Quantum isometry groups of noncommutative manifolds associated to groupC∗-algebras

Abstract

Abstract. Let Γ be a finitely generated discrete group. The standard spectral triple on the group C * -algebra C * (Γ) is shown to admit the quantum group of orientation preserving isometries. This leads to new examples of compact quantum groups. In particular the quantum isometry group of the C * -algebra of the free group on n-generators is computed and turns out to be a quantum group extension of the quantum permutation group A 2n of Wang. The quantum groups of orientation and real structure preserving isometries are also considered and construction of the Laplacian for the standard spectral triple on C * (Γ) discussed.From the early days of the theory of spectral triples a prominent role was played by an example of a spectral triple on a group C * -algebra C * (Γ), where Γ is a (discrete) group equipped with a fixed length function l : Γ → R + (it appears already in Connes' original paper [Co 2 ]). When Γ is finitely generated, properties of the natural word-length function on Γ and its associated spectral triple reflect deep combinatorial geometric aspects of Γ -so for example summability of the triple corresponds to the growth conditions, Rieffel regularity of the spectral triple is closely related to the Haagerup's Rapid Decay property ([OR]).Nowadays spectral triples are viewed as objects best suited to describe noncommutative (compact, Riemannian) manifolds. Thus, especially in the mathematical physics literature, the focus seems to be on investigating spectral triples on deformations of function algebras of classical spaces. In recent work of Goswami and the first named author ([Go 1 ], [BG 2 ], [Go 2 ]), the notion of a quantum isometry group of a spectral triple, understood as the universal compact quantum group acting on the corresponding noncommutative manifold in the way compatible with the spectral triple structure was introduced. It was shown that if the spectral triple is sufficiently well-behaved, the quantum isometry group exists, and in the classical situation the definition of 'compatible' actions given in [Go 1 ] describes precisely the relevant group actions preserving the Riemannian metric. Moreover Goswami and the first named author investigated quantum isometry groups of classical manifolds and their noncommutative deformations. In [BGS] together with Goswami we studied quantum isometry groups of spectral triples on AF C * -algebras introduced in [CI]. Already there the nature of the problem becomes more combinatorial and many connections with quantum permutation groups ([Wan], [BBC 1 ]), or more general quantum symmetry group of finite graphs ([Bic]) appear.The aim of this paper is the initiation of the study of quantum isometry groups of spectral triples on group C * -algebras. We begin by introducing basic notations Permanent address of the second named author:

Quantum isometry groups for dihedral group D2(2n+1)

“…We first introduce some basic notations and terminologies of spectral triples on group C * -algebras, and the details can be found in [3,7,13].…”

“…The following notion on quantum families of orientation preserving isometries was introduced in [10,13].…”

“…It is easy to check that the conditions in Theorem 2.2 are satisfied, so that the quantum group of orientation preserving isometries QISO + (C(Γ ), l 2 (Γ ), D (Γ ) ) exists. In [13], we know that QISO + (C(Γ ), l 2 (Γ ), D (Γ ) ) depend on the initial choice of generating set. We will further denote it by QISO + (Γ , S).…”

“…Moreover, Bhowmick and Skalski [13] studied quantum isometry groups of spectral triples on the group C * -algebra. Nowadays, it comes to know that the nature of the problem in the mathematical physics literature has already become more combinatorial, and many connections with quantum permutation groups and general quantum symmetry groups of finite graphs have appeared [5,6,8,2].…”

“…In [14], Soltan et al defined the doubling procedure and shown that the quantum isometry group of S n is isomorphic to the finite quantum group obtained as doubling of the group algebra C [S n ] with standard cocommutative Hopf algebra structure. For this, it is important to compute some quantum isometry groups of spectral triples on group C * -algebras and also see whether the Hopf algebra structure on the quantum isometry group of the dihedral groups is related to the general 'doubling procedure' mentioned in [14] by Soltan et al It is our main goal in this paper that, by using the idea in [13,12], we would further investigate some quantum isometry groups of spectral triples on group C * -algebras of finite generated groups. The rest of the paper is organized as follows.…”