Comodules over weak multiplier bialgebras

Research output: Contribution to journalArticle

7 Citations (Scopus)

Abstract

This is a sequel paper of [Weak multiplier bialgebras, Trans. Amer. Math. Soc., in press] in which we study the comodules over a regular weak multiplier bialgebra over a field, with a full comultiplication. Replacing the usual notion of coassociative coaction over a (weak) bialgebra, a comodule is defined via a pair of compatible linear maps. Both the total algebra and the base (co)algebra of a regular weak multiplier bialgebra with a full comultiplication are shown to carry comodule structures. Kahng and Van Daele's integrals [The Larson-Sweedler theorem for weak multiplier Hopf algebras, in preparation] are interpreted as comodule maps from the total to the base algebra. Generalizing the counitality of a comodule to the multiplier setting, we consider the particular class of so-called full comodules. They are shown to carry bi(co)module structures over the base (co)algebra and constitute a monoidal category via the (co)module tensor product over the base (co)algebra. If a regular weak multiplier bialgebra with a full comultiplication possesses an antipode, then finite-dimensional full comodules are shown to possess duals in the monoidal category of full comodules. Hopf modules are introduced over regular weak multiplier bialgebras with a full comultiplication. Whenever there is an antipode, the Fundamental Theorem of Hopf Modules is proven. It asserts that the category of Hopf modules is equivalent to the category of firm modules over the base algebra.

Original languageEnglish
Article number1450037
JournalInternational Journal of Mathematics
Volume25
Issue number5
DOIs
Publication statusPublished - 2014

Fingerprint

Comodule
Bialgebra
Multiplier
Module
Coalgebra
Antipode
Monoidal Category
Algebra
Compatible Maps
Coaction
Multiplier Algebra
Linear map
Hopf Algebra
Theorem
Tensor Product
Preparation

Keywords

  • Antipode
  • Comodule
  • Duality
  • Hopf module
  • Integral
  • Monoidal category
  • Weak multiplier bialgebra

ASJC Scopus subject areas

  • Mathematics(all)

Cite this

Comodules over weak multiplier bialgebras. / Böhm, G.

In: International Journal of Mathematics, Vol. 25, No. 5, 1450037, 2014.

Research output: Contribution to journalArticle

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