The Dirac Operator on SU q (2)
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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.read more
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'Schwinger Model' on the Fuzzy Sphere
TL;DR: In this paper, the authors constructed a model of spinor fields interacting with specific gauge fields on fuzzy sphere and analyzed the chiral symmetry of this "Schwinger model", which introduced interaction between spinors and specific one parameter family of gauge fields.
Spectral gaps for twisted Dolbeault-Dirac operators over the irreducible quantum flag manifolds
TL;DR: In this paper , it was shown that tensoring the Laplace and Dolbeault Dirac operators of a K¨ahler structure with a negative Hermitian holomorphic module produces operators with spectral gaps around zero.
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Dirac operator on the quantum fuzzy four-sphere $ S_{qF}^{4} $
TL;DR: In this article, the q-deformed fuzzy Dirac and chirality operators on quantum fuzzy four-sphere were studied and the results showed that in each step their results have correct commutative limit in the limit case when the noncommutative parameter tends to infinity.
Book ChapterDOI
Prelude: A General Overview
TL;DR: A general overview of the various topics discussed in this volume, emphasizing the deep relations existing between them, can be found in this paper, followed by a brief historical account of the emergence of the concept of "quantization" both in physics and mathematics.
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Twisted Dirac Operator on Quantum SU(2) in Disc Coordinates
Ulrich Krähmer,Elmar Wagner +1 more
TL;DR: In this article, the quantum disc is used to define a noncommutative analogue of a dense coordinate chart and of left-invariant vector fields on quantum SU(2), which yields two twisted Dirac operators for different twists that are related by a gauge transformation.
References
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Book
Quantum Groups
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.
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Foundations of Quantum Group Theory
TL;DR: In this paper, the authors define Hopf algebras as "quasitriangular Hopf-algebraes" and introduce matrix quantum groups and bicrossproduct hopf alges.
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Elements of Noncommutative Geometry
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.
Journal ArticleDOI
Noncommutative geometry and reality
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.