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Gelfand–Kirillov dimensions of simple modules over twisted group algebras Ashish Guptaa, Umamaheswaran Arunachalamb a Department of Mathematics, Ramakrishna Mission Vivekananda Educational and Research Institute (RKMVERI), India b Harish-Chandra Research Institute, India
Russian version:
Izvestiya Rossiiskoi Akademii Nauk. Seriya Matematicheskaya, 2022, Volume 86, Issue 4, DOI: https://doi.org/10.4213/im9182 
Bibliographic databases:
    IntroductionLet $k$ be a field and let $k^\times$ denote the group $k \setminus \{0\}$. Let $\mathfrak q$ be an $n \times n$ multiplicatively antisymmetric matrix with entries in $k^\times$. This means that the entries (the so-called multi-parameters) $q_{ij}$ of $\mathfrak q$ satisfy $q_{ii} = 1$ and $q_{ji} = q_{ij}^{-1}.$ The rank $n$ quantum torus $\Lambda_{\mathfrak q}$ is the associative algebra generated over the field $k$ by the variables $X_1,\dots, X_n$ together with their inverses subject to the relations
$$
\begin{equation}
X_i X_j = q_{ij}X_j X_i \quad \forall\, 1 \leqslant i, j \leqslant n.
\end{equation}
\tag{1}
$$
These algebras play an important role in non-commutative geometry [1] and have found applications in the representation theory of torsion-free nilpotent groups [2]. For example, if $H$ is a finitely generated and torsion-free nilpotent group of class two with centre $\zeta H$, then the central localization $kH(k \zeta H \setminus \{0\})^{-1}$ is an algebra of this type. While many different aspects including irreducible and projective modules [3]–[9], automorphisms and derivations [10]–[14], Krull and global dimensions [2], [15]–[17], etc., of these algebras have been studied in the recent past (a survey appears in [18]), the focus of the present article is the Gelfand–Kirillov dimensions of simple modules over the quantum tori. Throughout the paper, by a module we mean a right module. The Gelfand–Kirillov dimension is a fundamental invariant of finitely generated modules over affine algebras. Let $M$ be a finitely generated module over a $k$-algebra $\mathcal A$ equipped with a finite-dimensional generating subspace, say, $M_0$. Let $\{a_0, \dots, a_{m - 1}, a_m\ (= 1)\}$ be a finite set of generators (including $1$) for the algebra $\mathcal A$. The latter is filtered by the sequence of finite-dimensional subspaces
$$
\begin{equation*}
\mathcal A_0=k, \qquad \mathcal A_1=\sum_{i=0}^m k a_i, \qquad \mathcal A_l=\mathcal A_1^l\quad (l \geqslant 2).
\end{equation*}
\notag
$$
This increasing filtration of the algebra $\mathcal A$ gives rise to the increasing filtration $\{M_0 \mathcal A_l\}_{l \in \mathbb N}$ of the module $M$. The Gelfand–Kirillov dimension (GK dimension) of the module $M$ is a measure of the growth of $M$ and is defined as
$$
\begin{equation*}
\mathscr{GK}(M)=\lim \sup_{l \to \infty} \frac{\log (\dim_k(M_0 \mathcal A_l))}{\log l}.
