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\title{Azumaya locus over certain quantum symplectic spaces II}
\alttitle{Lieu d'Azumaya sur certains espaces symplectiques quantiques II}

\author{\firstname{Snehashis} \lastname{Mukherjee}}
\address{Stat-Math Unit, Indian Statistical Institute, Bangalore Center, 8th Mile, Mysore Rd, RVCE Post, Bengaluru, Karnataka 560059, India}
\email{tutunsnehashis@gmail.com}

\subjclass{16D60, 16D70, 16S85}

\keywords{\kwd{Quantum Heisenberg algebra} \kwd{polynomial identity algebra} \kwd{Smith normal form} \kwd{eigenvalue analysis} \kwd{simple modules} \kwd{Azumaya locus}}
\altkeywords{\kwd{AlgÃ¨bre de Heisenberg quantique} \kwd{algÃ¨bre Ã&nbsp; identitÃ© polynomiale} \kwd{forme normale de Smith} \kwd{analyse des valeurs propres} \kwd{modules simples} \kwd{lieu d'Azumaya}}

\thanks{The author is grateful to the Indian Statistical Institute, Bangalore Centre for financial support to carry out this research}

\begin{abstract}
This article undertakes an exploration of a particular variant of multiparameter quantum symplectic algebras, focusing specifically on the quantum Heisenberg algebra at the roots of unity. In this context, the algebra undergoes a transformation into Polynomial Identity algebra, where the dimensions of the simple modules are restricted by their PI degree. We conduct an extensive examination of all possible maximal dimensional simple modules associated with this algebra. Additionally, we present a condition that is both necessary and sufficient for the attainment of maximal dimensional simple modules, thereby facilitating the classification of its Azumaya locus.
\end{abstract}

\begin{altabstract}
Cet article explore une variante particuliÃ¨re des algÃ¨bres symplectiques quantiques multiparamÃ©triques, en se concentrant spÃ©cifiquement sur l'algÃ¨bre de Heisenberg quantique aux racines de l'unitÃ©. Dans ce contexte, l'algÃ¨bre subit une transformation en algÃ¨bre Ã&nbsp; identitÃ© polynomiale, oÃ¹ les dimensions des modules simples sont restreintes par leur degrÃ© PI. Nous effectuons un examen approfondi de tous les modules simples de dimension maximale possibles associÃ©s Ã&nbsp; cette algÃ¨bre. En outre, nous prÃ©sentons une condition qui est Ã&nbsp; la fois nÃ©cessaire et suffisante pour l'obtention de modules simples de dimension maximale, facilitant ainsi la classification de son lieu d'Azumaya.
\end{altabstract}

\begin{document}

\maketitle

\section{Introduction}

Unless otherwise stated throughout the paper a module means a right module, $q$ is a primitive $l$-th root of unity, $l$ being odd and $\mathbb{K}$ is an algebraically closed field.

A fundamental invariant of a noncommutative prime Polynomial Identity algebra $R$ is its Azumaya locus $\mathcal{AL}(R) \mk\subset\mk \mspect Z(R)$, the dense open subset of $\mspect Z(R)$ which parametrizes the maximal dimensional simple $R$-modules. More precisely, $\mathfrak{m} \in \mathcal{AL}(R)$ if and only if $\mathfrak{m}R$ is the annihilator in $R$ of a simple $R$-module V with $\dime_\mathbb{K}V = n$, where $n$ is the PI-degree of $ R$, the maximal $\mathbb{K}$-dimension of simple $R$-modules i.e.
\[
\mathcal{AL}(R)\coloneqq\braces[\big]{\mathfrak{m}\in \mspect Z(R):\ \exists V\ \text{simple},\ V\mathfrak{m}=0,\ \dime_{\mathbb{K}}V=n }.
\]
Also $\mathfrak{m}\in \mathcal{AL}(R) \iff R/\mathfrak{m}R\cong M_n(\mathbb{K}) \iff R/\mathfrak{m}R$ semisimple. Also if $\mathfrak{m} \in \mspect Z(R)\setminus \mathcal{AL}(R)$ we have $\dime_{\mathbb{K}}(R/\mathfrak{m}R) \geq n^2$.

In 2001, Horton introduced the multiparameter version of quantum symplectic space $K_{n,\Gamma}^{P,Q}$ in~\cite{aj} where $\Gamma \coloneqq (\gamma_{ij})$ is an $n \times n$ multiplicatively antisymmetric matrix over $\mathbb{K}$, that is, $ \gamma_{ii}=1$ and $\gamma_{ij}\gamma_{ji}=1$ for all $1 \leq i,j\leq n$ and let $ P\eqqcolon (p_1,\dots, p_n) $ and $ Q\coloneqq(q_1,\dots,q_n)$ be $n$-tuples of elements of $\mathbb{K}\setminus\{0\}$ such that $ p_i \neq q_i $ for each $ i \in \{1,\dots,n\} $. These algebras encompass the graded quantized Weyl algebras, the quantized Heisenberg space, and a class of algebras introduced by Oh. Natural tori are known to act as automorphisms on these algebras. The research of Brown, Goodearl, and Lenagan has established that understanding the prime and primitive ideals of the algebras hinges on identifying the prime ideals that remain invariant under these automorphisms. Horton has successfully identified these invariant prime ideals, which not only builds upon the earlier findings of Oh in~\cite{oo,ooo}, but also extends the work of GÃ³mez-Torrecillas, El Kaoutit, and Benyakoub in~\cite{torr,torr2} using distinct methodologies. Moreover, in the final section, she demonstrates how this achievement leads to a comprehensive determination of all primitive ideals.

