Alphabeta MathQuantum Groups at Roots of Unity and Tilting Quotients: Examples and Counterexamples
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1 · Prerequisites
Abstract Algebra
- Algebraic Closure, Embeddings, and Separability
- 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 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
- Tensor Products of Modules
- The Field of Fractions and Localisation
Braid Groups
Categorification and Geometric Representation Theory
- Bialgebras, Convolution, and Antipode Identities
- Coalgebras, Counits, and the Fundamental Coalgebra Theorem
- Divided Powers, Integral Forms, and Quantum Highest Weights
- Quantized Enveloping Algebras and Quantum Serre Relations
- Quantum Groups at Roots of Unity and Tilting Quotients
- Tensor Coherence and Algebraic Descent
Coxeter Groups
Lie Theory
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- 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
Real Analysis
- Constant Rank, Submersions, Immersions and Regular Level Sets
- Construction of the Real Numbers via Cauchy Sequences
- Construction of the Real Numbers via Dedekind Cuts
- Countability and Uncountability
- Foundations of the Real Numbers for Analysis
- Roots, Rational Powers, and Classical Inequalities
- Suprema and Infima
2 · Logical flowchart
3 · Items
Example
The rank-one small quantum group at a primitive fifth root has dimension 125
Counterexample