scholarly journals Quantum symmetry groups of C *-algebras equipped with orthogonal filtrations

2012 ◽  
Vol 106 (5) ◽  
pp. 980-1004 ◽  
Author(s):  
Teodor Banica ◽  
Adam Skalski
Keyword(s):  
Symmetry Groups ◽  
C Algebras ◽  
Quantum Symmetry

scholarly journals Projective limits of quantum symmetry groups and the doubling construction for Hopf algebras

2014 ◽  
Vol 17 (02) ◽  
pp. 1450012 ◽  
Author(s):  
Adam Skalski ◽  
Piotr M. Sołtan
Keyword(s):  
Hopf Algebras ◽  
Discrete Group ◽  
Inductive Limit ◽  
Symmetric Groups ◽  
Projective Limit ◽  
Symmetry Groups ◽  
C Algebras ◽  
Quantum Symmetry ◽  
Doubling Construction ◽  
Projective Limits

The quantum symmetry group of the inductive limit of C*-algebras equipped with orthogonal filtrations is shown to be the projective limit of the quantum symmetry groups of the C*-algebras appearing in the sequence. Some explicit examples of such projective limits are studied, including the case of quantum symmetry groups of the duals of finite symmetric groups, which do not fit directly into the framework of the main theorem and require further specific study. The investigations reveal a deep connection between quantum symmetry groups of discrete group duals and the doubling construction for Hopf algebras.

scholarly journals Two-parameter families of quantum symmetry groups

2011 ◽  
Vol 260 (11) ◽  
pp. 3252-3282 ◽  
Author(s):  
Teodor Banica ◽  
Adam Skalski
Keyword(s):  
Symmetry Groups ◽  
Quantum Symmetry ◽  
Two Parameter

scholarly journals Quantum Symmetries of Graph C*-algebras

2018 ◽  
Vol 61 (4) ◽  
pp. 848-864 ◽  
Author(s):  
Simon Schmidt ◽  
Moritz Weber
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Operator Algebras ◽  
Automorphism Groups ◽  
Undirected Graphs ◽  
C Algebras ◽  
Quantum Symmetry ◽  
Quantum Symmetries ◽  
Multiple Edges ◽  
Definition Of

AbstractThe study of graph C*-algebras has a long history in operator algebras. Surprisingly, their quantum symmetries have not yet been computed. We close this gap by proving that the quantum automorphism group of a finite, directed graph without multiple edges acts maximally on the corresponding graph C*-algebra. This shows that the quantum symmetry of a graph coincides with the quantum symmetry of the graph C*-algebra. In our result, we use the definition of quantum automorphism groups of graphs as given by Banica in 2005. Note that Bichon gave a different definition in 2003; our action is inspired from his work. We review and compare these two definitions and we give a complete table of quantum automorphism groups (with respect to either of the two definitions) for undirected graphs on four vertices.

Classical and quantum symmetry groups of a free‐fall particle

1985 ◽  
Vol 26 (1) ◽  
pp. 55-61
Author(s):  
Toshihiro Iwai ◽  
See‐Gew Rew
Keyword(s):  
Free Fall ◽  
Symmetry Groups ◽  
Quantum Symmetry

scholarly journals Invariance of KMS states on graph C*-algebras under classical and quantum symmetry

2021 ◽  
pp. 1-17
Author(s):  
Soumalya Joardar ◽  
Arnab Mandal
Keyword(s):  
Automorphism Group ◽  
Automorphism Groups ◽  
Inverse Temperature ◽  
Circulant Graphs ◽  
Connected Graphs ◽  
Kms States ◽  
C Algebras ◽  
Kms State ◽  
Quantum Symmetry ◽  
Strongly Connected

Abstract We study the invariance of KMS states on graph $C^{\ast }$ -algebras coming from strongly connected and circulant graphs under the classical and quantum symmetry of the graphs. We show that the unique KMS state for strongly connected graphs is invariant under the quantum automorphism group of the graph. For circulant graphs, it is shown that the action of classical and quantum automorphism groups preserves only one of the KMS states occurring at the critical inverse temperature. We also give an example of a graph $C^{\ast }$ -algebra having more than one KMS state such that all of them are invariant under the action of classical automorphism group of the graph, but there is a unique KMS state which is invariant under the action of quantum automorphism group of the graph.

scholarly journals Quantum symmetry of graph C∗-algebras associated with connected graphs

2018 ◽  
Vol 21 (03) ◽  
pp. 1850019 ◽  
Author(s):  
Soumalya Joardar ◽  
Arnab Mandal
Keyword(s):  
Automorphism Group ◽  
Directed Graph ◽  
Automorphism Groups ◽  
Multiple Edge ◽  
Connected Graphs ◽  
C Algebras ◽  
Underlying Graph ◽  
Quantum Symmetry ◽  
Quantum Symmetries ◽  
Funct Anal

We define a notion of quantum automorphism groups of graph [Formula: see text]-algebras for finite, connected graphs. Under the assumption that the underlying graph does not have any multiple edge or loop, the quantum automorphism group of the underlying directed graph in the sense of Banica [Quantum automorphism groups of homogeneous graphs, J. Funct. Anal. 224 (2005) 243–280] (which is also the symmetry object in the sense of [S. Schmidt and M. Weber, Quantum symmetry of graph [Formula: see text]-algebras, arXiv:1706.08833 ] is shown to be a quantum subgroup of quantum automorphism group in our sense. Quantum symmetries for some concrete graph [Formula: see text]-algebras have been computed.

scholarly journals Quantum symmetries on noncommutative complex spheres with partial commutation relations

2018 ◽  
Vol 21 (04) ◽  
pp. 1850028
Author(s):  
Simeng Wang
Keyword(s):  
Quantum Groups ◽  
Unitary Groups ◽  
Symmetry Groups ◽  
Orthogonal Groups ◽  
Commutation Relations ◽  
Quantum Symmetry ◽  
Quantum Symmetries ◽  
Free Independence

We introduce the notion of noncommutative complex spheres with partial commutation relations for the coordinates. We compute the corresponding quantum symmetry groups of these spheres, and this yields new quantum unitary groups with partial commutation relations. We also discuss some geometric aspects of the quantum orthogonal groups associated with the mixture of classical and free independence discovered by Speicher and Weber. We show that these quantum groups are quantum symmetry groups on some quantum spaces of spherical vectors with partial commutation relations.

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