work is supported by National Natural Science Foundation of China (11931009, 12271265, 12101316, 12071191, 12261131498). C. Bai is also supported by the Fundamental Research Funds for the Central Universities and Nankai Zhide Foundation. X. Gao is also supported by Innovative Fundamental Research Group Project of Gansu Province(23JRRA684). Y. Zhang is also supported by China Scholarship Council to visit University of Southern California and he thanks Professor Susan Montgomery for hospitality during the visit.

Published 30 August 2023