Alphabeta MathCategory O Finiteness Duality and Blocks
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1 · Prerequisites
- Abelian Categories
- Absolute and Conditional Convergence; Rearrangement; Products
- Algebraic Closure, Embeddings, and Separability
- Algebraic Extensions, Extension Degree, and Finite Fields
- Arc Length and Rectifiable Curves
- Artinian Rings and Length
- Binary Operations, Monoids, Groups and Subgroups
- Bounded Variation and the Riemann–Stieltjes Integral
- Cardinal Arithmetic, Cofinality and the Alephs
- Cartan Subalgebras and Root Space Decompositions
- Categories, Functors and Natural Transformations
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Chains, Antichains, Sperner and Dilworth
- Compactness in Metric Spaces
- Completeness, Completion, and Uniform Continuity
- Complex Coordinate and Path Transport
- Complex Differentiability and the Cauchy–Riemann Equations
- Complex Power Series and Analytic Functions
- Congruences, the Integers Modulo n and the Chinese Remainder Theorem
- Connectedness
- Construction of the Natural Numbers
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Continuity, IVT, EVT, and Uniform Continuity
- Contour Integration
- Cosets, Index and Lagrange's Theorem
- Countability and Uncountability
- Cyclic Groups and Direct Products
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Divisibility, Greatest Common Divisors and Bézout's Identity
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Filters and Ultrafilters
- Finite Counting, Factorials and Binomial Coefficients
- Finite Weyl Invariants, Bruhat Order, and Kostant Harmonics
- Foundations of the Real Numbers for Analysis
- Free Modules, Exact Sequences, Projective and Injective Modules
- Fundamental Trigonometric Identities
- Goursat's Theorem and Cauchy's Theorem in a Convex Domain
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Group Homomorphisms and the Isomorphism Theorems
- Harish Chandra Isomorphism Casimir and Central Characters
- Highest Weight Theory for Complex Semisimple Lie Algebras
- Holomorphic Functions of Several Complex Variables
- Homomorphisms Between Verma Modules and Linkage
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- Inner Product Spaces, Gram-Schmidt, Projections and Adjoints
- Lie Algebra Representations, Enveloping Algebras, and PBW
- Limits and Colimits
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- Line Integrals and the Gradient Theorem
- Linear Independence, Bases and Dimension
- Linear Transformations, Rank-Nullity and Quotient Spaces
- Matrices, the Matrix of a Linear Map, and Change of Basis
- Metric Spaces
- Mixed Partials, Taylor Formulae, and Extrema
- Modules, Submodules, Quotient Modules and the Isomorphism Theorems
- Monotone Functions, Discontinuities, and Continuity Sets
- Monotone Sequences, Bolzano-Weierstrass, and Cauchy Completeness
- Noetherian Rings and Hilbert Basis
- Normal Subgroups and Quotient Groups
- Order, Zorn's Lemma, and the Axiom of Choice
- Ordinal Arithmetic and the First Uncountable Ordinal
- Ordinals, Cardinals, and Transfinite Recursion
- Partitions of Unity and Paracompactness
- pi: the Equivalent Characterizations
- Polynomial Rings, the Division Algorithm and Roots
- Power Series and Real-Analytic Functions
- Preadditive and Additive Categories and Biproducts
- Primes, Euclid's Lemma and the Fundamental Theorem of Arithmetic
- Properties of the Integral and the Working FTC
- Reflective Subcategories and the Adjoint Functor Theorems
- Relations, Functions, and Quotients
- Rings, Subrings, Integral Domains and Fields
- Rⁿ as a Normed Space; Vector-Valued Functions
- Root Systems, Dynkin Diagrams, and the Cartan-Killing Classification
- Roots, Rational Powers, and Classical Inequalities
- Semisimple Lie Algebras, Cohomology, and Levi Theory
- Separation Axioms: the Hierarchy
- Sequences and Limits
- Sequences and Series of Functions; Uniform Convergence
- Series: Convergence and the Nonnegative Tests
- Simple Field Extensions and the Construction of the Complex Numbers
- Sine, Cosine, and the Definition of Pi
- Solvable and Nilpotent Lie Algebras
- Splitting Fields
- Subobject Lattices Generators and the Grothendieck Axioms
- Subspaces, Products, and Quotients
- Suprema and Infima
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tensor Products of Modules
- 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 Determinant of a Linear Operator, Cofactors and Cramer's Rule
- The Exponential Function
- The Fundamental Theorem of Finite Abelian Groups
- The Fundamental Theorems of Calculus
- The Galois Correspondence
