Rota-Baxter operators on $\mathrm{sl(2,\mathbb {C})}$ sl (2,C) and solutions of the classical Yang-Baxter equation

Abstract

We explicitly determine all Rota-Baxter operators (of weight zero) on $\mathrm{sl(2,\mathbb {C})}$ sl (2,C) under the Cartan-Weyl basis. For the skew-symmetric operators, we give the corresponding skew-symmetric solutions of the classical Yang-Baxter equation in $\mathrm{sl(2,\mathbb {C})}$ sl (2,C), confirming the related study by Semenov-Tian-Shansky. In general, these Rota-Baxter operators give a family of solutions of the classical Yang-Baxter equation in the six-dimensional Lie algebra $\mathrm{sl(2,\mathbb {C})}\ltimes _{{\rm ad}^{\ast }} \mathrm{sl(2,\mathbb {C})}^{\ast }$ sl (2,C)⋉ ad * sl (2,C)*. They also give rise to three-dimensional pre-Lie algebras which in turn yield solutions of the classical Yang-Baxter equation in other six-dimensional Lie algebras.