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THE HYPEROCTAHEDRAL QUANTUM GROUP
, 2007
"... Abstract. We consider the hypercube in R n, and show that its quantum symmetry group is a qdeformation of On at q = −1. Then we consider the graph formed by n segments, and show that its quantum symmetry group is free in some natural sense. ..."
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Cited by 14 (11 self)
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Abstract. We consider the hypercube in R n, and show that its quantum symmetry group is a qdeformation of On at q = −1. Then we consider the graph formed by n segments, and show that its quantum symmetry group is free in some natural sense.
Quantum permutation groups: a survey
"... Abstract. This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum gr ..."
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Cited by 7 (6 self)
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Abstract. This is a presentation of recent work on quantum permutation groups. Contains: a short introduction to operator algebras and Hopf algebras; quantum permutation groups, and their basic properties; diagrams, integration formulae, asymptotic laws, matrix models; the hyperoctahedral quantum group, free wreath products, quantum automorphism groups of finite graphs, graphs having no quantum symmetry; complex Hadamard matrices, cocycle twists of the symmetric group, quantum groups acting on 4 points; remarks and comments. 1.
QUANTUM GROUPS AND HADAMARD MATRICES
, 2006
"... Abstract. To any complex Hadamard matrix we associate a quantum permutation group. The correspondence is not onetoone, but the quantum group encapsulates a number of subtle properties of the matrix. We investigate various aspects of the construction: compatibility to product operations, characteri ..."
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Cited by 6 (5 self)
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Abstract. To any complex Hadamard matrix we associate a quantum permutation group. The correspondence is not onetoone, but the quantum group encapsulates a number of subtle properties of the matrix. We investigate various aspects of the construction: compatibility to product operations, characterization of matrices which give usual groups, explicit computations for small matrices.
GROWTH ESTIMATES FOR DISCRETE QUANTUM GROUPS
, 2006
"... Abstract. We discuss the notion of growth for discrete quantum groups, with various explicit computations. Then we make the remark that several key estimates relating growth and random walks on discrete groups make sense for discrete quantum groups. Of particular interest here is Gromov’s estimate r ..."
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Cited by 4 (3 self)
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Abstract. We discuss the notion of growth for discrete quantum groups, with various explicit computations. Then we make the remark that several key estimates relating growth and random walks on discrete groups make sense for discrete quantum groups. Of particular interest here is Gromov’s estimate regarding polynomial growth, and we present a verification of the corresponding quantum group conjecture, for duals of classical Lie groups.
SPECTRAL ANALYSIS OF THE FREE ORTHOGONAL MATRIX
, 901
"... Abstract. We compute the spectral measure of the standard generators uij of the Wang algebra Ao(n). We show in particular that this measure has support [−2 / √ n + 2, 2 / √ n + 2], and that it has no atoms. The computation is done by using various techniques, involving the general Wang algebra Ao( ..."
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Cited by 3 (3 self)
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Abstract. We compute the spectral measure of the standard generators uij of the Wang algebra Ao(n). We show in particular that this measure has support [−2 / √ n + 2, 2 / √ n + 2], and that it has no atoms. The computation is done by using various techniques, involving the general Wang algebra Ao(F), a representation of SU q 2 due to Woronowicz, and several calculations with orthogonal polynomials.
Quantum groups acting on 4 points
 TEODOR BANICA, JULIEN BICHON, BENOÎT COLLINS, AND STEPHEN CURRAN
"... Abstract. We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group Q4. Our main tool is a new presentation for the algebra C(Q4), corresponding to an isomorphism of type Q4 ≃ SO−1(3). The quantum subgroups of Q4 are subject to a McKa ..."
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Abstract. We classify the compact quantum groups acting on 4 points. These are the quantum subgroups of the quantum permutation group Q4. Our main tool is a new presentation for the algebra C(Q4), corresponding to an isomorphism of type Q4 ≃ SO−1(3). The quantum subgroups of Q4 are subject to a McKay type correspondence, that we describe at the level of algebraic invariants. 1.
QUANTUM ISOMETRIES AND NONCOMMUTATIVE SPHERES
, 905
"... Abstract. We introduce and study two new examples of noncommutative spheres: the halfliberated sphere, and the free sphere. Together with the usual sphere, these two spheres have the property that the corresponding quantum isometry group is “easy”, in the representation theory sense. We present as ..."
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Abstract. We introduce and study two new examples of noncommutative spheres: the halfliberated sphere, and the free sphere. Together with the usual sphere, these two spheres have the property that the corresponding quantum isometry group is “easy”, in the representation theory sense. We present as well some general comments on the axiomatization problem, and on the “untwisted ” and “noneasy ” case.
REPRESENTATIONS OF QUANTUM PERMUTATION ALGEBRAS
, 2009
"... We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type π: As(n) → B(H). We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and ..."
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We develop a combinatorial approach to the quantum permutation algebras, as Hopf images of representations of type π: As(n) → B(H). We discuss several general problems, including the commutativity and cocommutativity ones, the existence of tensor product or free wreath product decompositions, and the Tannakian aspects of the construction. The main motivation comes from the quantum invariants of the complex Hadamard matrices: we show here that, under suitable regularity assumptions, the computations can be performed up to n = 6.
Documenta Math. 1349 Reduced Operator Algebras of TracePreserving Quantum Automorphism Groups
, 2013
"... Abstract. Let B be a finite dimensional C ∗algebra equipped with its canonical trace induced by the regular representation of B on itself. In this paper, we study various properties of the tracepreserving quantum automorphism group G of B. We prove that the discrete dual quantum group ̂ G has the ..."
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Abstract. Let B be a finite dimensional C ∗algebra equipped with its canonical trace induced by the regular representation of B on itself. In this paper, we study various properties of the tracepreserving quantum automorphism group G of B. We prove that the discrete dual quantum group ̂ G has the property of rapid decay, the reduced von Neumann algebra L ∞ (G) has the Haagerup property and is solid, and that L ∞ (G) is (in most cases)aprime type II1factor. As applications of these and other results, we deduce the metric approximation property, exactness, simplicity and uniqueness oftracefor the reduced C ∗algebra Cr(G), and the existence of a multiplierbounded approximate identity for the convolution algebra L 1 (G).