\end{equation*}
\notag
$$
Although this definition seems to depend upon the choice of a generating subspace of $M$ (namely, $M_0$) and a set of algebra generators of $\mathcal A$ (namely, $\{a_0, \dots, a_{m-1}, sa_m\}$), the GK dimension actually turns out to be independent of such a choice. Though the definition is somewhat technical, the GK dimension of modules turns out to be a useful and important invariant. Moreover, as is well known, for a wide range of algebras including many quantum groups the GK dimension of finitely generated modules is an integer [19]. For modules over the algebras we are studying the GK dimension is related to another important invariant for modules, namely, the Krull dimension [16]. For further details on the GK dimension we refer the reader to the excellent references [20] and [21]. The question of GK dimension of simple modules over the algebras $\Lambda_{\mathfrak q}$ was first considered in Sec. 6 of [16], where it was shown that every simple module over a hereditary (of global dimension one) quantum torus $\Lambda_{\mathfrak q}$ of rank $n$ has the GK dimension $n-1$. The same problem remains open for other values of the global dimension of $\Lambda_{\mathfrak q}$. The same problem for some other classes of algebras was considered in [22]–[25]. For $D$-modules this question is particularly important [26], [27]. The evidence thus abounds in support of the fact that the determination of GK dimensions of simple modules is a substantial step in the study of various classes of algebras including quantum groups and rings of differential operators. The structure of $\Lambda_{\mathfrak q}$ and its modules can vary substantially depending on the matrix of multi-parameters $\mathfrak q$. For example, if $\mathfrak q$ has entries that generate a subgroup of $k^\times$ of maximal possible (torsion-free) rank, namely, $\frac{n(n -1)}{2}$, then the ring $\Lambda_{\mathfrak q}$ is a simple hereditary Noetherian domain [16], whereas, in the case when this same subgroup has rank zero, it is a module-finite algebra over its centre. We note that, in all these cases, the algebra $\Lambda_{\mathfrak q}$ has the same GK dimension. The situation thus calls for having an additional invariant which is convenient enough to work with and at the same time well-suited to our goal of studying the simple modules of quantum torus algebras. This invariant is provided by the Krull dimension of non-commutative rings which, in our case, coincides with another dimensional invariant, namely, the global dimension [16]. 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 $\Lambda_{\mathfrak q}$-modules was established, assuming that $\Lambda_{\mathfrak q}$ has the Krull dimension $n-1$ and is itself simple. In the current article we show that this dichotomy remains intact even when this simplicity assumption is lifted. Theorem 1. Let $\Lambda_{\mathfrak q}$ be an $n$-dimensional quantum torus algebra with the Krull dimension $n-1$. The following dichotomy holds for the GK dimension of a simple $\Lambda_{\mathfrak q}$-module $M$: where $\mathcal Z(\Lambda_{\mathfrak q})$ stands for the centre of $\Lambda_{\mathfrak q}$. Remark 1. Under the additional hypothesis that $\Lambda_{\mathfrak q}$ is a simple ring, it was shown in [28] that for a simple $\Lambda_{\mathfrak q}$-module $M$ either $\mathscr{GK}(M) = 1$ or $\mathscr{GK}(M) = n - 1$. We note that this simplicity hypothesis means that the centre $\mathcal Z(\Lambda_{\mathfrak q})$ of the ring $\Lambda_{\mathfrak q}$ is $k$ [16]. In general, it follows from Theorem A of [2] and Lemma 1.1 of [14] that a quantum torus $\Lambda_{\mathfrak q}$ of rank $n$ having the Krull dimension $n-1$ can have the centre isomorphic to an (ordinary) Laurent polynomial ring over $k$ in up to $n-2$ variables. There is thus a lack of generality in assuming that $\Lambda_{\mathfrak q}$ is simple. This