Let $\mathbb{K}$ be a field and let $\mathbb{K}^*$ denote the group $\mathbb{K}\setminus\{0\}$. Let $q \in \mathbb{K}^*$ with $q^2 \neq 1$. Setting $ p_i = q^2$ and $q_i = 1$ with $\gamma_{i,j} = q$ whenever $i &lt; j$ in $K_{n,\Gamma}^{P,Q}$ gives rise to the coordinate ring of quantum Heisenberg algebra $F_q(n)$. The coordinate ring $F_q(n)$ of the quantum Heisenberg algebra is the algebra generated over the field $\mathbb{K}$ by the variables $x_1,\dots,x_n,y_1, \dots, y_n$ subject to the relations
\[
\begin{aligned}
&amp; x_ix_j=q^{-1}x_jx_i						&amp;&amp; \forall i&lt;j,\\
&amp; y_iy_j=qy_jy_i 							&amp;&amp; \forall i&lt;j,\\
&amp; x_iy_j=qy_jx_i 							&amp;&amp; \forall i\neq j,\\
&amp; x_iy_i=y_ix_i+\sum_{l&lt;i}(1-q^{2})y_lx_l 	&amp;&amp; \forall i.
\end{aligned}
\]
The algebra $F_q(n)$ is an affine Noetherian domain. The family of ordered monomials
\begin{equation*}\label{kbasis}
\braces[\big]{x_1^{a_1}y_1^{b_1}\cdots x_n^{a_n}y_n^{b_n} \ : \ a_i,b_i\in \mathbb{Z}_{\geq 0}}
\end{equation*}
is a $\mathbb{K}$-basis of $F_q(n)$~\cite[Proposition~2.5]{aj}. Moreover, $F_q(n)$ is an iterated skew polynomial ring.

The quantum Heisenberg space was first introduced by Faddeev, Reshetikhin, and Takhtajan in~\cite{fa} and Jacobsen and Zhang studied $F_q(n)$ in~\cite{jaz} where $ \mathbb{K}=\mathbb{C}$ and $q$ is a root of unity. In~\cite{jaz}, they considered generators $z_0,\dots,z_{n-1},z_0^*,\dots,z_{n-1}^*$ for $F_q(n)$. Setting $z_i = y_{n-i}$ and $z_i^*=x_{n-i}$ for each $i$ gives the equivalent algebra above. In their article, they delved into the algebraic properties, including the center and degree, as well as the representation theory of $F_q(N)$ under the condition that $q$ is a primitive root of unity. It's worth noting that this algebra fits neatly within the broader framework outlined by Procesi and De Concini~\cite{di}. This framework not only provides a means to compute the center and degree but also unveils a Poisson structure that is potentially linked to the representation theory via the De Concini--Kac--Procesi conjecture~\cite{di2}. They had successfully validated the validity of this conjecture within the context of their investigation.

The algebra $F_q(n)$ shares a close relationship with the high-dimensional quantum Heisenberg algebra~\cite{koro}, as well as with the quantum symplectic space $\mathcal{O}_q(\mathfrak{sp}\mathbb{K}^{2n})$. Given this connection, it is reasonable to anticipate that many characteristics of these algebras will manifest within $F_q(n)$, and indeed they do. The present author has classified the Azumaya locus of quantum symplectic space $\mathcal{O}_q(\mathfrak{sp}\mathbb{K}^{2n})$ assuming $q$ is an odd root of unity~\cite{sn}.

It is noteworthy that recent investigations by Brown and Yakimov~\cite{bry} have provided valuable insights into the correlation between the Azumaya locus $\mathcal{AL}(R)$ and the discriminant ideal $D_{l}$ of a prime PI algebra $R$. Their main theorem~\cite[Main Theorem]{bry} establishes, under mild assumptions, the alignment of the zero set of the discriminant ideal $D_l\parens[\big]{Z/Z(R),\tr}$ of a prime affine polynomial identity (PI) algebra $R$ with the zero set of the modified discriminant ideal $MD_l\parens[\big]{Z/Z(R),\tr}$ of $R$. They offer an explicit description of this set in terms of the dimensions of the irreducible representations of $R$.

Additionally, Brown and Yakimov prove that when $l$ is the square of the PI-degree of $R$, this zero set precisely complements the Azumaya locus of $R$. They leverage this description to classify the Azumaya loci of the multiparameter quantized Weyl algebras at roots of unity.

The purpose of this paper is to give a complete description of the Azumaya loci of the quantum Heisenberg algebra at roots of unity. We first give a necessary and sufficient condition for obtaining maximal dimensional simple modules and then describe the Azumaya loci. Since the Azumaya locus corresponds to maximal dimensional simple modules, in this article we will focus on constructing maximal dimensional simple modules over this algebra for the non generic case. For an infinite-dimensional non-commutative algebra, the classification of its simple modules poses a formidable challenge in general. It is noteworthy that, despite the significance of studying the representation theory of quantum algebras, no comprehensive method has been established to date.

The existing examples of successfully classified algebras primarily revolve around generalized Weyl algebras, as seen in works such as~\cite{ba1,ba2,ba3,ba5,ba6,ba7,ba8,ba9}. Additionally, Ore extensions with Dedekind rings as coefficient rings have been explored in~\cite{ba4}. The classification of finite-dimensional simple $R$-modules for a specific class of iterated skew polynomial rings, denoted as $R=A[y;\alpha][x;\alpha^{-1},\delta]$, where $A$ is an affine domain over an algebraically closed field, was achieved by Jordan in~\cite{dj}. In this article, we present a novel approach to constructing simple modules over $F_q(n)$.

\subsection{Some Important Results:} Let us define for each $i$, $1 \leq i \leq n$, $\Omega_i\coloneqq\sum_{k \leq i}(1-q^{2})y_kx_k$. These elements are going to play a crucial role throughout the article.

\begin{rema}[{\cite[Lemma~2.1]{aj}}]
Note that for all $1 \leq i \leq n$, $\Omega_i$ is a normal element of $F_q(n)$. More precisely, we have:
\begin{enumerate}
\item For all $i$, $j$ with $ 1 \leq i &lt; j \leq n$, $\Omega_ix_j=q^{-2}x_j\Omega_i$ and $\Omega_iy_j=q^{2}y_j\Omega_i$.
\item For all $i$, $1 \leq j \leq i \leq n$, $\Omega_ix_j=x_j\Omega_i$ and $\Omega_iy_j=y_j\Omega_i$.
\item For all $i$, $j$ with $1 \leq i,j \leq n$, $\Omega_i\Omega_j=\Omega_j\Omega_i$.
\end{enumerate}
\end{rema}

