Alphabeta MathRational Cherednik Algebras and Their Category O
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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
- Maschke's Theorem, Complete Reducibility and the Structure of k[G]
- 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
- The Group Algebra and Representations of Finite Groups
Commutative Algebra
Complex Analysis
- 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
- Contour Integration
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Holomorphic Functions of Several Complex Variables
- Isolated Singularities and Laurent Series
- The Holomorphic Inverse Function Theorem and Weierstrass Preparation
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Riemann Sphere and Möbius Transformations
- The Winding Number and the Global Cauchy Theorem
Differential Geometry
Group Theory
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- 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
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
- 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 Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The Total Derivative in ℝᵐ → ℝⁿ
- Topology of ℝ
Set Theory
2 · Logical flowchart
3 · Items
Lemma
A logarithmic connection determines meridian eigenvalues from its residue
Definition
Reflection data and parameters for a rational Cherednik algebra
Definition
The rational Cherednik algebra H_t,c(W,h)
Lemma
The rational Cherednik commutator satisfies Jacobi compatibility
Theorem
The PBW theorem for rational Cherednik algebras
Definition
Category O for a rational Cherednik algebra
Definition
Standard modules for rational Cherednik category O
Theorem
Standard modules lie in rational Cherednik category O
Definition
The polynomial representation of a rational Cherednik algebra
Lemma
Every rational Cherednik category-O object has finite length
Lemma
Dunkl operators realize the polynomial standard module
Lemma
Finite modules with a connection on a complex polynomial open are vector bundles
Lemma
Dunkl localization gives flat connections on the regular locus
Definition
The KZ monodromy functor and the reflection Hecke algebra
Lemma
The standard-module KZ connection has the reflection residue
Theorem
Cherednik category O has a standard-filtered projective generator
Lemma
Hecke relations specialize in the generalized-standard family
Theorem
KZ monodromy of standard modules satisfies the reflection Hecke relations
Theorem
KZ monodromy on category O factors through the reflection Hecke algebra
Lemma
Nilpotent induced Cherednik connections have uniform moderate growth on generic lines
Lemma
Uniformly bounded separate rationality implies rationality
Theorem
Localized Cherednik monodromy is fully faithful and its image is closed under subquotients
Lemma
Cherednik costandards have orthogonality and the standard character
Theorem
Cherednik BGG reciprocity and the KZ projective rank
Theorem
KZ identifies the torsion quotient with Hecke modules under the Hecke-rank hypothesis
Sources
- Pavel Etingof and Xiaoguang Ma, Lecture Notes on Cherednik Algebras
- Victor Ginzburg, Nicolas Guay, Eric Opdam, and Raphael Rouquier, On the category O for rational Cherednik algebras
- Pavel Etingof and Victor Ginzburg, Symplectic reflection algebras, Calogero–Moser space, and deformed Harish-Chandra homomorphism
- Ginzburg, Guay, Opdam and Rouquier, On the category O for rational Cherednik algebras
- Etingof and Ma, Lecture Notes on Cherednik Algebras