| In recent years,the applications of operad theory have attracted extensive attention of modern mathematiciansin structural theory,cohomology theory and deformation theory of algebra.Poisson structures also play an increasingly important role in differential geometry,algebraic representation theory and so on.In this thesis,we mainly study the trunction ideal of Poisson operad,the Gelfand-Kirillov dimension of Poisson operad quotient,the classification of the small Gelfand-Kirillov dimension of Poisson operad quotient and explore the irreducible decomposition method of Poisson operad.The main structure of this thesis is as follows:In the first chapter,we introduce the research background and significance of some related work,and give the main problems and results of this study.In the second chapter,we mainly introduce two definitions of operad,and review the ideal of operad and the trunction ideal of operad.In the third chapter,we firstly study Lie algebra and Lie operad.Secondly,we introduce Poisson algebra and Poisson operad,and give the basic properties of Poisson operad.Finally,it is proved that the trunction ideal chain of Poisson operad is a special descending chain.In the fourth chapter,we firstly give the concept of Gelfand-Kirillov dimension and some relevant conclusions.Secondly,we calculate the Gelfand-Kirillov dimension of Poisson operad quotient by the relation between ideal and the trunction ideal.Finally,we study the classification of the quotient of Poisson operad when the dimension of Gelfand-Kirillov is less than or equal to four,and pointed out that the quotient of Poisson operad is not unique when the dimension of small Gelfand-Kirillov is greater than four.In the fifth chapter,we firstly give an algorithm to find the common invariant subspace of many linear transformations in linear space,whose dimension is specified.Based on this algorithm,we take Pois(3)as an example to understand the steps and methods of solving the irreducible decomposition of Poisson operad. |
PDF Full Text Request