Multiplier bialgebras in braided monoidal categories

Böhm, Gabriella; Lack, Stephen

doi:10.1016/j.jalgebra.2014.11.002

Cited by 6 publications

(33 citation statements)

References 9 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…subject to compatibility conditions in [5]. A similar explanation of the monoidality of the category of modules over a (nice enough) weak multiplier bimonoid A is possible, in fact, but the treatment is technically more involved.…”

Section: T T T T T T Tmentioning

confidence: 98%

“…Remark 4.13. In [5], monoidality of the category of modules over a (nice enough) multiplier bimonoid A in a braided monoidal category C was explained by the structure of the induced endofunctor A(−) on C. Namely, it was shown to carry the structure of a multiplier bimonad; a generalization of bimonad (which is another name for opmonoidal monad). Recall that a multiplier bimonad on a monoidal category is an endofunctor T equipped with a morphism T 0 : T (I) → I and natural transformations…”

Section: T T T T T T Tmentioning

confidence: 99%

See 1 more Smart Citation

Weak Multiplier Bimonoids

Böhm

Gómez-Torrecillas

Lack

2017

Appl Categor Struct

Self Cite

View full text Add to dashboard Cite

ABSTRACT. Based on the novel notion of 'weakly counital fusion morphism', regular weak multiplier bimonoids in braided monoidal categories are introduced. They generalize weak multiplier bialgebras over fields [4] and multiplier bimonoids in braided monoidal categories [5]. Under some assumptions the so-called base object of a regular weak multiplier bimonoid is shown to carry a coseparable comonoid structure; hence to possess a monoidal category of bicomodules. In this case, appropriately defined modules over a regular weak multiplier bimonoid are proven to constitute a monoidal category with a strict monoidal forgetful type functor to the category of bicomodules over the base object.Braided monoidal categories considered include various categories of modules or graded modules, the category of complete bornological spaces, and the category of complex Hilbert spaces and continuous linear transformations.

show abstract

Section: T T T T T T Tmentioning

confidence: 98%

Section: T T T T T T Tmentioning

confidence: 99%

Weak Multiplier Bimonoids

Böhm

Gómez-Torrecillas

Lack

2017

Appl Categor Struct

Self Cite

View full text Add to dashboard Cite

show abstract

“…Multiplier bimonoids in braided monoidal categories are defined as compatible pairs of counital fusion morphisms [3]. Thus it is not too surprising that the first step in our simplicial characterization of multiplier bimonoids is a simplicial treatment of counital fusion morphisms.…”

Section: A Simplicial Description Of Multiplier Bimonoidsmentioning

confidence: 99%

“…Finally, applying the above construction to the braided monoidal category (C) rev = C rev we obtain a simplicial set M 4 . [3] in a braided monoidal category C consists of a fusion morphism t 1 in C and a fusion morphism t 2 in C rev with a common counit e : A → I such that the diagrams…”

Section: Counital Fusion Morphismsmentioning

confidence: 99%

“…For many applications it is important to work with Hopf algebras and bialgebras not only over fields but, more generally, in braided monoidal categories which are different from the symmetric monoidal category of vector spaces. The formulation of multiplier bialgebras in braided monoidal categories is our longstanding project initiated in [3].…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

A simplicial approach to multiplier bimonoids

Böhm¹,

Lack²

2017

Bull. Belg. Math. Soc. Simon Stevin

Self Cite

View full text Add to dashboard Cite

Although multiplier bimonoids in general are not known to correspond to comonoids in any monoidal category, we classify them in terms of maps from the Catalan simplicial set to another suitable simplicial set; thus they can be regarded as (co)monoids in something more general than a monoidal category (namely, the simplicial set itself). We analyze the particular simplicial maps corresponding to that class of multiplier bimonoids which can be regarded as comonoids.

show abstract

Multiplier Hopf Monoids

Böhm

Lack

2016

Algebr Represent Theor

Self Cite

View full text Add to dashboard Cite

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 provedwhich states an equivalence between the base category and the category of Hopf modules over a multiplier Hopf monoid.

show abstract

Multiplier bialgebras in braided monoidal categories

Abstract: Abstract. Multiplier bimonoids (or bialgebras) in arbitrary braided monoidal categories are defined. They are shown to possess monoidal categories of comodules and modules. These facts are explained by the structures carried by their induced functors.

Cited by 6 publications

References 9 publications

Weak Multiplier Bimonoids

Weak Multiplier Bimonoids

A simplicial approach to multiplier bimonoids

Multiplier Hopf Monoids

Contact Info

Product

Resources

About