Alphabeta MathEuler-Twisted Hall Products, Green Coproducts, and Quantum Groups
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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 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
- 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
Algebraic Geometry
Braid Groups
Categorification and Geometric Representation Theory
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
Combinatorics
Commutative Algebra
Coxeter Groups
- Bipartite Coxeter Elements and Ordered Root Complexes
- Canonical Roots, Signs, and Faithful Reflections
- Conjugacy in Sₙ, Generation, and the Simplicity of Aₙ
- Coxeter Presentations, Exchange, and Reduced Word Theorems
- Finite Coxeter Diagrams and Complete Classification
- Finite Reflection Length and Orthogonal Moved Spaces
- Noncrossing Partition Lattices and Kreweras Complements
- Permutation Statistics, Inversions and Eulerian Numbers
- Real Forms and Reflection Geometry
- Symmetric Groups, Cycle Decomposition and the Sign Homomorphism
- Tits Cones, Chambers, and Parabolic Stabilizers
Functional Analysis
Homological Algebra
Lie Theory
- Affine Lie Algebras and Loop Central Extensions
- Finite Weyl Invariants, Bruhat Order, and Kostant Harmonics
- Harish Chandra Isomorphism Casimir and Central Characters
- Integrable Highest Weight Kac Moody Modules
- Kac Moody Algebras from Generalized Cartan Matrices
- Lie Algebra Representations, Enveloping Algebras, and PBW
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Diagonalisation and the Minimal Polynomial
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- Eigenvalues, Eigenvectors and the Characteristic Polynomial
- Exterior Powers, Orientation and Hodge Duality
- Gaussian Elimination, Elementary Matrices and Reduced Row Echelon Form
- 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
- 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
- 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
- Fundamental Trigonometric Identities
- Limits of Real Functions
- limsup, liminf, and Subsequential Limits
- 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 Riemann Integral: Definition and Integrability
- 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
2 · Logical flowchart
3 · Items
Definition
The Hall-normalized extended positive Borel of a loopless quiver
Lemma
A symmetric Gaussian alternating sum vanishes in the Serre range
Lemma
A zero Yoneda splice factors through the middle object
Lemma
Low-degree exact sequences for Yoneda Ext in both variables
Definition
Finite-field hereditary Hall categories
Lemma
Finite-dimensional representations of an acyclic quiver form a hereditary finitary Hall category
Lemma
Finite loopless quiver nilpotent representation categories are hereditary and finitary
Lemma
The hereditary Euler form is additive on the Grothendieck group
Lemma
Exact-sequence counts equal endpoint automorphism counts times subobject counts
Definition
The Euler-twisted Hall algebra
Lemma
The Euler-twisted Hall product is well-defined
Definition
The Green Hall completion and coproduct
Lemma
The Euler-twisted Hall product is associative and unital
Lemma
Green coproduct coefficients are finite
Lemma
The Green cross–square counting identity
Definition
The extended Hall algebra and composition subalgebra
Lemma
The extended Hall torus action is well-defined
Lemma
The Green Hall coproduct is coassociative
Lemma
Products of Green coproducts converge coefficientwise
Theorem
Green’s coproduct is an algebra map into the twisted tensor completion
Proposition
The Green Hall pairing is nondegenerate and adjoint to product and coproduct
Corollary
The extended Green coproduct is an algebra map
Lemma
The extended Green and repository Drinfeld–Jimbo coproduct conventions agree on simple generators
Proposition
The Ringel assignment is a bialgebra homomorphism onto the extended composition subalgebra
Theorem
Prime-power specializations of the integral quantum half have a nondegenerate Green form
Lemma
Finite Serre word forms descend and have classical PBW rank bounds
Lemma
Green positivity specializes the integral half into the composition algebra
Lemma
Reflection classifies finite-Dynkin quiver indecomposables over a finite field
Lemma
Finite-Dynkin Hall algebras are generated by vertex simples
Lemma
Finite-field descent and restriction of Schur quiver representations
Lemma
Explicit affine D and E flag families give exponentially many indecomposables
Lemma
The full Hall algebra is simple-generated exactly in componentwise finite Dynkin type
Lemma
Rank-one skew kernels in a numerical quantum Serre algebra
Lemma
The nondegenerate integral specialization has a quantum Casimir
Lemma
A minimal numerical Serre kernel obeys the quantum Casimir constraint
Theorem
The prime-power numerical Serre half is the integral specialization
Theorem
The Ringel map embeds the extended positive Borel and is full Hall exactly in finite Dynkin type
Sources
- Olivier Schiffmann, Lectures on Hall Algebras, arXiv:math/0611617v2
- Claus Michael Ringel, The Hall Algebra Approach to Quantum Groups, E.L.A.M. Lectures (1993)
- Claus Michael Ringel, Hall algebras and quantum groups, Inventiones Mathematicae 101 (1990), 583–591
- Barry Mitchell, Theory of Categories, Chapter VII, §§4–5
- Charles A. Weibel, An Introduction to Homological Algebra, §3.4, Yoneda Ext
- Barry Mitchell, Theory of Categories, Chapter VII, §§2–5
- Matthew B. Young, Degenerate versions of Green’s theorem for Hall modules, arXiv:1810.08238v2
- Barry Mitchell, Theory of Categories, Chapter VII, §5
- Kyeonghoon Jeong, Seok-Jin Kang and Masaki Kashiwara, Crystal Bases for Quantum Generalized Kac-Moody Algebras, arXiv:math/0305390
- Masaki Kashiwara, On crystal bases of the q-analogue of universal enveloping algebras
- Olivier Schiffmann, Lectures on Hall Algebras
- Claus Michael Ringel, The Hall Algebra Approach to Quantum Groups, E.L.A.M. Lectures
- Istvan Heckenberger and Stefan Kolb, On the Bernstein-Gelfand-Gelfand resolution for Kac-Moody algebras and quantized enveloping algebras, §3.4 (route context)
- G. Lusztig, Introduction to Quantum Groups, §§1.2 and 38.1
- George Lusztig, Introduction to Quantum Groups, chapter 6, sections 6.1.1–6.1.2
- Toshiyuki Tanisaki, Characters of integrable highest weight modules over a quantum group, section 4
- Alexander Kleshchev, Lectures on Infinite Dimensional Lie Algebras, Proposition 9.3.4 (classical relation-module model)
- Istvan Heckenberger and Stefan Kolb, On the Bernstein-Gelfand-Gelfand resolution for Kac-Moody algebras and quantized enveloping algebras, section 3.4
- George Lusztig, Introduction to Quantum Groups, chapter 6
- Claus Michael Ringel, Hall algebras and quantum groups, E.L.A.M. Lectures (1993)