- #1
- 22,183
- 3,324
- Author: Brian Hall
- Title: Lie Groups, Lie Algebras, and Representations: An Elementary Introduction
- Amazon link https://www.amazon.com/dp/1441923136/?tag=pfamazon01-20
- Level: Grad
Table of Contents:
Code:
[LIST]
[*] General Theory
[LIST]
[*] Matrix Lie Groups
[LIST]
[*] Definition of a Matrix Lie Group
[LIST]
[*] Counterexamples
[/LIST]
[*] Examples of Matrix Lie Groups
[LIST]
[*] The general linear groups GL(n;R) and GL(n; C)
[*] The special linear groups SL(n;R) and SL(n; C)
[*] The orthogonal and special orthogonal groups, O(n) and SO(n)
[*] The unitary and special unitary groups, U(n) and SU(n)
[*] The complex orthogonal groups, O(n; C) and SO(n; C)
[*] The generalized orthogonal and Lorentz groups
[*] The symplectic groups Sp(n; R), Sp(n;C), and Sp(n)
[*] The Heisenberg group H
[*] The groups R*, C*, S^1, R, and R^n
[*] The Euclidean and Poincare groups E(n) and P(n; 1)
[/LIST]
[*] Compactness
[LIST]
[*] Examples of compact groups
[*] Examples of noncompact groups
[/LIST]
[*] Connectedness
[*] Simple Connectedness
[*] Homomorphisms and Isomorphisms
[LIST]
[*] Example: SU(2) and SO(3)
[/LIST]
[*] The Polar Decomposition for SL(n; R) and SL(n; C)
[*] Lie Groups
[*] Exercises
[/LIST]
[*] Lie Algebras and the Exponential Mapping
[LIST]
[*] The Matrix Exponential
[*] Computing the Exponential of a Matrix
[LIST]
[*] Case 1: X is diagonalizable
[*] Case 2: X is nilpotent
[*] Case 3: X arbitrary
[/LIST]
[*] The Matrix Logarithm
[*] Further Properties of the Matrix Exponential
[*] The Lie Algebra of a Matrix Lie Group
[LIST]
[*] Physicists' Convention
[*] The general linear groups
[*] The special linear groups
[*] The unitary groups
[*] The orthogonal groups
[*] The generalized orthogonal groups
[*] The symplectic groups
[*] The Heisenberg group
[*] The Euclidean and Poincare groups
[/LIST]
[*] Properties of the Lie Algebra
[*] The Exponential Mapping
[*] Lie Algebras
[LIST]
[*] Structure constants
[*] Direct sums
[/LIST]
[*] The Complexification of a Real Lie Algebra
[*] Exercises
[/LIST]
[*] The Baker—Campbell-HausdorfF Formula
[LIST]
[*] 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
[*] Group Versus Lie Algebra Homomorphisms
[*] Covering Groups
[*] Subgroups and Subalgebras
[*] Exercises
[/LIST]
[*] Basic Representation Theory
[LIST]
[*] Representations
[*] Why Study Representations?