was partly remedied in [29], where it was shown that, if $\Lambda_{\mathfrak q}$ has the Krull dimension $n-1$ but is not necessarily a simple ring and if $M$ is a simple $\Lambda_{\mathfrak q}$-module, then $\mathscr{GK}(M) = 1$ or $\mathscr{GK}(M) \geqslant \mathscr{GK}( \Lambda_{\mathfrak q}) - \mathscr{GK}(\mathcal Z(\Lambda_{\mathfrak q})) - 1$. In Theorem 1, we gave an exact determination of the GK dimension of all possible simple $\Lambda_{\mathfrak q}$-modules for a quantum torus $\Lambda_{\mathfrak q}$ with the Krull (or global) dimension $n -1$. Carrying out a study of simple modules over quantum polynomials, we often find it useful to impose some kind of an independence assumption on the multi-parameters $q_{ij}$. For example, in the consideration of simple modules in [5] and [8] it is assumed that the multi-parameters are in general position, that is, generate a subgroup of $k^\times$ of maximal rank. Evidently, we may present an $n$-dimensional quantum torus $\Lambda_{\mathfrak q}$ as a skew Laurent polynomial ring over the subring $\Lambda'$ generated by the variables $X_1^{\pm 1}, \dots, X_{n -1}^{\pm 1}$. Thus, $\Lambda_{\mathfrak q} = \Lambda'[X_n^{\pm 1}; \sigma]$, where $\sigma$ is a scalar automorphism of $\Lambda'$ given by $X_i \mapsto p_iX_i$ for $p_i \in k^\times$. In the following, we denote the set of GK dimensions of simple modules over a given quantum torus algebra $\Lambda_{\mathfrak q}$ by $\mathscr V(\Lambda_{\mathfrak q})$. Our next theorem expresses $\mathscr V(\Lambda_{\mathfrak q})$ in terms of $\mathscr V(\Lambda')$, assuming an independence condition for the multi-parameters (see Theorem 2). Definition 1. (i) The $\lambda$-group $\mathscr G(\Lambda_{\mathfrak q})$ of a quantum torus algebra $\Lambda_{\mathfrak q}$ is defined as the subgroup of $k^\times$ generated by the multi-parameters $q_{ij}$. (ii) Given a scalar automorphism $\sigma$ of $\Lambda_{\mathfrak q}$, we define $\mathscr H_\sigma$ as the subgroup of $k^\times$ generated by the scalars $p_i$ ($i = 1, \dots, n$). We can now state our second theorem. Theorem 2. Let $\Lambda_{\mathfrak{q}}$ be an $n$-dimensional quantum torus algebra; consider the skew-Laurent extension Here $\mathscr V(\Lambda_\mathfrak q) + 1$ stands for the set $\{u + 1 \mid u \in \mathscr V(\Lambda_\mathfrak q)\}$. Remark 2. Theorem 2 does not necessarily mean that for each value $d$ in the set We provide some examples illustrating our two theorems at the end of Sec. 3.2. This paper completes the story, so to speak, for the GK dimension of simple modules over the quantum torus $\Lambda_{\mathfrak q}$ in the case when $\mathrm{K}.\dim(\Lambda_{\mathfrak q}) = n - 1$, initiated in [28] and continued in [29]. In [16], the corresponding problem for $\mathrm{K}.\dim(\Lambda_{\mathfrak q}) = 1$ was considered and answered. Our work thus naturally leads to the following question. Question. Let $\Lambda_{\mathfrak q}$ be an $n$-dimensional quantum torus algebra with Krull dimension either $2$ or $n-2$. Does there exist a non-holonomic simple $\Lambda_{\mathfrak q}$-module with GK dimension $n-1$? If yes, then how can such a simple module be constructed? What are the possible values of the GK dimensions of simple $\Lambda_{\mathfrak q}$-modules? § 1. The $n$-dimensional quantum torusIn this section, we recall some known facts concerning the quantum torus algebras that we need in the development of our results to follow. 1.1. Twisted group algebra structureAs already noted, the algebras $\Lambda_{\mathfrak q}$ have the structure of a twisted group algebra $k \ast A$ of a free Abelian group of rank $n$ over $k$. We briefly recall this kind of a structure and refer the reader to [30] for further details. For a given group $G$ a $k$-algebra $R$ is said to be a twisted group algebra of $G$ over $k$ if $R$ contains, as $k$-basis, a copy $\overline{G}: = \{ \bar{g} \mid g \in G \} $ of $G$ such that the multiplication in $R$ satisfies the conditions
