Mathematical logic, algebra and number theory

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

M. E. Goncharov

Sobolev Institute of Mathematics, Academician Koptyug avenue, 4, 630090, Novosibirsk, Russia

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 $sl_2(\mathbb C)$.

Keywords: Lie bialgebra, Rota—Baxter operator, classical Yang—Baxter equation, general linear Lie algebra.

UDC: 512.554

MSC: 17B38

Received August 14, 2023, published February 14, 2024

Language: English

DOI: 10.33048/semi.2024.21.007