An operator approach to the rational solutions of the classical Yang-Baxter equation

TL;DR

This paper introduces a new operator-based method to construct rational solutions of the classical Yang-Baxter equation, extending previous operator forms and providing a systematic approach for Lie algebras.

Contribution

It defines $ ext{O}$-operators linked to various representations and generalizes the construction of rational CYBE solutions using these operators.

Findings

Defined $ ext{O}$-operators for Lie algebra representations.

Constructed rational solutions from $ ext{O}$-operators associated to coadjoint and trivial product representations.

Extended the operator approach to Lie algebras with invariant bilinear forms.

Abstract

Motivated by the study of the operator forms of the constant classical Yang-Baxter equation given by Semonov-Tian-Shansky, Kupershmidt and the others, we try to construct the rational solutions of the classical Yang-Baxter equation with parameters by certain linear operators. The fact that the rational solutions of the CYBE for the simple complex Lie algebras can be interpreted in term of certain linear operators motivates us to give the notion of $\mathcal O$-operators such that these linear operators are the $\mathcal O$-operators associated to the adjoint representations. Such a study can be generalized to the Lie algebras with nondegenerate symmetric invariant bilinear forms. Furthermore we give a construction of a rational solution of the CYBE from an $\mathcal O$-operator associated to the coadjoint representation and an arbitrary representation with a trivial product in the representation space respectively.