Linear Commuting Maps On Finite-dimensional Simple Lie Algebras And Their Subalgebras

An important way of studying the algebraic structure of a Lie algebra is to describe connections of its internal elements by the actions of its linear maps. Recently,it has become a new hot topic on studying linear commuting maps of a Lie algebra. The main purpose of this essay is to determine all the commuting linear maps on finite-dimensional simple Lie algebras or their subalgebras. In this way their internal structures become more clear. The essay is divided into the following four chapters:Chapter One is to show out basic notations and important results about finite di-mensional simple Lie algebras and their subalgebras, which are necessary for the following chapters.Chapter Two aims to determine the linear commuting maps on parabolic subalgebras of finite-dimensional simple Lie algebras. At first, we introduce the concept of commuting map. Then analyzing actions of commuting maps on the basis elements of these parabolic subalgebras, we decide all the linear commuting maps of these subalgebras. It is shown that all commuting maps should be scalar multiplication maps. As corollaries, the commuting automorphisms and derivations are determined.Chapter Three is mainly to decide the linear commuting maps on the Lie algebra of strictly upper triangular matrices over commutative rings. If the commutative ring is an algebraically closed field of characteristic zero, such Lie algebra is isomorphic to the maximal nilpotent Lie algebra of the finite-dimensional simple Lie algebra of type A. We determine the linear commuting maps by analyzing their actions on basis elements of such Lie algebra. It is shown that any commuting map of such Lie algebra should be a sum of a scalar multiplication map, an extremal map and a central map. By introducing the concept of graded diagonal automorphisms and graded diagonal derivations, we decide the automorphisms and derivations of such Lie algebra.Chapter Four is devoted to determining the automorphisms and derivations of nilpo-tent Lie algebras of finite-dimensional simple Lie algebras over commutative rings. We determine the concrete forms of the automorphisms and derivations. If the commutative rings are algebraically closed fields of characteristic zero, the above Lie algebras are isomor-phic to the maximal nilpotent Lie algebras of finite-dimensional simple Lie algebras. So the concrete forms of automorphisms and derivations of these maximal nilpotent Lie algebras are determined.