REPRESENTATIONS OF THE n-DIMENSIONAL QUANTUM TORUS

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 K.dim(O-q((F-x)(n))) = n - 1, where K.dim stands for the Krull dimension. In this case, if M is a simple O-q((F-x)(n))-module, then GK-dim(M) = 1 or GK-dim(M) >= GK-dim(O-q((F-x)(n)) - GK-dim(Z(O-q((F-x)(n)))) - 1, where Z(C) stands for the center of an algebra C. We also show that there always exists a simple F * A-module satisfying the above inequality.