Alphabeta MathQuantum Groups at Roots of Unity and Tilting Quotients
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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
- 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
- Finite Fields and Cyclotomic Extensions
- 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
- Semidirect Products, Automorphism Groups and Split Extensions
- Simple Field Extensions and the Construction of the Complex Numbers
- Splitting Fields
- Tensor Products of Modules
- The Field of Fractions and Localisation
- The Fundamental Theorem of Finite Abelian Groups
- The Galois Correspondence
- The Group Algebra and Representations of Finite Groups
Algebraic Geometry
Algebraic Topology
Categorification and Geometric Representation Theory
- Balanced Triples and Global Crystal Bases
- Bialgebras, Convolution, and Antipode Identities
- Coalgebras, Counits, and the Fundamental Coalgebra Theorem
- Comodules, Matrix Coefficients, and Coalgebra Duality
- Divided Powers, Integral Forms, and Quantum Highest Weights
- Finite Hopf Integrals, Frobenius Duality, and Maschke
- Hopf Ideals, Finite Duals, and Basic Constructions
- Hopf Modules, Coinvariants, and the Fundamental Theorem
- Kashiwara Operators and Crystal Lattices
- Quantized Enveloping Algebras and Quantum Serre Relations
- Tensor Coherence and Algebraic Descent
- Tensor-Product Crystals and Highest-Weight Crystal Uniqueness
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- 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
- Subobject Lattices Generators and the Grothendieck Axioms
- Tensor and Fusion Categories
- The Diagram Lemmas in an Abelian Category
- Universal Properties, Representables and the Yoneda Lemma
Combinatorics
Commutative Algebra
- Artinian Rings and Length
- Integral Extensions and Going Up
- Krull Dimension and Height Theorems
- Localisation of Modules and Support
- Noether Normalisation and Nullstellensatz
- Noetherian Rings and Hilbert Basis
- Prime Spectra and Radicals
- Valuation Rings and Discrete Valuation Rings
- Zariski Topology on Prime Spectra
Complex Analysis
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Holomorphic Functions of Several Complex Variables
- The Holomorphic Inverse Function Theorem and Weierstrass Preparation
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Winding Number and the Global Cauchy Theorem
Coxeter Groups
- Affine Reflections, Coroot Translations, and Alcoves
- Canonical Roots, Signs, and Faithful Reflections
- Conjugacy in Sₙ, Generation, and the Simplicity of Aₙ
- Coxeter Presentations, Exchange, and Reduced Word Theorems
- Crystallographic Root Lattices and Weyl Group Interfaces
- Finite Coxeter Diagrams and Complete Classification
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- Real Forms and Reflection Geometry
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
Differential Geometry
- Distributions Integral Manifolds and the Frobenius Theorem
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- Rank Theorems and Embedded Submanifolds
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- Smooth Partitions of Unity and Exhaustions
- Smooth Vector Bundles and Sections
- Tangent Cotangent and the Differential
- Vector Fields Flows and Lie Derivatives
Functional Analysis
- Bounded Linear Operators and Quotient Spaces
- Complex Norms and Realification
- Complex Subspace Realification
- Finite Dimensional Normed Spaces and Riesz Lemma
- Hilbert Space Geometry and Riesz Representation
- Locally Convex Spaces and Continuous Separation
- Normed and Banach Spaces
- The Analytic Hahn Banach Theorem
Homological Algebra
Lie Theory
- Cartan Subalgebras and Root Space Decompositions
- Category O Finiteness Duality and Blocks
- Finite Weyl Invariants, Bruhat Order, and Kostant Harmonics
- Harish Chandra Isomorphism Casimir and Central Characters
- Highest Weight Theory for Complex Semisimple Lie Algebras
- Homomorphisms Between Verma Modules and Linkage
- Integrable Highest Weight Kac Moody Modules
- Kac Moody Algebras from Generalized Cartan Matrices
- Lie Algebra Representations, Enveloping Algebras, and PBW
- Lie Groups, Invariant Fields, and the Exponential Map
- Lie Subgroups, Actions, and Homogeneous Spaces
- Root Systems, Dynkin Diagrams, and the Cartan-Killing Classification
- Semisimple Lie Algebras, Cohomology, and Levi Theory
- Solvable and Nilpotent Lie Algebras
