Metrics and Pairs of Left and Right Connections on Bimodules

Abstract

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.

arXiv comment16 pages, LaTeX, nofigures

References in corpus (4)

• Quantum group gauge theory on quantum spaces
• Gravity from Noncommutative Geometry
• LINEAR CONNECTIONS ON EXTENDED SPACE-TIME
• Metric On Quantum Spaes

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• Riemannian Geometry of Noncommutative Surfaces
• Gauge theories on quantum spaces
• On the Koszul formula in noncommutative geometry
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