The following results are used in later parts of the article.
\begin{lemm}
Direct computations yield the following results for $F_q(n)$:
\begin{enumerate}
\item \label{lemm_2_1} $\Omega_i=x_iy_i-q^2y_ix_i$ for $1 \leq i \leq n$.
\item \label{lemm_2_2} $x_iy_i=y_ix_i+\Omega_{i-1}$ for $1 \leq i \leq n$.
\item \label{lemm_2_3} $x_i^ky_i=y_ix_i^k+\frac{1-q^{2k}}{1-q^2}\Omega_{i-1}x_i^{k-1}$ holds for $k \geq 1$ and for each $i$ with $1 \leq i \leq n$.
\item \label{lemm_2_4} $y_i^kx_i=x_iy_i^k-\frac{1-q^{-2k}}{1-q^{-2}}\Omega_{i-1}y_i^{k-1}$ holds for $k \geq 1$ and for each $i$ with $1 \leq i \leq n$.
\end{enumerate}
\end{lemm}

\begin{proof}
\eqref{lemm_2_1}--\eqref{lemm_2_2} follows from definition of the algebra. The identities~\eqref{lemm_2_3} and~\eqref{lemm_2_4} can be proved by induction on $k$ and using the relation~\eqref{lemm_2_2}.
\end{proof}

Let $A$ be an algebra and $M$ be a right $A$-module and $S\subset A$ be a right Ore set. The submodule
\[
\tors_{S}(M)\coloneqq\braces[\big]{m\in M : ms=0 \ \ \text{for some $s\in S$}}
\]
is called the $S$-torsion submodule of $M$. The module $M$ is said to be $S$-torsion if tor$_{S}(M)=M$ and $S$-torsion-free if tor$_{S}(M)=0$. If the Ore set $S$ is generated by $x\in A$, we simply say that the $S$-torsion/torsion-free module $M$ is $x$-torsion/torsion-free.

A non zero element $x$ of an algebra ${A}$ is called a normal element if $x{A}={A}x$. So for each $i=1,\dots,n$, $\Omega_i$, $x_1$ and $y_1$ are normal in $F_q(n)$. Clearly if $x$ is a normal element of $A$, then the set $\braces[\big]{x^i \ : \ i\geq 0}$ is an Ore set generated by $x$. The next lemma is obvious.

\begin{lemm}\label{itn}
Suppose that $A$ is an algebra, $x$ is a normal element of $A$ and $M$ is a simple $A$-module. Then either $Mx=0$ (if $M$ is $x$-torsion) or the map $x_{M} \colon M \to M$, $m \mapsto mx$ is a bijection (if $M$ is $x$-torsion-free).
\end{lemm}

The above lemma says that the action of a normal element on a simple module is either trivial or invertible.

Jacobsen and Zhang's computation has yielded the center of $F_q(n)$, producing the following outcome.

\begin{theo}[{\cite[Proposition~5.5]{jaz}}]\label{cen}
The center of $F_q(n)$ is generated by the elements $x_1^ay_1^b$ where $a+b=l$, $x_i^l$, $y_i^l$ for $i=2,\dots,n$.
\end{theo}

Also the following result will be useful.

\begin{lemm}[{\cite[Lemma~3.9]{jaz}}]\label{vimp}
For each $i=1,\dots,n$, we have $\Omega_i^l=(1-q^{2})^l\sum_{k \leq i}y_k^lx_k^l$.
\end{lemm}

The arrangement of the paper is as follows: Section~\ref{section_2} provides an introduction to Polynomial Identity algebras, with a focus on establishing the main properties. Theorem~\ref{finite} asserts that $F_q(n)$ qualifies as a prime PI algebra, accompanied by a discussion on Hanyal's findings concerning its PI degree. Section~\ref{section_3} provides a comprehensive exposition on the explicit structure of all potential maximal dimensional simple modules over $F_q(n)$. The key finding in this section is encapsulated in the following theorem:
\begin{theo}[Theorem~\ref{main1}] %%%%% A
Let $N$ be a\/ $\Omega_2,\dots,\Omega_{n-1}$ torsion-free simple $F_q(n)$-module. Then $N$ is maximal dimensional simple $F_q(n)$-module.
\end{theo}
Moving on to Section~\ref{section_4}, the PI degree computation of the prime factor of $F_q(n)$ is presented. Notably, we present the subsequent result:
\begin{theo}[Theorem~\ref{main2}] %%%%%% B
Under the roots of unity assumption for any $p$ with $2 \leq p \leq n-1$, the factor algebra $F_q(n)/\angles{\Omega_p}$ is a prime PI ring and
\[
\pideg F_q(n)/\angles{\Omega_p} = l^{n-2}.
\]
\end{theo}
Additionally, we establish that the set of maximal dimensional simple modules introduced in Section~\ref{section_3} forms the entirety of such modules. In particular we prove:
\begin{theo}[Theorem~\ref{main}] %%%%% C
$N$ is a maximal dimensional simple $F_q(n)$-module if and only if $N$ is $\Omega_2,\dots,\Omega_{n-1}$ torsion-free simple $F_q(n)$-module.
\end{theo}
Leveraging Theorem~\ref{cen} and Lemma~\ref{vimp}, along with the necessary and sufficient condition derived for maximal dimensional simple modules in Section~\ref{section_4}, Section~\ref{section_5} culminates in determining the Azumaya loci of quantum Heisenberg algebra.

\section{Preliminaries}\label{section_2}

In this section, we recall relevant information about the Polynomial Identity algebra and we prove that $F_q(n)$ is a PI algebra which enables us to comment on the $\mathbb{K}$-dimension of simple modules over $F_q(n)$ at roots of unity.

A ring $R$ is said to be a Polynomial Identity (PI) ring if $R$ satisfies some monic polynomial $f\in \mathbb{Z}\langle x_1,\dots,x_k\rangle$ i.e., $f(r_1,\dots,r_k)=0$ for all $r_i\in R$. The minimal degree of a PI ring $R$ is the least degree of all monic polynomial identities for $R$. PI rings cover a large class of rings including commutative rings. Commutative rings satisfy the polynomial identity $x_1x_2-x_2x_1$ and therefore have minimal degree~$2$. Let us recall a result which provides a sufficient condition for a ring to be~PI.

\begin{prop}[{\cite[Corollary~13.1.13]{mcr}}]\label{f}
If $R$ is a ring that is a finitely generated module over a commutative subring, then $R$ is a PI ring.
\end{prop}

