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229
Convexity and commuting Hamiltonians
 Bull. London Math. Soc.,14
, 1982
"... A wellknown result of Schur [9] asserts that the diagonal elements (al,..., an) of annxn Hermitian matrix A satisfy a system of linear inequalities involving the eigenvalues (Xi,..., Xn). In geometric terms, regarding a and k as points in R " and allowing the symmetric group £ „ to act by permutati ..."
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Cited by 162 (1 self)
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A wellknown result of Schur [9] asserts that the diagonal elements (al,..., an) of annxn Hermitian matrix A satisfy a system of linear inequalities involving the eigenvalues (Xi,..., Xn). In geometric terms, regarding a and k as points in R " and allowing the symmetric group £ „ to act by permutation of coordinates, this result
Spiders for rank 2 Lie algebras
 Commun. Math. Phys
, 1996
"... Abstract. A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or grouplike object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point o ..."
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Cited by 62 (1 self)
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Abstract. A spider is an axiomatization of the representation theory of a group, quantum group, Lie algebra, or other group or grouplike object. It is also known as a spherical category, or a strict, monoidal category with a few extra properties, or by several other names. A recently useful point of view, developed by other authors, of the representation theory of sl(2) has been to present it as a spider by generators and relations. That is, one has an algebraic spider, defined by invariants of linear representations, and one identifies it as isomorphic to a combinatorial spider, given by generators and relations. We generalize this approach to the rank 2 simple Lie algebras, namely A2, B2, and G2. Our combinatorial rank 2 spiders yield bases for invariant spaces which are probably related to Lusztig’s canonical bases, and they are useful for computing quantities such as generalized 6jsymbols and quantum link invariants. Their definition originates in definitions of the rank 2 quantum link invariants that were discovered independently by the author and Francois Jaeger. 1.
Quantum Field Theory on Noncommutative Spacetimes and the
 Persistence of Ultraviolet Divergences,” hepth/9812180; “Quantum Field Theory on the Noncommutative Plane with E(q)(2) Symmetry,” hepth/9904132
"... We study properties of a scalar quantum field theory on twodimensional noncommutative spacetimes. Contrary to the common belief that noncommutativity of spacetime would be a key to remove the ultraviolet divergences, we show that field theories on a noncommutative plane with the most natural Heis ..."
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Cited by 51 (1 self)
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We study properties of a scalar quantum field theory on twodimensional noncommutative spacetimes. Contrary to the common belief that noncommutativity of spacetime would be a key to remove the ultraviolet divergences, we show that field theories on a noncommutative plane with the most natural Heisenberglike commutation relations among coordinates or even on a noncommutative quantum plane with Eq(2)symmetry have ultraviolet divergences, while the theory on a noncommutative cylinder is ultraviolet finite. Thus, ultraviolet behaviour of a field theory on noncommutative spaces is sensitive to the topology of the spacetime, namely to its compactness. We present general arguments for the case of higher spacetime dimensions and as well discuss the symmetry transformations of physical states on noncommutative spacetimes.
Conformally equivariant quantization: Existence and uniqueness
"... We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudoRiemannian manifold (M,g). In other words, we establish a canonical isomorphism between the spaces of polynomials on T ∗ M and of differential operators on tensor de ..."
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Cited by 41 (5 self)
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We prove the existence and the uniqueness of a conformally equivariant symbol calculus and quantization on any conformally flat pseudoRiemannian manifold (M,g). In other words, we establish a canonical isomorphism between the spaces of polynomials on T ∗ M and of differential operators on tensor densities over M, both viewed as modules over the Lie algebra o(p + 1,q + 1) where p + q = dim(M). This quantization exists for generic values of the weights of the tensor densities and compute the critical values of the weights yielding obstructions to the existence of such an isomorphism. In the particular case of halfdensities, we obtain a conformally invariant starproduct.
Lagrangian reduction and the double spherical pendulum
 ZAMP
, 1993
"... This paper studies the stability and bifurcations of the relative equilibria of the double spherical pendulum, which has the circle as its symmetry group. This example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian re ..."
