Commuting Maps On Kac-Moody Algebras And Their Subalgebras

An important way to study its structure of an algebra is to describe relations of its elements via its linear transformations. Its automorphisms and derivations are usual linear transformations. In recent years, the commuting mapping is also used to study the structure of associative algebras and Lie algebras. The main goal of this thesis is to determine the commuting mappings on a Kac-Moody algebra and its subalgebra, and so their internal structures arc described. The thesis is divided into the following five chapters:In Chapter One, we introduce the basic concepts and notations on Kac-Moody algebra and their subalgebras.In Chapter Two, we introduce nonlinear strongly commuting mappings on parabol-ic subalgebras of finite dimensional simple Lie algebras. Firstly, we give the concept of nonlinear strongly commuting mapping. Then, by analysis of actions of nonlinear strong-ly commuting mappings on a fixed base of the parabolic subalgebras, we prove that any nonlinear strongly commuting mapping is a scalar multiplication map.In Chapter Three, we determine the commuting automorphisms and commuting deriva-tions on Kac-Moody algebras. Further more, we prove that all commuting automorphisms constitute a group, which is a normal subgroup of the corresponding automorphism group. And all commuting derivations constitute a Lie algebra, which is a Lie ideal of the corre-sponding derivation Lie algebra.In Chapter Four, we determine the commuting automorphisms and commuting deriva-tions on Borel subalgebras of Kac-Moody algebras. Further more, we prove that all com-muting automorphisms constitute a group, which is a normal subgroup of the corresponding automorphism group. And all commuting derivations constitute a Lie algebra, which is a Lie ideal of the corresponding derivation Lie algebra.In Chapter Five, we determine the commuting derivations of the positive part on non-twisted affine Kac-Moody algebras of type Al, Dl and El. By analysis of actions of commuting derivations on a fixed base of the positive part, we prove that any commuting derivation of the positive part is a zero mapping.