Restricted Enveloping Algebras Of Restricted Poisson Algebras

Poisson algebra originates from Hamilton's mechanical system.In the past 50 years,Poisson algebra was always a common research field of algebras and geometries,which plays import.ant roles in theoretical physics,differential geometry,representation theory of algebras,and so on.The properties of Poisson algebras over a field of positive characteristic are different from ones over a field of zero characteristic.In order to develop the deformation quantizat,ion theory over positive characteristic,Bezrukavnikov and Kaledin introduced the notion of restricted Poisson algebras in 1998.In this thesis,we will study restricted Poisson algebras from the viewpoint of representation theory of algebras.Inspired by the restricted Lie modules,we introduce the definitions of restricted Poisson modules and restricted Poisson enveloping algebras,and investigate their basic properties.This thesis consists of the following three parts.In the first chapter,we mainly introduce the background,significance and development.In addition,we also introduce our main questions and results in this thesis.In Chapter 2,we mainly recall some related notions of restricted Lie algebras,restricted Lie Rinehart algebras and their rest,ricted enveloping algebras.It is worth to mention the Poisson enveloping algebra can be constructed by the ones of the corresponding Lie-Rinehart algebra.In Chapter 3,we firstly recall the notions of Poisson algebra,Poisson modules and Poisson enveloping algebras and some wellknown results.Secondly,we introduce the definition of restricted Poisson modules and study its basic properties.Finally,we constructed the corresponding restricted Poisson enveloping algebra and give its another equivalent definition by smash product.Furthermore,the universal property of restricted Poisson enveloping algebras are obtained.