A Finite Quantum Symmetry of M(3,C)

The 27-dimensional Hopf algebra A(F), defined by the exact sequence of quantum groups A(SL(2,C))->A(SL_q(2))->A(F), q^3=1, is studied as a finite quantum group symmetry of the matrix algebra M(3,C), describing the color sector of Alain Connes' formulation of the Standard Model. The duality with the Hopf algebra H,investigated in a recent work by Robert Coquereaux, is established and used to define a representation of H on M(3,C) and two commuting representations of H on A(F).Comment: Amslatex, 17 pages, only Reference [DHS] modifie

Metrics and Pairs of Left and Right Connections on Bimodules

Properties of metrics and pairs consisting of left and right connections are studied on the bimodules of differential 1-forms. Those bimodules are obtained from the derivation based calculus of an algebra of matrix valued functions, and an SL\sb q(2,\IC)-covariant calculus of the quantum plane plane at a generic $q$ and the cubic root of unity. It is shown that, in the aforementioned examples, giving up the middle-linearity of metrics significantly enlarges the space of metrics. A~metric compatibility condition for the pairs of left and right connections is defined. Also, a compatibility condition between a left and right connection is discussed. Consequences entailed by reducing to the centre of a bimodule the domain of those conditions are investigated in detail. Alternative ways of relating left and right connections are considered.Comment: 16 pages, LaTeX, nofigure

\\ C;( CONNECTIONS ON BIMODULES IN NONCOMMUTATIVE GEOMETRY

Abstract Properties of pairs consisting of left and right connections are studied on the bimodules of differential I-forms. An example is given in terms of an S Lq(2, C) covariant calculus of the quantum plane at a generic q and at the cubic root of unity. A compatibility condition between a left and right connection is discussed. Consequences entailed by reducing to the centre of a bimodule the domain of those conditions are investigated. Alternative ways of relating left and right connections are considered