[Paper Review] Regular multiplier Hopf algebroids. Basic theory and examples

Thomas Timmermann, Alfons Van Daele|arXiv (Cornell University)|Jul 2, 2013

Algebraic structures and combinatorial modelsMathematics33 references7 citations

TL;DR

This paper introduces regular multiplier Hopf algebroids as non-unital generalizations of Hopf algebroids and weak multiplier Hopf algebras, establishing that bijectivity of two canonical maps is equivalent to the existence of an antipode. It characterizes antipode invertibility and provides foundational theory with illustrative examples.

ABSTRACT

Multiplier Hopf algebroids are algebraic versions of quantum groupoids that generalize Hopf algebroids to the non-unital case and weak (multiplier) Hopf algebras to non-separable base algebras. The main structure maps of a multiplier Hopf algebroid are a left and a right comultiplication. We show that bijectivity of two associated canonical maps is equivalent to the existence of an antipode, discuss invertibility of the antipode, and present some examples and special cases.

Motivation & Objective

• To develop a foundational theory for multiplier Hopf algebroids as algebraic analogues of quantum groupoids in the non-unital setting.
• To generalize Hopf algebroids and weak multiplier Hopf algebras by extending them to non-separable base algebras.
• To establish conditions under which an antipode exists in this generalized framework.
• To investigate the invertibility of the antipode in the context of multiplier Hopf algebroids.
• To present concrete examples and special cases illustrating the theory in action.

Proposed method

• Introduce the structure of a multiplier Hopf algebroid via left and right comultiplication maps on a non-unital algebra.
• Define two canonical maps associated with the comultiplication and analyze their bijectivity as a key structural condition.
• Prove that bijectivity of these canonical maps is equivalent to the existence of an antipode.
• Analyze the antipode's invertibility using algebraic conditions derived from the canonical maps.
• Construct examples of regular multiplier Hopf algebroids, including special cases such as groupoid algebras and weak multiplier Hopf algebras.
• Use categorical and algebraic techniques to generalize known results from unital and separable settings to the non-unital, non-separable case.

Experimental results

Research questions

• RQ1 What conditions ensure the existence of an antipode in a multiplier Hopf algebroid?
• RQ2 How does the bijectivity of the canonical maps relate to the antipode's existence?
• RQ3 In what ways can the theory of multiplier Hopf algebroids generalize Hopf algebroids and weak multiplier Hopf algebras?
• RQ4 What are the structural implications of antipode invertibility in this framework?
• RQ5 What are representative examples and special cases of regular multiplier Hopf algebroids?

Key findings

• The existence of an antipode in a multiplier Hopf algebroid is equivalent to the bijectivity of two canonical maps derived from the comultiplication.
• The antipode is invertible if and only if specific algebraic conditions on the canonical maps are satisfied, generalizing known results from unital settings.
• The framework successfully extends Hopf algebroids and weak multiplier Hopf algebras to non-unital and non-separable base algebras.
• Examples include groupoid algebras and weak multiplier Hopf algebras, which are shown to fit naturally within the new framework.
• The theory provides a consistent algebraic foundation for quantum groupoids in broader, non-unital contexts.
• The canonical maps serve as central tools for characterizing structural properties such as the existence and invertibility of the antipode.