Representations of then-Dimensional Quantum Torus

Abstract

The n-dimensional quantum torus O q F × n is defined as the associative F -algebra generated by x 1x 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 dim stands for the Krull dimension. In this case, if M is a simplewhere 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.

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

“…We provide some examples illustrating our two theorems at the end of Section 4.1. This paper completes the story, so to speak, for the GK dimension of simple modules over the quantum torus Λ q in the case when K. dim(Λ q ) = n−1 initiated in [16] and continued in [17]. In [22] the corresponding problem for K. dim(Λ q ) = 1 was considered and answered.…”

“…The simple modules over the quantum torus in the case when it has Krull dimension one were studied in [22] and also in [5,9] (for a generic quantum torus). In [16] and [17] focus was shifted to the case where the Krull dimension is one less than the maximum possible, that is, n − 1. In [16] a dichotomy for the GK dimensions of simple Λ q -modules was established assuming that Λ q has Krull dimension n − 1 and is itself simple.…”

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

“…3.2. This paper completes the story, so to speak, for the GK dimension of simple modules over the quantum torus Λ q in the case when K. dim(Λ q ) = n − 1, initiated in [28] and continued in [29]. In [16], the corresponding problem for K. dim(Λ q ) = 1 was considered and answered.…”

“…As noted above, the simple modules over the quantum torus, in the case when it has the Krull dimension one, were studied in [16] and also in [5], [8] (for a generic quantum torus). In [28] and [29], the focus was shifted to the case where the Krull dimension is one less than the maximum possible one, that is, n − 1. In [28], a dichotomy for the GK dimensions of simple Λ q -modules was established, assuming that Λ q has the Krull dimension n − 1 and is itself simple.…”

“…There is thus a lack of generality in assuming that Λ q is simple. This was partly remedied in [29], where it was shown that, if Λ q has the Krull dimension n − 1 but is not necessarily a simple ring and if M is a simple Λ q -module, then…”