Levi-Civita Connections on the Quantum Groups SL_{q}(N), O_{q}(N) and Sp_{q}(N)

Abstract

For bicovariant differential calculi on quantum groups various notions on connections and metrics (bicovariant connections, invariant metrics, the compatibility of a connection with a metric, Levi-Civita connections) are introduced and studied. It is proved that for the bicovariant differential calculi on $SL_{q}(N)$, $O_{q}(N)$ and $Sp_{q}(N)$ from the classification of Schmuedgen, K. and Schueler, A. there exist unique Levi-Civita connections.