Multiplier Hopf Monoids

G. Böhm, Stephen Lack

Research output: Contribution to journalArticle

Abstract

The notion of multiplier Hopf monoid in any braided monoidal category is introduced as a multiplier bimonoid whose constituent fusion morphisms are isomorphisms. In the category of vector spaces over the complex numbers, Van Daele’s definition of multiplier Hopf algebra is re-obtained. It is shown that the key features of multiplier Hopf algebras (over fields) remain valid in this more general context. Namely, for a multiplier Hopf monoid A, the existence of a unique antipode is proved — in an appropriate, multiplier-valued sense — which is shown to be a morphism of multiplier bimonoids from a twisted version of A to A. For a regular multiplier Hopf monoid (whose twisted versions are multiplier Hopf monoids as well) the antipode is proved to factorize through a proper automorphism of the object A. Under mild further assumptions, duals in the base category are shown to lift to the monoidal categories of modules and of comodules over a regular multiplier Hopf monoid. Finally, the so-called Fundamental Theorem of Hopf modules is proved — which states an equivalence between the base category and the category of Hopf modules over a multiplier Hopf monoid.

Original languageEnglish
Pages (from-to)1-46
Number of pages46
JournalAlgebras and Representation Theory
DOIs
Publication statusAccepted/In press - Jul 27 2016

Fingerprint

Monoids
Multiplier
Monoid
Multiplier Algebra
Antipode
Monoidal Category
Hopf Algebra
Module
Factorise
Comodule
Morphism
Complex number
Morphisms
Automorphism
Vector space
Isomorphism
Fusion
Equivalence
Valid
Theorem

Keywords

  • Braided monoidal category
  • Hopf algebra
  • Multiplier

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Multiplier Hopf Monoids. / Böhm, G.; Lack, Stephen.

In: Algebras and Representation Theory, 27.07.2016, p. 1-46.

Research output: Contribution to journalArticle

Böhm, G. ; Lack, Stephen. / Multiplier Hopf Monoids. In: Algebras and Representation Theory. 2016 ; pp. 1-46.
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