Quantum isometry group of dual of finitely generated discrete groups - II

Abstract

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 .

Quantum isometry groups of dual of finitely generated discrete groups and quantum groups

“…Morphisms in C τ are CQG morphisms intertwining the respective actions. For the proof the reader is referred to Lemma 2.16 of [18]. Corollary 2.15 It follows from Lemma 2.14 that there is a universal object, say (Q τ , α τ ) in C τ and (Q τ , α τ ) ∼ = Q(Γ, S).…”

“…From [4] we also came to know about Q (Z n * Z n * · · Z n ) k copies except the case n = 4. We discuss the n = 4 case in [18].…”

“…Moreover, Q( C * r (Z × Z 2 ), F ) can also be identified with the quantum group K + 2 (for more details see Section 5 of [2]). We also give yet another description of K + n in [18].…”

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

“…Moreover, {A i,j } i,j≥0 is a sub-filtration of {U n } n≥0 in the sense of [8] so that by Corollary 2.11 of that paper, QISO( A ⋊ β B) is a quantum subgroup of QISO(C * (Z 9 ⋊ β Z 3 ), τ, {U n } n≥0 ). Now by the first computation in Section 5 of [31], C(QISO(C * (Z 9 ⋊ β Z 3 ), τ, {U n } n≥0 )) is isomorphic to C * (Z 9 ⋊ β Z 3 ) ⊕ C * (Z 9 ⋊ β Z 3 ), so it has the vector space dimension equal 27 + 27 = 54. Therefore, the vector space dimension of C (QISO( A ⋊ β B)) is no greater than 54.…”

“…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].…”