The double triangle algebra(DTA) associated to an ADE graph is considered. A description of its bialgebra structure based on a reconstruction approach is given. This approach takes as initial data the representation theory of the DTA as given by Ocneanu's cell calculus. It is also proved that the resulting DTA has the structure of a weak *-Hopf algebra. As an illustrative example, the case of the graph A 3 is described in detail.e-print archive: http://lanl.arXiv.org/abs/hep-th/0401140

We consider the algebra of N × N matrices as a reduced quantum plane on which a finitedimensional quantum group H acts. This quantum group is a quotient of Uq(sl(2, C)), q being an N -th root of unity. Most of the time we shall take N = 3; in that case dim(H) = 27. We recall the properties of this action and introduce a differential calculus for this algebra: it is a quotient of the Wess-Zumino complex. The quantum group H also acts on the corresponding differential algebra and we study its decomposition in terms of the representation theory of H. We also investigate the properties of connections, in the sense of non commutative geometry, that are taken as 1-forms belonging to this differential algebra. By tensoring this differential calculus with usual forms over space-time, one can construct generalized connections with covariance properties with respect to the usual Lorentz group and with respect to a finitedimensional quantum group.

Using the fact that the algebra M3(C l ) of 3 × 3 complex matrices can be taken as a reduced quantum plane, we build a differential calculus Ω(S) on the quantum space S defined by the algebra, where M is a space-time manifold. This calculus is covariant under the action and coaction of finite dimensional dual quantum groups. We study the star structures on these quantum groups and the compatible one in M3(C l ). This leads to an invariant scalar product on the later space. We analyse the differential algebra Ω(M3(C l )) in terms of quantum group representations, and consider in particular the space of 1-forms on S since its elements can be considered as generalized gauge fields.