Differentially Divided Poissonian Algebra And Comodule

In this paper,we propose a definition of differential graded Poisson coalgebras,which can be regard as a “combination” of a differential graded coalgebras and a graded Poisson coalgebras.At first,we review some basic definitions of graded coalgebras and graded Lie-coalgebras.At that basis,we extend the definitions of differential graded coalgebras and graded Poisson coalgebras.Then we give the definition of differential graded Poisson coalgebras in the combination of them.After giving the definition,we have a specific discussion about the basic properties of differential graded Poisson coalgebras.For example,let(C,?,?)be any differential graded Poisson coalgebra,then(Ccop,?cop,?cop,?cop)also is a differential graded Poisson coalgebra;let(C,?C,?D,?C)and(D,?D,?D,?D)be any differential graded Poisson coalgebras,then(C ? D,?,?,?)is also a differential graded Poisson coalgebra,the category of differential graded Poisson coalgebras is a symmetric monoidal category.Then the comodule structure of differential graded Poisson coalgebras is naturally noticed.We first give the definition of differential graded Poisson comodules on the basis of coalgebras structure,and it is dual to the module structure of differential graded Poisson algebras.Then we discuss the properties of differential graded Poisson comodule.The situation of comodules is as similar as coalgebras.At the end of this article,we introduce the universal enveloping coalgebras of differential graded Poisson coalgebras and give some basic properties of it.