Alphabeta MathFinitary Ringel–Hall Algebras
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Cosets, Index and Lagrange's Theorem
- Divisibility, Euclidean Domains, Principal Ideal Domains and Unique Factorisation
- Group Actions, Orbits, Stabilisers and Cayley's Theorem
- Ideals, Quotient Rings and the Isomorphism Theorems for Rings
- 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
- Splitting Fields
- Symmetric Polynomials and the Fundamental Theorem of Symmetric Functions
Category Theory
- Abelian Categories
- Adjunctions Units and Counits
- Categories, Functors and Natural Transformations
- Exactness and the Member Calculus
- Limits and Colimits
- Preadditive and Additive Categories and Biproducts
- Reflective Subcategories and the Adjoint Functor Theorems
- Subobject Lattices Generators and the Grothendieck Axioms
- The Diagram Lemmas in an Abelian Category
Homological Algebra
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
Number Theory
Real Analysis
Set Theory
- Cardinal Arithmetic, Cofinality and the Alephs
- Construction of the Natural Numbers
- 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
Symmetric Groups and Symmetric Functions
2 · Logical flowchart
3 · Items
Definition
Small finitary abelian categories for Hall multiplication
Lemma
Flags lift uniquely through a quotient in an abelian category
Definition
Nilpotent representations of the Jordan quiver
Definition
The untwisted Hall algebra of a finitary abelian category
Lemma
Jordan-quiver categories are abelian; finite-field models are finitary
Lemma
Jordan-quiver isomorphism classes are indexed by partitions
Lemma
Untwisted Hall structure constants and products are finite
Theorem
The untwisted Hall product is associative and unital
Theorem
The generic Jordan-quiver Hall algebra is the symmetric-function ring
Definition
Hall–Littlewood scalar product and monic functions
Lemma
Automorphism counts for nilpotent Jordan representations
Theorem
Jordan Hall classes are Hall–Littlewood functions with the transported Green pairing
Sources
- Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2, §§1.1 and 1.3
- Matthew B. Young, Degenerate versions of Green’s theorem for Hall modules, arXiv:1810.08238v2, §2.1
- Saunders Mac Lane, Categories for the Working Mathematician, VIII.3
- Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2, §1.3
- Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2, §2.1
- S. Treil, Linear Algebra Done Wrong, Chapter 9, Theorems 4.2–4.3
- Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2, §§2.1–2.4
- I. G. Macdonald, Symmetric Functions and Hall Polynomials, second edition, III §4 (4.11), VI §1 introductory characterization
- Olivier Schiffmann, Lectures on Hall Algebras, §2.3, Lemma 2.8
- Olivier Schiffmann, Lectures on Hall Algebras, §§2.3–2.4, Lemma 2.8 and (2.9)
- I. G. Macdonald, Symmetric Functions and Hall Polynomials, III §§2–4 and VI §1