Alphabeta MathMoment Maps and Symplectic Reduction
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Composition Series, the Jordan–Hölder Theorem and Solvable Groups
- Cosets, Index and Lagrange's Theorem
- Cyclic Groups and Direct Products
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Finite Averaging and Character-Theory Prerequisites
- Free Modules, Exact Sequences, Projective and Injective Modules
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Normal Subgroups and Quotient Groups
- Polynomial Rings, the Division Algorithm and Roots
- Rings, Subrings, Integral Domains and Fields
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Sylow's Theorems, p-Groups and Nilpotent Groups
- Tensor Products of Modules
- The Fundamental Theorem of Algebra
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
Category Theory
Differential Geometry
- Euclidean Ordinary Differential Equations with Smooth Dependence
- Hamiltonian Mechanics and Completely Integrable Systems
- Integration of Forms and the General Stokes Theorem
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Riemannian Metrics Length Distance and Volume
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Symplectic Manifolds, Moser Stability, and Darboux–Weinstein Theory
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
- The Exterior Derivative and Cartan Calculus
- Vector Fields Flows and Lie Derivatives
Functional Analysis
Homological Algebra
Lie Theory
Linear Algebra
- Complexification, Realification and Real Structures
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- The Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Measure Theory
Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Fubini and Change of Variables
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- Power Series and Real-Analytic Functions
- Properties of the Integral and the Working FTC
- Regular Surfaces and Surface Integrals
- Rⁿ as a Normed Space; Vector-Valued Functions
- Roots, Rational Powers, and Classical Inequalities
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Representation Theory of Groups
Set Theory
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Hausdorff via the Diagonal
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group of the Circle
- The Seifert–van Kampen Theorem
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Urysohn's Lemma and the Tietze Extension Theorem
2 · Logical flowchart
3 · Items
Definition
The coadjoint representation, action and orbits
Definition
Symplectic and Hamiltonian Lie-group actions
Definition
Moment map, component Hamiltonians and infinitesimal moment maps
Proposition
The infinitesimal generator of a symplectic action is symplectic
Proposition
Moment map components generate the negative infinitesimal action
Proposition
For connected groups, equivariance is equivalent to the moment-map Poisson bracket identity
Lemma
The nonequivariance defect of an infinitesimal moment map is a constant Lie-algebra two-cocycle
Proposition
Moment maps for one action form an affine space over coadjoint-fixed covectors
Corollary
Semisimple Hamiltonian actions have a unique equivariant moment map when one exists
Proposition
Whitehead's second lemma removes the infinitesimal equivariance obstruction for semisimple actions
Theorem
Noether's conservation law for Hamiltonian actions
Proposition
Equivariant symplectomorphisms preserve moment maps up to a coadjoint-fixed covector
Proposition
The cotangent lift of an action is Hamiltonian with the tautological moment map
Lemma
The tautological cotangent moment map is equivariant
Definition
The Kirillov--Kostant--Souriau form on a coadjoint orbit
Lemma
The KKS formula is independent of the Lie-algebra representatives
Theorem
Coadjoint orbits are symplectic manifolds
Proposition
The coadjoint-orbit inclusion is an equivariant moment map
Proposition
Products and opposites of symplectic moment maps
Lemma
The differential of the moment map and the orbit-orthogonal identity
Proposition
Regularity of a moment map is equivalent to local freeness
Proposition
The moment level is invariant under the coadjoint stabilizer
Lemma
The characteristic kernel on a regular moment level
Lemma
An invariant horizontal form on a free proper quotient descends uniquely
Theorem
Marsden--Weinstein--Meyer symplectic reduction
Corollary
Zero-level symplectic reduction and the dimension formula
Proposition
The dimension of a regular reduced space at a nonzero value
Proposition
Invariant Hamiltonians descend to reduced Hamiltonians
Proposition
Reduction commutes with products
Theorem
Reduction in stages for free proper regular actions
Proposition
The shifting trick identifies reduction at a value with a zero reduction
Proposition
Compact-group symplectic actions admit an invariant compatible almost-complex structure
Proposition
A compact-group moment map can be averaged to an equivariant one when the affine obstruction vanishes
Remark
Nonregular or nonfree symplectic quotients need not be manifolds
Remark
Convexity and toric classification for Hamiltonian torus actions
False statement
Every symplectic action is Hamiltonian
False statement
An infinitesimal moment map is automatically equivariant
False statement
Moment maps are unique without normalization
False statement
The cotangent-lift moment map has a plus sign under the library fundamental-field convention
False statement
Every value of a moment map gives a smooth symplectic quotient
False statement