Left Symmetric Algebra

This paper briefly tells us the basic theory of left-symmetric algebra(LSA) over an algebraically closed field with characteristic 0.There are six chapters. In chapter 1, by comparing left-invariant affine structure on Lie group and left-symmetric structure on lie algebra, we relate the famous Milnor question with left-symmetric algebra, which shows the meaning of studying the left-symmetric algebra.In chapter 2,the main algebraic theory of LSA is given,including the definition and basic properties of radical,derivation,inner derivation.Besides,we introduce four kinds of left-symmetric algebras with additional conditions,they are:complete left symmetric algebra,Novikov algebra,bi-symmetric algebra and filiform left symmetric algebra.At the end of the chapter,main results of sub-adjacent lie algebra are given.In chapter 3,we explicitly discuss the conditions for the existence of left-symmetric structure and list the known main results of classification.In chapter 4, we discuss the cohomology theory of left symmetric algebra and its application.The algebras mentioned in the previous chapters are mainly finite- dimensional,in chapter 5,we discuss the left symmetric structures on several important infinite-dimensional lie algebra.they are:Witt algebra,Virasoro algebra,super-Virasoro,W algebra W(2,2) and conformal current-type lie algebra, Schrodinger-Virasoro algebra.we will discuss their relation with LSAs.In the last chapter,we give a survey of the related fields.