Alphabeta MathDrinfeld Doubles and Yetter–Drinfeld Module Constructions — Examples
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1 · Prerequisites
Abstract Algebra
- Binary Operations, Monoids, Groups and Subgroups
- Chain Conditions, Semisimple Modules and the Wedderburn–Artin Theorem
- Cosets, Index and Lagrange's Theorem
- 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, 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
- Tensor Products of Modules
Categorification and Geometric Representation Theory
- Bialgebras, Convolution, and Antipode Identities
- Coalgebras, Counits, and the Fundamental Coalgebra Theorem
- Comodules, Matrix Coefficients, and Coalgebra Duality
- Drinfeld Doubles and Yetter–Drinfeld Module Constructions
- Finite Hopf Integrals, Frobenius Duality, and Maschke
- Hopf Ideals, Finite Duals, and Basic Constructions
- Hopf Module Tensor Products and Rigid Duality
- Hopf Modules, Coinvariants, and the Fundamental Theorem
- Quasitriangular Hopf Algebras and Braided Module Categories
- Tensor Coherence and Algebraic Descent
Linear Algebra
- Determinants of Matrices over a Commutative Ring
- Dual Spaces, Bilinear and Quadratic Forms, and Sylvester's Law of Inertia
- 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
- Vector Spaces, Linear Subspaces, Span and Direct Sums
Real Analysis
2 · Logical flowchart
3 · Items
Counterexample
The full dual of an infinite group algebra has no transpose coproduct
Example
The Drinfeld double of a finite group algebra
Example
A nonabelian group makes the double cross-relations noncommutative
Sources
- Kalle Kytölä, Introduction to Hopf algebras and representations, lecture notes, §§4.3–4.4
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik, Tensor Categories, §7.14
- Kalle Kytölä, Introduction to Hopf algebras and representations, lecture notes, §4.3, Example 4.21 and Exercise 42
- G. A. García and C. Vay, Simple modules of small quantum groups at dihedral groups, Doc. Math. 29 (2024), §3.1.3
- Kalle Kytölä, Introduction to Hopf algebras and representations, lecture notes, Exercise 42