Quantum Isometry Groups

Abstract

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A note on the injectivity of actions of compact quantum groups on a class of $C^{\ast }$-algebras

“…Moreover, by that corollary and Theorem 2.5, we get a unitary representation of Qθ which implements α −θ . But by the generalities of RWK (or, more general cocycle-twisted) deformation of CQG as in the Chapter 7 of [9], we conclude that α = (α −θ ) θ is unitarily implemented too, where the corresponding Hilbert space and unitary essentially remain the same. In particular, α is injective.…”

Quantum Symmetries of the Twisted Tensor Products of C*-Algebras

“…The quantum isometry group (in the sense of [9]) of the C * -algebra A was studied in [10]. For more details, we refer to Chapter 5 of [21]. Indeed, we can fix a spectral triple on A coming from the family constructed by Christensen and Ivan ( [14]).…”

“…Soon after this, the theory of quantum symmetries was extended to finite metric spaces and finite graphs by Banica, Bichon and their collaborators (see [3,4,13], and more recently [41] and [23]), who uncovered several interesting connections to combinatorics, representation theory and free probability ( [37], [43], [5] and the references therein). The next breakthrough came through the work of Goswami and his coauthors ( [9,20]), who introduced the concept of quantum isometry groups associated to a given spectral triple á la Connes, viewed as a noncommutative differential manifold (for a general description of Goswami's theory we refer to a recent book [21], another introduction to the subject of quantum symmetry groups may be found in the lecture notes [1]). Among examples fitting in the Goswami's framework were the spectral triples associated with the group C * -algebras of discrete groups, whose quantum isometry groups were first studied in [12], and later analyzed for example in [6], [7] and [31].…”

“…Later, when the theory became to be viewed as one of the instances of the noncommutative mathematics à la Connes ([16]), there was a hope that by analogy with the classical situation the quantum groups might arise as quantum symmetries of physical objects appearing in the quantum field theory. It is worth mentioning that Goswami's theory was in particular applied to compute the quantum isometry group of the finite spectral triple corresponding to the standard model in particle physics ( [16], [15]), for which we refer to Chapter 9 of [21].…”

Hopf coactions on odd spheres

“…The second author of the present article and his collaborators (including Bhowmick, Skalski and others) approached the problem from a geometric perspective and formulated an analogue of the Riemannian isometry groups in the framework of (compact) quantum groups acting on C * -algebras. We refer the reader to [GB16] and the references therein for a comprehensive account of the theory of quantum isometry groups.…”