L-operators And Lb-equation On Leibniz Algebras

In our paper,we mainly study when Lie algebras are generalized to Leibniz algebras,the operators and the equation which corresponding to O-operators and Classcial Yang-Baxter equation(CYBE) in Lie algebras,that is,the L-operators and Lb-equation on Leibniz algebras.We give the definitions of L-operators and Lb-equation and we also research the relations between them.At the same time,by studying the relations between Lie algebras and pre-Lie algebras,we give the notion of quasi-pre-Lie algebra,and we also study the relations between the quasi-pre-Lie algebra and Lb-equation.Furthermore,we get two subsidiary conclusions when we study the bimodules and dual bimodules of Leibniz algebras:one is we use the bimodule to get a necessary and sufficient condition about matched pair construction by two Leibniz algebras and the other is we use the dual bimodule to get an important nondegenerate skew-symmetric bilinear form on Leibniz algebras.