Covariant Bimodules and Differential Calculi on Quantum Groups of Type B, C, D

Non-Commutative differential calculi on quantum groups were first studied by S. L. Woronowicz [W1], [W2]. His seminal paper [W2] provides a general framework for such calculi. Following the general ideas of A. Connes the basic objects in this approach are differential forms rather than vector fields. It seems that there is no functorial way to construct differential calculi for general quantum groups. In the meantime concrete examples of covariant differential calculi on quantum spaces have been constructed by several authors such as [CS], [J], [PW], [W1], [W2] and others, but only very few papers (cf. e.g. [PW]) are concerned with the classification of such calculi. In this work we present some results which classify all bicovariant bimodules and bicovariant differential calculi on quantum groups of type B, C and D. The crucial assumption is that the differentials of the coordinate functions form a basis for the left module of differential forms. The result on classification are Theorem 5 and 6. The latter may be rephrased by saying that there is up to isomorphism a unique N -dimensional bicovariant differential calculus on quantum groups of type Bn, Cn, Dn provided that the deformation parameter q is not a root of unity and that N ≥ 3, where N = 2n+ 1 for Bn and N = 2n for Cn and Dn . In case C1 (N = 2, Spq(1) = SLq2(2)) there are precisely two non-isomorphic 4-dimensional bicovariant differential calculi which correspond to the 4D± calculi on SUq(2) constructed by Woronowicz [W2]. Details of proofs and additional related results will appear elsewhere.