$$
\begin{equation}
\overline g_1 \overline g_2=\gamma(g_1, g_2)\overline{g_1g_2},
\end{equation}
\tag{2}
$$
where $\gamma\colon G \times G \rightarrow k^{\times}$. The associativity of the multiplication means that the twisting function $\gamma$ is a $2$-cocycle, that is, For a subgroup $H$ of $G$ the $k$-linear span of $\overline{H} := \{\bar {h} \mid h \in H \}$ is a subalgebra of $R$ which is a twisted group algebra of $H$ over $k$ with the defining cocycle equal to the restriction of $\gamma$ to $H \times H$. This subalgebra will be denoted by $k \ast H$. If $G$ is Abelian, it is known that the centre of $k \ast G$ is of the form $k \ast Z$ for a suitable subgroup $Z \leqslant G$ (for example, see Lemma 1.1 of [14]). 1.2. Commutative (sub-)twisted group algebras and the Krull dimensionIn the case of the quantum tori $\Lambda_q = k \ast A$, the subgroups $B$ for which the subalgebra $k \ast B$ is commutative play an important role. For example, the following fact was conjectured in [16] and shown in [31]. Theorem 3. Given a quantum torus algebra $k \ast A$, the supremum of the ranks of subgroups $B \leqslant A$ such that the corresponding (sub-)twisted group algebra $k \ast B$ is commutative is equal both to the Krull and the global dimensions of the algebra $k \ast A$. The following fact is a rewording of Theorem 3 of [31]. Proposition 1. Suppose that a quantum torus algebra $k \ast A$ has a finitely generated module with GK dimension $m$. Then $A$ has a subgroup $B$ with rank equal to $\operatorname{rk}(A)- m$ for which the corresponding (sub-)twisted group algebra $k \ast B$ is commutative. The following interesting corollary is a clear consequence of combining the last two results when recalling from [16] that the GK dimension of a finitely generated $k \ast A$-module is a non-negative integer. Corollary 1. The GK dimension of a finitely generated $k \ast A$-module $M$ satisfies the condition where $\mathrm{K.dim}(k \ast A)$ stands for the Krull dimension of the algebra $k \ast A$. 1.3. Ore subsets and the subgroups of finite index in $A$Using the fact that $A \cong \mathbb Z^n$ is an ordered group (with the lexicographic order), it is not difficult to show that $k \ast A$ is a domain and every unit in this ring has the form $\mu \bar a$ for $\mu \in k^{\times}$ and $a \in A$. Let $M$ be a finitely generated $k \ast A$-module. If $A_0$ is a subgroup of $A$ with finite index, then it is not difficult to see that $M$ is a finitely generated $k \ast A_0$-module. We recall that, in a ring $R$, an element $r \in R$ normalizes a subring $S$ if $rS = Sr$. The algebra $k \ast A$ is a finite normalizing extension of its subalgebra $k \ast A_0$, that is, it is generated over the latter by a finite number of normalizing elements, namely, the images in $k \ast A$ of a transversal $T$ of $A_0$ in $A$. As is well-known (see, for example, [16]), for a subgroup $B \leqslant A$ the set of non-zero elements of the (sub-)twisted group algebra $k \ast B$ is an Ore subset in $k \ast A$. 1.4. Crossed products resulting from Ore localizationA more general structure than a twisted group algebra is a crossed product $R \ast G$ of a group $G$ over a ring $R$. Here the ground ring $R$ need not be a field and the scalars in $R$ need not be central in $R \ast G$. As in the case of twisted group algebras, a copy $\bar G$ of $G$ contained in $R \ast G$ is an $R$-module basis, and the multiplication of the basis elements is defined exactly as in (2) above. However, as already remarked, the scalars in $R$ need not commute with the basis elements $\bar g$, but the relation $\bar g r= \sigma_g(r)\bar g$ holds for $r \in R$ and $\sigma_g \in \operatorname{Aut}$. We refer the interested reader to the text [32] for further details concerning crossed products. Crossed products are not