Alphabeta MathKLR Algebras, Induction, and Quantum-Serre Categorification — Examples
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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
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
- Tensor and Fusion Categories
- Universal Properties, Representables and the Yoneda Lemma
Combinatorics
Commutative Algebra
Coxeter Groups
Group Theory
Homological Algebra
Lie Theory
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
- 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
2 · Logical flowchart
3 · Items
Example
The KLR algebras in root degrees αᵢ and 2αᵢ
Example
The two-vertex KLR polynomial representation
Example
The double-edge (e=2) KLR braid correction
Example
The nil-Hecke divided-power projective in rank two
Counterexample
Degenerate parameters Qᵢⱼ destroy the faithful polynomial-representation route
Sources
- Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups I, Representation Theory 13 (2009) 309-347, arXiv:0803.4121v2
- Alexander Kleshchev, Representation theory of symmetric groups and related Hecke algebras, Bulletin of the AMS 47 (2010) 419-481 (survey), arXiv:0909.4844
- Mikhail Khovanov and Aaron D. Lauda, A diagrammatic approach to categorification of quantum groups II, Transactions of the AMS 363 (2011) 2685-2700, arXiv:0804.2080v1
- Raphael Rouquier, 2-Kac-Moody Algebras, arXiv:0812.5023v3 (66 pages)