Lie Groups, Lie Algebras, and Representations An Elementary Introduction

Lie Groups, Lie Algebras, and Representations An Elementary Introduction / by Brian Hall.

This book provides an introduction to Lie groups, Lie algebras, and repre­ sentation theory, aimed at graduate students in mathematics and physics. Although there are already several excellent books that cover many of the same topics, this book has two distinctive features that I hope will make it a...

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Bibliographic Details
Main Author: Hall, Brian (Author)
Corporate Author: SpringerLink (Online service)
Format: eBook
Language:English
Published: New York, NY : Springer New York : Imprint: Springer, 2003.
Edition:1st ed. 2003.
Series:Graduate Texts in Mathematics, 222
Springer eBook Collection.
Subjects:
Algebra.
Group theory.
Topological groups.
Lie groups.
Physics.
Electronic resources (E-books)
Online Access:Click to view e-book
Holy Cross Note:Loaded electronically.
Electronic access restricted to members of the Holy Cross Community.
Table of Contents:
  • I General Theory
  • 1 Matrix Lie Groups
  • 2 Lie Algebras and the Exponential Mapping
  • 3 The Baker-Campbell-Hausdorff Formula
  • 4 Basic Representation Theory
  • II Semisimple Theory
  • 5 The Representations of SU(3)
  • 6 Semisimple Lie Algebras
  • 7 Representations of Complex Semisimple Lie Algebras
  • 8 More on Roots and Weights
  • A A Quick Introduction to Groups
  • A.1 Definition of a Group and Basic Properties
  • A.2 Examples of Groups
  • A.2.1 The trivial group
  • A.2.2 The integers
  • A.2.4 Nonzero real numbers under multiplication
  • A.2.5 Nonzero complex numbers under multiplication
  • A.2.6 Complex numbers of absolute value 1 under multiplication
  • A.2.7 The general linear groups
  • A.2.8 Permutation group (symmetric group)
  • A.3 Subgroups, the Center, and Direct Products
  • A.4 Homomorphisms and Isomorphisms
  • A.5 Quotient Groups
  • A.6 Exercises
  • B Linear Algebra Review
  • B.1 Eigenvectors, Eigenvalues, and the Characteristic Polynomial
  • B.2 Diagonalization
  • B.3 Generalized Eigenvectors and the SN Decomposition
  • B.4 The Jordan Canonical Form
  • B.5 The Trace
  • B.6 Inner Products
  • B.7 Dual Spaces
  • B.8 Simultaneous Diagonalization
  • C More on Lie Groups
  • C.1 Manifolds
  • C.1.1 Definition
  • C.1.2 Tangent space
  • C.1.3 Differentials of smooth mappings
  • C.1.4 Vector fields
  • C.1.5 The flow along a vector field
  • C.1.6 Submanifolds of vector spaces
  • C.1.7 Complex manifolds
  • C.2 Lie Groups
  • C.2.1 Definition
  • C.2.2 The Lie algebra
  • C.2.3 The exponential mapping
  • C.2.4 Homomorphisms
  • C.2.5 Quotient groups and covering groups
  • C.2.6 Matrix Lie groups as Lie groups
  • C.2.7 Complex Lie groups
  • C.3 Examples of Nonmatrix Lie Groups
  • C.4 Differential Forms and Haar Measure
  • D Clebsch-Gordan Theory for SU(2) and the Wigner-Eckart Theorem
  • D.1 Tensor Products of sl(2; ?) Representations
  • D.2 The Wigner-Eckart Theorem
  • D.3 More on Vector Operators
  • E Computing Fundamental Groups of Matrix Lie Groups
  • E.1 The Fundamental Group
  • E.2 The Universal Cover
  • E.3 Fundamental Groups of Compact Lie Groups I
  • E.4 Fundamental Groups of Compact Lie Groups II
  • E.5 Fundamental Groups of Noncompact Lie Groups
  • References.

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