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23
Free product formulae for quantum permutation groups
 J. Math. Inst. Jussieu
"... Abstract. Associated to a finite graph X is its quantum automorphism group G(X). We prove a formula of type G(X ∗ Y) = G(X) ∗w G(Y), where ∗w is a free wreath product. Then we discuss representation theory of free wreath products, with the conjectural formula µ(G ∗w H) = µ(G) ⊠ µ(H), where µ is t ..."
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Cited by 15 (14 self)
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Abstract. Associated to a finite graph X is its quantum automorphism group G(X). We prove a formula of type G(X ∗ Y) = G(X) ∗w G(Y), where ∗w is a free wreath product. Then we discuss representation theory of free wreath products, with the conjectural formula µ(G ∗w H) = µ(G) ⊠ µ(H), where µ is the associated spectral measure. This is verified in two situations: one using free probability techniques, the other one using planar algebras.
Quantum group of isometries in classical and noncommutative geometry
 Comm. Math. Phys
"... We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then pro ..."
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Cited by 14 (8 self)
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We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold. Our formulation accommodates spectral triples which are not of type II. We give explicit description of quantum isometry groups of commutative and noncommutative tori, and in this context, obtain the quantum double torus defined in [7] as the universal quantum group of holomorphic isometries of the noncommutative torus. 1
Integration over quantum permutation groups
 J. Funct. Anal
"... Abstract. We find a combinatorial formula for the Haar measure of quantum permutation groups. This leads to a dynamic formula for laws of diagonal coefficients, explaining the Poisson/free Poisson convergence result for characters. ..."
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Cited by 12 (10 self)
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Abstract. We find a combinatorial formula for the Haar measure of quantum permutation groups. This leads to a dynamic formula for laws of diagonal coefficients, explaining the Poisson/free Poisson convergence result for characters.
Quantum isometry groups : examples and computations
"... In this followup of [4], where quantum isometry group of a noncommutative manifold has been defined, we explicitly compute such quantum groups for a number of classical as well as noncommutative manifolds including the spheres and the tori. It is also proved that the quantum isometry group of an is ..."
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Cited by 8 (8 self)
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In this followup of [4], where quantum isometry group of a noncommutative manifold has been defined, we explicitly compute such quantum groups for a number of classical as well as noncommutative manifolds including the spheres and the tori. It is also proved that the quantum isometry group of an isospectral deformation of a (classical or noncommutative) manifold is a suitable deformation of the quantum isometry group of the original (undeformed) manifold. 1
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 automorphism groups of vertextransitive graphs of order
 11, J. Algebraic Combin
"... Abstract. We study quantum automorphism groups of vertextransitive graphs having less than 11 vertices. With one possible exception, these can be obtained from cyclic groups Zn, symmetric groups Sn and quantum symmetric groupsQn, by using various product operations. The exceptional case is that of ..."
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Cited by 7 (5 self)
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Abstract. We study quantum automorphism groups of vertextransitive graphs having less than 11 vertices. With one possible exception, these can be obtained from cyclic groups Zn, symmetric groups Sn and quantum symmetric groupsQn, by using various product operations. The exceptional case is that of the Petersen graph, and we present some questions about it.
ON THE STRUCTURE OF QUANTUM PERMUTATION GROUPS
"... Abstract. The quantum permutation group of the set Xn = {1,...,n} corresponds to the Hopf algebra Aaut(Xn). This is an algebra constructed with generators and relations, known to be isomorphic to C(Sn) forn ≤ 3, and to be infinite dimensional for n ≥ 4. In this paper we find an explicit representati ..."
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Cited by 7 (7 self)
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Abstract. The quantum permutation group of the set Xn = {1,...,n} corresponds to the Hopf algebra Aaut(Xn). This is an algebra constructed with generators and relations, known to be isomorphic to C(Sn) forn ≤ 3, and to be infinite dimensional for n ≥ 4. In this paper we find an explicit representation of the algebra Aaut(Xn), related to Clifford algebras. For n =4the representation is faithful in the discrete quantum group sense.
GRAPHS HAVING NO QUANTUM SYMMETRY
"... Abstract. We consider circulant graphs having p vertices, with p prime. To any such graph we associate a certain number k, that we call type of the graph. We prove that for p>> k the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism ..."
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Cited by 6 (5 self)
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Abstract. We consider circulant graphs having p vertices, with p prime. To any such graph we associate a certain number k, that we call type of the graph. We prove that for p>> k the graph has no quantum symmetry, in the sense that the quantum automorphism group reduces to the classical automorphism group.
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.
Quantum isometry groups of the Podles sphere
"... For µ ∈ [0, 1], c ≥ 0, We identify the quantum group SOµ(3) as the universal object in the category of compact quantum groups acting ‘by orientation and volume preserving isometries ’ in the sense of [8] on the natural spectral triple on the Podles sphere S 2 µ,c constructed by Dabrowski, D’Andrea, ..."
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Cited by 4 (4 self)
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For µ ∈ [0, 1], c ≥ 0, We identify the quantum group SOµ(3) as the universal object in the category of compact quantum groups acting ‘by orientation and volume preserving isometries ’ in the sense of [8] on the natural spectral triple on the Podles sphere S 2 µ,c constructed by Dabrowski, D’Andrea, Landi and Wagner in [12]. Moreover, we explicitly compute such universal quantum groups for another class of spectral triples on S2 µ,c ( c> 0) constructed by Chakraborty and Pal ([9]). 1