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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 (19 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 22 (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 (13 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 13 (5 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 13 (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
Braided Lie algebras and bicovariant differential calculi over coquasitriangular Hopf algebras
 J. Algebra
"... We show that if gΓ is the quantum tangent space (or quantum Lie algebra in the sense of Woronowicz) of a bicovariant first order differential calculus over a coquasitriangular Hopf algebra (A,r), then a certain extension of it is a braided Lie algebra in the category of Acomodules. This is used to ..."
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Cited by 9 (1 self)
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We show that if gΓ is the quantum tangent space (or quantum Lie algebra in the sense of Woronowicz) of a bicovariant first order differential calculus over a coquasitriangular Hopf algebra (A,r), then a certain extension of it is a braided Lie algebra in the category of Acomodules. This is used to show that the Woronowicz quantum universal enveloping algebra U(gΓ) is a bialgebra in the braided category of Acomodules. We show that this algebra is quadratic when the calculus is inner. Examples with this unexpected property include finite groups and quantum groups with their standard differential calculi. We also find a quantum Lie functor for coquasitriangular Hopf algebras, which has properties analogous to the classical one. This functor gives trivial results on standard quantum groups Oq(G), but reasonable ones on examples closer to the classical case, such as the cotriangular Jordanian deformations. In addition, we show that split braidedLie algebras define ‘generalisedLie algebras ’ in a different sense of deforming the adjoint representation. We construct these and their enveloping algebras for Oq(SLn), recovering the Witten algebra for n = 2. Introduction.
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.
Advances in quantum and braided geometry
 Proc. XXI ICGTMP (Quantum groups volume), Goslar 1996, Heron
"... We demonstrate our recent general results on the Casimir construction and moduli space of all bicovariant calculi by means of some detailed examples, including finitedifference and 2jet cacluli on R n and full details of the Casimir construction of the 4D calculus of SUq(2). We likewise demonstrate ..."
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Cited by 4 (3 self)
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We demonstrate our recent general results on the Casimir construction and moduli space of all bicovariant calculi by means of some detailed examples, including finitedifference and 2jet cacluli on R n and full details of the Casimir construction of the 4D calculus of SUq(2). We likewise demonstrate our previous general constructions with T. Brzezinski of quantum group gauge theory with examples of such nonuniversal differential calculi on spacetime. We outline a notion of quantum homotopy of a quantum space. We indicate a possible application to classical integrable systems. 1