The Dirac Operator on SU q (2)
TL;DR: In this article, a 3+summable spectral triple Open Image in new window over the quantum group SUq(2) which is equivariant with respect to a left and a right action was constructed.
Abstract: We construct a 3+-summable spectral triple Open image in new window over the quantum group SUq(2) which is equivariant with respect to a left and a right action of Open image in new window The geometry is isospectral to the classical case since the spectrum of the operator D is the same as that of the usual Dirac operator on the 3-dimensional round sphere. The presence of an equivariant real structure J demands a modification in the axiomatic framework of spectral geometry, whereby the commutant and first-order properties need be satisfied only modulo infinitesimals of arbitrary high order.
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Book•
15 Jun 2006
TL;DR: In this paper, a course on non-commutative geometry from the non-computative point of view was presented at the 1997 Summer School on Non-Commutative Geometry and Applications at the European Mathematical Society (EMS) at Monsaraz and Lisboa, Portugal, September 1-10, 1997.
Abstract: This is the introduction and bibliography for lecture notes of a course given at the Summer School on Noncommutative Geometry and Applications, sponsored by the European Mathematical Society, at Monsaraz and Lisboa, Portugal, September 1-10, 1997 In the published version, an epilogue of recent developments and many new references from 1998-2006 have been added
1 Commutative geometry from the noncommutative point of view
2 Spectral triples on the Riemann sphere
3 Real spectral triples, the axiomatic foundation
4 Geometries on the noncommutative torus
5 The noncommutative integral
6 Quantization and the tangent groupoid
7 Equivalence of geometries
8 Action functionals
9 Epilogue: new directions
220 citations
TL;DR: The assumption that space-time is a noncommutative space formed as a prod- uct of a continuous four dimensional manifold times a nite space predicts, almost uniquely, the existence of a single point in space as discussed by the authors.
Abstract: The assumption that space-time is a noncommutative space formed as a prod- uct of a continuous four dimensional manifold times a nite space predicts, almost uniquely,
123 citations
TL;DR: In this article, a semigroup of inner fluctuations is introduced for noncommutative spectral models beyond the Standard Model, which has a key application in non-convex spectral models.
105 citations
TL;DR: In this paper, the authors formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or non-commutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly non commutative) manifold satisfying certain regularity assumptions.
Abstract: We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold satisfying certain regularity assumptions. The idea of ‘quantum families’ (due to Woronowicz and Soltan) are relevant to our construction. A number of explicit examples are given and possible applications of our results to the problem of constructing quantum group equivariant spectral triples are discussed.
100 citations
TL;DR: In this paper, a quantum generalization of the notion of the group of Riemannian isometries for a compact manifold is introduced, by introducing a natural notion of smooth and isometric action by a compact quantum group on a non-commutative manifold described by spectral triples.
Abstract: We formulate a quantum generalization of the notion of the group of Riemannian isometries for a compact Riemannian manifold, by introducing a natural notion of smooth and isometric action by a compact quantum group on a classical or noncommutative manifold described by spectral triples, and then proving the existence of a universal object (called the quantum isometry group) in the category of compact quantum groups acting smoothly and isometrically on a given (possibly noncommutative) manifold satisfying certain regularity assumptions. In fact, we identify the quantum isometry group with the universal object in a bigger category, namely the category of `quantum families of smooth isometries', defined along the line of Woronowicz and Soltan. We also construct a spectral triple on the Hilbert space of forms on a noncommutative manifold which is equivariant with respect to a natural unitary representation of the quantum isometry group. We give explicit description of quantum isometry groups of commutative and noncommutative tori, and in this context, obtain the quantum double torus defined in \cite{hajac} as the universal quantum group of holomorphic isometries of the noncommutative torus.
88 citations
Cites background from "The Dirac Operator on SU q (2)"
...There has been a lot of activity in this direction recently, see, for example, the articles by Chakraborty and Pal ([5]), Connes ([7]), Landi et al ([8]) and the references therein....
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References
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04 Nov 1994
TL;DR: In this paper, the authors introduce the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions and present the quantum groups attached to SL2 as well as the basic concepts of the Hopf algebras.
