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

摘要

We explicitly determine all Rota-Baxter operators (of weight zero) on$sl(2,C)$ under the Cartan-Weyl basis. For the skew-symmetric operators, wegive the corresponding skew-symmetric solutions of the classical Yang-Baxterequation in $sl(2,C)$, confirming the related study by Semenov-Tian-Shansky. Ingeneral, these Rota-Baxter operators give a family of solutions of theclassical Yang-Baxter equation in the 6-dimensional Lie algebra $sl(2,C)\ltimes_{{\rm ad}^{\ast}} sl(2,C)^{\ast}$. They also give rise to 3-dimensionalpre-Lie algebras which in turn yield solutions of the classical Yang-Baxterequation in other 6-dimensional Lie algebras.