Deformation of Spectral Triples and Their 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.

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QUANTUM ISOMETRY GROUPS OF SYMMETRIC GROUPS

International Journal of Mathematics ◽

10.1142/s0129167x12500747 ◽

2012 ◽

Vol 23 (07) ◽

pp. 1250074 ◽

Cited By ~ 9

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.

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