李代数上保持问题的研究
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摘要
所谓动力系统就是由状态空间及其上的自映射所构成的系统,从代数学角度看,动力系统是一个具有有序态射特征的范畴.代数结构对动力系统的意义不仅表现为代数系统结构的复杂性与多样性可以形成更丰富的动力系统,而且其本身也是研究和刻画动力系统的基本工具之一.代数结构对动力系统的刻画涵盖了相空间,自映射以及动力系统本身.因此,探寻动力系统中具有基本意义的常见的代数系统及其上的映射具有重要的意义.
典型群,李代数,李型群及矩阵代数是常见的代数系统.本学位论文在利用根系理论,李型单群理论和矩阵方法的基础上,重点对李代数和矩阵代数的一些映射进行了研究.本学位论文共分为七章:
第一章是绪论部分,介绍了论文的选题意义,选题背景,课题的研究现状,论文的结构及主要研究方法.
第二章研究了复单李代数上的一些线性的和非线性的保持问题.矩阵代数上保持问题的研究已基本成熟,许多学者积极倡导把保持问题做到其它抽象的代数系统上.李代数上保持问题的研究至今仍停留在典型李代数阶段,抽象李代数上保持问题的研究尚无人涉足,这正是本章研究的主要动机.这一章综合利用李型单群中的一些重要结果,李代数的根系理论,根空间分解理论以及轨道技巧等研究了任意复单代数上的保可解性,保李括积,保ad-幂零理想,保标准抛物子代数的映射,得到了如下结论:
(1)复单李代数上的一个可逆线性映射是双向保可解性的当且仅当它可以写成一个内自同构,一个图自同构,一个标量乘法映射和一个对角自同构的乘积.
(2)复单李代数上的一个非线性可逆映射是保李括积的当且仅当它可以写成一个内自同构,一个图自同构,一个对角自同构和一个由复数域的自同构诱导的双射的乘积.
(3)复单李代数的Borel子代数上的一个非线性可逆映射(?)是保ad-幂零理想的当且仅当(?)=(?)_(?)·(?)_～,这里φ是根系(?)的Dynkin图的一个对称,(?)_(?)是由φ扩充得到的偏序集(?)+上的一个自同构,～是Borel子代数的一个等价关系.
(4)复单李代数上的一个非线性可逆映射是保标准抛物子代数的当且仅当它可以写成一个由Dynkin图的对称诱导的可逆线性映射和一个保格的可逆映射的乘积;复单李代数上的一个可逆线性映射是保标准抛物子代数的当且仅当它可以写成一个由根系(?)的Dynkin图的对称诱导的可逆线性映射和一个由容许集诱导的可逆线性映射的乘积.
第三章分别讨论了辛代数和D_m型正交代数的极大幂零子代数上的保括积零的线性映射.自同构问题一直是典型李代数研究的一个重要课题,而研究保可换(在这里即保括积零)线性映射是线性保持问题中一类很重要的问题.自然产生的一个问题是典型李代数上保括积零的线性映射与其李自同构的区别有多大.为此本章研究了辛代数和正交代数的极大幂零子代数上的保括积零的线性映射,这也就把文献[]中的结果由特殊线性李代数的极大幂零子代数分别推广到了辛代数和D_m型正交代数的极大幂零子代数上.由于B_m型正交代数的根系与辛代数的根系对偶,因此由本章所得的结果也不难知道B_m型正交代数上相应的结果.这就使得关于典型李代数的极大幂零子代数上保括积零线性映射的研究比较完善.这一章的主要结论如下:
(1)设m≥4,(?)是辛代数sp(2m,F)的一个极大幂零子代数的双向保括积零的线性映射当且仅当(?)能够写成一个标量乘法映射,一个内自同构,一个极端映射,一个推广的对角自同构和一个中心映射的乘积.
(2)设m≥5,(?)是正交代数o(2m,F)的一个极大幂零子代数的双向保括积零的线性映射当且仅当(?)能够写成一个标量乘法映射,一个图自同构,一个内自同构,一个次中心自同构,一个极端映射,一个推广的对角自同构和一个中心映射的乘积.
这一章得到了三种既不是李自同构也不是李反自同构的保括积零的线性映射,其中的极端映射与李自同构有很大的区别.