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Cited by 40 (20 self)
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This paper studies the stability and bifurcations of the relative equilibria of the double spherical pendulum, which has the circle as its symmetry group. This example as well as others with nonabelian symmetry groups, such as the rigid body, illustrate some useful general theory about Lagrangian reduction. In particular, we establish a satisfactory global theory of Lagrangian reduction that is consistent with the classical local Routh theory for systems with an abelian symmetry group. 1
Symplectic structures associated to LiePoisson groups
 Commun. Math. Phys
, 1994
"... The LiePoisson analogues of the cotangent bundle and coadjoint orbits of a Lie group are considered. For the natural Poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are described. Thus the construction of the Kirillov symplectic form i ..."
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Cited by 40 (4 self)
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The LiePoisson analogues of the cotangent bundle and coadjoint orbits of a Lie group are considered. For the natural Poisson brackets the symplectic leaves in these manifolds are classified and the corresponding symplectic forms are described. Thus the construction of the Kirillov symplectic form is generalized for LiePoisson groups.
TwoDimensional Directional Wavelets and the ScaleAngle Representation
, 1995
"... The twodimensional continuous wavelet transform (CWT), derived from a square integrable representation of the similitude group of IR 2 , is characterized by a rotation parameter, in addition to the usual translations and dilations. This enables it to detect edges and directions in images, provide ..."
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Cited by 29 (6 self)
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The twodimensional continuous wavelet transform (CWT), derived from a square integrable representation of the similitude group of IR 2 , is characterized by a rotation parameter, in addition to the usual translations and dilations. This enables it to detect edges and directions in images, provided a directional wavelet is used. First we review the general properties of the 2D CWT and describe several classes of wavelets, including the directional ones. Then we turn to the problem of wavelet calibration. We show, in particular, how the reproducing kernel may be used for defining and evaluating the scale and angle resolving power of a wavelet. Finally we illustrate the usefulness of the scaleangle representation of the CWT on the problem of disentangling a train of damped plane waves. UCLIPT9503 May 1995 3 Supported by ONR (Office of Naval Research), Grant Nr.N00149310561 and by ARPA (Advanced Research Project Agency), Grant Nr.MDA 9729310013 1. Introduction The wavele...
Lattice WessZuminoWitten Model and Quantum
, 1993
"... Quantum groups play a role of symmetries of integrable theories in two dimensions. They may be detected on the classical level as PoissonLie symmetries of the corresponding phase spaces. We discuss specifically the WessZuminoWitten conformally invariant quantum field model combining two chiral pa ..."
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Cited by 26 (1 self)
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Quantum groups play a role of symmetries of integrable theories in two dimensions. They may be detected on the classical level as PoissonLie symmetries of the corresponding phase spaces. We discuss specifically the WessZuminoWitten conformally invariant quantum field model combining two chiral parts which describe the left and rightmoving degrees of freedom. On one hand side, the quantum group plays the role of the symmetry of the chiral components of the theory. On the other hand, the model admits a lattice regularization (in the Minkowski space) in which the current algebra symmetry of the theory also becomes quantum, providing the simplest example of a quantum group symmetry coupling spacetime and internal degrees of freedom. We develop a free field approach to the representation theory of the lattice sl(2)based current algebra and show how to use it to rigorously construct an exact solution of the quantum SL(2) WZW model on lattice. 1.
Modular localization and Wigner particles
 Rev. Math. Phys
"... Dedicated to Huzihiro Araki on the occasion of his seventieth birthday Abstract. We propose a framework for the free field construction of algebras of local observables which uses as an input the BisognanoWichmann relations and a representation of the Poincaré group on the oneparticle Hilbert spac ..."
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Cited by 22 (4 self)
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Dedicated to Huzihiro Araki on the occasion of his seventieth birthday Abstract. We propose a framework for the free field construction of algebras of local observables which uses as an input the BisognanoWichmann relations and a representation of the Poincaré group on the oneparticle Hilbert space. The abstract real Hilbert subspace version of the TomitaTakesaki theory enables us to bypass some limitations of the Wigner formalism by introducing an intrinsic spacetime localization. Our approach works also for continuous spin representations to which we associate a net of von Neumann algebras on spacelike cones with the ReehSchlieder property. The positivity of the energy in the representation turns out to be equivalent to the isotony of the net, in the spirit of Borchers theorem. Our procedure extends to other spacetimes homogeneous under a group of geometric transformations as in the case of conformal symmetries and de Sitter spacetime. 1.