We now define the PI degree of prime PI-algebras. This definition will suffice because the algebra covered in this article is prime. We now recall one of the fundamental results from the Polynomial Identity theory.
\begin{theo}[Posner's Theorem {\cite[Theorem~13.6.5]{mcr}}]\label{pos}
Let $A$ be a prime PI ring with centre $Z(A)$ and minimal degree $d$. Let $S=Z(A)\setminus\{0\}$, $Q=AS^{-1}$ and $F=Z(A)S^{-1}$. Then $Q$ is a central simple algebra with centre $F$ and $\dime_{F}(Q)=\parens[\big]{\frac{d}{2}}^2$.
\end{theo}
\begin{defi}
The PI degree of a prime PI ring $A$ with minimal degree $d$ is $\pideg(A)=\frac{d}{2}$.
\end{defi}

The following result provides an important link between the PI degree of a prime affine PI algebra over an algebraically closed field and the $\mathbb{K}$-dimension of its irreducible representations (cf.~\cite[Theorem~I.13.5, Lemma~III.1.2]{brg}).

\begin{prop}\label{sim}
Let $A$ be a prime affine PI algebra over an algebraically closed field $\mathbb{K}$, with $\pideg(A) = n$ and $V$ be a simple $A$-module. Then $V$ is a vector space over $\mathbb{K}$ of dimension $t$, where $t \leq n$, and $A/\ann_A(V) \cong M_t(\mathbb{K})$. Moreover, the upper bound\/ $\pideg(A)$ is attained by some simple $A$-modules.
\end{prop}

Now in the root of unity setting, the above proposition yields the following result.

\begin{theo}\label{finite}
$F_q(n)$ is a PI ring.
\end{theo}
\begin{proof}
Consider the polynomial algebra $P= \mathbb{K}[x_i^l,y_i^l: 1\leq i\leq n]$. Note $F_q(n)$ becomes a finitely generated module over $P$. Hence the assertion follows from~\cite[Corollary~13.1.13]{mcr}.
\end{proof}

From Theorem~\ref{finite} and~\cite[Theorem~I.13.5]{brg}, we conclude if $M$ is a $F_q(n)$-simple module, then $\dime_{\mathbb{K}}M \leq \pideg\parens[\big]{F_q(n)}$.

The following few results will be useful.

\begin{prop}[{\cite[Proposition~7.1]{di}}]\label{quan}
Let $\qmatrix =(q_{ij})$ be an $n \times n$ multiplicatively antisymmetric matrix over $\mathbb{K}$ and $\mathcal{O}_{\qmatrix }(\mathbb{K}^n)$ denotes the quantum affine space generated by $x_1,\dots,x_n$ satisfying the relations $x_ix_j=q_{ij}x_jx_i$ for $i&lt;j$. Suppose that $q_{ij}=q^{h_{ij}}$ for all $i,j$, where $q \in \mathbb{K}^*$ is a primitive $l$-th root of unity and the $h_{ij} \in \mathbb{Z}$. Let $h$ be the cardinality of the image of the homomorphism
\[
\mathbb{Z}^n \overset{(h_{ij})}{\longrightarrow} \mathbb{Z}^n \overset{\pi}{\longrightarrow} (\mathbb{Z}/l\mathbb{Z})^n,
\]
where $\pi$ denotes the canonical epimorphism. Then
\[
\pideg\parens[\big]{\mathcal{O}_{\qmatrix }(\mathbb{K}^n)} = \pideg(\mathcal{O}_{\qmatrix }\parens[\big]{\mathbb{(K^*)}^n} = \sqrt{h}.
\]
\end{prop}

It is well known that a skew-symmetric matrix over $\mathbb{Z}$ such as our matrix $H\coloneqq(h_{ij})$ can be brought into a $2\times 2$ block diagonal form (commonly known as skew normal form) by a unimodular matrix $W\in GL_n(\mathbb{Z})$. Then $H$ is congruent to a block diagonal matrix of the form:
\[
WHW^{t}=\diagonal\parens[\Bigg]{
\begin{pmatrix}
0&amp;h_1\\
-h_1&amp;0
\end{pmatrix},\dots,
\begin{pmatrix}
0&amp;h_s\\
-h_s&amp;0
\end{pmatrix},\mathbf{0}_{n-2s}},
\]
where $\mathbf{0}_{n-2s}$ is the square matrix of zeros of dimension equals $\dime(\ker H)$, so that $2s=\ran(H)=n-\dime(\ker H)$ and $h_i\mid h_{i+1}\in \mathbb{Z}\setminus\{0\}$, $\forall 1\leq i \leq s-1$. The nonzero $h_1,h_1,\dots,h_s,h_s$ (each occurs twice) are called the invariant factors of $H$. The following result simplifies the calculation of $h$ in the statement of Proposition~\ref{quan} by the properties of the integral matrix $H$, namely the dimension of its kernel, along with its invariant factors and the value of $l$.

\begin{lemm}[{\cite[Lemma~5.7]{ar}}]\label{mainpi}
Take $1\neq q\in \mathbb{K}^*$, a primitive $l$-th root of unity. Let $H$ be a skew symmetric integral matrix associated to\/ $\qmatrix $ with invariant factors $h_1,h_1,\dots,h_s,h_s$. Then PI degree of $\mathcal{O}_{\qmatrix }(\mathbb{K}^n)$ is given as
\[
\pideg\parens[\big]{\mathcal{O}_{\qmatrix }(\mathbb{K}^n)}=\pideg\parens[\big]{\mathcal{O}_{\qmatrix }\parens[\big]{\mathbb{(K^*)}^n}}=\prod_{i=1}^{\frac{n-\dime(\ker H)}{2}}\frac{l}{\gcdi(h_i,l)},
\]
where $\mathcal{O}_{\qmatrix }\parens[\big]{\mathbb{(K^*)}^n}$ is the quantum torus generated by $x_1^{\pm1},\dots,x_n^{\pm1}$ with relations $x_ix_j=q_{ij}x_jx_i$ for $i&lt;j$.
\end{lemm}

An easy corollary to Lemma~\ref{mainpi} is the following:

\begin{coro}\label{PI}
$\pideg\parens[\big]{\mathcal{O}_{\qmatrix }(\mathbb{K}^n)}=\pideg(\mathcal{O}_{\qmatrix }\parens[\big]{\mathbb{(K^*)}^n} \leq l^{\frac{\ran H}{2}}$.
\end{coro}

In~\cite[Example~5.4]{han}, Hanyal proved the following:
\[
\pideg\parens[\big]{F_q(n)}=
\begin{dcases}
l^{n-1} 											&amp;\text{when $l$ is odd},\\
\frac{l^{n-1}}{2^{n-2}} 							&amp;\text{when $l \in 4\mathbb{Z}$},\\
\frac{l^{n-1}}{2^{\lfloor\frac{n-1}{2}\rfloor}} 	&amp;\text{when $l \in 2\mathbb{Z}$ but $l \notin 4\mathbb{Z}$}.
\end{dcases}
\]

\section{Construction and classification of maximal dimensional simple modules}\label{section_3}

From now on a module means a right module, $q$ is a primitive $l$-th root of unity, $l$ being odd and $\mathbb{K}$ is an algebraically closed field.

