The Dirac operator on SU_q(2)
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In this paper, a 3^+ summable spectral triple (A(SU_q(2)),H,D) over the quantum group SU_q (2) which is equivariant with respect to a left and a right action was constructed.Abstract:
We construct a 3^+ summable spectral triple (A(SU_q(2)),H,D) over the quantum group SU_q(2) which is equivariant with respect to a left and a right action of U_q(su(2)). 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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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.
Book
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.
Book
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.