- Tensor Product Multiplicities and Littlewood Richardson
- The BGG Resolution
- The Weyl Kac Character Formula
- Verma Modules and Shapovalov Forms
- Weyl Character and Multiplicity Formulas
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
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- 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
Number Theory
Real Analysis
- Absolute and Conditional Convergence; Rearrangement; Products
- Arc Length and Rectifiable Curves
- Bounded Variation and the Riemann–Stieltjes Integral
- 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
- 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
- Picard-Lindelöf and First-Order Ordinary Differential Equations
- 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 Fundamental Theorems of Calculus
- The Inverse and Implicit Function Theorems
- The Inverse Function Theorem Completed
- The Logarithm and General Powers
- The Riemann Integral in Rᵐ and Jordan Content
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Scheme Theory
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
- 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
- Function Space Topologies and the Exponential Law
- Hereditary and Productive Behaviour of the Separation Axioms
- Homotopy and Homotopy Equivalence
- Metric Spaces
- Partitions of Unity and Paracompactness
- Separation Axioms: the Hierarchy
- Subspaces, Products, and Quotients
- The Fundamental Group
- The Topology of Euclidean Space
- Topological Spaces and Continuity
- Uniform Spaces: the Three Definitions
2 · Logical flowchart
3 · Items
Definition
De Concini–Kac, Lusztig, and small quantum-group forms at roots of unity
Lemma
Affine closest pairs and downward translation from a lower facet
Lemma
Higher divided quantum Serre relations from a skew derivation
Lemma
Integral facet displacements with closed-alcove dominant representatives
Lemma
Saturated Weyl filtrations of finite dimensional integral based modules
Lemma
The full divided-power integral form is a Hopf form
Lemma
Quantum Lucas and generation of specialized divided powers
Lemma
Simply-laced braid symmetries preserve the full divided-power form
Definition
The full toral-weight Lusztig module category
Definition
Tilting objects and negligible morphisms in a root-of-unity category
Lemma
The root-specialized toral quotient by group-like generators
Lemma
The rank-one root-power quotient is a finite-dimensional Hopf algebra
Theorem
Integral convex root PBW bases in finite simply-laced type
Lemma
Full divided powers force root-of-unity weight reflection symmetry
Lemma
Braid-stable PBW decompositions of the small ordinary image
Proposition
The rank-one De Concini–Kac to Lusztig specialization map is not injective
Theorem
Quantum Frobenius and its kernel in the good odd-order regime
Theorem
The full toral-weight root category is braided, rigid and spherical
Definition
Integral Weyl and dual Weyl families at a root of unity
Lemma
Root specialization of integral Weyl lattices and their characters
Lemma
Weyl quantum dimensions and affine Weyl signs
Theorem
Tensor stability and dual stability of root-of-unity tilting modules
Lemma
Weyl flag summands, standard-costandard orthogonality and tilting uniqueness
Lemma
Finite highest weight truncations from integral lowest-highest tensors
Theorem
The negligible ideal and semisimple trace quotient of root tiltings
Theorem
Affine Weyl linkage from central monodromy traces
Theorem
Existence and highest weight classification of root-of-unity tiltings
Lemma
Simplicity of Weyl modules in the closed bottom alcove
Theorem
Strong affine linkage for root Weyl modules and tilting sections
Theorem
Outside-alcove indecomposable tiltings are negligible
Theorem
The simply-laced large-odd-order tilting fusion quotient
Sources
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych, and Victor Ostrik, Tensor Categories
- Pavel Etingof and Mykola Semenyakin, A Brief Introduction to Quantum Groups
- Stephen F. Sawin, Quantum Groups at Roots of Unity and Modularity
- George Lusztig, Introduction to Quantum Groups
- Henning Haahr Andersen and Catharina Stroppel, Fusion Rings for Quantum Groups
- Etingof, Gelaki, Nikshych and Ostrik, Tensor Categories
- Volodymyr Mazorchuk, Lectures on Algebraic Categorification
- H. H. Andersen, J. C. Jantzen and W. Soergel, Representations of quantum groups at a p-th root of unity and of semisimple groups in characteristic p, Asterisque 220 (1994)