Multiplier Hopf monoids

TL;DR

This paper generalizes the concept of multiplier Hopf algebras to braided monoidal categories, establishing foundational properties like the existence of an antipode and the Fundamental Theorem of Hopf modules in this broader setting.

Contribution

It introduces multiplier Hopf monoids in braided monoidal categories, extending classical multiplier Hopf algebra theory beyond vector spaces.

Findings

Existence of a unique antipode for multiplier Hopf monoids.

Antipode factorization through automorphisms in regular cases.

Equivalence between the base category and Hopf modules category.

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.