Multiplier Hopf monoids

Gabriella Böhm, Stephen Lack*

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    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
    Volume20
    Issue number1
    DOIs
    Publication statusPublished - 2017

    Keywords

    • Hopf algebra
    • Multiplier
    • Braided monoidal category

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