only generalizations of twisted group algebras but also arise as suitable localizations of the former rings. A crossed product arises in our paper in this very latter form. For example, the Ore localization $(k \ast A)(k \ast B \setminus \{0\})^{-1}$ is a crossed product $D \ast A/B$, where $B \leqslant A$ and $D$ stands for the quotient division ring of the Noetherian domain $k \ast B$. § 2. The GK dimension of finitely generated $\Lambda_{\mathfrak q}$-modulesThe GK dimension is a particularly well-behaved dimension for the finitely generated modules over the algebras we are studying. One reason for this is that the Hilbert–Samuel machinery, which works for almost commutative algebras, can be adapted for the algebras $\Lambda_{\mathfrak q}$ as well (Sec. 5 of [16]). For example, we have the following assertion. In [31], the finitely generated modules over crossed products of a free Abelian group of finite rank over a division ring $D$ were studied with group-theoretic applications in mind. A dimension for finitely generated modules which was shown to coincide with the Gelfand–Kirillov dimension (measured relative to $D$) was introduced and studied. We aim to employ this dimension in our investigation of simple modules over the algebras we are considering here which are special cases of the aforementioned crossed products. We state below its definition and some key properties, which were established in [31]. This dimension is used in conjunction with an appropriate notion of a critical module, which we discuss below. Definition 2 (see [31]). Let $M$ be a finitely generated $\Lambda_{\mathfrak{q}}$-module. The dimension $\dim M$ of $M$ is the maximum of $r$, where $0 \leqslant r \leqslant n$, such that, for some subset $\mathcal I := \{i_1, i_2, \dots, i_r\}$ of the indexing set $\{1, \dots, n\}$, $M$ is not a torsion module as a $\Lambda_{\mathfrak q, \mathcal I}$-module, where $\Lambda_{\mathfrak q, \mathcal I}$ stands for the subalgebra of $\Lambda_{\mathfrak q}$ generated by the variables $X_i$ for $i \in \mathcal I$ and their inverses. Remark 3. In [31] it was shown that the dimension $\dim M$ of $M$, in the sense of the last definition, coincides with the GK dimension of $M$. Moreover, the notion of dimension in Definition 2 was introduced in [31] more generally for finitely generated modules over crossed products of a free Abelian group over a division ring. These crossed products include the Ore localizations of the algebras $\Lambda_{\mathfrak{q}}$ discussed in Sec. 1.4. The following facts from [31], which we state for the algebras $\Lambda_{\mathfrak q}$, were shown for more general crossed products. Lemma 1. Let $M$ be a finitely generated $\Lambda_{\mathfrak{q}}$-module with GK dimension $d$ and let $\Lambda_1$ be the subalgebra of $\Lambda_{\mathfrak{q}}$ generated by the variables $\{ X_{i_i}, X_{i_2}, \dots, X_{i_d} \}$ and their inverses. Then a $\Lambda_1$-submodule which is free of infinite rank cannot be embedded in $M$. Definition 3. A non-zero $\Lambda_{\mathfrak q}$-module $N$ is said to be critical if for every non-zero submodule $N$ of $M$ we have Proposition 3 (see Proposition 2.5 of [31]). Every non-zero $\Lambda_{\mathfrak{q}}$-module contains a finitely generated critical submodule. § 3. The dichotomy result3.1. Proof of Theorem 1As in § 1, we may write $\Lambda_\mathfrak q$ as a twisted group algebra $\Lambda_\mathfrak q : = k \ast A$ for a free Abelian group $A$ with rank $n$. Moreover, let $\bar {a}$ stand for the image of $a \in A$ in $\Lambda_\mathfrak q$. It is not difficult to see that $\mathcal Z(\Lambda_{\mathfrak q})$ has the form $k \ast Z$ for a suitable subgroup $Z$ of $A$ (see, for example, Lemma 1.1 of [14]). Since (see, for example, [16]), the latter alternative in the assertion of the theorem then reads
$$
\begin{equation*}
\mathscr{GK}(M)=\operatorname{rk}(A)-\operatorname{rk}(Z)-1.
