# Lie Groups, Lie Algebras, and Representations by Hall

• Geometry

## For those who have used this book

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[*] General Theory
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[*] Matrix Lie Groups
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[*] Definition of a Matrix Lie Group
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[*] Counterexamples
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[*] Examples of Matrix Lie Groups
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[*] 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)
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[*] Compactness
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[*] Examples of compact groups
[*] Examples of noncompact groups
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[*] Connectedness
[*] Simple Connectedness
[*] Homomorphisms and Isomorphisms
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[*] Example: SU(2) and SO(3)
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[*] The Polar Decomposition for SL(n; R) and SL(n; C)
[*] Lie Groups
[*] Exercises
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[*] Lie Algebras and the Exponential Mapping
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[*] The Matrix Exponential
[*] Computing the Exponential of a Matrix
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[*] Case 1: X is diagonalizable
[*] Case 2: X is nilpotent
[*] Case 3: X arbitrary
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[*] The Matrix Logarithm
[*] Further Properties of the Matrix Exponential
[*] The Lie Algebra of a Matrix Lie Group
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[*] 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
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[*] Properties of the Lie Algebra
[*] The Exponential Mapping
[*] Lie Algebras
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[*] Structure constants
[*] Direct sums
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[*] The Complexification of a Real Lie Algebra
[*] Exercises
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[*] The Baker—Campbell-HausdorfF Formula
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[*] 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
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[*] Basic Representation Theory
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[*] Representations
[*] Why Study Representations?
[*] Examples of Representations
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[*] The standard representation
[*] The trivial 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
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[*] 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
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[*] Semisimple Theory
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[*] The Representations of SU(3)
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[*] 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
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[*] Semisimple Lie Algebras
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[*] 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
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[*] The Cartan subalgebra
[*] The roots
[*] Inner products of roots
[*] The Weyl group
[*] Positive roots
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[*] Uniqueness Results
[*] Exercises
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[*] Representations of Complex Semisimple Lie Algebras
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[*] Integral and Dominant Integral Elements
[*] The Theorem of the Highest Weight
[*] Constructing the Representations I: Verma Modules
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[*] Verma modules
[*] Irreducible quotient modules
[*] Finite-dimensional quotient modules
[*] The sl(2; C) case
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[*] Constructing the Representations II: The Peter-Weyl Theorem
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[*] The Peter-Weyl theorem
[*] The Weyl character formula
[*] Constructing the representations
[*] Analytically integral versus algebraically integral elements
[*] The SU(2) case
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[*] Constructing the Representations III: The Borel-Weil Construction
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[*] The complex-group approach
[*] The setup
[*] The strategy
[*] The construction
[*] The SL(2; C) case
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[*] Further Results
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[*] Duality
[*] The weights and their multiplicities
[*] The Weyl character formula and the Weyl dimension formula
[*] The analytical proof of the Weyl character formula
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[*] Exercises
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[*] More on Roots and Weights
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[*] Abstract Root Systems
[*] Duality
[*] Bases and Weyl Chambers
[*] Integral and Dominant Integral Elements
[*] Examples in Rank Two
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[*] The root systems
[*] Connection with Lie algebras
[*] The Weyl groups
[*] Duality
[*] Positive roots and dominant integral elements
[*] Weight diagrams
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[*] Examples in Rank Three
[*] The Root Systems of the Classical Lie Algebras
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[*] The orthogonal algebras so(2n; C)
[*] The orthogonal algebras so(2n + 1; C)
[*] The symplectic algebras sp(n; C)
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[*] Dynkin Diagrams and the Classification
[*] The Root Lattice and the Weight Lattice
[*] Exercises
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[*] Quick Introduction to Groups
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[*] Definition of a Group and Basic Properties
[*] Examples of Groups
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[*] 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
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[*] Subgroups, the Center, and Direct Products
[*] Homomorphisms and Isomorphisms
[*] Quotient Groups
[*] Exercises
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[*] Linear Algebra Review
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[*] 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
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[*] More on Lie Groups
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[*] Manifolds
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[*] Definition
[*] Tangent space
[*] Differentials of smooth mappings
[*] Vector fields
[*] The flow along a vector field
[*] Submanifolds of vector spaces
[*] Complex manifolds
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[*] Lie Groups
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[*] Definition
[*] The Lie algebra
[*] The exponential mapping
[*] Homomorphisms
[*] Quotient groups and covering groups
[*] Matrix Lie groups as Lie groups
[*] Complex Lie groups
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[*] Examples of Nonmatrix Lie Groups
[*] Differential Forms and Haar Measure
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[*] Clebsch-Gordan Theory for SU(2) and the Wigner-Eckart Theorem
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[*] Tensor Products of sl(2; C) Representations
[*] The Wigner-Eckart Theorem
[*] More on Vector Operators
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[*] Computing Fundamental Groups of Matrix Lie Groups
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[*] 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
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[*] References
[*] Index
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