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30
Symmetries of a generic coaction
 Math. Ann
, 1999
"... Abstract. If B is C ∗algebra of dimension 4 ≤ n < ∞ then the finite dimensional irreducible representations of the compact quantum automorphism group of B, say Gaut ( ̂ B), have the same fusion rules as the ones of SO(3). As consequences, we get (1) a structure result for Gaut ( ̂ B) in the case ..."
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Cited by 23 (18 self)
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Abstract. If B is C ∗algebra of dimension 4 ≤ n < ∞ then the finite dimensional irreducible representations of the compact quantum automorphism group of B, say Gaut ( ̂ B), have the same fusion rules as the ones of SO(3). As consequences, we get (1) a structure result for Gaut ( ̂ B) in the case where B is a matrix algebra (2) if n ≥ 5 then the dual ̂ Gaut ( ̂ B) is not amenable (3) if n ≥ 4 then the fixed point subfactor
Quantum automorphism groups of homogeneous graphs
 J. Funct. Anal
"... Let X be a finite graph, with edges colored and possibly oriented, such that an oriented edge and a nonoriented one cannot have same color. The universal Hopf algebra H(X) coacting on X is in general non commutative, infinite dimensional, bigger than the algebra of functions on the usual symmetry g ..."
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Cited by 23 (11 self)
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Let X be a finite graph, with edges colored and possibly oriented, such that an oriented edge and a nonoriented one cannot have same color. The universal Hopf algebra H(X) coacting on X is in general non commutative, infinite dimensional, bigger than the algebra of functions on the usual symmetry group G(X). For a graph with no edges Tannakian duality makes H(X) correspond to a TemperleyLieb algebra. We study some versions of this correspondence.
Compact quantum groups
, 1999
"... We study the concept of coamenability for a compact quantum group. Several conditions are derived that are shown to be equivalent to it. Some consequences of coamenability that we obtain are faithfulness of the Haar integral and automatic normboundedness of positive linear functionals on the quan ..."
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Cited by 22 (3 self)
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We study the concept of coamenability for a compact quantum group. Several conditions are derived that are shown to be equivalent to it. Some consequences of coamenability that we obtain are faithfulness of the Haar integral and automatic normboundedness of positive linear functionals on the quantum group’s Hopf ∗algebra (neither of these properties necessarily holds without coamenability).
Integration over compact quantum groups
"... Abstract. We find a combinatorial formula for the Haar functional of the orthogonal and unitary quantum groups. As an application, we consider diagonal coefficients of the fundamental representation, and we investigate their spectral measures. ..."
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Cited by 18 (16 self)
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Abstract. We find a combinatorial formula for the Haar functional of the orthogonal and unitary quantum groups. As an application, we consider diagonal coefficients of the fundamental representation, and we investigate their spectral measures.
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.
Fusion rules for representations of compact quantum groups
"... The compact quantum groups are objects which generalise at the same time the compact groups, the duals of discrete groups and the q−deformations (with q> 0) of classical compact Lie groups. A compact quantum group is an abstract object which may be described by (is by definition the dual of) the alg ..."
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Cited by 12 (6 self)
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The compact quantum groups are objects which generalise at the same time the compact groups, the duals of discrete groups and the q−deformations (with q> 0) of classical compact Lie groups. A compact quantum group is an abstract object which may be described by (is by definition the dual of) the algebra of “continuous functions
Quantum groups and FussCatalan algebras
 Comm. Math. Phys
"... Abstract. The categories of representations of compact quantum groups of automorphisms of certain inclusions of finite dimensional C ∗algebras are shown to be isomorphic to the categories of FussCatalan diagrams. ..."
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Cited by 9 (5 self)
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Abstract. The categories of representations of compact quantum groups of automorphisms of certain inclusions of finite dimensional C ∗algebras are shown to be isomorphic to the categories of FussCatalan diagrams.
Subfactors associated to compact Kac algebras, preprint
"... We construct inclusions of the form (B0 ⊗ P) G ⊂ (B1 ⊗ P) G, where G is a compact quantum group of Kac type acting on an inclusion of finite dimensional C ∗algebras B0 ⊂ B1 and on a II1 factor P. Under suitable assumptions on the actions of G, this is a subfactor, whose Jones tower and standard inv ..."
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Cited by 8 (7 self)
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We construct inclusions of the form (B0 ⊗ P) G ⊂ (B1 ⊗ P) G, where G is a compact quantum group of Kac type acting on an inclusion of finite dimensional C ∗algebras B0 ⊂ B1 and on a II1 factor P. Under suitable assumptions on the actions of G, this is a subfactor, whose Jones tower and standard invariant can be computed by using techniques of A. Wassermann. The subfactors associated to subgroups of compact groups, to projective representations of compact groups, to finite quantum groups, to finitely generated discrete groups, to vertex models and to spin models are of this form. 1.
Amenability and coamenability for algebraic quantum groups
, 2002
"... We define concepts of amenability and coamenability for algebraic quantum groups in the sense of A. Van Daele [23]. We show that coamenability of an algebraic quantum group always implies amenability of its dual. Various necessary and/or sufficient conditions for amenability or coamenability are o ..."
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Cited by 8 (1 self)
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We define concepts of amenability and coamenability for algebraic quantum groups in the sense of A. Van Daele [23]. We show that coamenability of an algebraic quantum group always implies amenability of its dual. Various necessary and/or sufficient conditions for amenability or coamenability are obtained. Coamenability is shown to have interesting consequences for the modular theory in the case that the algebraic quantum group is of compact type.