TY - JOUR

T1 - Multiplier Hopf monoids

AU - Böhm, Gabriella

AU - Lack, Stephen

PY - 2017

Y1 - 2017

N2 - 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.

AB - 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.

KW - Hopf algebra

KW - Multiplier

KW - Braided monoidal category

UR - http://www.scopus.com/inward/record.url?scp=84980042039&partnerID=8YFLogxK

UR - http://purl.org/au-research/grants/arc/DP130101969

UR - http://purl.org/au-research/grants/arc/FT110100385

U2 - 10.1007/s10468-016-9630-7

DO - 10.1007/s10468-016-9630-7

M3 - Article

AN - SCOPUS:84980042039

SN - 1386-923X

VL - 20

SP - 1

EP - 46

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

IS - 1

ER -