Alphabeta MathVertex Modules Intertwiners And Fusion
About this page
Pipeline-generated
How statement and proof provenance work
The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.
- Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
- AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
- AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.
These labels describe origin, not correctness: citations and verification chips remain separate evidence.
1 · Prerequisites
Abstract Algebra
- 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 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
- The Field of Fractions and Localisation
Category Theory
- Abelian Categories
- Braided and Symmetric Monoidal Categories
- Categories, Functors and Natural Transformations
- Duality and Rigidity in Monoidal Categories
- Limits and Colimits
- Monoidal Categories and Monoidal Functors
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- Tensor and Fusion Categories
Combinatorics
Commutative Algebra
Complex Analysis
- Analytic Continuation, Monodromy, and Riemann Surfaces
- Analyticity of Holomorphic Functions; Liouville and Morera
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Conformal Mapping, Branches, and the Schwarz Lemma
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Simply Connected Plane Domains: the Grand Equivalence
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Winding Number and the Global Cauchy Theorem
Linear Algebra
Quantum Algebra and Representation Theory
Quantum Field Theory Mathematics
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Bounded Variation and the Riemann–Stieltjes Integral
- 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
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Mixed Partials, Taylor Formulae, and Extrema
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- pi: the Equivalent Characterizations
- Power Series and Real-Analytic Functions
- 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
- Sine, Cosine, and the Definition of Pi
- Suprema and Infima
- The Cantor Set, Baire Category, and Measure Zero in ℝ
- The Complex Exponential and Euler's Formula
- The Derivative and the Mean Value Theorems
- The Exponential Function
- The Formal Laurent Series Field ℝ((t⁻¹)): Cauchy Complete, Non-Archimedean, Not Complete
- The Fundamental Theorems of Calculus
- The Logarithm and General Powers
- 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
String Theory
Topology
- Applications of the Fundamental Group
- Compactness
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Connectedness
- Countability Axioms and Cardinal Functions
- Covering Spaces and Lifting
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Fundamental Group of the Circle
- The Topology of Euclidean Space
- Topological Spaces and Continuity
2 · Logical flowchart
3 · Items
Definition
Strongly graded generalized module in the HLZ setting
Definition
Quasi-finite-dimensional generalized module
Definition
Contragredient module of a generalized module
Lemma
Opposite fields under formal coordinate inversion
Definition
C1-cofinite generalized module in the ungraded HLZ convention
Definition
Logarithmic intertwining operator
Theorem
The restricted dual with opposite action is a contragredient module
Definition
P(z)-intertwining map
Theorem
Strongly graded contragredient duality for Möbius and conformal modules
Lemma
Finite Jacobi quotient with a bounded exterior dual and C1-cofinite leaves
Lemma
Parameter-uniform convergence of formal Euler series
Theorem
Multipoint analytic extension with a bounded exterior dual
Definition
HLZ convergence and extension property
Definition
Intertwining-operator space and fusion multiplicity
Lemma
Finite generation of the multipoint Jacobi quotient
Definition
P(z)-tensor product in a module category
Theorem
Logarithmic intertwiners and P(z)-maps correspond
Definition
P(z)-compatibility subspace and local grading restriction
Lemma
Pole bounds for the Jacobi quotient at a finite divisor
Lemma
Logarithmic reconstruction and parallel transport of P(z)-products
Theorem
C1-cofinite ordinary intertwining spaces are finite dimensional
Theorem
Compatibility subspace yields the P(z)-tensor product
Definition
P(1)-based braided monoidal structure from HLZ intertwiners
Theorem
Strongly graded P(z) products from restricted-dual images