\subsection{Simple modules of type \texorpdfstring{$M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$}{M1(underline mu(I,J), underline lambda(I))}}\label{sub}

For any two subsets $I$ and $J$ of indices $[1,n]$, let $\underline{\mu}(I,J)\coloneqq(\mu_1,\dots,\mu_n)\in \mathbb{K}^n$ such that $\mu_i=0$ iff $i\in I\cap J$ and let $\underline{\lambda}(I)\coloneqq(\lambda_1,\dots,\lambda_n)\in \mathbb{({K}^*)}^{n-1}\times \mathbb{K}$ such that $\lambda_i=q^2\lambda_{i-1}$ if $i\in I$. Given such $\underline{\mu}(I,J)$ and $\underline{\lambda}(I)$, let $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$ be the $\mathbb{K}$-vector space with basis consisting of
\[
\braces[\big]{e(\underline{a}) \ \vert \ \underline{a}=(a_1,\dots,a_{n-1}), \ 0\leq a_i \leq l-1, \ 1\leq i\leq n-1}.
\]
Let $\epsilon_i(\pm 1)$ denote the vector $(0,\dots,{\pm 1},\dots,0)$ whose $i$-th coordinate is $\pm 1$ and the remaining are zero. Define the $F_q(n)$-module structure on the $\mathbb{K}$-space $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$ by the action of each generator on the basis vectors as follows: for $1\leq i \leq n$,
\begin{align*}
e(\underline{a})x_i&amp;=
\begin{dcases}
\mu_1\mathbb{A}(\underline{a};1) \, e(\underline{a})		&amp; i=1,\\
\mu_i\mathbb{A}(\underline{a};i) \, 
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(+1)}		&amp; i\in [2,n]\setminus I,\\
\mu_i^{-1}\mathbb{A}(\underline{a};i)\frac{-(1- q^{-2a_{i-1}})\lambda_{i-1}}{1-q^{-2}}
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(-1)}		&amp; i\in I\setminus (I\cap J),\\
\mathbb{A}(\underline{a};i)\frac{-(1- q^{-2a_{i-1}})\lambda_{i-1}}{1-q^{-2}}
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(-1)}		&amp; a_{i-1}\neq 0, \ i\in I\cap J,\\
0														&amp; a_{i-1}=0, \ i\in I\cap J,
\end{dcases}
\\
e(\underline{a})y_i&amp;=
\begin{dcases}
\frac{\mu^{-1}\lambda_1}{1-q^2}\mathbb{B}(\underline{a};1) \, e(\underline{a})		&amp; i=1,\\
\mu_i^{-1} \mathbb{B}(\underline{a};i)\frac{\lambda_{i}- q^{2a_{i-1}}\lambda_{i-1}}{1-q^{2}}
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(-1)}									&amp; i\in [2,n]\setminus I,\\
\mu_i \mathbb{B}(\underline{a};i) \,
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(+1)}									&amp; i\in I\setminus (I\cap J),\\
\mathbb{B}(\underline{a};i) \, e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(+1)}	&amp; a_{i-1}\neq l-1, \ i\in I\cap J,\\
0																					&amp; a_{i-1}=l-1, \ i\in I\cap J,
\end{dcases}
\end{align*}
where $\lambda_0=0$ and $\oplus$ is addition in the additive group $\bigoplus_{i=1}^{n-1}\mathbb{Z}/l\mathbb{Z}$ and the scalars $\mathbb{A}(\underline{a};i)$ and $\mathbb{B}(\underline{a};i)$ are given by
\begin{align*}
\mathbb{A}(\underline{a};i)&amp;=
\begin{dcases}
\prod_{\substack{j=1 \\j \not\in I}}^{n-1} q^{a_j}
\prod_{\substack{j=1 \\ j \in I}}^{n-1} q^{-a_j} 		&amp; i=1,\\
\prod_{j=1}^{i-2}q^{-a_j}							&amp; i\in[2,n]\setminus I,\\
\prod_{j=1}^{i-2}q^{-a_j}
{\prod_{\substack{j=i \\j \not\in I}}^{n-1}} q^{2a_j}
{\prod_{\substack{j=i \\ j \in I}}^{n-1}}q^{-2a_j}	&amp; i\in I,
\end{dcases}
\\
\mathbb{B}(\underline{a};i)&amp;=
\begin{dcases}
\prod_{\substack{j=1 \\ j \not\in I}}^{n-1} q^{a_j}
\prod_{\substack{j=1 \\ j \in I}}^{n-1} q^{-a_j}		&amp; i=1,\\
\prod_{j=1}^{i-2}q^{a_j}
{\prod_{\substack{j=i \\ j \not\in I}}^{n-1}} q^{2a_j}
{\prod_{\substack{j=i \\ j \in I}}^{n-1}} q^{-2a_j}	&amp; i\in [2,n]\setminus I,\\
\prod_{j=1}^{i-2}q^{a_j}								&amp; i \in I.
\end{dcases}
\end{align*}
In order to establish the well-definedness we need to check that the $\mathbb{K}$-endomorphisms of $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$ defined by the above rules satisfy the defining relations of $F_q(n)$. Indeed we can compute
\begin{align*}
e(\underline{a})x_iy_i&amp;=
\begin{dcases}
\frac{\lambda_1}{1-q^2}\mathbb{A}(\underline{a};1) \, \mathbb{B}(\underline{a};1) \, e(\underline{a})	&amp; i=1,\\
\mathbb{A}(\underline{a};i) \, \mathbb{B}\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(+1);i}
\frac{\lambda_i-q^{2a_{i-1}+2}\lambda_{i-1}}{1-q^2}e(\underline{a}) \,								&amp; i\in [2,n] \setminus I,\\
\mathbb{A}(\underline{a};i) \, \mathbb{B}\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(-1);i}
\frac{-(1-q^{-2a_{i-1}})\lambda_{i-1}}{1-q^{-2}}e(\underline{a})										&amp; a_{i-1}\neq 0, \ i\in I,\\
0																									&amp; a_{i-1}=0, \ i\in I,
\end{dcases}
\\
e(\underline{a})y_ix_i&amp;=
\begin{dcases}
\frac{\lambda_1}{1-q^2}\mathbb{A}(\underline{a};1) \, \mathbb{B}(\underline{a};1) \, e(\underline{a})	&amp; i=1,\\
\mathbb{A}\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(-1);i} \,
\mathbb{B}(\underline{a};i)\frac{\lambda_i-q^{2a_{i-1}}\lambda_{i-1}}{1-q^2}e(\underline{a}),			&amp; i\in [2,n]\setminus I,\\
\mathbb{A}\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(+1);i} \,
\mathbb{B}(\underline{a};i)\frac{-(1-q^{-2a_{i-1}-2})\lambda_{i-1}}{1-q^{-2}}e(\underline{a})			&amp; a_{i-1}\neq l-1, \ i\in I,\\
0																									&amp; a_{i-1}=l-1, \ i\in I,\\
\end{dcases}
\end{align*}
so that for $i=1$
\[
e(\underline{a})(x_1y_1-y_1x_1)=0,
\]
for each $i\in [2,n]\setminus I$:
\[
e(\underline{a})(x_iy_i-y_ix_i)={\prod_{\substack{j=i \\ j \not\in I}}^{n-1}} q^{2a_j}
{\prod_{\substack{j=i \\ j \in I}}^{n-1}} q^{-2a_j} q^{2a_{i-1}} \lambda_{i-1}e(\underline{a})
\]
and for each $i\in I$
\[
e(\underline{a})(x_iy_i-y_ix_i)={\prod_{\substack{j=i \\ j \not\in I}}^{n-1}} q^{2a_j}
{\prod_{\substack{j=i \\ j \in I}}^{n-1}} q^{-2a_j} q^{-2a_{i-1}} \lambda_{i-1}e(\underline{a}).
\]
Now for $i\in [2,n]\setminus I$:
\[
e(\underline{a})\parens*{\sum_{k=1}^{i-1}(1-q^2)y_kx_k}={\prod_{\substack{j=i \\ j \not\in I}}^{n-1}} q^{2a_j}
{\prod_{\substack{j=i \\ j \in I}}^{n-1}} q^{-2a_j} q^{2a_{i-1}} \lambda_{i-1}e(\underline{a}).
\]
and for $i \in I$
\[
e(\underline{a})\parens*{\sum_{k=1}^{i-1}(1-q^2)y_kx_k}={\prod_{\substack{j=i \\ j \not\in I}}^{n-1}} q^{2a_j}
{\prod_{\substack{j=i \\ j \in I}}^{n-1}} q^{-2a_j} q^{-2a_{i-1}} \lambda_{i-1}e(\underline{a}).
\]
The remaining relations can be easily verified. Therefore the above action on the $\mathbb{K}$-space $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$ is well defined. Now we have the following result:

