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First order calculi with values in right-universal bimodules. (English) Zbl 0878.16017
Budzyński, Robert (ed.) et al., Quantum groups and quantum spaces. Lectures delivered during the minisemester, Warsaw, Poland, December 1, 1995. Warszawa: Polish Academy of Sciences, Inst. of Mathematics, Banach Cent. Publ. 40, 171-184 (1997).
As a starting point a concept is used in accordance to which differential calculi investigated recently in the context of quantum groups and noncommutative geometry (see, for example, Pittner, 1995) may be treated as derivations of an algebra with values in a bimodule (Bourbaki, 1989). It is shown that the differential calculi on a unital associative algebra with universal right bimodule (Cuntz, Quillen, 1995) may be considered as a generalization of previously studied constructions (Pusz, Woronowicz, 1989; Wess, Zumino, 1990). It is stated that in such a language the results under consideration may be described and handled in a more natural and simple way. As a by-product, here an intrinsic, coordinate-free and bases independent (i.e., covariant) generalization is obtained of the first order noncommutative differential calculi with partial derivatives (Borowiec, Kharchenko, Oziewicz, 1994; Borowiec, Kharchenko, 1995).
For the entire collection see [Zbl 0865.00041].
Reviewer: A.A.Bogush (Minsk)

MSC:
16W25 Derivations, actions of Lie algebras
46L85 Noncommutative topology
46L87 Noncommutative differential geometry
16D20 Bimodules in associative algebras
17B37 Quantum groups (quantized enveloping algebras) and related deformations
16U80 Generalizations of commutativity (associative rings and algebras)
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