Rota-Baxter operators on sl(2, ℂ) and solutions of the classical Yang-Baxter equation

We explicitly determine all Rota-Baxter operators (of weight zero) on 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 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 sl(2, ℂ) ⋉_ad* sl(2, ℂ)*. They also give rise to threedimensional pre-Lie algebras which in turn yield solutions of the classical Yang- Baxter equation in other six-dimensional Lie algebras.