Abstract: Here is an introduction to the theory of quantum groups with emphasis on the spectacular connections with knot theory and Drinfeld's recent fundamental contributions. It presents the quantum groups attached to SL2 as well as the basic concepts of the theory of Hopf algebras. Coverage also focuses on Hopf algebras that produce solutions of the Yang-Baxter equation and provides an account of Drinfeld's elegant treatment of the monodromy of the Knizhnik-Zamolodchikov equations.
5,966 citations
"The Dirac Operator on SU q (2)" refers background in this paper
...We recall that there is another convention for the generators of Uq(su(2)) in widespread use: see [19], for instance....
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Book•
01 Apr 2000TL;DR: In this paper, the authors define Hopf algebras as "quasitriangular Hopf-algebraes" and introduce matrix quantum groups and bicrossproduct hopf alges.
Abstract: Introduction 1. Definition of Hopf algebras 2. Quasitriangular Hopf algebras 3. Quantum enveloping algebras 4. Matrix quantum groups 5. Quantum random walks and combinatorics 6. Bicrossproduct Hopf algebras 7. Quantum double and double cross products 8. Lie bialgebras and Poisson brackets 9. Representation theory 10. Braided groups and q-deformation References Symbols Indexes.
2,219 citations
"The Dirac Operator on SU q (2)" refers methods in this paper
...Here we follow Majid’s “lexicographic convention” [23, 24] (where, with c = −qb, d = a, a factor of q is needed to restore alphabetical order)....
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Book•
22 Dec 2011
1,164 citations
"The Dirac Operator on SU q (2)" refers background or methods in this paper
...The handy compendium [21] gives both versions, denoting by Ŭq(su(2)) the version which we adopt here....
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...13) embeds the prehilbert space V densely in the Hilbert space Hψ, and the representation πψ extends to the GNS representation of C(SUq(2)) on Hψ, as described by the Peter-Weyl theorem [21, 32]....
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...1) coincide with those of Ŭq(su(2)) in [21], after exchange of e and f (see Remark 2....
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...We begin with the known representation theory [21] of Uq(su(2))....
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...We recall [21] that A has a vector-space basis consisting of matrix elements of its irreducible corepresentations, { tmn : 2l ∈ N, m, n = −l, ....
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Book•
23 Oct 2000
TL;DR: In this article, a wide range of topics including sources of non-commutative geometry, fundamentals of Non-Commutative topology, K-theory and Morita equivalance, non-commodity integrodifferential calculus, noncommutativity Riemannian spin manifolds, commutative geometrics, tori, second quantization, quantum field theory, and pseudodifferential operators are discussed.
Abstract: This volume covers a wide range of topics including sources of noncommutative geometry; fundamentals of noncommutative topology; K-theory and Morita equivalance; noncommutative integrodifferential calculus; noncommutative Riemannian spin manifolds; commutative geometrics; tori; proof of Connes' Riemannian spin theorem; second quantization; noncommutative quantum field theory; C*-algebras; Hopf algebras; Clifford algebras; Moyal algebras; pseudodifferential operators; the commutative Chern character theorem.
918 citations
TL;DR: The notion of real structure in spectral geometry was introduced in this paper, motivated by Atiyah's KR•theory and by Tomita's involution J. It allows us to remove two unpleasant features of the Connes-Lott description of the standard model, namely, the use of bivector potentials and the asymmetry in the Poincare duality and in the unimodularity condition.
Abstract: We introduce the notion of real structure in our spectral geometry. This notion is motivated by Atiyah’s KR‐theory and by Tomita’s involution J. It allows us to remove two unpleasant features of the ‘‘Connes–Lott’’ description of the standard model, namely, the use of bivector potentials and the asymmetry in the Poincare duality and in the unimodularity condition.
617 citations
"The Dirac Operator on SU q (2)" refers background or methods in this paper
...Therefore, the equivariant J we shall use does not intertwine the spin representation of A(SUq(2)) with its commutant, and it is not possible to satisfy all the desirable properties of a real spectral triple as set forth in [8, 15]....
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...However, here is where we part company with the axiom scheme for real spectral triples proposed in [8]....
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...We shall see that by requiring equivariance of J it is not possible to satisfy all usual properties of a real spectral triple like in [8] or [15]....
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