Comodules over weak multiplier bialgebras

TL;DR

This paper extends the theory of comodules over weak multiplier bialgebras, introducing new structures, interpreting integrals as comodule maps, and establishing a fundamental theorem for Hopf modules in this setting.

Contribution

It defines a new notion of comodules via compatible linear maps, explores their monoidal category structure, and proves a fundamental theorem relating Hopf modules to firm modules.

Findings

Total and base algebras carry comodule structures.

Integrals are interpreted as comodule maps.

Fundamental theorem of Hopf modules established.

Abstract

This is a sequel paper of arXiv:1306.1466 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 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.