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29
Liberation of orthogonal lie groups
 Adv. Math
"... Abstract. We show that under suitable assumptions, we have a onetoone correspondence between classical groups and free quantum groups, in the compact orthogonal case. We classify the groups under correspondence, with the result that there are exactly 6 of them: On, Sn, Hn, Bn, S ′ n, B ′ n. We inv ..."
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Cited by 48 (17 self)
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Abstract. We show that under suitable assumptions, we have a onetoone correspondence between classical groups and free quantum groups, in the compact orthogonal case. We classify the groups under correspondence, with the result that there are exactly 6 of them: On, Sn, Hn, Bn, S ′ n, B ′ n. We investigate the representation theory aspects of the correspondence, with the result that for On, Sn, Hn, Bn, this is compatible with the BercoviciPata bijection. Finally, we discuss some more general classification problems in the compact orthogonal case, notably with the construction of a new quantum group.
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 37 (22 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.
Classification results for easy quantum groups
 Pacific J. Math
"... Abstract. We study the orthogonal quantum groups satisfying the “easiness ” assumption axiomatized in our previous paper, with the construction of some new examples, and with some partial classification results. The conjectural conclusion is that the easy quantum groups consist of the previously kno ..."
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Cited by 23 (10 self)
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Abstract. We study the orthogonal quantum groups satisfying the “easiness ” assumption axiomatized in our previous paper, with the construction of some new examples, and with some partial classification results. The conjectural conclusion is that the easy quantum groups consist of the previously known 14 examples, plus of an hypothetical multiparameter “hyperoctahedral series”, related to the complex reflection groups H s n = Zs ≀ Sn. We discuss as well the general structure, and the computation of asymptotic laws of characters, for the new quantum groups that we construct.
Integration over the Pauli quantum group
"... Abstract. We prove that the Pauli representation of the quantum permutation algebra A(S4) is faithful. This provides the second known model for a free quantum algebra. We use this model for performing some computations, with the result that at level of laws of diagonal coordinates, the Lebesgue meas ..."
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Cited by 15 (12 self)
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Abstract. We prove that the Pauli representation of the quantum permutation algebra A(S4) is faithful. This provides the second known model for a free quantum algebra. We use this model for performing some computations, with the result that at level of laws of diagonal coordinates, the Lebesgue measure appears between the Dirac mass and the free Poisson law.
Algebraic quantum permutation groups
 Goswami, D.: Quantum Group of isometries in Classical and Non Commutative Geometry
"... Abstract. We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If K is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra K n: this is a refinement of Wang’s universality theorem f ..."
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Cited by 15 (2 self)
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Abstract. We discuss some algebraic aspects of quantum permutation groups, working over arbitrary fields. If K is any characteristic zero field, we show that there exists a universal cosemisimple Hopf algebra coacting on the diagonal algebra K n: this is a refinement of Wang’s universality theorem for the (compact) quantum permutation group. We also prove a structural result for Hopf algebras having a nonergodic coaction on the diagonal algebra K n, on which we determine the possible group gradings when K is algebraically closed and has characteristic zero. 1.
Reduced Operator Algebras of TracePreserving Quantum Automorphism Groups
 DOCUMENTA MATH.
, 2013
"... 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 ..."
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Cited by 14 (1 self)
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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).
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 13 (7 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 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 12 (8 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 11 (6 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.
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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Cited by 11 (8 self)
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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.