Lie代数的分类和识别(影印版)

本书的目的是为将Lie代数和Lie群应用于解决 科学和工程中出现的问题的研究人员和实践者提供 工具。作者解决了用一种更合适的基来表示在任意 基上得到的Lie代数的问题，在这种基中Lie代数的 所有基本特征都是直接可见的。这包括实现直和分 解、识别根和Levi分解、计算零根和Casimir不变 量。每种算法都给出了实例。 对于低维Lie代数，这使得完全识别给定的Lie 代数成为可能。作者提供了一个代表性列表，列出 了所有维数小于或等于6的Lie代数及其重要性质， 包括它们的Casimir不变量。该列表的排序方式， 使识别变得容易，只使用Lie代数的与基无关的性 质。他们还描述了某些具有完全或部分分类的任意 有限维的幂零和可解Lie代数类，并详细讨论了它 们的构造和性质。 本书的内容基于先前散布在期刊文章中的材料 ，其中许多文章由作者之一或两位作者与合作者共 同撰写。本书的读者应该熟悉入门水平的Lie代数 理论。

Preface Acknowledgements Part 1 General Theory Chapter 1 Introduction and Motivation Chapter 2 Basic Concepts 2.1 Definitions 2.2 Levi theorem 2.3 Classification of complex simple Lie algebras 2.4 Chevalley cohomology of Lie algebras Chapter 3 Invariants of the Coadjoint Representation of a Lie Algebra 3.1 Casimir operators and generalized Casimir invariants 3.2 Calculation of generalized Casimir invariants using the infinitesimal method 3.3 Calculation of generalized Casimir invariants by the method of moving frames Part 2 Recognition of a Lie Algebra Given by Its Structure Constants Chapter 4 Identification of Lie Algebras through the Use of Invariants 4.1 Elementary invariants 4.2 More sophisticated invariants Chapter 5 Decomposition into a Direct Sum 5.1 General theory and criteria 5.2 Algorithm 5.3 Examples Chapter 6 Levi Decomposition. Identification of the Radical and Levi Factor 6.1 Original algorithm 6.2 Modified algorithm 6.3 Examples Chapter 7 The Nilradical of a Lie Algebra 7.1 General theory 7.2 Algorithm 7.3 Examples 7.4 Identification of the nilradical using the Killing form Part 3 Nilpotent, Solvable and Levi Decomposable Lie Algebras Chapter 8 Nilpotent Lie Algebras 8.1 Maximal Abelian ideals and their extensions 8.2 Classification of low-dimensional nilpotent Lie algebras Chapter 9 Solvable Lie Algebras and Their Nilradicals 9.1 General structure of a solvable Lie algebra 9.2 General procedure for classifying all solvable Lie algebras with a given nilradical 9.3 Upper bound on the dimension of a solvable extension of a given nilradical 9.4 Particular classes of nilradicals and their solvable extensions 9.5 Vector fields realizing bases of the coadjoint representation of a solvable Lie algebra Chapter 10 Solvable Lie Algebras with Abelian Nilradicals 10.1 Basic structural theorems 10.2 Decomposability properties of the solvable Lie algebras 10.3 Solvable Lie algebras with centers of maximal dimension 10.4 Solvable Lie algebras with one nonnilpotent element and an n-dimensional Abelian nilradical 10.5 Solvable Lie algebras with two nonnilpotent elements and n-dimensional Abelian nilradical 10.6 Generalized Casimir invariants of solvable Lie algebras with Abelian nilradicals Chapter 11 Solvable Lie Algebras with Heisenberg Nilradical 11.1 he Heisenberg relations and the Heisenberg algebra 11.2 Classification of solvable Lie algebras with nilradical h (m)