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Matrix algebras converge to the sphere for quantum Gromov-Hausdorff distance
- Mem. Amer. Math. Soc
"... Abstract. On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2-sphere (or to other spaces). There is often careful bookkeeping with lengths, which suggests that one is dealing with “quantum metric spaces”. We show how t ..."
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Cited by 33 (5 self)
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Abstract. On looking at the literature associated with string theory one finds statements that a sequence of matrix algebras converges to the 2-sphere (or to other spaces). There is often careful bookkeeping with lengths, which suggests that one is dealing with “quantum metric spaces”. We show how to make these ideas precise by means of Berezin quantization using coherent states. We work in the general setting of integral coadjoint orbits for compact Lie groups. On perusing the theoretical physics literature which deals with string theory and related parts of quantum field theory, one finds in many scattered places assertions that the complex matrix algebras, Mn, converge to the two-sphere, S 2, (or to related spaces) as n goes to infinity. Here S 2 is viewed as synonymous with the algebra C(S 2) of continuous complex-valued functions on S 2 (of which S 2 is the maximal-ideal space). Approximating the sphere by matrix algebras is attractive for the following reason. In trying to carry out quantum field theory on S 2 it is natural to try to proceed by approximating S 2 by finite spaces. But “lattice ” approximations coming from choosing a finite set of points in S 2 break the very important symmetry of the action of SU(2) on S 2 (via SO(3)). But SU(2) acts naturally on the matrix algebras, in a way coherent with its action on S 2, as we will recall below. So it is natural to use them to approximate C(S 2). In this setting the matrix algebras are often referred to as “fuzzy spheres”. (See [33], [34], [17], [22], [24] and references therein.) When using the approximation of S 2 by matrix algebras, the precise sense of convergence is usually not explicitly specified in the literature. Much of the literature is at a largely algebraic level, with indications that the notion of convergence which is intended involves how structure constants and important formulas change as n grows. See, for
Order-unit quantum Gromov-Hausdorff distance
- J. Funct. Anal
, 2003
"... Abstract. We introduce a new distance distoq between compact quantum metric spaces. We show that distoq is Lipschitz equivalent to Rieffel’s distance distq, and give criteria for when a parameterized family of compact quantum metric spaces is continuous with respect to distoq. As applications, we sh ..."
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Cited by 20 (5 self)
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Abstract. We introduce a new distance distoq between compact quantum metric spaces. We show that distoq is Lipschitz equivalent to Rieffel’s distance distq, and give criteria for when a parameterized family of compact quantum metric spaces is continuous with respect to distoq. As applications, we show that the continuity of a parameterized family of quantum metric spaces induced by ergodic actions of a fixed compact group is determined by the multiplicities of the actions, generalizing Rieffel’s work on noncommutative tori and integral coadjoint orbits of semisimple compact connected Lie groups; we also show that the θ-deformations of Connes and Landi are continuous in the parameter θ. 1.
Poisson boundary of the dual of SUq(n)
- COMM. MATH. PHYS
, 2004
"... We prove that for any non-trivial product-type action α of SUq(n) (0 < q < 1) on an ITPFI factor N, the relative commutant (Nα) ′ ∩ N is isomorphic to the algebra L ∞ (SUq(n)/Tn−1) of bounded measurable functions on the quantum flag manifold SUq(n)/Tn−1. This is equivalent to the computation ..."
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Cited by 12 (3 self)
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We prove that for any non-trivial product-type action α of SUq(n) (0 < q < 1) on an ITPFI factor N, the relative commutant (Nα) ′ ∩ N is isomorphic to the algebra L ∞ (SUq(n)/Tn−1) of bounded measurable functions on the quantum flag manifold SUq(n)/Tn−1. This is equivalent to the computation of the Poisson boundary of the dual discrete quantum group ̂ SUq(n). The proof relies on a connection between the Poisson integral and the Berezin transform. Our main technical result says that a sequence of Berezin transforms defined by a random walk on the dominant weights of SU(n) converges to the identity on the quantum flag manifold. This is a q-analogue of some known results on quantization of coadjoint orbits of Lie groups.
Field-Theoretic Weyl Deformation Quantization of Enlarged Poisson Algebras
, 2008
"... C∗-algebraic Weyl quantization is extended by allowing also degenerate presymplectic forms for the Weyl relations with infinitely many degrees of freedom, and by starting out from enlarged classical Poisson algebras. A powerful tool is found in the construction of Poisson algebras and non-commutativ ..."
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C∗-algebraic Weyl quantization is extended by allowing also degenerate presymplectic forms for the Weyl relations with infinitely many degrees of freedom, and by starting out from enlarged classical Poisson algebras. A powerful tool is found in the construction of Poisson algebras and non-commutative twisted Banach-∗-algebras on the stage of measures on the not locally compact test function space. Already within this frame strict deformation quantization is obtained, but in terms of Banach-∗-algebras instead of C∗-algebras. Fourier transformation and representation theory of the measure Banach-∗algebras are combined with the theory of continuous projective group representations to arrive at the genuine C∗-algebraic strict deformation quantization in the sense of Rieffel and Landsman. Weyl quantization is recognized to depend in the first step functorially on the (in general) infinite dimensional, pre-symplectic test function space; but in the second step one has to select a family of representations, indexed by the deformation parameter �. The latter ambiguity is in the present investigation connected with the choice of a folium of states, a structure, which does not necessarily require a Hilbert space representation.
Quantum flag manifolds as quotients of degenerate quantized universal enveloping algebras
, 2014
"... Let g be a semi-simple Lie algebra with fixed root system, and Uqpgq the quantization of its universal enveloping algebra. Let S be a subset of the simple roots of g. We show that the defining relations for Uqpgq can be slightly modified in such a way that the resulting algebra Uqpg;Sq allows a homo ..."
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Let g be a semi-simple Lie algebra with fixed root system, and Uqpgq the quantization of its universal enveloping algebra. Let S be a subset of the simple roots of g. We show that the defining relations for Uqpgq can be slightly modified in such a way that the resulting algebra Uqpg;Sq allows a homomorphism onto (an extension of) the algebra PolpGq{KS,qq of functions on the quantum flag manifold Gq{KS,q corresponding to S. Moreover, this homomorphism is equivariant with respect to a natural adjoint action of Uqpgq on Uqpg;Sq and the standard action of Uqpgq on PolpGq{KS,qq.
Your file Votre rtiterence
, 2009
"... Study of the coadjoint orbits of the Poincare group in 2 + 1 dimensions and their coherent states ..."
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Study of the coadjoint orbits of the Poincare group in 2 + 1 dimensions and their coherent states
Noncommutative boundaries of q-deformations
"... The study of random walks on duals of compact quantum groups was initiated by Masaki Izumi in [I1]. The motivation was to compute the relative commutant of the fixed point algebra for product-type actions of compact quantum groups. In the classical case such actions are always minimal, that is, the ..."
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The study of random walks on duals of compact quantum groups was initiated by Masaki Izumi in [I1]. The motivation was to compute the relative commutant of the fixed point algebra for product-type actions of compact quantum groups. In the classical case such actions are always minimal, that is, the commutant is trivial. For quantum groups this is not so. It turns out that
