Classification Of 3-dimensional Rota-Baxter Algebras(Coalgebras) And Bihom-Lie Bialgebras

In this paper,in chapter 2 and 3,our aim is to determine the Rota-Baxter operators of weight 0 on the 3-dimensional Lie algebras whose derived algebra's dimension is 2.Thus,by[31],we obtain the Rota-Baxter operators on all the 3-dimensional Lie algebras.Furthermore,we give the corresponding solutions of the classical Yang-Baxter equation in the 6-dimensional Lie algebras g×ad*g*and the induced left-symmetric algebra structures from the Rota-Baxter operators on g.In chapter 4,we mainly study Rota-Baxter oper-ators(of weight zero)on pre-Lie coalgebras.At the same time,we study the invertible Rota-Baxter operators on pre-Lie coalgebras.So we give some fundamental results on both invertible Rota-Baxter relations and derivations on pre-Lie coalgebras.In chapter 5,motivated by Yau's work on Hom-Lie algebras and the Hom-Yang-Baxter equation[40],we introduce a two deformed parameters version of the Yang-Baxter equation,called the Classical Bihom-Yang-Baxter equation(CBHYBE),and give the relations between the Bihom-Lie bialgebras and CBHYBE.Several classes of the solutions of the CBHYBE are constructed,which come from Bihom-Lie bialgebras.Finally,we study coboundary and quasi-triangular Bihom-Lie bialgebras.