We show that the bicovariant first order differential calculi on a factorisable semisimple quantum group are in 1-1 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.

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 one-to-one correspondence, between the module structure of the 1-forms and the metric torsionfree connections on it. In the commutative limit the connection remains as a shadow of the algebraic structure of the 1-forms.

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.

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) Yang-Mills theory (this differs from the U(1) Maxwell theory in noncommutative geometry), including the moduli spaces of flat connections. We show that the Yang-Mills action has a simple form in terms of Wilson loops in the permutation group, and we discuss aspects of the quantum theory. 1

This review is based on two lectures given at the 2000 TMR school in Torino ∗. We discuss two main themes: i) Moyal-type deformations of gauge theories, as emerging from M-theory and open string theories, and ii) the noncommutative geometry of finite groups, with the explicit example of Z2, and its application to Kaluza-Klein gauge theories on discrete internal spaces.

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 co-quasitriangular Hopf algebra (A,r), then a certain extension of it is a braided Lie algebra in the category of A-comodules. This is used to show that the Woronowicz quantum universal enveloping algebra U(gΓ) is a bialgebra in the braided category of A-comodules. 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 co-quasitriangular 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 braided-Lie algebras define ‘generalised-Lie 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.

L.P.T.H.E.-ORSAY 95/78 ESI-preprint 285 1Laboratoire associ'e au Centre National de la Recherche Scientifique- URA D0063 1 Abstract 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 Introduction In [23], J.L. Koszul described a powerful algebraic version of differential geometry in terms of a commutative associative algebra C, C-modules and connections ("derivation laws") on these modules. For the applications to differential geometry, C is the algebra of smooth functions on a manifold and the C-modules are modules of smooth sections of smooth vector bundles over the manifold. The fact that classical differential geometry admits such an algebraic formulation is at the very origin of the idea of noncommutative differential geometry. Historically, the motivation of noncommutative geometry was the development of quantum theory [12]. In noncommutative geometry, one replaces the commutative associative algebra C by an associative algebra A which is not assumed to be commutative. However this replacement raises several problems which will be discussed in this lecture.

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 2-jet 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

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Discretized nonabelian gauge theories living on finite group spaces G are defined by means of a geometric action ∫ Tr F ∧ ∗F. This technique is extended to obtain discrete versions of the Born-Infeld action. The discretizations are in 1-1 correspondence with differential calculi on finite groups. A consistency condition for duality invariance of the discretized field equations is derived for discretized U(1) actions S[F] living on a 4-dimensional abelian G. Discretized electromagnetism satisfies this condition and therefore admits duality rotations. Yang-Mills and Born-Infeld theories are also considered on product spaces M D × G, and we find the corresponding field theories on M D after Kaluza-Klein reduction on the G discrete internal spaces. We examine in some detail the case G = ZN, and discuss the limit N → ∞. A self-contained review on the noncommutative differential geometry of finite groups is included.