Weak multiplier bialgebras

TL;DR

This paper introduces weak multiplier bialgebras, a non-unital generalization of weak bialgebras with a multiplier-valued comultiplication, and explores their algebraic structures and relations to existing theories.

Contribution

It proposes the concept of weak multiplier bialgebras, analyzes their base algebras as coseparable co-Frobenius coalgebras, and establishes their module categories as monoidal, connecting to prior weak multiplier Hopf algebra frameworks.

Findings

Base algebras carry coseparable co-Frobenius coalgebra structures

Modules form a monoidal category via tensor products

Relation to existing weak multiplier Hopf algebras is clarified

Abstract

A non-unital generalization of weak bialgebra is proposed with a multiplier-valued comultiplication. Certain canonical subalgebras of the multiplier algebra (named the `base algebras') are shown to carry coseparable co-Frobenius coalgebra structures. Appropriate modules over a weak multiplier bialgebra are shown to constitute a monoidal category via the (co)module tensor product over the base algebra. The relation to Van Daele and Wang's (regular and arbitrary) weak multiplier Hopf algebra is discussed.