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Geometry Lie Groups, Lie Algebras, and Representations by Hall

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  1. Jan 26, 2013 #1

    micromass

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    Table of Contents:
    Code (Text):

    [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: May 6, 2017
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  3. Jan 28, 2013 #2

    Fredrik

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    Staff Emeritus
    Science Advisor
    Gold Member

    This is an excellent presentation of Lie groups, Lie algebras and their representations for people who don't know differential geometry. To people who do know differential geometry, a Lie group is (roughly) a group that's also a smooth manifold, and a Lie algebra is a vector space with a Lie bracket. There's a Lie algebra associated with each Lie group, because there's a natural way to define a Lie bracket on the tangent space at the identity.

    Hall avoids all that, by using simple definitions of the terms "lie group" and "lie algebra" that are equivalent to the standard ones when the group is a subgroup of ##\operatorname{GL}(n;\mathbb C)## (the group of invertible complex n×n matrices). Since essentially all the interesting examples of Lie groups are (isomorphic to) matrix groups, there's no good reason to not do this.
     
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