\begin{theo}\label{linindep}
The module $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$ is a simple $F_q(n)$-module of dimension $l^{n-1}$.
\end{theo}
\begin{proof}
Let $P$ be a nonzero submodule of $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$. We claim that $P$ contains a basis vector of the form $e(a_1,\dots,a_{n-1})$. Indeed, any member $p\in P$ is a finite $\mathbb{K}$-linear combination of such vectors, i.e.,
\[
p\coloneqq\sum_{\text{finite}} \lambda_k e\parens[\big]{a_1^{(k)},\dots,a_{n-1}^{(k)}}
\]
for some $\lambda_k\in \mathbb{K}$. Suppose there exist two non-zero coefficients, say, $\lambda_u$, $\lambda_v$. We can choose the largest index $1\leq r\leq n-1$ such that $a_r^{(u)}\neq a_r^{(v)}$ in ${\mathbb{Z}}/{l\mathbb{Z}}$. Now the following two cases may arise:

%~ J'utiliserais plutÃ´t ": " que " --- ", car Ã§a me donne l'impression que le tiret fait partie du "r \notin I", mais c'est une question de goÃ»t.
\begin{case}[$r \notin I$]
Then the vectors $e\parens[\big]{a_1^{(u)},\dots,a_{n-1}^{(u)}}$ and $e\parens[\big]{a_1^{(v)},\dots,a_{n-1}^{(v)}}$ are eigenvectors of $\Omega_{r}$ associated with the eigenvalues
\[
\Lambda_u=\prod_{\substack{j=r+1 \\ j \not\in I}}^{n-1} q^{2a_j^{(u)}}
{\prod_{\substack{j=r+1 \\ j \in I}}^{n-1}} q^{-2a_j^{(u)}} q^{2a_r^{(u)}} \lambda_{r} \qquad\text{and}\qquad \Lambda_v={\prod_{\substack{j=r+1 \\ j \not\in I}}^{n-1}} q^{2a_j^{(v)}}
{\prod_{\substack{j=r+1 \\ j \in I}}^{n-1}} q^{-2a_j^{(v)}} q^{2a_r^{(v)}} \lambda_{r}
\]
respectively. We claim that $\Lambda_u\neq \Lambda_v$. Indeed,
\[
\Lambda_u=\Lambda_v\implies q^{2a_r^{(u)}}\lambda_{r}=q^{2a_r^{(v)}}\lambda_{r} \implies q^{2\parens*{a_r^{(u)}-a_r^{(v)}}}=1 \implies l\mid \parens[\big]{a_r^{(u)}-a_r^{(v)}},
\]
which is a contradiction.
\end{case}

