Two-dimensional Double Poisson Algebras,Double Lie Algebras And Double Associative Algebras

In this thesis,by using the relationships between Rota-Baxter operators,the average operators and the double algebras,we give all double Lie algebras,double Poisson algebras,double commutative algebras and double associative algebras on the two-dimensional complex vector space.For the two-dimensional complex vector space V,we first determine all Rota-Baxter operators on the quadratic associative algebra End(V).Based on the equivalence between the double Lie algebras and the quadratic Rota-Baxter associative algebras,we give all double Lie algebras on the two-dimensional vector space V.On this basis,we give all double Poisson algebras on the two-dimensional vector space V.Then,for the two-dimensional vector space V,we determine all average operators on the quadratic associative algebra End(V).Based on the equivalence between double commutative algebras and quadratic average associative algebras,we give all double commutative algebras on the two-dimensional vector space V.Finally,for the two-dimensional vector space V,we determine all conjugate average operators on the quadratic associative algebra End(V).Based on the equivalence between double associative algebras and quadratic conjugate average associative algebras,we give all double associative algebras on the two-dimensional vector space V,and show that the two-dimensional double associative algebras can not give all the two-dimensional double Lie algebras by commutators.