16 research outputs found
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 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