[*] Examples of Representations
[LIST]
[*] The standard representation
[*] The trivial representation
[*] The adjoint representation
[*] Some representations of SU(2)
[*] Two unitary representations of SO(3)
[*] A unitary representation of the reals
[*] The unitary representations of the Heisenberg group
[/LIST]
[*] 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
[/LIST]
[/LIST]
[*] Semisimple Theory
[LIST]
[*] The Representations of SU(3)
[LIST]
[*] 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
[/LIST]
[*] Semisimple Lie Algebras
[LIST]
[*] 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
[LIST]
[*] The Cartan subalgebra
[*] The roots
[*] Inner products of roots
[*] The Weyl group
[*] Positive roots
[/LIST]
[*] Uniqueness Results
[*] Exercises
[/LIST]
[*] Representations of Complex Semisimple Lie Algebras
[LIST]
[*] Integral and Dominant Integral Elements
[*] The Theorem of the Highest Weight
[*] Constructing the Representations I: Verma Modules
[LIST]
[*] Verma modules
[*] Irreducible quotient modules
[*] Finite-dimensional quotient modules
[*] The sl(2; C) case
[/LIST]
[*] Constructing the Representations II: The Peter-Weyl Theorem
[LIST]
[*] The Peter-Weyl theorem
[*] The Weyl character formula
[*] Constructing the representations
[*] Analytically integral versus algebraically integral elements
[*] The SU(2) case
[/LIST]
[*] Constructing the Representations III: The Borel-Weil Construction
[LIST]
[*] The complex-group approach
[*] The setup
[*] The strategy
[*] The construction
[*] The SL(2; C) case
[/LIST]
[*] Further Results
[LIST]
[*] Duality
[*] The weights and their multiplicities
[*] The Weyl character formula and the Weyl dimension formula
[*] The analytical proof of the Weyl character formula
[/LIST]
[*] Exercises
[/LIST]
[*] More on Roots and Weights
[LIST]
[*] Abstract Root Systems
[*] Duality
[*] Bases and Weyl Chambers
[*] Integral and Dominant Integral Elements
[*] Examples in Rank Two
[LIST]
[*] The root systems
[*] Connection with Lie algebras
[*] The Weyl groups
[*] Duality
[*] Positive roots and dominant integral elements
[*] Weight diagrams
[/LIST]
[*] Examples in Rank Three
[*] Additional Properties
[*] The Root Systems of the Classical Lie Algebras
[LIST]
[*] The orthogonal algebras so(2n; C)
[*] The orthogonal algebras so(2n + 1; C)
[*] The symplectic algebras sp(n; C)
[/LIST]
[*] Dynkin Diagrams and the Classification
[*] The Root Lattice and the Weight Lattice
[*] Exercises
[/LIST]
[/LIST]
[*] Quick Introduction to Groups
[LIST]
[*] Definition of a Group and Basic Properties
[*] Examples of Groups
[LIST]
[*] The trivial group
[*] The integers
[*] The reals and R^n
[*] Nonzero real numbers under multiplication
[*] Nonzero complex numbers under multiplication
[*] Complex numbers of absolute value 1 under multiplication
[*] The general linear groups
[*] Permutation group (symmetric group)
[*] Integers mod n
[/LIST]
[*] Subgroups, the Center, and Direct Products
[*] Homomorphisms and Isomorphisms
[*] Quotient Groups
[*] Exercises
[/LIST]
[*] Linear Algebra Review
[LIST]
[*] Eigenvectors, Eigenvg^lues, and the Characteristic Polynomial
[*] Diagonalization
[*] Generalized Eigenvectors and the SN Decomposition
[*] The Jordan Canonical Form
[*] The Trace
[*] Inner Products
[*] Dual Spaces
[*] Simultaneous Diagonalization
[/LIST]
[*] More on Lie Groups
[LIST]
[*] Manifolds
[LIST]
[*] Definition
[*] Tangent space
[*] Differentials of smooth mappings
[*] Vector fields
[*] The flow along a vector field
[*] Submanifolds of vector spaces
[*] Complex manifolds
[/LIST]
[*] Lie Groups
[LIST]
[*] Definition
[*] The Lie algebra
[*] The exponential mapping
[*] Homomorphisms
[*] Quotient groups and covering groups
[*] Matrix Lie groups as Lie groups
[*] Complex Lie groups
[/LIST]
[*] Examples of Nonmatrix Lie Groups
[*] Differential Forms and Haar Measure
[/LIST]
[*] Clebsch-Gordan Theory for SU(2) and the Wigner-Eckart Theorem
[LIST]
[*] Tensor Products of sl(2; C) Representations
[*] The Wigner-Eckart Theorem
[*] More on Vector Operators
[/LIST]
[*] Computing Fundamental Groups of Matrix Lie Groups
[LIST]
[*] The Fundamental Group
[*] The Universal Cover
[*] Fundamental Groups of Compact Lie Groups I
[*] Fundamental Groups of Compact Lie Groups II
[*] Fundamental Groups of Noncompact Lie Groups
[/LIST]
[*] References
[*] Index
[/LIST]
Last edited by a moderator: