Algebraic quantum transformation groupoids of compact type

TL;DR

This paper introduces algebraic quantum transformation groupoids of compact type, establishing a duality with discrete type groupoids, and provides examples from Fell bundles and quantum quotient spaces.

Contribution

It defines a new class of measured multiplier Hopf *-algebroids called algebraic quantum transformation groupoids of compact type and explores their duality with discrete type groupoids.

Findings

Established a Pontrjagin-like duality between compact and discrete algebraic quantum transformation groupoids.

Constructed examples from Fell bundles and quantum quotient spaces.

Revealed the connection to Van Daele's algebraic quantum groups.

Abstract

In this work, we introduce a class of Timmermann's measured multiplier Hopf *-algebroids called algebraic quantum transformation groupoids of compact type. Each object in this class admits a Pontrjagin-like dual called an algebraic quantum transformation groupoid of discrete type. This compact/discrete duality in the framework of algebraic quantum transformation groupoids recover the one between compact and discrete Van Daele's algebraic quantum groups. Among the non-trivial examples of algebraic quantum transformation groupoids of compact type, we give constructions arising from Fell bundles and quantum quotient spaces.