本文關(guān)鍵詞：三角代數(shù)上的幾類映射的研究　出處：《陜西師范大學(xué)》2016年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 三角代數(shù) 非線性廣義Lie導(dǎo)子 零點(diǎn)ξ-Lie高階弱可導(dǎo)映射 Lie不變映射 非線性(m n)-Lie中心化子 非線性(m n)-高階可導(dǎo)映射

【摘要】：本文用代數(shù)的結(jié)構(gòu)性質(zhì)及代數(shù)分解方法研究了三角代數(shù)上的一些映射.所討論的映射包括:三角代數(shù)上的非線性廣義Lie導(dǎo)子,零點(diǎn)ξ-Lie弱可導(dǎo)和零點(diǎn)ξ-Lie高階弱可導(dǎo)映射,Lie不變映射和非線性(m,n)-Lie中心化子,非線性(m,n)-可導(dǎo)和非線性(m,n)-高階可導(dǎo)映射.全文共分四章,主要內(nèi)容如下:第一章介紹了本文選題的意義及背景,并回顧了國(guó)內(nèi)外學(xué)者關(guān)于此課題的研究進(jìn)展和成果,給出了后幾章將用到的一些概念和結(jié)論.第二章研究了三角代數(shù)上的非線性廣義Lie導(dǎo)子,證明了三角代數(shù)上的每一個(gè)非線性廣義Lie導(dǎo)子都是一個(gè)可加的廣義導(dǎo)子與一個(gè)在交換子上為零的中心值映射的和.此外,我們給出了三角代數(shù)上的零點(diǎn)ξ-Lie弱可導(dǎo)映射和零點(diǎn)ξ-Lie高階弱可導(dǎo)映射的一般形式.第三章研究了三角代數(shù)上關(guān)于內(nèi)導(dǎo)子空間Lie不變的線性映射,證明了此類映射都是一個(gè)Lie導(dǎo)子與一個(gè)中心元乘以恒等映射的和.同時(shí),我們刻畫了|(m-n)(m + n)|-無(wú)撓的三角代數(shù)上的非線性(m,n)-Lie中心化子.第四章研究了三角代數(shù)上的非線性(m,n)-可導(dǎo)和非線性(m,n)-高階可導(dǎo)映射,證明了|m+n| 無(wú)撓的三角代數(shù)上的非線性(m,n)-可導(dǎo)映射和非線性(m,n)-高階可導(dǎo)映射分別是導(dǎo)子和高階導(dǎo)子.本文得到的結(jié)果有:(1)設(shè)u是一個(gè)三角代數(shù)且滿足πA(Z(u))= 和πB(Z(u))=Z(B).若δ是u上的一個(gè)非線性廣義Lie導(dǎo)子,f是與δ相關(guān)的非線性映射,則在u上分別存在兩個(gè)可加的廣義導(dǎo)子φ和g,以及一個(gè)到u的中心且在交換子上為零的映射ξ使得對(duì)任意的x ∈u,有δ(x)=φ(x)+ ξ(x)和f(x)= g(x)+ ξ(x).(2)設(shè)u是數(shù)域F上的一個(gè)三角代數(shù).若d是u上的一個(gè)零點(diǎn)ξ Lie(ξ ≠ 1)弱可導(dǎo)映射,則在u上存在一個(gè)導(dǎo)子δ和一個(gè)中心元λ使得對(duì)任意的x ∈ u,有d(x)= δ(x)+ λx.(3)設(shè)u是數(shù)域F上的一個(gè)三角代數(shù).若D ={dk}k∈N是u上的一個(gè)零點(diǎn)ξ-Lie(ξ ≠ 1)高階弱可導(dǎo)映射且dk(1)= 0((?)k ∈ N+),則D是高階導(dǎo)子.(4)設(shè)u是一個(gè)三角代數(shù)且滿足πA(Z(u))=Z(A)和πB(Z(u))=Z(B),φ是u上的一個(gè)R-線性映射.若ID(u)是關(guān)于φ的一個(gè)Lie不變子空間,則在u上存在一個(gè)Lie導(dǎo)子δ和一個(gè)中心元λ使得對(duì)任意的x ∈ u,有φ(x)= δ(x)+λx.(5)設(shè) m,n 是固定的整數(shù)且(m+n)(m-n)≠ 0,u 是一個(gè)|(m+n)(m-n)|-無(wú)撓的三角代數(shù)且滿足πA(Z(u))= Z(A)和πB(Z(u))= Z(B).若L是u上的一個(gè)非線性(m,n)-Lie中心化子,則存在一個(gè)中心元λ和一個(gè)到u的中心且在交換子上為零的映射ξ使得對(duì)任意的x ∈u,有L(有= λx + ξ(x).(6)設(shè)m和n是固定的整數(shù)且m + n ≠ 0,u是一個(gè)|m + n|-無(wú)撓的三角代數(shù).若d是u上的一個(gè)非線性(m,n)-可導(dǎo)映射,則d是一個(gè)導(dǎo)子.(7)設(shè)m和n是固定的整數(shù)且m + n ≠ 0,u是一個(gè)|m + n|-無(wú)撓的三角代數(shù).若D = {dk}k∈N是u上的一個(gè)非線性(m,n)-高階可導(dǎo)映射,則D是一個(gè)高階導(dǎo)子.
[Abstract]:In this paper, we use the structural properties of algebras and the algebraic decomposition method to study some mappings on triangular algebras, including: nonlinear generalized Lie derivations on triangular algebras. 00:00 尉 -Lie weakly differentiable and 00:00 尉 -lie higher-order weakly differentiable mappings lie invariant maps and nonlinear mechnium-lie centralizers, nonlinear mechnium-derivable and nonlinear mechnian. The thesis is divided into four chapters. The main contents are as follows: chapter 1 introduces the significance and background of this topic and reviews the research progress and achievements of domestic and foreign scholars on this topic. In the second chapter, the nonlinear generalized Lie derivations on triangular algebras are studied. It is proved that every nonlinear generalized Lie derivation on a triangular algebra is the sum of an additive generalized derivation and a central value mapping with zero on the commutator. In this paper, we give the general forms of 00:00 尉 -Lie weakly differentiable mappings and 00:00 尉 -lie higher-order weak differentiable mappings on triangular algebras. In chapter 3, we study the Lie invariance of inner derivation spaces on triangular algebras. Mapping. It is proved that all such mappings are the sum of a Lie derivation and a central element multiplied by identity mappings. At the same time, we characterize the nonlinearity m on the torsion-free trigonometric algebra. In chapter 4th, we study the nonlinear mechnium-derivable and the nonlinear mm-nng-higher-order differentiable mappings over triangular algebras. In this paper, we prove the nonlinearity of mng-derivable mapping and nonlinear mm-on triangular-algebras without torsion. The results obtained in this paper are as follows: 1) Let u be a trigonometric algebra and satisfy 蟺 A ~ (1) Z ~ ((1)) = and 蟺 B ~ (+). If 未 is a nonlinear generalized Lie derivation on u. If f is a nonlinear mapping related to 未, then there are two additive generalized derivations 蠁 and g on u, and a 尉 to the center of u and zero on commutator such that for any x 鈭，

本文編號(hào)：1425733