Generalizations of the classical Yang-Baxter equation and O-operators

TL;DR

This paper explores generalized operators related to the classical Yang-Baxter equation, extending the concept of O-operators and examining their connections to various algebraic structures in integrable systems.

Contribution

It introduces a broader class of operators linked to CYBE, generalizing O-operators, and discusses their relationships with differential and Rota-Baxter operators.

Findings

Extended the framework of O-operators to new classes of operators.

Established connections between $\\calo$-operators, differential operators, and Rota-Baxter operators.

Provided insights into the algebraic structures underlying integrable systems.

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 (QYBE), 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 O-operators. In [3], the O-operators were in turn interpreted as tensor solutions of CYBE by enlarging the Lie algebra. The purpose of this paper is to extend this study to a more general class of operators that were recently introduced [4] in the study of Lax pairs in integrable systems. Relationship between $\calo$-operators, relative differential operators and Rota-Baxter operators are also discussed.