Research On Preserver Problems On Lie Algebras

A dynamical system is a phase space with a self-map.In algebraic words,a dynamical system is a category with the feature of ordered maps.In fact,in studying dynamical system, the phase space,the self-map and the dynamical system itself are limned by algebraic structure.Therefore,it is necessary to investigate the common algebraic systems and their maps with the basic significance for dynamical system.Classical groups,Lie algebras and matrix algebras are the familiar algebraic systems. By the theory of root systems,some results of Chevalley groups and some skill of matrix computation,this thesis focuses on some maps on Lie algebras and matrix algebras.This dissertation consists of seven chapters.In Chapter 1,the author introduces the background,the significance and the development of the subject chosen.Besides these,the main methods employed and the structure of this thesis are explained.In Chapter 2,using some results of Chevalley groups,the theory of root systems and root space decomposition,the author characterizes some preserver maps on complex simple Lie algebras.Let g be an arbitrary complex simple Lie algebra.The main results in Chapter 2 are as follows.(1)An invertible linear map on g preserves solvability in both directions if and only if it can be decomposed into the product of an inner automorphism,a graph automorphism,a diagonal automorphism and a scalar multiplication map.(2)A bijective map(without linearity assumption)on g preserves Lie products if and only if it is a composition of an inner automorphism,a graph automorphism,a diagonal automorphism and a bijective map extended by an automorphism of the complex field.(3)A bijective map(?)(without linearity assumption)on a Borel subalgebra of(?)preserves ad-nilpotent ideals if and only if(?)takes the form:(?)=(?)·(?)_～,whereφis a symmetry of the Dynkin diagram of(?),(?)_φis an automorphism of the partial ordering set (?)~+extended fromφ,and～is an equivalence relation of the Borel subalgebra.(4)An invertible linear map on(?)preserves standard parabolic subalgebras if and only if it can be decomposed into the product of an invertible linear map induced by a symmetry of the Dynkin diagram of(?)and an invertible linear map induced by a permissible set.A bijective(without linear assumption)map on(?)preserves standard parabolic subalgebras if and only if it can be decomposed into the product of an invertible linear map induced by a symmetry of the Dynkin diagram of(?)and an invertible map on(?)preserving lattices.In Chapter 3,the author determines the linear maps preserving zero Lie brackets the maximal nilpotent subalgebras of the symplectic algebra and the orthogonal algebra,respectively. The main results in Chapter 3 are as follows. (1)Let m≥4.A linear map(?)on a maximal nilpotent subalgebra of the symplectic algebra preserves zero Lie brackets in both directions if and only if(?)can be decomposed into the product of a scalar multiplication map,an inner automorphism,an extremal map,a generalized diagonal automorphism and a central map.(2)Let m≥5.A linear map(?)on a maximal nilpotent subalgebra of the orthogonal algebra preserves zero Lie brackets in both directions if and only if(?)can be decomposed into the product of a scalar multiplication map,a graph automorphism,an inner automorphism, a sub-central automorphism,an extremal map,a generalized diagonal automorphism and a central map.In Chapter 4,based on the results of Chapter 3,the author characterizes the linear maps that preserve subalgebras consisting of nilpotent elements on the symplectic algebra and the orthogonal algebras.It is proved that an invertible linear mapφon the symplectic algebra or the orthogonal algebra preserves subalgebras consisting of nilpotent elements if and only ifφcan be decomposed into the product of a scalar multiplication map,an inner automorphism,a graph automorphism and a generalized diagonal automorphism.In Chapter 5,the author studies the linear maps preserving square-zero matrices on upper triangular matrix algebras and strictly upper triangular matrix algebras.Let T_n(F) (resp.,N_n(F))be the upper triangular matrix algebra(resp.,strictly upper triangular matrix) algebra over a field F.Firstly,nine types of standard nonsingular linear maps on N_n(F) preserving square-zero matrices in both directions are constructed,then each nonsingular linear map on N_n(F)preserving square-zero matrices in both directions is characterized by using these maps.Finally,nonsingular linear maps on T_n(F)preserving square-zero matrices in both directions and nonsingular linear maps on N_n(F)preserving zero products of matrices in both directions are determined as applications of the result about square-zero preservers on N_n(F).In Chapter 6,the author discusses some local maps,biderivations and Lie triple derivations of upper triangular matrix algebras over commutative rings.Let R be a commutative ring with identity,T_n(R)the upper triangular matrix algebra over R.In this chapter,it is proved that every local Lie derivation of T_n(R)is a Lie derivation and every biderivation of T_n(R)is the sum of an inner biderivation and an extremal biderivation.When 2 is a unit in R,then every local Lie automorphism of T_n(R)is a Lie automorphism and every Lie triple derivation of T_n(R)can be written as the sum of an inner derivation and a central Lie derivation.Core conclusions are summarized in the last chapter,accompanied with the direction of the future research.