\end{equation*}
\notag
$$
Let $P$ be the annihilator of $M$ in $k \ast Z$. The action of $k \ast Z$ on $M$ gives an embedding
$$
\begin{equation*}
k \ast Z)/P \hookrightarrow \operatorname{End}_{\Lambda_{\mathfrak q}}(M).
\end{equation*}
\notag
$$
Since $\operatorname{End}_{\Lambda_{\mathfrak q}}(M)$ is a division ring, it follows that $P$ is a prime ideal of $k \ast Z$. As is well known (see, for example, Proposition 9.4.21 of [16]), the quantum torus algebra $k \ast A$ satisfies the Nullstellensatz and, in particular, $\operatorname{End}_{\Lambda_{\mathfrak q}}(M)$ is algebraic over $k$. Hence, so is $(k \ast Z)/P$. Since a commutative affine algebraic domain is a field [33], it follows that $P$ is a maximal ideal of $k \ast Z$. We write $K = (k \ast Z)/P$ and $Q = P\Lambda_{\mathfrak q}$. Clearly, $M$ is a simple $\Lambda_{\mathfrak q}/Q$-module. By Chapter 1, Lemmas 1.3 and 1.4 of [32], the $k$-algebra $\Lambda_{\mathfrak q}/Q$ is a twisted group algebra $K \ast A/Z$ of $A/Z$ over $K$ with a transversal $T$ for $Z$ in $A$ yielding a $K$-basis as the set $\{ \bar t + Q \mid t \in T\}$. Moreover, the elements $\zeta + Q$, where $\zeta \in k \ast Z$, constitute a copy of $K$ in $\Lambda_{\mathfrak q}/Q$. We note that the group-theoretic commutator $[\bar{t_1} + Q, \bar{t_2} + Q]$ with values in the group of units of $\Lambda_{\mathfrak q}/Q$ satisfies the following equation for any $t_1, t_2 \in T$:
$$
\begin{equation}
[\overline{t_1}+Q, \overline{t_2}+Q]=[\overline{t_1},\overline{t_2}\,] +Q \in k^\times+Q.
\end{equation}
\tag{3}
$$
Now, in view of Proposition 5.1(c) of [20], we have
$$
\begin{equation*}
\mathscr{GK}_{k \ast A}(M) = \mathscr{GK}_{K \ast A/Z}(M),
\end{equation*}
\notag
$$
where the notation $\mathscr{GK}_R(M)$ for a suitable ring $R$ stands for the GK dimension of $M$ viewed as a (right) $R$-module. In the last equation, both in left- and right-hand sides, the GK dimension is measured relative to $k$. Since $K$ is finitely generated and algebraic over $k$, it follows that $[K : k] < \infty$ and, in view of Lemma 2(ii) of [34], it suffices to determine the possible values of the GK dimension of the simple $K \ast A/Z$-module $M$ measured relative to $K$. Our main point in passing to the algebra $K \ast A/Z$ is that, as a $K$-algebra, it is central, that is, has the centre $K$. Indeed, as we already saw in § 1, the centre of $K \ast A/Z$ is of the form $K \ast Y$ for a subgroup $Y$ of $A/Z$. If the image $\bar t + Q$ of some coset $tZ \in Y$ centralizes all elements of $K \ast A/Z$, then, using (3), we have
$$
\begin{equation*}
[\overline t, \overline {t_1}\,]+Q=1+Q \quad \forall\, t_1 \in T.
