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Covariant differential calculi on quantum spaces and on quantum groups. (English. Abridged French version) Zbl 0795.17019
The paper announces results concerning the classification of left covariant differential calculi on some quantum spaces (quantum hyperboloid, quantum plane, quantum complex plane) and of bicovariant differential calculi on some quantum groups \((ax+b\) group, quantum groups of type \(B,C,D)\). All these algebras are finitely generated. The results, being true under the main assumption that the differentials of the generators of the corresponding algebra form a basis of the left module of first order differential forms, may be summarized as follows:
For the quantum spaces, there are precisely two (quantum hyperboloid and quantum plane) or one (complex quantum plane) nonisomorphic left covariant differential calculi for “nonspecial” values of the deformation parameter. For the quantum groups of type \(B,C,D\), let \(N \geq 3\), where the classical analoga are groups of \(N\times N\)-matrices, and let the deformation parameter be not a root of unity. Then there exists up to isomorphism one bicovariant differential calculus. For the \(ax+b\) group, there are up to isomorphism two bicovariant differential calculi, if the deformation parameter is different from \(\pm 1\).
A detailed exposition of classification results of the same authors for the quantum groups of types \(A,B,C\) and \(D\) is being published in Commun. Math.Phys.

17B37 Quantum groups (quantized enveloping algebras) and related deformations
46L85 Noncommutative topology
46L87 Noncommutative differential geometry