We study quantum isometry groups, denoted by Q(Γ, S), of spectral triples on C * r (Γ) for a finitely generated discrete group Γ coming from the word-length metric with respect to a symmetric generating set S. We first prove a few general results about Q(Γ, S) including:• For a group Γ with polynomial growth property, the dual of Q(Γ, S) has polynomial growth property provided the action of Q(Γ, S) on C * r (Γ) has full spectrum. • Q(Γ, S) ∼ = QISO(Γ, d) for any discrete abelian group Γ, where d is a suitable metric on the dual compact abelian groupΓ.We then carry out explicit computations of Q(Γ, S) 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 Z k 2 or Z l 4 for some k, l in the direct product decomposition into cyclic subgroups.
We define a notion of quantum automorphism groups of graph C * -algebras for finite, connected graphs. Under the assumption that the underlying graph does not have any multiple edge or loop, the quantum automorphism group of the underlying directed graph in the sense of T. Banica ([1]) (which is also the symmetry object in the sense of [11]) is shown to be a quantum subgroup of quantum automorphism group in our sense. Quantum symmetries for some concrete graph C * -algebras have been computed. Example:1. If we take a space of n points X n then the quantum automorphism group of the C * -algebra
As a continuation of the programme of [19], we carry out explicit computations of Q(Γ, S), the quantum isometry group of the canonical spectral triple on C * r (Γ) coming from the word length function corresponding to a finite generating set S, for several interesting examples of Γ not covered by the previous work [19]. These include the braid group of 3 generators, Z * n 4 etc. Moreover, we give an alternative description of the quantum groups H + s (n, 0) and K + n (studied in [6], [4]) in terms of free wreath product. In the last section we give several new examples of groups for which Q(Γ) turns out to be a doubling of C * (Γ). ⊗ respectively. We'll use the leg-numbering notation. Let Q be a unital C * -algebra. Consider the multiplier algebra M(K(H)⊗Q) which has two natural embeddings into M(K(H)⊗Q⊗Q). The first one is obtained by extending the map x → x ⊗ 1 and the second one is obtained by composing this map with the flip on the last two factors. We will write ω 12 and ω 13 for the images of an element ω ∈ M(K(H)⊗Q) under these two maps respectively. We'll denote the Hilbert C * -module by H⊗Q obtained by the completion of H ⊗ Q with respect to the norm induced by the Q valued inner product Compact quantum groups and free wreath productLet us recall the basic notions of compact quantum groups, then actions on C * -algebra and free wreath product by quantum permutation groups.Definition 2.1 A compact quantum group (CQG for short) is a pair (Q, ∆), where Q is a unital C * -algebra and ∆ : Q → Q⊗Q is a unital C * -homomorphism satisfying two conditions :Sometimes we may denote the CQG (Q, ∆) simply as Q, if ∆ is understood from the context. Definition 2.3 A unitary (co) representation of a CQG (Q, ∆) on a Hilbert space H is a C-linear map from H to the Hilbert module H⊗Q such thatGiven such a unitary representation we have a unitary elementŨ belonging to M(K(H)⊗Q) given bỹHere we state Proposition 6.2 of [20] which will be useful for us.Proposition 2.4 If a unitary representation of a CQG leaves a finite dimensional subspace of H, then it'll also leave its orthogonal complement invariant.Remark 2.5 It is known that the linear span of matrix elements of a finite dimensional unitary representation form a dense Hopf *-algebra Q 0 of (Q, ∆), on which an antipode κ and co-unit ǫ are defined.Definition 2.6 We say that CQG (Q, ∆) acts on a unital C * -algebra B if there is a unital C * -homomorphism (called action) α : B → B⊗Q satisfying the following :Definition 2.7 The action is said to be faithful if the * -algebra generated by the setRemark 2.8 Given an action α of a CQG Q on a unital C * -algebra B, we can always find a norm-dense, unital * -subalgebra B 0 ⊆ B such that α| B0 : B 0 → B 0 ⊗ Q 0 is a Hopf-algebraic co-action. Moreover, α is faithful if and only if the * -algebra generated by {(f ⊗ id)α(b) ∀ f ∈ B * 0 , ∀ b ∈ B 0 } is the whole of Q 0 .
We give a notion of quantum automorphism group of graph C * -algebra without sink at critical inverse temperature. This is defined to be the universal object of a category of CQG's having a linear action in the sense of [11] and preserving the KMS state at critical inverse temperature. We show that this category for a certain KMS state at critical inverse temperature coincides with the category introduced in [11] for a class of graphs. We also introduce an orthogonal filtration on Cuntz algebra with respect to the unique KMS state and show that the category of CQG's preserving the orthogonal filtration coincides with the category introduced in this paper.
Let A be a commutative unital algebra over an algebraically closed field k of characteristic = 2, whose generators form a finite-dimensional subspace V , with no nontrivial homogeneous quadratic relations. Let Q be a Hopf algebra that coacts on A inner-faithfully, while leaving V invariant. We prove that Q must be commutative when either: (i) the coaction preserves a nondegenerate bilinear form on V ; or (ii) Q is co-semisimple, finite-dimensional, and char(k) = 0.
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