\begin{case}[$r \in I$]
Then the vectors $e\parens[\big]{a_1^{(u)},\dots,a_{n-1}^{(u)}}$ and $e\parens[\big]{a_1^{(v)},\dots,a_{n-1}^{(v)}}$ are eigenvectors of $\Omega_{r}$ associated with the eigenvalues
\[
\Lambda_u=
{\prod_{\substack{j=r+1 \\ j \not\in I}}^{n-1}} q^{2a_j^{(u)}}
{\prod_{\substack{j=r+1 \\ j \in I}}^{n-1}} q^{-2a_j^{(u)}} q^{-2a_r^{(u)}} \lambda_{r} \qquad\text{and}\qquad \Lambda_v={\prod_{\substack{j=r+1 \\ j \not\in I}}^{n-1}} q^{2a_j^{(v)}}
{\prod_{\substack{j=r+1 \\ j \in I}}^{n-1}} q^{-2a_j^{(v)}} q^{-2a_r^{(v)}} \lambda_{r}
\]
respectively. We claim that $\Lambda_u\neq \Lambda_v$. Indeed,
\[
\Lambda_u=\Lambda_v\implies q^{-2a_r^{(u)}}\lambda_{r}=q^{-2a_r^{(v)}}\lambda_{r} \implies q^{2\parens*{a_r^{(u)}-a_r^{(v)}}}=1 \implies l\mid \parens[\big]{a_r^{(u)}-a_r^{(v)}},
\]
which is a contradiction.

Now $p\Omega_r-\Lambda_up$ is a non zero element in $P$ of smaller length than $p$. Hence by induction, it follows that $P$ contains a basis vector of the form $e(a_1,\dots,a_{n-1})$ and hence $P=M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$ by the actions of $x_i$ and $y_i$ for all $1\leq i \leq n$. This completes the proof.\qedhere
\end{case}
\let\qed\relax
\end{proof}

\begin{coro}\label{anncoro}
Let $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$ be a simple $F_q(n)$-module. Then
\begin{enumerate}
\item The set $I$ consists of indices $i\in [2,n]$ so that $x_i$ act as nilpotent operator on $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$.
\item The set $J$ consists of indices $j\in [2,n]$ so that $y_j$ act as nilpotent operator on $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$.
\item The module $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$ is\/ $\Omega _i$-torsionfree for all $1\leq i\leq n-1$.
\item For each $i \in [1,n]\setminus I$, the elements $x_i^{l}-\mu_i^{l}$ and $y_i^{l}-\mu_i^{-l}(1-q^2)^{-l}(\lambda_i^{l}-\lambda_{i-1}^{l})$ are in the annihilator of\/ $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$.
\item For each $i\in I\setminus (I\cap J)$, the elements $x_i^{l}$ and $y_{i}^{l}-\mu_i^{l}$ are in the annihilator of $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$.
\item For each $i\in I\cap J$, the elements $x_i^{l}$ and $y_i^{l}$ are in the annihilator of $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$.
\end{enumerate}
\end{coro}

\subsection{Simple modules of type \texorpdfstring{$M_2\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$}{M2(underline mu(I,J), underline lambda(I), delta1)}}

For any two subsets $I$ and $J$ of indices $[1,n]$, let $\underline{\mu}(I,J)\coloneqq(\mu_1,\dots,\mu_{n-1})\in \mathbb{K}^{n-1}$ such that $\mu_{i-1}=0$ iff $i\in I\cap J \cap [3,n]$, $\underline{\lambda}(I)\coloneqq(\lambda_1,\dots,\lambda_{n-1})\in \mathbb{({K}^*)}^{n-2}\times \mathbb{K}$ such that $\lambda_i=q^2\lambda_{i-1}$ if $i\in I$ and $\delta_1 \in \mathbb{K}^*$. Given such $\underline{\mu}(I,J)$, $\underline{\lambda}(I)$ and $\delta_1$, let $M_2\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$ be the $\mathbb{K}$-vector space with basis consisting~of
\[
\braces[\big]{e(\underline{a}) \ \vert \ \underline{a}=(a_1,\dots,a_{n-1}), \ 0\leq a_i \leq l-1, \ 1\leq i\leq n-1}.
\]
Let $\epsilon_i(\pm 1)$ denote the vector $(0,\dots,{\pm 1},\dots,0)$ whose $i$-th coordinate is $\pm 1$ and the remaining are zero. Define the $F_q(n)$-module structure on the $\mathbb{K}$-space $M_2\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$ by the action of each generator on the basis vectors as follows: for $1\leq i \leq n$,
\begin{align*}
e(\underline{a})x_i&amp;=
\begin{dcases}
\mu_1e\parens[\big]{\underline{a}\oplus\epsilon_{1}(+1)}	&amp; i=1,\\
\mu_1\delta_1^{-1}\frac{\lambda_1}{1-q^2}\mathbb{A}(\underline{a};i)\,
e\parens[\big]{\underline{a}\oplus\epsilon_{1}(+1)}		&amp; i=2,\\
\mu_{i-1}\mathbb{A}(\underline{a};i)\,
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(+1)}		&amp; i\in [3,n]\setminus I,\\
\mu_{i-1}^{-1}\mathbb{A}(\underline{a};i)\frac{-(1- q^{-2a_{i-1}})\lambda_{i-1}}{1-q^{-2}}
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(-1)}		&amp; i\in I\setminus (I\cap J),\\
\mathbb{A}(\underline{a};i)\frac{-(1- q^{-2a_{i-1}})\lambda_{i-1}}{1-q^{-2}}
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(-1)}		&amp; a_{i-1}\neq 0, \ i\in I\cap J,\\
0														&amp; a_{i-1}=0, \ i\in I\cap J,
\end{dcases}
\\
e(\underline{a})y_i&amp;=
\begin{dcases}
0														&amp; i=1,\\
\mu_1^{-1}\delta_1\mathbb{B}(\underline{a};i)\,
e\parens[\big]{\underline{a}\oplus\epsilon_{1}(-1)}		&amp; i=2,\\
\mu_{i-1}^{-1} \mathbb{B}(\underline{a};i)\frac{\lambda_{i-1}- q^{2a_{i-1}}\lambda_{i-2}}{1-q^{2}}
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(-1)}		&amp; i\in [3,n]\setminus I,\\
\mu_{i-1} \mathbb{B}(\underline{a};i)\,
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(+1)}		&amp; i\in I\setminus (I\cap J),\\
\mathbb{B}(\underline{a};i)\,
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(+1)}		&amp; a_{i-1}\neq l-1,\ i\in I\cap J,\\
0														&amp; a_{i-1}=l-1, \ i\in I\cap J,
\end{dcases}
\end{align*}
where $\oplus$ is addition in the additive group $\bigoplus_{i=1}^{n-1}\mathbb{Z}/l\mathbb{Z}$ and the scalars $\mathbb{A}(\underline{a};i)$ and $\mathbb{B}(\underline{a};i)$ are given~by
\begin{align*}
\mathbb{A}(\underline{a};i)&amp;=
\begin{dcases}
q^{a_1}													&amp; i=2,\\
\prod_{j=1}^{i-2}q^{-a_j}								&amp; i\in[3,n]\setminus I,\\
\prod_{j=1}^{i-2}q^{-a_j}
{\prod_{\substack{j=i \\j \not\in I}}^{n-1}} q^{2a_j}
{\prod_{\substack{j=i \\ j \in I}}^{n-1}}q^{-2a_j}		&amp; i\in I,
\end{dcases}
\\
\mathbb{B}(\underline{a};i)&amp;=
\begin{dcases}
q^{1-a_1}\prod_{\substack{j=2 \\j \notin I}}^{n-1}q^{-2a_j}
\prod_{\substack{j=2 \\j \in I}^{n-1}}q^{2a_j}			&amp; i=2,\\
\prod_{j=1}^{i-2}q^{a_j}
\prod_{\substack{j=i \\ j \not\in I}}^{n-1} q^{2a_j}
\prod_{\substack{j=i \\ j \in I}}^{n-1} q^{-2a_j}			&amp; i\in [3,n]\setminus I,\\
\prod_{j=1}^{i-2}q^{a_j}									&amp; i \in I.
\end{dcases}
\end{align*}
One can easily verify the defining relations similarly as in Subsection~\ref{sub}.
\begin{theo}
$M_2\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$ is a simple $F_q(n)$-module of dimension $l^{n-1}$.
\end{theo}
\begin{proof}
The calculations are similar to Theorem~\ref{linindep}. Note that for $e(\underline{a}) \neq e(\underline{b})$, if $a_1 \neq b_1$ and $a_2=b_2,\dots,a_{n-1}=b_{n-1}$, then the vectors are eigenvectors corresponding to the operator $x_1y_2$ with distinct eigenvalues.
\end{proof}

