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

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.