Hopf algebras and antipodes

Depends on

• Bialgebras and bialgebra maps
• Convolution of linear maps is associative and unital

Used by

• A module and comodule need not satisfy Hopf compatibility Counterexample
• A quasi-bialgebra need not be quasi-Hopf Counterexample
• Finite Drinfeld double: convention and candidate multiplication Definition
• Hopf ideals and Hopf subalgebras Definition
• Left and right integrals and integral functionals Definition
• Left-left Yetter–Drinfeld module Definition
• Leg notation and quasitriangular Hopf algebras Definition
• Right-right Hopf modules and coinvariants Definition
• The left dual of a finite-dimensional Hopf module Definition
• A finite calculation separating convolution from composition Example
• An invertible Yang–Baxter solution that is not a universal R-matrix Example
• The canonical R-matrix for the finite double of the group of order two Example
• The diagonal tensor action of ℂ[C₂] on trivial and sign modules Example
• The free Hopf module over the polynomial Hopf algebra Example
• The regular Hopf module and its projection Example
• The trivial universal R-matrix for the group algebra of a two-element group Example
• A commutative or cocommutative Hopf algebra has involutive antipode Lemma
• Counit normalization and the Yang–Baxter identity for a quasitriangular structure Lemma
• Double coalgebra, multiplicativity, and antipode Lemma
• Double crossing laws and associativity Lemma
• Group algebras, finite-group function algebras, and polynomial Hopf algebras Lemma
• Right duals and the antipode twist on biduals Lemma
• The coinvariant projection of a right-right Hopf module Lemma
• The finite dual is a right regular Hopf module Lemma
• The rank-one root-power quotient is a finite-dimensional Hopf algebra Lemma
• The Taft algebra: PBW basis and Hopf structure Lemma
• Quasi-Hopf and weak-Hopf extensions change different axioms Remark
• Evaluation, coevaluation, and triangle identities for left duals Theorem
• Finite Hopf algebras have one-dimensional integral spaces and bijective antipode Theorem
• Finite-dimensional Hopf algebras are Frobenius Theorem
• Hopf quotients, kernels of Hopf maps, and tensor products Theorem
• Hopf–Maschke theorem by integral averaging Theorem
• Modules over the finite double are left-left Yetter–Drinfeld modules Theorem
• The dual of a finite-dimensional Hopf algebra is a Hopf algebra Theorem
• The fundamental theorem of right-right Hopf modules Theorem
• Uniqueness and anti-multiplicativity and anti-comultiplicativity of the antipode Theorem

Dependency tree · two levels

8 results within two dependency steps of this one, each drawn at its shortest distance from it. An arrow runs from a result to what uses it, so the chart reads left to right and ends at this result, which carries a heavier outline. Every node is a link to that result. Click elsewhere on the chart to enlarge it.

Sources

• Hans-Jürgen Schneider, Lectures on Hopf algebras, notes by Sonia Natale (standard reference, not scraped)
• Pavel Etingof and Mykola Semenyakin, A brief introduction to quantum groups (standard reference, not scraped)