Gelfand-Kirillov dimensions of simple modules over twisted group algebras $k \ast A$

Abstract

For the n-dimensional multiparameter quantum torus algebra Λ q over a field k defined by a multiplicatively antisymmetric matrix q = (q ij ) we show that in the case when the torsion-free rank of the subgroup of k × generated by the q ij is large enough there is a characteristic set of values (possibly with gaps) from 0 to n that can occur as the Gelfand-Kirillov dimension of simple modules. The special case when K. dim(Λ q ) = n − 1 and Λ q is simple studied in A. Gupta, GK-dimensions of simple modules over K[X ±1 , σ], Comm. Algebra, 41(7) (2013), 2593-2597 is considered without assuming simplicity and it is shown that a dichotomy still holds for the GK dimension of simple modules.