Lie Groups, Lie Algebras, and... | Buy

Summary

This book addresses Lie groups, Lie algebras, and representation theory. In order to keep the prerequisites to a minimum, the author restricts attention to matrix Lie groups and Lie algebras. This approach keeps the discussion concrete, allows the reader to get to the heart of the subject quickly, and covers all of the most interesting examples. The book also introduces the often-intimidating machinery of roots and the Weyl group in a gradual way, using examples and representation theory as motivation. The text is divided into two parts. The first covers Lie groups and Lie algebras and the relationship between them, along with basic representation theory. The second part covers the theory of semisimple Lie groups and Lie algebras, beginning with a detailed analysis of the representations of SU(3). The author illustrates the general theory with numerous images pertaining to Lie algebras of rank two and rank three, including images of root systems, lattices of dominant integral weights, and weight diagrams. This book is sure to become a standard textbook for graduate students in mathematics and physics with little or no prior exposure to Lie theory. Brian Hall is an Associate Professor of Mathematics at the University of Notre Dame.

Author Biography

Brian C. Hall is an Associate Professor of Mathematics at the University of Notre Dame.

Preface
General Theory
Matrix Lie Groups
Definition of a Matrix Lie Group
Examples of Matrix Lie Groups
Compactness
Connectedness
Simple Connectedness
Homomorphisms and Isomorphisms
(Optional) The Polar Decomposition for $ {SL}(n; {R})$ and $ {SL}(n; {C})$
Lie Groups
Exercises
Lie Algebras and the Exponential Mapping
The Matrix Exponential
Computing the Exponential of a Matrix
The Matrix Logarithm
Further Properties of the Matrix Exponential
The Lie Algebra of a Matrix Lie Group
Properties of the Lie Algebra
The Exponential Mapping
Lie Algebras
The Complexification of a Real Lie Algebra
Exercises
The Baker--Campbell--Hausdorff Formula
The Baker--Campbell--Hausdorff Formula for the Heisenberg Group
The General Baker--Campbell--Hausdorff Formula
The Derivative of the Exponential Mapping
Proof of the Baker--Campbell--Hausdorff Formula
The Series Form of the Baker--Campbell--Hausdorff Formula
Lie Algebra Versus Lie Group Homomorphisms
Covering Groups
Subgroups and Subalgebras
Exercises
Basic Representation Theory
Representations
Why Study Representations?
Examples of Representations
The Irreducible Representations of $ {su}(2)$
Direct Sums of Representations
Tensor Products of Representations
Dual Representations
Schur's Lemma
Group Versus Lie Algebra Representations
Complete Reducibility
Exercises
Semisimple Theory
The Representations of $ {SU}(3)$
Introduction
Weights and Roots
The Theorem of the Highest Weight
Proof of the Theorem
An Example: Highest Weight $( 1,1) $
The Weyl Group
Weight Diagrams
Exercises
Semisimple Lie Algebras
Complete Reducibility and Semisimple Lie Algebras
Examples of Reductive and Semisimple Lie Algebras
Cartan Subalgebras
Roots and Root Spaces
Inner Products of Roots and Co-roots
The Weyl Group
Root Systems
Positive Roots
The $ {sl}(n;{C})$ Case
Uniqueness Results
Exercises
Representations of Complex Semisimple Lie Algebras
Integral and Dominant Integral Weights
The Theorem of the Highest Weight
Constructing the Representations I: Verma Modules
Constructing the Representations II: The Peter--Weyl Theorem
Constructing the Representations III: The Borel--Weil Construction
Further Results
Exercises
More on Roots and Weights
Abstract Root Systems
Duality
Bases and Weyl Chambers
Integral and Dominant Integral Weights
Examples in Rank 2
Examples in Rank 3
Additional Properties
The Root Systems of the Classical Lie Algebras
Dynkin Diagrams and the Classification
The Root Lattice and the Weight Lattice
Exercises
A Quick Introduction to Groups
Definition of a Group and Basic Properties
Some Examples of Groups
Subgroups, the Center, and Direct Products
Homomorphisms and Isomorphisms
Quotient Groups
Exercises
Linear Algebra Review
Eigenvectors, Eigenvalues, and the Characteristic Polynomial
Diagonalization
Generalized Eigenvectors and the SN Decomposition
The Jordan Canonical Form
The Trace
Inner Products
Dual Spaces and Weights
More on Lie Groups
Manifolds
Lie Groups
Examples of Non-matrix Lie Groups
Differential Forms and Haar Measure
Clebsch--Gordan theory for ${SU}(2)$ and the Wigner--Eckart Theorem
Tensor Products of ${sl}(2; {C})$ Representations
The Wigner--Eckart Theorem
Computing Fundamental Groups of Matrix Lie Groups
The Fundamental Group
The Universal Cover
Fundamental Groups of Compact Lie Groups I
Fundamental Groups of Compact Lie Groups II
Fundamental Groups of Non-compact Lie Groups

Bibliography
Table of Contents provided by Publisher. All Rights Reserved.

Supplemental Materials

What is included with this book?

The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.

The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.