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\(\mathrm{GL}_{r,s}(n)\)-covariant differential calculi on the quantum \(n\)-space. (English) Zbl 1473.17037

In this paper, the authors develop differential calculi on a quantum \(n\)-space covariant with respect to the action of the quantum group \(GL_{r,s}(n)\), considering two cases of noncommutativity and commutativity of the coordinates of quantum \(n\)-space with the matrix entries of a matrix in the quantum group. They obtain the relations of differentials with the coordinates in terms of \(r/s\) when \(d^2=0\) and \(d^3=0\) and noncommutative parameters of the quantum \(n\)-space. They are also show that the ratio \(r/s\) has to be equal to square of one of the two primitive cubic roots of the unity when \(d^3=0\).

MSC:

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