第四章利用第三章的结果刻画了辛代数和正交代数的保幂零元子代数的线性映射,这在一定程度上把文献[]的结果从特殊线性李代数推广到了辛代数和正交代数上,同时也为下一步研究典型李代数上保幂零子代数的线性映射打下了基础.在该章证明了辛代数或正交代数上的一个可逆线性映射φ是保幂零元子代数的当且仅当φ可以写成一个标量乘法映射,一个内自同构,一个图自同构和一个推广的对角自同构的乘积.
第五章研究了上三角和严格上三角矩阵代数上的保平方零矩阵的可逆线性映射.在线性保持问题中,最基础的问题是研究保秩映射,保幂等映射,保幂零(平方零)映射,这是由于很多其它的保持问题都可以化归为或借助于这些保持映射的结果解决.在这一章首先构造了域F上严格上三角矩阵代数N_n(F)的九种双向保平方零矩阵的可逆线性映射,然后利用这些映射详细刻画了N_n(F)的每一个双向保平方零矩阵的线性映射,最后利用这一结果分别刻画了N_n(F)的保矩阵乘积零的可逆线性映射和上三角矩阵代数T_n(F)的双向保平方零矩阵的可逆线性映射.
矩阵代数上的线性保持映射大多数被证明是一些同构或反同构的乘积,很少有例外的情形发生.而这一章得到了五种既不是自同构又不是反自同构的N_n(F)的保平方零矩阵的线性映射.
第六章研究了可换环上上三角矩阵代数的一些局部映射,双导子和李三次导子.上世纪九十年代,许多学者开始研究一类新的保持问题——局部映射,其目的在于研究线性映射的性质在什么条件下能由它们的局部性质来确定.双导子和李三次导子的研究起源于近几十年来学者们所关注的一个问题,即一个映射在什么条件下成为导子.在这一章得到了如下结论:
含单位元的可换环R上的上三角矩阵代数T_n(R)的每一个局部李导子是李导子,每一个双导子都可以写成一个内双导子与一个极端双导子的和;当2是R中的可逆元时,T_n(R)每一个局部李自同构是李自同构,每一个李三次导子都可以写成一个内导子与一个中心李导子的和.
由此可见T_n(R)的李自同构和李导子在一定意义下完全由他们的局部作用决定,T_n(R)上存在不是内双导子的双导子.
第七章总结了本学位论文的核心结论,并提出了今后进一步的研究目标.
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.
典型群,李代数,李型群及矩阵代数是常见的代数系统.本学位论文在利用根系理论,李型单群理论和矩阵方法的基础上,重点对李代数和矩阵代数的一些映射进行了研究.本学位论文共分为七章:
第一章是绪论部分,介绍了论文的选题意义,选题背景,课题的研究现状,论文的结构及主要研究方法.
第二章研究了复单李代数上的一些线性的和非线性的保持问题.矩阵代数上保持问题的研究已基本成熟,许多学者积极倡导把保持问题做到其它抽象的代数系统上.李代数上保持问题的研究至今仍停留在典型李代数阶段,抽象李代数上保持问题的研究尚无人涉足,这正是本章研究的主要动机.这一章综合利用李型单群中的一些重要结果,李代数的根系理论,根空间分解理论以及轨道技巧等研究了任意复单代数上的保可解性,保李括积,保ad-幂零理想,保标准抛物子代数的映射,得到了如下结论:
(1)复单李代数上的一个可逆线性映射是双向保可解性的当且仅当它可以写成一个内自同构,一个图自同构,一个标量乘法映射和一个对角自同构的乘积.
(2)复单李代数上的一个非线性可逆映射是保李括积的当且仅当它可以写成一个内自同构,一个图自同构,一个对角自同构和一个由复数域的自同构诱导的双射的乘积.
(3)复单李代数的Borel子代数上的一个非线性可逆映射(?)是保ad-幂零理想的当且仅当(?)=(?)_(?)·(?)_～,这里φ是根系(?)的Dynkin图的一个对称,(?)_(?)是由φ扩充得到的偏序集(?)+上的一个自同构,～是Borel子代数的一个等价关系.
