Results 1  10
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13
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 8 (6 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.
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.
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 4 (1 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.
Invariants of the halfliberated orthogonal group
 Ann. Inst. Fourier
"... Abstract. The halfliberated orthogonal group O ∗ n appears as intermediate quantum group between the orthogonal group On, and its free version O + n. We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by usi ..."
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Cited by 3 (3 self)
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Abstract. The halfliberated orthogonal group O ∗ n appears as intermediate quantum group between the orthogonal group On, and its free version O + n. We discuss here its basic algebraic properties, and we classify its irreducible representations. The classification of representations is done by using a certain twistingtype relation between O ∗ n and Un, a non abelian discrete group playing the role of weight lattice for O ∗ n, and a number of methods inspired from the theory of Lie algebras. We use these results for showing that the discrete quantum group dual to O ∗ n has polynomial growth.
A NOTE ON FREE QUANTUM GROUPS
, 710
"... Abstract. We study the free complexification operation for compact quantum groups, G → G c. We prove that, with suitable definitions, this induces a onetoone correspondence between free orthogonal quantum groups of infinite level, and free unitary quantum groups satisfying G = G c. ..."
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Cited by 3 (3 self)
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Abstract. We study the free complexification operation for compact quantum groups, G → G c. We prove that, with suitable definitions, this induces a onetoone correspondence between free orthogonal quantum groups of infinite level, and free unitary quantum groups satisfying G = G c.
Fusion rules for quantum reflection groups
 J. Noncommut. Geom
"... Abstract. We find the fusion rules for the quantum analogues of the complex reflection groups H s n = Zs ≀Sn. The irreducible representations can be indexed by the elements of the free monoid N ∗s, and their tensor products are given by formulae which remind the ClebschGordan rules (which appear at ..."
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Cited by 3 (3 self)
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Abstract. We find the fusion rules for the quantum analogues of the complex reflection groups H s n = Zs ≀Sn. The irreducible representations can be indexed by the elements of the free monoid N ∗s, and their tensor products are given by formulae which remind the ClebschGordan rules (which appear at s = 1).
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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Cited by 2 (2 self)
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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.
Stochastic aspects of easy quantum groups, Probab. Theory Related Fields 149
, 2011
"... Abstract. We consider several orthogonal quantum groups satisfying the “easiness” assumption axiomatized in our previous paper. For each of them we discuss the computation of the asymptotic law of Tr(u k) with respect to the Haar measure, u being the fundamental representation. For the classical gro ..."
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Cited by 2 (2 self)
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Abstract. We consider several orthogonal quantum groups satisfying the “easiness” assumption axiomatized in our previous paper. For each of them we discuss the computation of the asymptotic law of Tr(u k) with respect to the Haar measure, u being the fundamental representation. For the classical groups On, Sn we recover in this way some wellknown results of Diaconis and Shahshahani.
SIMPLE COMPACT QUANTUM GROUPS I
, 810
"... Abstract. The notion of simple compact quantum group is introduced. As nontrivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups Bu(Q) for Q ∈ GL(n, C) satisfying Q ¯ Q = ±In, n ≥ 2; (b) The q ..."
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Cited by 1 (0 self)
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Abstract. The notion of simple compact quantum group is introduced. As nontrivial (noncommutative and noncocommutative) examples, the following families of compact quantum groups are shown to be simple: (a) The universal quantum groups Bu(Q) for Q ∈ GL(n, C) satisfying Q ¯ Q = ±In, n ≥ 2; (b) The quantum automorphism groups Aaut(B, τ) of finite dimensional C∗algebras B endowed with the canonical trace τ when dim(B) ≥ 4, including the quantum permutation groups Aaut(Xn) on n points (n ≥ 4); (c) The standard deformations Kq of simple compact Lie groups K and their twists K u q, as well as Rieffel’s deformation KJ. 1.
FREE BESSEL LAWS
, 710
"... Abstract. We introduce and study a remarkable family of real probability measures πst, that we call free Bessel laws. These are related to the free Poisson law π via the formulae πs1 = π ⊠s and π1t = π ⊞t. Our study includes: definition and basic properties, analytic aspects (supports, atoms, densit ..."
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Abstract. We introduce and study a remarkable family of real probability measures πst, that we call free Bessel laws. These are related to the free Poisson law π via the formulae πs1 = π ⊠s and π1t = π ⊞t. Our study includes: definition and basic properties, analytic aspects (supports, atoms, densities), combinatorial aspects (functional transforms, moments, partitions), and a discussion of the relation with random matrices and quantum groups.