Lectures on Lie Groups

Description

This volume consists of nine lectures on selected topics of Lie group theory. We provide the readers a concise introduction as well as a comprehensive 'tour of revisiting' the remarkable achievements of S Lie, W Killing, É Cartan and H Weyl on structural and classification theory of semi-simple Lie groups, Lie algebras and their representations; and also the wonderful duet of Cartans' theory on Lie groups and symmetric spaces.With the benefit of retrospective hindsight, mainly inspired by the outstanding contribution of H Weyl in the special case of compact connected Lie groups, we develop the above theory via a route quite different from the original methods engaged by most other books.We begin our revisiting with the compact theory which is much simpler than that of the general semi-simple Lie theory; mainly due to the well fittings between the Frobenius-Schur character theory and the maximal tori theorem of É Cartan together with Weyl's reduction (cf. Lectures 1-4). It is a wonderful reality of the Lie theory that the clear-cut orbital geometry of the adjoint action of compact Lie groups on themselves (i.e. the geometry of conjugacy classes) is not only the key to understand the compact theory, but it actually already constitutes the central core of the entire semi-simple theory, as well as that of the symmetric spaces (cf. Lectures 5-9). This is the main reason that makes the succeeding generalizations to the semi-simple Lie theory, and then further to the Cartan theory on Lie groups and symmetric spaces, conceptually quite natural, and technically rather straightforward. Contents Preface Lecture 1 Linear Groups and Linear Representations 1. Basic Concepts and Definitions 2. A Brief Overview 3. Compact Groups, Haar Integral and the Averaging Method 3.1. Haar integral of functions defined on compact groups 3.2. Existence of invariant inner (resp. Hermitian) product 4. Frobenius–Schur Orthogonality and the Character Theory 5. Classification of Irreducible Complex Representations of S3 6. L2(G) and Concluding Remarks Lecture 2 Lie Groups and Lie Algebras 1. One-Parameter Subgroups and Lie Algebras 2. Lie Subgroups and the Fundamental Theorem of Lie 3. Lie Homomorphisms and Simply Connected Lie Groups 4. Adjoint Actions and Adjoint Representations Lecture 3 Orbital Geometry of the Adjoint Action 1. Bi-Invariant Riemannian Structure on a Compact Connected Lie Group and the Maximal Tori Theorem of ´E. Cartan 2. Root System and Weight System 3. Classification of Rank 1 Compact Connected Lie Groups Lecture 4 Coxeter Groups, Weyl Reduction and Weyl Formulas 1. Geometry of Coxeter Groups 2. Geometry of (W, h) and the Root System 3. The Volume Function and Weyl Integral Formula 4. Weyl Character Formula and Classification of Complex Irreducible Representations Lecture 5 Structural Theory of Compact Lie Algebras 1. Characterization of Compact Lie Algebras 2. Cartan Decomposition and Structural Constants of Compact Lie Algebras Lecture 6 Classification Theory of Compact Lie Algebras and Compact Connected Lie Groups 1. Classification of Simple Compact Lie Algebras 2. Classification of Geometric Root Patterns 3. Classical Compact Lie Groups and Their Root Systems Lecture 7 Basic Structural Theory of Lie Algebras 1. Jordan Decomposition: A Review and an Overview 1.1. Jordan decomposition 1.2. A review and an overview 1.2.1. Basic definitions 1.2.2. Examples of solvable (resp. nilpotent) Lie algebras 1.2.3. Some remarkable generalizations of Theorem 1 in the setting of Lie algebras and their linear representations 2. Nilpotent Lie Algebras and Solvable Lie Algebras 2.1. Nilpotent Lie algebras 2.1.1. Theorem of Engel 2.1.2. Generalization of Jordan decomposition for nilpotent Lie algebras, primary decomposition 2.2. Solvable Lie algebras 2.2.1. Theorem of Lie 2.2.2. Cartans criterion of solvability 3. Semi-Simple Lie Algebras Lecture 8 Classification Theory of Complex Semi-Simple Lie Algebras 1. Cartan Subalgebras and Cartan Decompositions for the General Case of an Infinite Field k 2. On the Structure of Cartan Decomposition of Complex Semi-Simple Lie Algebras Lecture 9 Lie Groups and Symmetric Spaces, the Classification of Real Semi-Simple Lie Algebras and Symmetric Spaces 1. Real Semi-Simple Lie Algebras 2. Lie Groups and Symmetric Spaces 3. Orthogonal Involutive Lie Algebras 3.1. Examples of orthogonal involutive Lie algebras 3.2. On the direct sum decomposition of orthogonal involutive Lie algebra 4. Classification Theory of Symmetric Spaces and Real Semi-Simple Lie Algebras 4.1. A brief summary and overview 4.2. On the automorphism group of a simple compact Lie algebra and the conjugacy classes of involutive automorphisms 5. Concluding Remarks