(4)复单李代数上的一个非线性可逆映射是保标准抛物子代数的当且仅当它可以写成一个由Dynkin图的对称诱导的可逆线性映射和一个保格的可逆映射的乘积;复单李代数上的一个可逆线性映射是保标准抛物子代数的当且仅当它可以写成一个由根系(?)的Dynkin图的对称诱导的可逆线性映射和一个由容许集诱导的可逆线性映射的乘积.
第三章分别讨论了辛代数和D_m型正交代数的极大幂零子代数上的保括积零的线性映射.自同构问题一直是典型李代数研究的一个重要课题,而研究保可换(在这里即保括积零)线性映射是线性保持问题中一类很重要的问题.自然产生的一个问题是典型李代数上保括积零的线性映射与其李自同构的区别有多大.为此本章研究了辛代数和正交代数的极大幂零子代数上的保括积零的线性映射,这也就把文献[]中的结果由特殊线性李代数的极大幂零子代数分别推广到了辛代数和D_m型正交代数的极大幂零子代数上.由于B_m型正交代数的根系与辛代数的根系对偶,因此由本章所得的结果也不难知道B_m型正交代数上相应的结果.这就使得关于典型李代数的极大幂零子代数上保括积零线性映射的研究比较完善.这一章的主要结论如下:
(1)设m≥4,(?)是辛代数sp(2m,F)的一个极大幂零子代数的双向保括积零的线性映射当且仅当(?)能够写成一个标量乘法映射,一个内自同构,一个极端映射,一个推广的对角自同构和一个中心映射的乘积.
(2)设m≥5,(?)是正交代数o(2m,F)的一个极大幂零子代数的双向保括积零的线性映射当且仅当(?)能够写成一个标量乘法映射,一个图自同构,一个内自同构,一个次中心自同构,一个极端映射,一个推广的对角自同构和一个中心映射的乘积.
这一章得到了三种既不是李自同构也不是李反自同构的保括积零的线性映射,其中的极端映射与李自同构有很大的区别.
第四章利用第三章的结果刻画了辛代数和正交代数的保幂零元子代数的线性映射,这在一定程度上把文献[]的结果从特殊线性李代数推广到了辛代数和正交代数上,同时也为下一步研究典型李代数上保幂零子代数的线性映射打下了基础.在该章证明了辛代数或正交代数上的一个可逆线性映射φ是保幂零元子代数的当且仅当φ可以写成一个标量乘法映射,一个内自同构,一个图自同构和一个推广的对角自同构的乘积.
第五章研究了上三角和严格上三角矩阵代数上的保平方零矩阵的可逆线性映射.在线性保持问题中,最基础的问题是研究保秩映射,保幂等映射,保幂零(平方零)映射,这是由于很多其它的保持问题都可以化归为或借助于这些保持映射的结果解决.在这一章首先构造了域F上严格上三角矩阵代数N_n(F)的九种双向保平方零矩阵的可逆线性映射,然后利用这些映射详细刻画了N_n(F)的每一个双向保平方零矩阵的线性映射,最后利用这一结果分别刻画了N_n(F)的保矩阵乘积零的可逆线性映射和上三角矩阵代数T_n(F)的双向保平方零矩阵的可逆线性映射.
矩阵代数上的线性保持映射大多数被证明是一些同构或反同构的乘积,很少有例外的情形发生.而这一章得到了五种既不是自同构又不是反自同构的N_n(F)的保平方零矩阵的线性映射.
第六章研究了可换环上上三角矩阵代数的一些局部映射,双导子和李三次导子.上世纪九十年代,许多学者开始研究一类新的保持问题——局部映射,其目的在于研究线性映射的性质在什么条件下能由它们的局部性质来确定.双导子和李三次导子的研究起源于近几十年来学者们所关注的一个问题,即一个映射在什么条件下成为导子.在这一章得到了如下结论:
含单位元的可换环R上的上三角矩阵代数T_n(R)的每一个局部李导子是李导子,每一个双导子都可以写成一个内双导子与一个极端双导子的和;当2是R中的可逆元时,T_n(R)每一个局部李自同构是李自同构,每一个李三次导子都可以写成一个内导子与一个中心李导子的和.
由此可见T_n(R)的李自同构和李导子在一定意义下完全由他们的局部作用决定,T_n(R)上存在不是内双导子的双导子.
第七章总结了本学位论文的核心结论,并提出了今后进一步的研究目标.
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.
引文
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