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\(\varvec{GL_{r,s}(n)}\)-Covariant Differential Calculi on the Quantum n-Space

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Abstract

Covariant differential calculi on a quantum n-space with invariance under the action of \(GL_{r,s}(n)\) are constructed in the two cases of noncommutativity and commutativity of the coordinates with the matrix entries of the two-parametric quantum group. We show that the noncommutative parameters of the quantum n-space have to satisfy some relations in terms of s without specifying their exact amounts in the first case whilst they have to be equal to s in the second case. The commutation relations among differential forms of the coordinates for both differential calculi \(d^2=0\) and \(d^3=0\) are obtained in terms of r / s as well as noncommutative parameters of the quantum n-space. It is also shown that the ratio r / s has to be equal to square of one of the two primitive cubic roots of the unity for differential calculus \(d^3=0\) in both cases.

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Authors and Affiliations

  1. Department of Theoretical Physics and Astrophysics, Faculty of Physics, University of Tabriz, Tabriz, 51666-16471, Iran

    Hossein Fakhri & Samaneh Laheghi

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  1. Hossein Fakhri
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  2. Samaneh Laheghi
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Correspondence to Hossein Fakhri.

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Communicated by Michaela Vancliff

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Fakhri, H., Laheghi, S. \(\varvec{GL_{r,s}(n)}\)-Covariant Differential Calculi on the Quantum n-Space. Adv. Appl. Clifford Algebras 29, 52 (2019). https://doi.org/10.1007/s00006-019-0968-x

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