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29
Classification of Bicovariant DIFFERENTIAL CALCULI
, 1996
"... We show that the bicovariant first order differential calculi on a factorisable semisimple quantum group are in 11 correspondence with irreducible representations V of the quantum group enveloping algebra. The corresponding calculus is constructed and has dimension dimV 2. The differential calculi ..."
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Cited by 25 (18 self)
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We show that the bicovariant first order differential calculi on a factorisable semisimple quantum group are in 11 correspondence with irreducible representations V of the quantum group enveloping algebra. The corresponding calculus is constructed and has dimension dimV 2. The differential calculi on a finite group algebra CG are also classified and shown to be in correspondence with pairs consisting of an irreducible representation V and a continuous parameter in CP dim V −1. They have dimension dimV. For a classical Lie group we obtain an infinite family of nonstandard calculi. General constructions for bicovariant calculi and their quantum tangent spaces are also obtained.
Lectures on graded differential algebras and noncommutative geometry
, 1999
"... These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments. ..."
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Cited by 24 (3 self)
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These notes contain a survey of some aspects of the theory of graded differential algebras and of noncommutative differential calculi as well as of some applications connected with physics. They also give a description of several new developments.
Differential calculi and linear connections
 J. Math. Phys
, 1996
"... A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a onetoone correspondence, between the module structure of ..."
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Cited by 21 (14 self)
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A method is proposed for defining an arbitrary number of differential calculi over a given noncommutative associative algebra. As an example the generalized quantum plane is studied. It is found that there is a strong correlation, but not a onetoone correspondence, between the module structure of the 1forms and the metric torsionfree connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1forms.
Noncommutative geometry and physics: a review of selected recent results
 Class. Quantum Grav
, 2000
"... This review is based on two lectures given at the 2000 TMR school in Torino ∗. We discuss two main themes: i) Moyaltype deformations of gauge theories, as emerging from Mtheory and open string theories, and ii) the noncommutative geometry of finite groups, with the explicit example of Z2, and its ..."
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Cited by 14 (6 self)
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This review is based on two lectures given at the 2000 TMR school in Torino ∗. We discuss two main themes: i) Moyaltype deformations of gauge theories, as emerging from Mtheory and open string theories, and ii) the noncommutative geometry of finite groups, with the explicit example of Z2, and its application to KaluzaKlein gauge theories on discrete internal spaces.
Some Aspects of Noncommutative Differential Geometry
"... We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finall ..."
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Cited by 12 (2 self)
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We discuss in some generality aspects of noncommutative differential geometry associated with reality conditions and with differential calculi. We then describe the differential calculus based on derivations as generalization of vector fields, and we show its relations with quantum mechanics. Finally we formulate a general theory of connections in this framework. 1
Electromagnetism and Gauge Theory on the Permutation Group S3
, 2002
"... Abstract Using noncommutative geometry we do U(1) gauge theory on the permutation group S3. Unlike usual lattice gauge theories the use of a nonAbelian group here as spacetime corresponds to a background Riemannian curvature. In this background we solve spin 0, 1/2 and spin 1 equations of motion, in ..."
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Cited by 11 (10 self)
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Abstract Using noncommutative geometry we do U(1) gauge theory on the permutation group S3. Unlike usual lattice gauge theories the use of a nonAbelian group here as spacetime corresponds to a background Riemannian curvature. In this background we solve spin 0, 1/2 and spin 1 equations of motion, including the spin 1 or ‘photon ’ case in the presence of sources, i.e. a theory of classical electromagnetism. Moreover, we solve the U(1) YangMills theory (this differs from the U(1) Maxwell theory in noncommutative geometry), including the moduli spaces of flat connections. We show that the YangMills action has a simple form in terms of Wilson loops in the permutation group, and we discuss aspects of the quantum theory. 1
LeviCivita Connections on the Quantum Groups SLq(N), Oq(N) and Spq(N)
, 1996
"... For bicovariant differential calculi on quantum groups various notions on connections and metrics (bicovariant connections, invariant metrics, the compatibility of a connection with a metric, LeviCivita connections) are introduced and studied. It is proved that for the bicovariant differential calc ..."
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Cited by 9 (1 self)
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For bicovariant differential calculi on quantum groups various notions on connections and metrics (bicovariant connections, invariant metrics, the compatibility of a connection with a metric, LeviCivita connections) are introduced and studied. It is proved that for the bicovariant differential calculi on SLq(N), Oq(N) and Spq(N) from the classification in [18] there exist unique LeviCivita connections.
Gravity on finite groups
 Commun. Math. Phys
"... Gravity theories are constructed on finite groups G. A selfconsistent review of the differential calculi on finite G is given, with some new developments. The example of a bicovariant differential calculus on the nonabelian finite group S3 is treated in detail, and used to build a gravitylike fiel ..."
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Cited by 8 (6 self)
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Gravity theories are constructed on finite groups G. A selfconsistent review of the differential calculi on finite G is given, with some new developments. The example of a bicovariant differential calculus on the nonabelian finite group S3 is treated in detail, and used to build a gravitylike field theory on S3.
Braided Lie algebras and bicovariant differential calculi over coquasitriangular Hopf algebras
, 2002
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