Algebra theory MOC

Hopf theory MOC

By “Hopf theory”¹ I mean the study of Hopf monoids and related constructions in the monoidal category K𝖬𝗈𝖽.² As such, the notation of string diagrams is useful, both for giving proofs and building intuition.

Algebras

• K-comonoid
• K-bimonoid
  • Quasitriangular K-bimonoid
• Hopf algebra

Examples

• Sweedler's large Hopf algebra, Sweedler's small Hopf algebra
• Drinfel'd double of the Taft algebra

Morphisms

• K-comonoid morphism
• K-bimonoid morphism
• Hopf algebra morphism

Internally

Subsets

• Coïdeal
• Biïdeal

Elements

• R-matrix
• Tracelike
• Casimir invariant of a Hopf K-algebra

Externally

• Convolution K-monoid
• Dual K-monoid of a K-comonoid
• Restricted dual of a K-monoid
• Restricted dual R-bimonoid

Representations

Internally

• H-invariant
• Annihilating coïdeal
• Bimonoid and Hopf annihilators

Externally

• Tensor product H-module
• H-hom module
• Dual H-module
• Adjoint action

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