Alphabeta MathDrinfeld Doubles and Yetter–Drinfeld Module Constructions
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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 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
- 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
- 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
Real Analysis
2 · Logical flowchart
3 · Items
Definition
Left-left Yetter–Drinfeld module
Definition
Finite Drinfeld double: convention and candidate multiplication
Lemma
Double crossing laws and associativity
Lemma
Double coalgebra, multiplicativity, and antipode
Theorem
Modules over the finite double are left-left Yetter–Drinfeld modules
Theorem
Finite double and its canonical quasitriangular structure
Sources
- G. A. García and C. Vay, Simple modules of small quantum groups at dihedral groups, Doc. Math. 29 (2024), §3.1.3
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik, Tensor Categories, §7.15
- Kalle Kytölä, Introduction to Hopf algebras and representations, lecture notes, §4.3
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik, Tensor Categories, §7.14
- Pavel Etingof, Shlomo Gelaki, Dmitri Nikshych and Victor Ostrik, Tensor Categories, §§7.14–7.15
- Kalle Kytölä, Introduction to Hopf algebras and representations, lecture notes, §4.3, Theorem 4.18