Rota-baxter Operators On Three Kinds Of 3-dimensional Lie Algebra And Jordan D-bialgebras In Low Dimensions

Lie algebra and Jordan algebra are important nonassociative algebras.Rota-Baxter operator is equivalent to the solution of classical Yang-Baxter equation in Lie algebra.In this paper we discuss the Rota-Baxter operators on three kinds of 3-dimensional Lie algebras except the 3-dimensional Abel Lie algebra,the Heisenberg Lie algebra and sl(2,C).We also discuss Jordan D-bialgebras in dimensions four and six.This paper is organized as follows:In section 1 we give some basic definitions and properties,such as the definitions of pre-Lie algebra,Rota-Baxter operator,O-operator,the classical Yang-Baxter equation,Jordan algebra and so on.At the same time we introduce the construction of Jordan D-bialgebra.In section 2 we compute the Rota-Baxter operators on 3-dimensional Lie algebras.First we find the classification of 3-dimensional Lie algebras on complex field,then we calculate Rota-Baxter operators on three kinds of 3-dimensional Lie algebras.We get the classification of Rota-Baxter operators on one kind of 3-dimensional Lie algebra,the classification of Rota-Baxter operators on one kind of 3-dimensional Lie algebra in a special case.We also get some examples of Rota-Baxter operators on one kind of 3-dimensional Lie algebra.In section 3 we get the solution of classical Yang-Baxter equation on the direct sum of Lie algebra and its representation space in tensor form using the Rota-Baxter operators on 3-dimensional Lie algebras in section 2.In section 4 we construct pre-Lie algebras on 3-dimensional Lie algebras using the Rota-Baxter operators on Lie algebras.In section 5 we give Jordan D-bialgebra in dimensions four and six according to the classification of Jordan algebra in dimensions two and three.First we find the construction of coboundary Jordan D-bialgebras and the classification of Jordan algebra in low dimensions,then we get the operations of coboundary Jordan D-bialgebra using the matched pair of Jordan algebra.