Quantum isometry groups of duals of free powers of cyclic groups

Details Quantum isometry groups of duals of free powers of cyclic groups Quantum isometry groups of duals of free powers of cyclic groups

2010-11-24

arxiv.org/abs/1011.5400v2

Mathematics

Operator Algebras

23 pages, in v2 some proofs are modified and expanded (notably that of Theorem 3.5), a few illustrations of the operations rel

Scientific paper

We study the quantum isometry groups of the noncommutative Riemannian manifolds associated to discrete group duals. The basic representation theory problem is to compute the law of the main character of the relevant quantum group, and our main result here is as follows: for the group Z_s^{*n}, with s>4 and n>1, half of the character follows the compound free Poisson law with respect to the measure $\underline{\epsilon}$/2, where $\epsilon$ is the uniform measure on the s-th roots of unity, and $\epsilon\to\underline{\epsilon}$ is the canonical projection map from complex to real measures. We discuss as well a number of technical versions of this result, notably with the construction of a new quantum group, which appears as a "representation-theoretic limit", at s equal to infinity.

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