\end{equation*}
\notag
$$
Since $[\overline t, \overline{t_1}\,] \in k^\times$, it follows that $[\bar t, \bar{t_1}] = 1$. Thus, if $\bar t + Q$ lies in the centre of $K \ast A/Z$, then $\bar t$ must be in the centre $k \ast Z$ of $k \ast A$. Since $t \in T$, this is possible only if $t = 1$. As already noted in § 1, by the theorem of Brookes [2], the dimension of $k \ast A$ is equal to the supremum of the ranks of the subgroups $B \leqslant A$ such that the subalgebra $k \ast B$ is commutative. In the present situation, this means the existence of a subgroup $B$ of $A$ with rank $n-1$ such that $k \ast B$ is commutative. When passing to $K \ast A/Z$, although the centre becomes equal to the base field, a small difficulty arises, namely, $A/Z$ need not be torsion-free. To overcome this, we may replace $A$ by a subgroup $A_0$ of finite index such that $A_0/Z$ is torsion-free. Then $B_0:=A_0 \cap B$ is a subgroup of $A_0$ with rank $n-1$ and, clearly, $k \ast B_0$ is a commutative subalgebra of $k \ast A_0$. Evidently, $k \ast B_0Z$ is commutative, and therefore $\operatorname{rk}(B_0Z) = \operatorname{rk}(B_0)$ by the preceding paragraph. Replacing $B_0$ by $B_0Z$ if necessary, we may assume that $B_0 \geqslant Z$. In view of (3), the subalgebra $K \ast B_0/Z$ of $K \ast A_0/Z$ is commutative. We obviously have
$$
\begin{equation*}
\operatorname{rk}(B_0/Z)=n-1-\operatorname{rk}(Z)=\operatorname{rk}(A_0/Z)-1= \operatorname{rk}(A/Z)-1.
\end{equation*}
\notag
$$
The last equation means that $K \ast A_0/Z$ is an $n-\operatorname{rk}(Z)-1$-dimensional quantum torus over $K$ with (Krull or global) dimension $n-\operatorname{rk}(Z)-1$. As a module over the subalgebra $K \ast A_0/Z$, the simple $K \ast A/Z$-module $M$ need not remain simple. However, since $K \ast A/Z$ is a finite normalizing extension of $K \ast A_0/Z$ (see § 1), it follows that the $K \ast A/Z$-module $M$ decomposes as a finite direct sum of simple $K \ast A_0/Z$-modules (see, for example, Exercise 15A.3 in [35]). We thus have as $K \ast A_0/Z$-modules. By Lemma 2.7 of [31],
$$
\begin{equation*}
\mathscr{GK}_{K \ast A/Z}(M)=\mathscr{GK}_{K \ast A_0/Z}(M).
\end{equation*}
\notag
$$
Moreover, the GK dimension of a finite direct sum of modules is the maximum of the GK dimensions of the summands (Proposition 5.1 of [20]). In view of these remarks, to establish the theorem, it suffices to show that, if $F$ is a field and $F \ast \mathbb Z^r$ is a twisted group algebra with centre $F$ and the dimension equal to $r-1$, then the following dichotomy holds for any simple $F \ast \mathbb Z^r$-module $N$:
$$
\begin{equation}
\mathscr{GK}(N)=1 \quad \text{or} \quad \mathscr{GK}(N)=r-1.
\end{equation}
\tag{4}
$$
But this is precisely the content of Theorem 2.1 of [28]. 3.2. Proof of Theorem 2We write $\Lambda^\ast$ for ${\Lambda^\ast}_{\mathfrak{q}, \sigma}$ and $\Lambda$ for $\Lambda_{\mathfrak{q}}$. Take into account Proposition 3; let $N$ be a finitely generated critical $\Lambda$-submodule of $M.$ Consider the $\Lambda^\ast$-submodule $N'$ of $M$ generated by $N$:
$$
\begin{equation}
N^\prime :=N\Lambda^\ast=\sum_{i \in \mathbb{Z}} N Y^i.
\end{equation}
\tag{5}
$$
Since $N$ is assumed to be critical, it follows that $N \ne 0$ and $N^\prime = M$. If the sum in (5) is direct, then $N \Lambda^\ast \cong N \displaystyle\otimes_{\Lambda} \Lambda^\ast$, and Lemma 2.4 of [31] gives Moreover, since the (left) $\Lambda$-module $\Lambda^\ast$ is free, it follows that $\Lambda^\ast$ is faithfully flat, and this implies that $N$ must be a simple $\Lambda$-module. Noting equation (6), we thus obtain
$$
\begin{equation}
\mathscr{GK}(M) \in \mathscr V (\Lambda_\mathfrak q)+1.
