scholarly journals A class of C∗-algebraic locally compact quantum groupoids part I. Motivation and definition

2018 ◽  
Vol 29 (04) ◽  
pp. 1850029 ◽  
Author(s):  
Byung-Jay Kahng ◽  
Alfons Van Daele
Keyword(s):  
Quantum Groups ◽  
Hopf Algebras ◽  
Idempotent Element ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Multiplier Hopf Algebras ◽  
Compact Quantum Groups

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 provide motivation and formulate the definition in the [Formula: see text]-algebra framework. Existence of a certain canonical idempotent element is required and it plays a fundamental role, including the establishment of the coassociativity of the comultiplication. This class contains locally compact quantum groups as a subclass.

A Note on the Antipode for Algebraic Quantum Groups

2012 ◽  
Vol 55 (2) ◽  
pp. 260-270
Author(s):  
L. Delvaux ◽  
A. Van Daele ◽  
Shuanhong Wang
Keyword(s):  
Hopf Algebra ◽  
Quantum Groups ◽  
Hopf Algebras ◽  
Fourth Power ◽  
General Situation ◽  
Locally Compact Quantum Groups ◽  
Algebraic Quantum ◽  
Multiplier Hopf Algebras ◽  
Compact Quantum Groups ◽  
Multiplier Hopf Algebra

AbstractRecently, Beattie, Bulacu ,and Torrecillas proved Radford's formula for the fourth power of the antipode for a co-Frobenius Hopf algebra.In this note, we show that this formula can be proved for any regular multiplier Hopf algebra with integrals (algebraic quantum groups). This, of course, not only includes the case of a finite-dimensional Hopf algebra, but also that of any Hopf algebra with integrals (co-Frobenius Hopf algebras). Moreover, it turns out that the proof in this more general situation, in fact, follows in a few lines from well-known formulas obtained earlier in the theory of regular multiplier Hopf algebras with integrals.We discuss these formulas and their importance in this theory. We also mention their generalizations, in particular to the (in a certain sense) more general theory of locally compact quantum groups. Doing so, and also because the proof of the main result itself is very short, the present note becomes largely of an expository nature.

scholarly journals Weak mixing for locally compact quantum groups

2016 ◽  
Vol 37 (5) ◽  
pp. 1657-1680 ◽  
Author(s):  
AMI VISELTER
Keyword(s):  
Quantum Groups ◽  
Von Neumann Algebras ◽  
Unitary Representations ◽  
Discrete Groups ◽  
Weak Mixing ◽  
Locally Compact ◽  
Von Neumann ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups ◽  
Weakly Mixing

We generalize the notion of weakly mixing unitary representations to locally compact quantum groups, introducing suitable extensions of all standard characterizations of weak mixing to this setting. These results are used to complement the non-commutative Jacobs–de Leeuw–Glicksberg splitting theorem of Runde and the author [Ergodic theory for quantum semigroups. J. Lond. Math. Soc. (2) 89(3) (2014), 941–959]. Furthermore, a relation between mixing and weak mixing of state-preserving actions of discrete quantum groups and the properties of certain inclusions of von Neumann algebras, which is known for discrete groups, is demonstrated.

scholarly journals The canonical central exact sequence for locally compact quantum groups

2016 ◽  
Vol 290 (8-9) ◽  
pp. 1303-1316 ◽  
Author(s):  
Paweł Kasprzak ◽  
Adam Skalski ◽  
Piotr Mikołaj Sołtan
Keyword(s):  
Exact Sequence ◽  
Quantum Groups ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups

scholarly journals From Quantum Groups to Groups

2013 ◽  
Vol 65 (5) ◽  
pp. 1073-1094 ◽  
Author(s):  
Mehrdad Kalantar ◽  
Matthias Neufang
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Locally Compact ◽  
Large Classes ◽  
Locally Compact Quantum Groups ◽  
Locally Compact Quantum Group ◽  
Recent Developments ◽  
Compact Quantum Groups ◽  
Quantum Version ◽  
Quantum Point

AbstractIn this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign to each locally compact quantum group 𝔾 a locally compact group 𝔾˜ that is the quantum version of point-masses and is an invariant for the latter. We show that “quantum point-masses” can be identified with several other locally compact groups that can be naturally assigned to the quantum group 𝔾. This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of 𝔾 are encoded by 𝔾˜; the latter, despite being a simpler object, can carry very important information about 𝔾.

scholarly journals A Note on Amenability of Locally Compact Quantum Groups

2014 ◽  
Vol 57 (2) ◽  
pp. 424-430 ◽  
Author(s):  
Piotr M. Sołtan ◽  
Ami Viselter
Keyword(s):  
Quantum Groups ◽  
Short Note ◽  
Locally Compact Groups ◽  
Compact Groups ◽  
Locally Compact ◽  
Locally Compact Quantum Groups ◽  
Compact Quantum Groups

AbstractIn this short note we introduce a notion called quantum injectivity of locally compact quantum groups, and prove that it is equivalent to amenability of the dual. In particular, this provides a new characterization of amenability of locally compact groups.

Sign in / Sign up

Export Citation Format

Share Document