# Groupoids

@article{Ibort2019Groupoids, title={Groupoids}, author={Alberto Ibort and Michael A. Rodriguez}, journal={An Introduction to Groups, Groupoids and Their Representations}, year={2019} }

This paper gives some new results for the theory of quantum groupoids. First, the authors recall the relevant notions and results of the theory of Li-Rinehart algebras and bialgebras. Then, they introduce some basics of the theory of bialgebroids. The examples of the universal enveloping agebras and jet spaces for Lie-Rinehart algebras are explicitly described. The quantum groupoids are then introduced. A suitable version of a quantum duality principle for quantum groupoids is also obtained…

## 115 Citations

### A class of C⁎-algebraic locally compact quantum groupoids Part II. Main theory

- MathematicsAdvances in Mathematics
- 2019

### A class of C*-algebraic locally compact quantum groupoids Part I. Motivation and definition

- Mathematics
- 2017

In this series of papers, we develop the theory of a class of locally compact quantum groupoids, which is motivated by the purely algebraic notion of weak multiplier Hopf algebras. In this Part I, we…

### QUOTIENTS OF ÉTALE GROUPOIDS AND THE ABELIANIZATIONS OF GROUPOID C*-ALGEBRAS

- MathematicsJournal of the Australian Mathematical Society
- 2020

Abstract In this paper, we introduce quotients of étale groupoids. Using the notion of quotients, we describe the abelianizations of groupoid C*-algebras. As another application, we obtain a simple…

### Ample groupoids, topological full groups, algebraic K-theory spectra and infinite loop spaces

- Mathematics
- 2022

. Inspired by work of Szymik and Wahl on the homology of Higman-Thompson groups, we establish a general connection between ample groupoids, topological full groups, algebraic K-theory spectra and…

### CANONICAL STRATIFICATION OF DEFINABLE GROUPOIDS

- Mathematics
- 2022

. Our aim is to precisely present a tame topology counterpart to canonical stratiﬁcation of a Lie groupoid. We consider a deﬁnable Lie groupoid in semialgebraic, subanalytic, or more generally, an…

### Separability idempotents in $C*$-algebras

- MathematicsJournal of Noncommutative Geometry
- 2018

In this paper, we study the notion of a separability idempotent in the C*-algebra framework. This is analogous to the notion in the purely algebraic setting, typically considered in the case of…

### Quantum field theoretic representation of Wilson surfaces. Part I. Higher coadjoint orbit theory

- MathematicsJournal of High Energy Physics
- 2022

This is the first of a series of two papers devoted to the partition function realization of Wilson surfaces in strict higher gauge theory. A higher version of the Kirillov-Kostant-Souriau theory of…

### Algebraic actions I. C*-algebras and groupoids

- Mathematics
- 2022

. We provide a framework for studying concrete C*-algebras associated with algebraic actions of semigroups: Given such an action, we construct an inverse semigroup, and we introduce conditions for…

### Construction of a $C^*$-algebraic quantum groupoid from a weak multiplier Hopf algebra

- Mathematics
- 2021

Van Daele and Wang developed a purely algebraic notion of weak multiplier Hopf algebras, which extends the notions of Hopf algebras, multiplier Hopf algebras, and weak Hopf algebras. With an…

### Stable finiteness of ample groupoid algebras, traces and applications

- Mathematics
- 2022

. In this paper we study stable ﬁniteness of ample groupoid algebras with applications to inverse semigroup algebras and Leavitt path algebras, recovering old results and proving some new ones. In…

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