\end{equation}
\tag{7}
$$
Clearly the assertion of the theorem holds true in this case. We are thus left with the possibility where the sum $\sum_{i \in \mathbb{Z}} NY^i$ fails to be direct. We know from Lemma 2.4 of [31] that, in this case, recalling that the dimension being referred to in this lemma coincides with the GK dimension measured relative to the ground field $k$. Let $d$ denote the common value in the last equation. By Definition 2 and the succeeding remark, there is a subset $I\,{=}\, \{i_1, \dots, i_d\}$ of the indexing set $\{1, \dots, n \}$ such that the module $N$ (and therefore $M$) is not $S: = \Lambda_{\mathfrak q}(I)\setminus \{0\}$-torsion, where $\Lambda_{\mathfrak q}(I)$ stands for the subalgebra of $\Lambda$ generated by the indeterminates $X_i^{\pm 1}$ for $i \in I$. Consider the Ore localization $\Lambda^\ast S^{-1}$. By the definition of $S$ we know that $M$ is not $S$-torsion, and hence the corresponding localization $MS^{-1}$ is non-zero. We note that the ring $\Lambda^\ast S^{-1}$ contains the quotient division ring $\mathscr D:=\Lambda_{\mathfrak q}(I)S^{-1}$. Using Lemma 1, we can readily see that $MS^{-1}$ is a finite-dimensional $\mathscr D$-space. We write $s=\dim_{\mathscr D} M S^{-1}$. We write and let $\mathscr G(I)$ denote the following subgroup of $\mathscr G(\Lambda_\mathfrak q)$:
$$
\begin{equation*}
\mathscr G(I) := \langle q_{kl} \mid k, l \in I \rangle.
\end{equation*}
\notag
$$
Next for every $j \in J$ we define
$$
\begin{equation*}
\mathscr G(I, j) := \langle q_{kj} \mid k \in I \rangle.
\end{equation*}
\notag
$$
We also need to refer to the subgroup $\mathscr H( I)$ of $\mathscr H_\sigma$ defined as follows: As in the hypothesis of the theorem, This is precisely the situation of Sec. 3.9 of [16] (where the “unlocalized” generators are the generators indexed by $J$) and, exactly as in that section, the following dependence relations must hold:
$$
\begin{equation}
p_j^s \in \langle \mathscr G(I), \mathscr G( I, j), \mathscr H( I) \rangle \quad \forall\, j \in J.
\end{equation}
\tag{9}
$$
By the assumption of the theorem, we have $\mathscr G(\Lambda_{\mathfrak q}) \cap \mathscr H_\sigma = 1$, and therefore $p_j^s \in \mathscr H(I)$. This means that
$$
\begin{equation*}
\operatorname{rk}(\mathscr H_\sigma) \leqslant |I|=d=\mathscr{GK}(M).
\end{equation*}
\notag
$$
It remains to show that $\operatorname{rk}(M)<n+1$. To this end, we suppose that $\operatorname{rk}(M)=n+1$. Since $\Lambda^\ast$ is a twisted group algebra, it follows from Definition 2 and Remark 3 that $M$ embeds a copy of the right regular module $\Lambda^\ast$. Since $M$ is simple, it follows that $M \cong \Lambda^\ast$. This means that $\Lambda^\ast$ is a division ring, which is clearly not true. Our proof is now complete. An example illustrating the theorem is in place. Example 1. Let $F$ be a field and let $\mathfrak q \in \mathrm{M}_n(F)$ be a multiplicatively antisymmetric matrix given by The authors are very grateful and indebted to the referee for kind suggestions which helped to improve the manuscript considerably. 
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