Rota-Baxter operators and non-skew-symmetric solutions of the classical Yang-Baxter equation on quadratic Lie algebras

TL;DR

This paper explores the relationship between Rota-Baxter operators and non-skew-symmetric solutions of the classical Yang-Baxter equation on quadratic Lie algebras, focusing on invariant solutions.

Contribution

It establishes connections between Rota-Baxter operators of non-zero weight and specific solutions of the classical Yang-Baxter equation on quadratic Lie algebras.

Findings

Identifies conditions for solutions where r + τ(r) is L-invariant.

Analyzes the structure of Rota-Baxter operators in relation to Yang-Baxter solutions.

Provides classifications of solutions under certain invariance conditions.

Abstract

We study possible connections between Rota-Baxter operators of non-zero weight and non-skew-symmetric solutions of the classical Yang-Baxter equation on finite-dimensional quadratic Lie algebras. The particular attention is made to the case when for a solution $r$ the element $r+\tau(r)$ is $L$-invariant.