On connection between Rota-Baxter operators and solutions of the classical Yang-Baxter equation with an ad-invariant symmetric part on general linear algebra

Abstract

In the paper, we find the connection between solutions of the classical Yang-Baxter equation with an ad-invariant symmetric part and Rota-Baxter operators of special type on a real general linear algebra $gl_n(\mathbb R)$. Using this connection, we classify solutions of the classical Yang-Baxter equation with an ad-invariant symmetric part on $gl_2(\mathbb C)$ using the classification of Rota-Baxter operators of nonzero weight on $gl_2(\mathbb C)$ and a classification of Rota-Baxter operators of weight 0 on $sl_2(\mathbb C)$