scholarly journals Topological automorphism groups of compact quantum groups

2018 ◽  
Vol 290 (1-2) ◽  
pp. 577-598
Author(s):  
Alexandru Chirvasitu ◽  
Issan Patri
Keyword(s):  
Quantum Groups ◽  
Automorphism Groups ◽  
Compact Quantum Groups ◽  
Topological Automorphism

scholarly journals The Frucht property in the quantum group setting

2021 ◽  
pp. 1-31
Author(s):  
T. Banica ◽  
J.P. McCarthy
Keyword(s):  
Finite Group ◽  
Automorphism Group ◽  
Quantum Groups ◽  
Automorphism Groups ◽  
Finite Graph ◽  
Classical Theorem ◽  
Group Setting ◽  
Finite Graphs ◽  
Negative Results ◽  
Compact Quantum Groups

Abstract A classical theorem of Frucht states that any finite group appears as the automorphism group of a finite graph. In the quantum setting, the problem is to understand the structure of the compact quantum groups which can appear as quantum automorphism groups of finite graphs. We discuss here this question, notably with a number of negative results.

scholarly journals RIESZ TRANSFORMS ON COMPACT QUANTUM GROUPS AND STRONG SOLIDITY

2021 ◽  
pp. 1-37
Author(s):  
Martijn Caspers
Keyword(s):  
Quantum Group ◽  
Quantum Groups ◽  
Riesz Transform ◽  
Wreath Products ◽  
Compact Quantum Group ◽  
Riesz Transforms ◽  
Markov Semigroup ◽  
Markov Semigroups ◽  
Free Products ◽  
Compact Quantum Groups

Abstract One of the main aims of this paper is to give a large class of strongly solid compact quantum groups. We do this by using quantum Markov semigroups and noncommutative Riesz transforms. We introduce a property for quantum Markov semigroups of central multipliers on a compact quantum group which we shall call ‘approximate linearity with almost commuting intertwiners’. We show that this property is stable under free products, monoidal equivalence, free wreath products and dual quantum subgroups. Examples include in particular all the (higher-dimensional) free orthogonal easy quantum groups. We then show that a compact quantum group with a quantum Markov semigroup that is approximately linear with almost commuting intertwiners satisfies the immediately gradient- ${\mathcal {S}}_2$ condition from [10] and derive strong solidity results (following [10]). Using the noncommutative Riesz transform we also show that these quantum groups have the Akemann–Ostrand property; in particular, the same strong solidity results follow again (now following [27]).

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
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