Lemma
Absolute exponential-polynomial expansions with real exponents
Lemma
Finite rational current actions and shifted partial evaluation
Lemma
Strong local grading and admission of transported coefficient modules
Lemma
Rational Ward identities for actual convergent products and iterates
Lemma
Weak partial-evaluation spectral expansions satisfy full P(z) compatibility
Lemma
Product-iterate factorization from compatible spectral partial evaluations
Lemma
Spanning by analytic nested universal tensor components with real weights
Lemma
Three-intertwiner tree expansions and extension of local associativity moves
Lemma
Logarithmic skew symmetry and the tensor unit maps
Theorem
Braided monoidal structure from HLZ analytic hypotheses
Sources
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part I, §2, Definitions 2.2, 2.5, 2.23 and 2.25, and Remarks 2.6, 2.24 and 2.28
- Christophe Nozaradan, Introduction to Vertex Algebras
- Tomoyuki Arakawa and Anne Moreau, Arc spaces and vertex algebras
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part VII
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part I
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part I, equations (2.57)–(2.67) and Lemma 2.22
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part II, Definition 3.10 and equation (3.26)
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part III, §4.1, Definition 4.2 and equations (4.3)–(4.5)
- Huang, Lepowsky and Zhang, Part I, Definition 2.32, equations (2.96)–(2.101), Theorem 2.34
- Huang, Differential equations and intertwining operators, §1; the bounded-exterior reduction here avoids a C1 assumption on the output dual
- Huang, Lepowsky and Zhang, Part VII, §11.2
- Yuto Moriwaki, Vertex operator algebra and parenthesized braid operad
- Huang, Lepowsky and Zhang, Part VII, Theorem 11.6, Remark 11.7 and the multipoint conclusion of Theorem 11.8
- Huang, Differential equations and intertwining operators, §§1–2; the proof here uses bounded exterior weights rather than assuming output-dual C1-cofiniteness
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part V
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part VII, §11.1, equations (11.1)–(11.8), and §11.2, Theorems 11.6 and 11.8, Remark 11.7
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part II
- Yi-Zhi Huang, Differential equations and intertwining operators, §1, Proposition 1.1 and Corollary 1.2
- Huang, Lepowsky and Zhang, Logarithmic tensor category theory, Part VII, §11.2
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part III
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part IV, Section 5.2, Proposition 5.33, P(z)-local grading restriction condition and Notation 5.43
- Huang, Differential equations and intertwining operators, Proposition 2.1, Lemma 2.2 and Theorem 2.3; the proof below spells out the finite-divisor multipoint extension
- Huang, Lepowsky and Zhang, Part VIII, §12.2, parallel transport; logarithmic reconstruction uses the L(0)-conjugation formula
- Yi-Zhi Huang, Differential equations and intertwining operators
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part IV
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part VIII, §12.2 and Theorem 12.15
- Huang, Lepowsky and Zhang, Part IV, §5.2, Proposition 5.33 and the dual-image construction
- Huang, Lepowsky and Zhang, Part V, Lemma 7.7, Proposition 7.8 and Proposition 7.9
- Huang, Lepowsky and Zhang, Logarithmic tensor category theory, IV
- Huang, Lepowsky and Zhang, Logarithmic tensor category theory, V
- Huang, Lepowsky and Zhang, Logarithmic tensor category theory, VI
- Huang, Lepowsky and Zhang, Part VI, Theorems 9.17 and 9.23; reconstructed local proof, no external theorem premise
- Huang, Lepowsky and Zhang, Part VI, Lemma 9.22 and Theorem 9.23; the proof below reconstructs the outer factorization
- Huang, Lepowsky and Zhang, Part VIII, Proposition 12.6 and Corollary 12.7
- Huang, Lepowsky and Zhang, Part VIII, Proposition 12.7, Corollary 12.8 and Proposition 12.9
- Huang, Lepowsky and Zhang, Logarithmic tensor category theory, II
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part VI, Assumption 10.1, Definition 9.28, Theorem 9.27 and Theorem 10.3
- Huang, Lepowsky, and Zhang, Logarithmic tensor category theory, Part VIII, Assumptions 12.1–12.2, Theorem 12.15, Corollary 12.16 and Remark 12.17