Spectral Triples and Quantum Isometry Groups on Group $$C^{*}$$ -Algebras

2016 ◽  
pp. 179-198
Author(s):  
Debashish Goswami ◽  
Jyotishman Bhowmick
Keyword(s):  
Isometry Groups ◽  
Spectral Triples ◽  
C Algebras ◽  
Quantum Isometry Groups

Quantum isometry groups of dual of finitely generated discrete groups and quantum groups

2017 ◽  
Vol 29 (03) ◽  
pp. 1750008 ◽  
Author(s):  
Debashish Goswami ◽  
Arnab Mandal
Keyword(s):  
Abelian Group ◽  
Direct Product ◽  
Polynomial Growth ◽  
Finitely Generated ◽  
Isometry Groups ◽  
Product Decomposition ◽  
Full Spectrum ◽  
Spectral Triples ◽  
Growth Property ◽  
Quantum Isometry Groups

We study quantum isometry groups, denoted by [Formula: see text], of spectral triples on [Formula: see text] for a finitely generated discrete group [Formula: see text] coming from the word-length metric with respect to a symmetric generating set [Formula: see text]. We first prove a few general results about [Formula: see text] including: • For a group [Formula: see text] with polynomial growth property, the dual of [Formula: see text] has polynomial growth property provided the action of [Formula: see text] on [Formula: see text] has full spectrum. •[Formula: see text] for any discrete abelian group [Formula: see text], where [Formula: see text] is a suitable metric on the dual compact abelian group [Formula: see text]. We then carry out explicit computations of [Formula: see text] for several classes of examples including free and direct product of cyclic groups, Baumslag–Solitar group, Coxeter groups etc. In particular, we have computed quantum isometry groups of all finitely generated abelian groups which do not have factors of the form [Formula: see text] or [Formula: see text] for some [Formula: see text] in the direct product decomposition into cyclic subgroups.

scholarly journals QUANTUM ISOMETRY GROUPS OF SYMMETRIC GROUPS

2012 ◽  
Vol 23 (07) ◽  
pp. 1250074 ◽  
Author(s):  
JAN LISZKA-DALECKI ◽  
PIOTR M. SOŁTAN
Keyword(s):  
Hopf Algebras ◽  
Isometry Group ◽  
Nearest Neighbor ◽  
Symmetric Groups ◽  
Geometric Interpretation ◽  
Group Algebras ◽  
Isometry Groups ◽  
Spectral Triples ◽  
Multiplier Hopf Algebras ◽  
Quantum Isometry Groups

We identify the quantum isometry groups of spectral triples built on the symmetric groups with length functions arising from the nearest-neighbor transpositions as generators. It turns out that they are isomorphic to certain "doubling" of the group algebras of the respective symmetric groups. We discuss the doubling procedure in the context of regular multiplier Hopf algebras. In the last section we study the dependence of the isometry group of Sn on the choice of generators in the case n = 3. We show that two different choices of generators lead to nonisomorphic quantum isometry groups which exhaust the list of noncommutative noncocommutative semisimple Hopf algebras of dimension 12. This provides noncommutative geometric interpretation of these Hopf algebras.

scholarly journals Some Counterexamples in the Theory of Quantum Isometry Groups

2010 ◽  
Vol 93 (3) ◽  
pp. 279-293 ◽  
Author(s):  
Jyotishman Bhowmick ◽  
Debashish Goswami
Keyword(s):  
Isometry Groups ◽  
Quantum Isometry Groups
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