The n-dimensional quantum torus O-q((F-x)(n)) is defined as the associative F-algebra generated by x(1), ... , x(n) together with their inverses satisfying the relations x(i)x(j) = q(ij)x(j)x(i), where q = (q(ij)). We show that the modules that are finitely generated over certain commutative sub-algebras B are B-torsion-free and have finite length. We determine the Gelfand-Kirillov dimensions of simple modules in the case when
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