Rota-baxter Operators On 6-dimensional Nilpotent Lie Algebras

In the Lie algebras cases,the Rota-Baxter operators with the weight 0 are the solutions of the classical Yang-Baxter equations and the Rota-Baxter operators with the weight 1 are the solutions of the deformed Yang-Baxter equations.In this paper,all classifications of Rota-Baxter operators with the weight 0 of six classses of six-dimensional nilpotent Lie algebras are determined.Meantime,we give all the solutions of the classical Yang-Baxter equations of L2,L3,L4,L19,L24,L26 these six series of algebras and their relatives,respectively.In this paper,we transform the solutions of such problems into solve solutions of the systems of multivariable quadratic equations,by observing the characteristics of the equations and classifying the equations and then we obtain all the solutions of the classical Yang-Baxter equations,that is,all the matrix forms of Rota-Baxter operators.