METRICS AND PAIRS OF LEFT AND RIGHT CONNECTIONS ON BIMODULES

Properties of metrics and pairs consisting of left and right connectio ns are studied on the bimodules of differential 1-forms. Those bimodul es are obtained from the derivation based calculus of an algebra of ma trix valued functions, and an SL(q)(2,C)-covariant calculus of the qua ntum plane at a generic q and the cubic root of unity. It is shown tha t, in the aforementioned examples, giving up the middle-linearity of m etrics significantly enlarges the space of metrics. A metric compatibi lity condition for the pairs of left and right connections is defined. Also, a compatibility condition between a left and right connection i s discussed. Consequences entailed by reducing to the center of a bimo dule the domain of those conditions are investigated in detail. Altern ative ways of relating left and right connections are considered. (C) 1996 American Institute of Physics.