Infinitesimal Deformation Of The Module Of Poisson Algebras And Compatible Poisson Algebras

In this thesis we mainly study deformation of Poisson algebras and compatible Poisson algebras.At the same time,we define the deformations of Poisson algebras and the infinitesimal deformation of the module of Poisson algebras,study the matched pair of compatible Poisson algebras,and discuss the relations between compatible Novikov-Poisson algebras and compatible transposed Poisson algebras.In the first part,we mainly introduce some basic concepts and conclusions related to this thesis.These include the definition of Poisson algebras,transposed Poisson algebras and so on.In the second part,we study the construction of the special Poisson algebras and transposed Poisson algebras.Firstly,we define the new algebraic operations on Poisson algebras and transposed Poisson algebras,and get the condition for the new operations to be Poisson algebras and transposed Poisson algebras.We construct the module of Poisson algebras and transposed Poisson algebras.We define infinitesimal deformation of the module of Poisson algebras.We discuss the equivalence conditions of two infinitesimal deformations.In the third part,we mainly study the Nijenhuis structure of Poisson algebras.We construct the new Poisson algebras by Nijenhuis operator of Poisson algebras.We define the Nijenhuis operator of Poisson algebras and Nijenhuis structure of Poisson algebras.We find the relations between the trivial deformation of Poisson algebras and the Nijenhuis structure.In the fourth part,we study compatible Poisson algebras.We give the definition of compatible Poisson algebras.We define the module of compatible Poisson algebras.We construct compatible Poisson algebras and compatible transposed Poisson algebras by using the derivation on compatible commutative associative algebras.In the fifth part,we study the matched pair of compatible Poisson algebras.We give the definition and the equivalent condition of compatible commutative associative algebras,and we give the definition and the equivalent condition of compatible Poisson algebras.In the sixth part,we mainly study the relations between compatible Novikov-Poisson algebras and compatible transposed Poisson algebras.Firstly,we give the definition of compatible Novikov-Poisson algebras and compatible transposed Poisson algebras.At the same time,we also give the equivalent condition of compatible Novikov-Poisson algebras and compatible transposed Poisson algebras.Finally,we construct compatible transposed Poisson algebras on compatible Novikov-Poisson algebras.