- The Holomorphic Inverse Function Theorem and Weierstrass Preparation
- The Identity Theorem, the Maximum Principle and the Open Mapping Theorem
- The Inverse and Implicit Function Theorems
- The Logarithm and General Powers
- The Riemann Integral: Definition and Integrability
- The Spectral Theorem, Positive Operators and Singular Value Decomposition
- The Topology of Euclidean Space
- The Total Derivative in ℝᵐ → ℝⁿ
- The Winding Number and the Global Cauchy Theorem
- The ZFC Axioms and the Basic Set Constructions
- Topological Spaces and Continuity
- Topology of ℝ
- Triangularisation, Generalised Eigenspaces and Jordan Canonical Form
- Vector Spaces, Linear Subspaces, Span and Direct Sums
- Verma Modules and Shapovalov Forms
2 · Logical flowchart
3 · Items
Definition
The classical BGG category O
Lemma
Noetherianity of the enveloping algebra
Lemma
Finite Borel-stable generators and weight flags
Proposition
The support description of category O with finite generation
Proposition
Verma and finite-dimensional weight modules belong to O
Theorem
Category O is abelian and extension closed among weight modules
Lemma
A nonzero O-object has a highest-weight vector
Theorem
The simple objects of O
Lemma
Finite filtrations by highest-weight quotients
Lemma
The center has finite-dimensional image on each O-object
Definition
Generalized central-character subcategories
Lemma
Generalized central-character summands
Theorem
Generalized central-character decomposition of O
Lemma
Finite weight-space detection of subquotients
Theorem
Every object of O has finite length
Proposition
Verma composition multiplicities are finite
Theorem
The strong linkage principle for Verma modules
Lemma
A Verma embedding implies strong linkage
Theorem
The BGG criterion for homomorphisms between Verma modules
Proposition
Finite-dimensional Hom spaces in O
Definition
Restricted Chevalley dual
Lemma
Restricted self-duality of simple highest-weight modules
Proposition
Restricted duality is exact and involutive on O
Definition
Standard and costandard objects
Proposition
The simple socle of a costandard object
Definition
The integral Weyl group of a weight
Lemma
Integral reflection linkage is an equivalence relation
Lemma
Simple extensions cannot cross linkage classes
Lemma
Splitting finite-length modules across separated simple classes
Theorem
Central-character summands refine into linkage blocks
Corollary
Duality preserves linkage blocks and block orthogonality
Lemma
Verma self-extensions in O split
Proposition
Finite-dimensional tensoring preserves O
Definition
The Grothendieck group and character of O
Proposition
Simple and standard bases of K0(O)
Sources
- Lecture 2 §3 Definition 3.1, pp.4–5
- §15.1 p.79, Noetherian parenthesis
- Lecture 2 §3 Proposition 3.6 and Lemma 3.7, p.5
- §15.1 Definition 15.1 and Lemma 15.3, p.79
- §2 Definition 2.1 and Lemma 2.2, pp.2–3
- Chen, Lecture 2 §3 Lemma 3.3 and Warning 3.4, p.5
- Chen, Lecture 8 §3 Theorem 3.9 proof, p.5: extension closure in the weight-module category
- §15.1 p.79, paragraph after Definition 15.1
- Lecture 6 §2 Proposition 2.2, p.5
- Chen, Lecture 2 §3 Proposition 3.6, p.5
- Sakellaridis, §2 Lemma 2.3, p.3
- §15.1 Corollary 15.4, p.80
- §15.1 Corollary 15.7, p.80
- §15.1 Corollary 15.7 and proof, p.80
- §6 Theorem 6.2(1), pp.9–10
- Etingof, §15.1 Lemma 15.9, p.81: finite weight-space detector method
- Chen, Lecture 6 §2 Corollary 2.3 and Theorem 2.4 proof, pp.5–6: finite Weyl-orbit labels
- Lecture 6 §2 Theorem 2.4 and proof, pp.5–6
- Pavel Etingof, Representations of Lie Groups, Lemma 15.9
- Pavel Etingof, Representations of Lie Groups, Theorem 20.13
- Pavel Etingof, Representations of Lie Groups, Corollary 20.14
- Pavel Etingof, Representations of Lie Groups, Theorem 15.11
- Lecture 6 §2 Proposition 2.6 and Lemma 2.7, p.6
- Lecture 8 §3 Constructions 3.1, 3.4, 3.7 and Lemmas 3.2, 3.8, pp.4–5
- Lecture 8 §3 Proposition 3.10, p.5
- Chen, Lecture 8 §3 Lemma 3.8 and Theorem 3.9, p.5
- Etingof, §20.4 Proposition 20.9, p.103; compare the different Cartan twist convention
- Lecture 8 §3 Definition 3.12, p.5
- Lecture 8 §3 Corollary 3.13, p.5
- Humphreys, §3.4 definitions and invariance identities, p.52; §4.9, p.83 (reflection-generated convention)
- §3.4 p.52, displayed definitions and invariance statements
- §1.13 pp.30–32 and §4.9 p.83, decomposed proof route
- §1.13 p.31, formal decomposition paragraph
- §1.13 pp.30–32 and §4.9 pp.83–84
- Humphreys, §4.9 Exercise, p.84
- Chen, Lecture 8 §3 Corollary 3.11, p.5
- §15.1 Exercise 15.6(i), p.80
- §2 Lemma 2.5, p.3
- §2 Definition 2.4 and paragraph after Lemma 2.5, p.3
- §6 Theorem 6.2(4) and proof, p.10