Alphabeta MathNoninvertible Symmetries and Anyonic Chains
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Free Groups and Presentations
- 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 over a Principal Ideal Domain and the Canonical Forms
- 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
- Tensor Products of Modules
- The Field of Fractions and Localisation
Braid Groups
Categorification and Geometric Representation Theory
Category Theory
- Abelian Categories
- Braided and Symmetric Monoidal Categories
- Categories, Functors and Natural Transformations
- Duality and Rigidity in Monoidal Categories
- Exactness and the Member Calculus
- Limits and Colimits
- Monoidal Categories and Monoidal Functors
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Strictification and Mac Lanes Coherence Theorem
- Subobject Lattices Generators and the Grothendieck Axioms
- Tensor and Fusion Categories
Combinatorics
Coxeter Groups
Differential Geometry
- Manifolds with Boundary Collars and Orientations
- Rank Theorems and Embedded Submanifolds
- Sard Theorem and Transversality
- Smooth Manifolds and Smooth Maps
- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Tensor Fields Exterior Algebra and Differential Forms
Functional Analysis
Lie Theory
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- 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
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Quantum Algebra and Representation Theory
Quantum Computing
Real Analysis
- 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
- Darboux, L'Hôpital, and Taylor's Theorem
- Foundations of the Real Numbers for Analysis
- 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
- Properties of the Integral and the Working FTC
- 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 Derivative and the Mean Value Theorems
- 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
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- Filters and Ultrafilters
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Relations, Functions, and Quotients
- The ZFC Axioms and the Basic Set Constructions
Topology
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Hereditary and Productive Behaviour of the Separation Axioms
- Metric Spaces
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Fixed-boundary anyonic fusion-path space
Definition
Tensor-invertible objects, fusion-ring units, and invertible operators
Lemma
Fusion paths give a basis of the fixed-boundary path space
Theorem
Unitary fusion paths form an orthonormal Hilbert basis
Definition
Frobenius–Schur sign in a normalized anyonic vacuum channel
Definition
Local anyonic channel projector
Lemma
The vacuum F-coefficient records the Frobenius–Schur sign
Lemma
The explicit Fibonacci model has a positive spherical pivotal structure
Proposition
The Fibonacci ring class is a unit but its simple object is not tensor-invertible
Theorem
Canonical F-move paths give the same channel operator
Theorem
Local channel projections are orthogonal idempotents
Proposition
Vacuum-channel anyonic projectors satisfy Temperley–Lieb relations
Lemma
All-label unitary Ising normal data and ribbon coherence
Lemma
The exact ell-four Ising gauge and parity chirality conversion
Theorem
Periodic Ising paths, two sectors, and the full line representation
Sources
- P. Etingof, S. Gelaki, D. Nikshych, and V. Ostrik, Tensor Categories
- Shu-Heng Shao, What’s Done Cannot Be Undone: TASI Lectures on Non-Invertible Symmetries
- David Penneys, Unitary Fusion Categories, Math 8800 lecture notes
- Andrei Bernevig and Titus Neupert, Topological Superconductors and Category Theory
- Chetan Nayak et al., Non-Abelian Anyons and Topological Quantum Computation, arXiv:0707.1889v2
- Joshua Chen, The Temperley-Lieb Categories and Skein Modules
- Jacob Siehler, Braided Near-Group Categories
- David Penneys, Unitary Fusion Categories
- Chetan Nayak et al., Non-Abelian Anyons and Topological Quantum Computation
- Shu-Heng Shao, What’s Done Cannot Be Undone