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An Elementary Introduction to Groups and Representations

By Hall, Brian C.

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Book Id: WPLBN0000660617
Format Type: PDF eBook
File Size: 1,015.03 KB
Reproduction Date: 2005
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Title: An Elementary Introduction to Groups and Representations  
Author: Hall, Brian C.
Language: English
Subject: Science., Mathematics, Logic
Publication Date:


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Hall, B. C. (n.d.). An Elementary Introduction to Groups and Representations. Retrieved from

Mathematics document containing theorems and formulas.

Excerpt: Outgrowth of a graduate course on Lie groups I taught at the University of Virginia in 1994. In trying to find a text for the course I discovered that books on Lie groups either presuppose a knowledge of differentiable manifolds or provide a mini-course on them at the beginning. Since my students did not have the necessary background on manifolds, I faced a dilemma: either use manifold techniques that my students were not familiar with, or else spend much of the course teaching those techniques instead of teaching Lie theory. To resolve this dilemma I chose to write my own notes using the notion of a matrix Lie group. A matrix Lie group is simply a closed subgroup of GL(n;C): Although these are often called simply matrix groups, my terminology emphasizes that every matrix group is a Lie group.

Table of Contents
Contents 1. Preface ii Chapter 1. Groups 1 1. De nition of a Group, and Basic Properties 1 2. Some Examples of Groups 3 3. Subgroups, the Center, and Direct Products 4 4. Homomorphisms and Isomorphisms 5 5. Exercises 6 Chapter 2. Matrix Lie Groups 9 1. De nition of a Matrix Lie Group 9 2. Examples of Matrix Lie Groups 10 3. Compactness 15 4. Connectedness 16 5. Simple-connectedness 18 6. Homomorphisms and Isomorphisms 19 7. Lie Groups 20 8. Exercises 22 Chapter 3. Lie Algebras and the Exponential Mapping 27 1. The Matrix Exponential 27 2. Computing the Exponential of a Matrix 29 3. The Matrix Logarithm 31 4. Further Properties of the Matrix Exponential 34 5. The Lie Algebra of a Matrix Lie Group 36 6. Properties of the Lie Algebra 40 7. The Exponential Mapping 44 8. Lie Algebras 46 9. The Complexi cation of a Real Lie Algebra 48 10. Exercises 50 Chapter 4. The Baker-Campbell-Hausdor Formula 53 1. The Baker-Campbell-Hausdor Formula for the Heisenberg Group 53 2. The General Baker-Campbell-Hausdor Formula 56 3. The Series Form of the Baker-Campbell-Hausdor Formula 63 4. Subgroups and Subalgebras 64 5. Exercises 65 Chapter 5. Basic Representation Theory 67 1. Representations 67 2. Why Study Representations? 69


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