author={Alberto Ibort and Michael A. Rodriguez},
  journal={An Introduction to Groups, Groupoids and Their Representations},
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… 

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

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

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


  • Fuyuta Komura
  • Mathematics
    Journal 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

. 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


. 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

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

  • R. Zucchini
  • Mathematics
    Journal 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

. 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

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

. 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



Hopf algebroids and quantum groupoids

We introduce the notion of Hopf algebroids, in which neither the total algebras nor the base algebras are required to be commutative. We give a class of Hopf algebroids associated to module algebras


This note presents some recent results on the new notion of quantum groupoid from both the perspective of Poisson geometry and Operator algebras. We shall only briefly present the major results known

The cyclic theory of Hopf algebroids

We give a systematic description of the cyclic cohomology theory of Hopf alge\-broids in terms of its associated category of modules. Then we introduce a dual cyclic homology theory by applying

Quantum Groups

This thesis consists of four papers. In the first paper we present methods and explicit formulas for describing simple weight modules over twisted generalized Weyl algebras. Under certain conditions

Hopf Algebroids and Their Cyclic Theory

The main objective of this thesis is to clarify concepts of generalised symmetries in noncommutative geometry (i.e., the noncommutative analogue of groupoids and Lie algebroids) and their associated


We review the extent to which the universal enveloping algebra of a Lie-Rinehart algebra resembles a Hopf algebra, and refer to this structure as a Rinehart bialgebra. We then prove a

Gerstenhaber Algebras and BV-Algebras in Poisson Geometry

Abstract:The purpose of this paper is to establish an explicit correspondence between various geometric structures on a vector bundle with some well-known algebraic structures such as Gerstenhaber

Integral theory for Hopf Algebroids

The theory of integrals is used to analyse the stru ture of Hopf algebroids [1, 6℄. We prove that the total algebra of a Hopf algebroid is a separable extension of the base algebra if and only if it

Lectures on Quantum Groups

Revised second edition. The text covers the material presented for a graduate-level course on quantum groups at Harvard University. Covered topics include: Poisson algebras and quantization,