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22
Charged Sectors, Spin and Statistics in Quantum Field Theory on Curved Spacetimes
, 1999
"... The first part of this paper extends the DoplicherHaagRoberts theory of superselection sectors to quantum field theory on arbitrary globally hyperbolic spacetimes. The statistics of a superselection sector may be defined as in flat spacetime and each charge has a conjugate charge when the spacetim ..."
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Cited by 30 (9 self)
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The first part of this paper extends the DoplicherHaagRoberts theory of superselection sectors to quantum field theory on arbitrary globally hyperbolic spacetimes. The statistics of a superselection sector may be defined as in flat spacetime and each charge has a conjugate charge when the spacetime possesses noncompact Cauchy surfaces. In this case, the field net and the gauge group can be constructed as in Minkowski spacetime. The second part of this paper derives spinstatistics theorems on spacetimes with appropriate symmetries. Two situations are considered: First, if the spacetime has a bifurcate Killing horizon, as is the case in the presence of black holes, then restricting the observables to the Killing horizon together with “modular covariance” for the Killing flow yields a conformally covariant quantum field theory on the circle and a conformal spinstatistics theorem for charged sectors localizable on the Killing horizon. Secondly, if the spacetime has a rotation and PT symmetry like the SchwarzschildKruskal black holes, “geometric modular action” of the rotational symmetry leads to a spinstatistics theorem for charged covariant sectors where the spin is defined via the SU(2)covering of the spatial rotation group SO(3).
Diagonal Crossed Products by Duals of QuasiQuantum Groups
 Rev. Math. Phys
, 1999
"... A twosided coaction δ: M → G ⊗M⊗G of a Hopf algebra (G, ∆, ǫ, S) on an associative algebra M is an algebra map of the form δ = (λ ⊗ idM) ◦ ρ = (idM ⊗ ρ) ◦ λ, where (λ, ρ) is a commuting pair of left and right Gcoactions on M, respectively. Denoting the associated commuting right and left actions ..."
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Cited by 25 (1 self)
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A twosided coaction δ: M → G ⊗M⊗G of a Hopf algebra (G, ∆, ǫ, S) on an associative algebra M is an algebra map of the form δ = (λ ⊗ idM) ◦ ρ = (idM ⊗ ρ) ◦ λ, where (λ, ρ) is a commuting pair of left and right Gcoactions on M, respectively. Denoting the associated commuting right and left actions of the dual Hopf algebra ˆ G on M by ⊳ and ⊲, respectively, we define the diagonal crossed product M ⊲ ⊳ ˆ G to be the algebra generated by M and ˆ G with relations given by ϕm = (ϕ (1) ⊲m ⊳ ˆ S −1 (ϕ (3)))ϕ (2), m ∈ M, ϕ ∈ ˆ G. We give a natural generalization of this construction to the case where G is a quasi–Hopf algebra in the sense of Drinfeld and, more generally, also in the sense of Mack and Schomerus (i.e., where the coproduct ∆ is nonunital). In these cases our diagonal crossed product will still be an associative algebra structure on M ⊗ ˆ G extending M ≡ M ⊗ ˆ1, even though the analogue of an ordinary crossed product M ⋊ ˆ G in general is not well defined as an associative algebra. Applications of our formalism include the field algebra constructions with quasiquantum
Graphs, groupoids and CuntzKrieger algebras
, 1996
"... We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions the ..."
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Cited by 25 (11 self)
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We associate to each locally finite directed graph G two locally compact groupoids G and G(?). The unit space of G is the space of onesided infinite paths in G, and G(?) is the reduction of G to the space of paths emanating from a distinguished vertex ?. We show that under certain conditions their C algebras are Morita equivalent; the groupoid C algebra C (G) is the CuntzKrieger algebra of an infinite f0; 1g matrix defined by G, and that the algebras C (G(?)) contain the C algebras used by Doplicher and Roberts in their duality theory for compact groups. We then analyse the ideal structure of these groupoid C algebras using the general theory of Renault, and calculate their Ktheory. 1 Introduction Over the past fifteen years many C algebras and classes of C algebras have been given groupoid models. Here we consider locally finite directed graphs, which may have infinitely many vertices, but only finitely many edges in and out of each vertex. We associate ...
Superselection structures for C*algebras with nontrivial center
, 1996
"... We present and proof some results in the context of superselection theory with the following conditions fixed: the observable algebra A has a nontrivial center Z and its relative commutant w.r.t. the field algebra F coincides with Z , i.e. we have A 0 " F = Z oe C 1l . In this frame we study Hi ..."
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Cited by 14 (4 self)
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We present and proof some results in the context of superselection theory with the following conditions fixed: the observable algebra A has a nontrivial center Z and its relative commutant w.r.t. the field algebra F coincides with Z , i.e. we have A 0 " F = Z oe C 1l . In this frame we study Hilbert C*systems with a compact group. We propose a generalization of the notion of an irreducible endomorphism and study the influence of the sector structure on Z . Finally we give several characterizations of the stabilizer of A.
Wavelet filters and infinitedimensional unitary groups, Wavelet analysis and applications
 AMS/IP Stud. Adv. Math. 25, Amer. Math. Soc
, 1999
"... Abstract. In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis which is based on representations of the C ∗algebra ON. A main tool in our analysis is the in ..."
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Cited by 13 (8 self)
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Abstract. In this paper, we study wavelet filters and their dependence on two numbers, the scale N and the genus g. We show that the wavelet filters, in the quadrature mirror case, have a harmonic analysis which is based on representations of the C ∗algebra ON. A main tool in our analysis is the infinitedimensional group of all maps T → U (N) (where U(N) is the group of all unitary NbyN matrices), and we study the extension problem from lowpass filter to multiresolution filter using this group.
Correspondences of ribbon categories
, 2006
"... Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories ..."
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Cited by 13 (3 self)
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Much of algebra and representation theory can be formulated in the general framework of tensor categories. The aim of this paper is to further develop this theory for braided tensor categories. Several results are established that do not have a substantial counterpart for symmetric tensor categories. In particular, we exhibit various equivalences involving categories of modules over algebras in ribbon categories. Finally we establish a correspondence of ribbon categories that can be applied to, and is in fact motivated by, the coset construction in conformal quantum field theory.
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.
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
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 12 (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.............................................
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 11 (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.