Biderivations And Left-symmetric Algebra Structures On The Planar Galilean Conformal Algebra

In this thesis,we mainly study the structure of the planar Galilean conformal algebra.Galilean conformal algebra(GCA)has been studied in the range of non-relativistic AdS/CFT conjectures in recent years.Planar Galilean conformal algebra is an extension of Galilean con-formal algebra(GCA).And the structures and representations of the planar Galilean conformal algebra has been studied by many scholars.Based on these,we study biderivations,linear com-muting maps and left-symmetric algebra structures on the planar Galilean conformal algebra.There are three chapters,as follows:In the first chapter,it is the introduction,which is divided into three sections.The back-ground of the research is introduced in the first section.In the second section,we give some basic knowledge.In order to simplify the writing,we give the symbol descriptions in the third section.There are three sections in the second chapter,we mainly study the biderivations and linear commuting maps on the planar Galilean conformal algebra.In Section 1,we mainly introduce some basic concepts and general conclusions.In Section 2,we determine that all the skew-symmetric biderivations of the planar Galilean conformal algebra g are inner bideriva-tions.Finally,in the last section,based on the result of skew-symmetric biderivations,we will give the form of each linear commuting map on the planar Galilean conformal algebra g,that is,they are standard.In the third chapter,we mainly determine the compatible left-symmetric algebra struc-tures on the planar Galilean conformal algebra with some natural grading conditions.In Sec-tion 1,we give the basic concepts and the results about left-symmetric algebraic structure of the twisted Heisenberg-Virasoro algebra which are used in our discussions.And we give the main results of this chapter.In the first subsection of Section 2,we classify the left-symmetric algebra structures on the centerless g under certain natural assumptions.In the second subsec-tion of Section 2,we determine the nontrivial central extensions of the left-symmetric algebras on the basis of the previous subsection.