### 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 language | English |
---|---|

Pages (from-to) | 1-46 |

Number of pages | 46 |

Journal | Algebras and Representation Theory |

DOIs | |

Publication status | Accepted/In press - Jul 27 2016 |

### Fingerprint

### Keywords

- Braided monoidal category
- Hopf algebra
- Multiplier

### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Algebras and Representation Theory*, 1-46. https://doi.org/10.1007/s10468-016-9630-7

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

Research output: Contribution to journal › Article

*Algebras and Representation Theory*, pp. 1-46. https://doi.org/10.1007/s10468-016-9630-7

}

TY - JOUR

T1 - Multiplier Hopf Monoids

AU - Böhm, G.

AU - Lack, Stephen

PY - 2016/7/27

Y1 - 2016/7/27

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 - Braided monoidal category

KW - Hopf algebra

KW - Multiplier

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

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

U2 - 10.1007/s10468-016-9630-7

DO - 10.1007/s10468-016-9630-7

M3 - Article

AN - SCOPUS:84980042039

SP - 1

EP - 46

JO - Algebras and Representation Theory

JF - Algebras and Representation Theory

SN - 1386-923X

ER -