Results 11  20
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105
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.
Canonical tensor product subfactors
"... Canonical tensor product subfactors (CTPS’s) describe, among other things, the embedding of chiral observables in twodimensional conformal quantum field theories. A new class of CTPS’s is constructed some of which are associated with certain modular invariants, thereby establishing the expected exi ..."
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Cited by 21 (6 self)
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Canonical tensor product subfactors (CTPS’s) describe, among other things, the embedding of chiral observables in twodimensional conformal quantum field theories. A new class of CTPS’s is constructed some of which are associated with certain modular invariants, thereby establishing the expected existence of the corresponding twodimensional theories. 1 Introduction and
A theory of dimension
 KTHEORY
, 1997
"... In which a theory of dimension related to the Jones index and based on the notion of conjugation is developed. An elementary proof of the additivity and multiplicativity of the dimension is given and there is an associated trace. Applications are given to a class of endomorphisms of factors and to ..."
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Cited by 17 (0 self)
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In which a theory of dimension related to the Jones index and based on the notion of conjugation is developed. An elementary proof of the additivity and multiplicativity of the dimension is given and there is an associated trace. Applications are given to a class of endomorphisms of factors and to the theory of subfactors. An important role is played by a notion of amenability inspired by the work of Popa.
Classification of twodimensional local conformal nets with c < 1 and 2cohomology vanishing for tensor categories
 Commun. Math. Phys
, 2004
"... We classify twodimensional local conformal nets with parity symmetry and central charge less than 1, up to isomorphism. The maximal ones are in a bijective correspondence with the pairs of ADE Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification ..."
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Cited by 17 (9 self)
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We classify twodimensional local conformal nets with parity symmetry and central charge less than 1, up to isomorphism. The maximal ones are in a bijective correspondence with the pairs of ADE Dynkin diagrams with the difference of their Coxeter numbers equal to 1. In our previous classification of onedimensional local conformal nets, Dynkin diagrams D2n+1 and E7 do not appear, but now they do appear in this classification of twodimensional local conformal nets. Such nets are also characterized as twodimensional local conformal nets with µindex equal to 1 and central charge less than 1. Our main tool, in addition to our previous classification results for onedimensional nets, is 2cohomology vanishing for certain tensor categories related to the Virasoro tensor categories with central charge less than 1.
A primer of Hopf algebras
 Insitut des Hautes Études Scientifiques, IHES/M/06/04
, 2006
"... Summary. In this paper, we review a number of basic results about socalled Hopf algebras. We begin by giving a historical account of the results obtained in the 1930’s and 1940’s about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to H ..."
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Cited by 13 (0 self)
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Summary. In this paper, we review a number of basic results about socalled Hopf algebras. We begin by giving a historical account of the results obtained in the 1930’s and 1940’s about the topology of Lie groups and compact symmetric spaces. The climax is provided by the structure theorems due to Hopf, Samelson, Leray and Borel. The main part of this paper is a thorough analysis of the relations between Hopf algebras and Lie groups (or algebraic groups). We emphasize especially the category of unipotent (and prounipotent) algebraic groups, in connection with MilnorMoore’s theorem. These methods are a powerful tool to show that some algebras are free polynomial rings. The last part is an introduction to the combinatorial aspects of polylogarithm functions and the corresponding multiple zeta values. 1 Introduction.............................................
Crossed products by endomorphisms, Vector Bundles and Group Duality
, 2003
"... We construct the crossed product A ⋊ E ̂ρ of a C*algebra A with centre Z by an endomorphism ρ, which is special in a weaker sense w.r.t. the notion introduced by Doplicher and Roberts. The notation ̂ρ denotes the tensor C*category of powers of ρ. We assign to ρ a geometrical invariant, representin ..."
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Cited by 13 (10 self)
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We construct the crossed product A ⋊ E ̂ρ of a C*algebra A with centre Z by an endomorphism ρ, which is special in a weaker sense w.r.t. the notion introduced by Doplicher and Roberts. The notation ̂ρ denotes the tensor C*category of powers of ρ. We assign to ρ a geometrical invariant, representing a cohomological obstruction to be special in the usual sense, and determining rank and first Chern class of the vector bundle E whose module of sections (contained in A ⋊ E ̂ρ) induces ρ on A. We prove that A is the fixed point C*algebra w.r.t. a Gaction on A ⋊ E ̂ρ, and characterize ̂ρ as the category of tensor powers of a suitable GHilbert Zbimodule (a socalled ’noncommutative pullback ’ of E). G is interpreted as the generally noncompact space of sections of a group bundle.
On charged fields with group symmetry and degeneracies of Verlinde’s matrix
 Poincaré (Phys. Théor.) 71, 359394 (1999) INVARIANTS FROM SUBFACTORS 37
"... We prove that the complete normal field net with compact symmetry group constructed by Doplicher and Roberts starting from a net of local observables in ≥ 2+1 space time dimensions and its set of localized (DHR) representations does not possess nontrivial DHR sectors. Whereas the superselection stru ..."
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Cited by 13 (6 self)
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We prove that the complete normal field net with compact symmetry group constructed by Doplicher and Roberts starting from a net of local observables in ≥ 2+1 space time dimensions and its set of localized (DHR) representations does not possess nontrivial DHR sectors. Whereas the superselection structure in 1+1 dimensions typically does not arise from a compact group, the DR construction is applicable to ‘degenerate sectors’, the existence of which (in the rational case) is equivalent to noninvertibility of Verlinde’s Smatrix. We prove Rehren’s conjecture that the enlarged theory is nondegenerate, which implies that every degenerate theory is an ‘orbifold ’ theory. Thus, the symmetry of a generic model ‘factorizes ’ into a group part and a pure quantum part which still must be clarified.
Superselection theory for subsystems
 COMM. MATH. PHYS
, 2000
"... An inclusion of observable nets satisfying duality induces an inclusion of canonical field nets. Any Bose net intermediate between the observable net and the field net and satisfying duality is the fixed–point net of the field net under a compact group. This compact group is its canonical gauge grou ..."
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Cited by 12 (4 self)
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An inclusion of observable nets satisfying duality induces an inclusion of canonical field nets. Any Bose net intermediate between the observable net and the field net and satisfying duality is the fixed–point net of the field net under a compact group. This compact group is its canonical gauge group if the occurrence of sectors with infinite statistics can be ruled out for the observable net and its vacuum Hilbert space is separable.
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 ..."
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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
Continuous fields of C*algebras arising from extensions of tensor C*categories
"... The notion of extension of a given C*category C by a C*algebra A is introduced. In the commutative case A = C(Ω), the objects of the extension category are interpreted as fiber bundles over Ω of objects belonging ..."
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Cited by 12 (10 self)
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The notion of extension of a given C*category C by a C*algebra A is introduced. In the commutative case A = C(Ω), the objects of the extension category are interpreted as fiber bundles over Ω of objects belonging