\subsection{Simple modules of type \texorpdfstring{$M_3\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$}{M3(underline mu(I,J), underline lambda(I), delta1)}}

For any two subsets $I$ and $J$ of indices $[1,n]$, let $\underline{\mu}(I,J)\coloneqq(\mu_1,\dots,\mu_{n-1})\in \mathbb{K}^{n-1}$ such that $\mu_{i-1}=0$ iff $i\in I\cap J \cap [3,n]$, $\underline{\lambda}(I)\coloneqq(\lambda_1,\dots,\lambda_{n-1})\in \mathbb{({K}^*)}^{n-2}\times \mathbb{K}$ and $\lambda_i=q^2\lambda_{i-1}$ if $i\in I$ and $\delta_1 \in \mathbb{K}^*$. Given such $\underline{\mu}(I,J)$, $\underline{\lambda}(I)$ and $\delta_1$, let $M_3\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$ be the $\mathbb{K}$-vector space with basis consisting of
\[
\braces[\big]{e(\underline{a}) \ \vert \ \underline{a}=(a_1,\dots,a_{n-1}), \ 0\leq a_i \leq l-1, \ 1\leq i\leq n-1}.
\]
Let $\epsilon_i(\pm 1)$ denote the vector $(0,\dots,{\pm 1},\dots,0)$ whose $i$-th coordinate is $\pm 1$ and the remaining are zero. Define the $F_q(n)$-module structure on the $\mathbb{K}$-space $M_3\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$ by the action of each generator on the basis vectors as follows: for $1\leq i \leq n$,
\begin{align*}
e(\underline{a})x_i&amp;=
\begin{dcases}
0														&amp; i=1,\\
\mu_1^{-1} \delta_1\mathbb{A}(\underline{a};i)\,
e\parens[\big]{\underline{a}\oplus\epsilon_{1}(-1)}		&amp; i=2,\\
\mu_i\mathbb{A}(\underline{a};i)\,
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(+1)}		&amp; i\in [3,n]\setminus I,\\
\mu_i^{-1}\mathbb{A}(\underline{a};i)\frac{-(1- q^{-2a_{i-1}})\lambda_{i-1}}{1-q^{-2}}
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(-1)}		&amp; i\in I\setminus (I\cap J),\\
\mathbb{A}(\underline{a};i)\frac{-(1- q^{-2a_{i-1}})\lambda_{i-1}}{1-q^{-2}}
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(-1)}		&amp; a_{i-1}\neq 0, \ i\in I\cap J,\\
0														&amp; a_{i-1}=0, \ i\in I\cap J,
\end{dcases}
\\
e(\underline{a})y_i&amp;=
\begin{dcases}
\mu_1e\parens[\big]{\underline{a}\oplus\epsilon_{1}(+1)}	&amp; i=1,\\
\delta_1^{-1}\frac{\lambda_1}{1-q^2}\mu_1\mathbb{B}(\underline{a};i)\,
e\parens[\big]{\underline{a}\oplus\epsilon_{1}(+1)}		&amp; i=2,\\
\mu_i^{-1} \mathbb{B}(\underline{a};i)\frac{\lambda_{i}- q^{2a_{i-1}}\lambda_{i-1}}{1-q^{2}}
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(-1)}		&amp; i\in [3,n]\setminus I,\\
\mu_i \mathbb{B}(\underline{a};i)\,
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(+1)}		&amp; i\in I\setminus (I\cap J),\\
\mathbb{B}(\underline{a};i)\,
e\parens[\big]{\underline{a}\oplus\epsilon_{i-1}(+1)}		&amp; a_{i-1}\neq l-1, \ i\in I\cap J,\\
0														&amp; a_{i-1}=l-1, \ i\in I\cap J,
\end{dcases}
\end{align*}
where $\oplus$ is addition in the additive group $\bigoplus_{i=1}^{n-1}\mathbb{Z}/l\mathbb{Z}$ and the scalars $\mathbb{A}(\underline{a};i)$ and $\mathbb{B}(\underline{a};i)$ are given~by
\begin{align*}
\mathbb{A}(\underline{a};i)&amp;=
\begin{dcases}
q^{-a_1} {\prod_{\substack{j=2 \\j \not\in I}}^{n-1}}
q^{2a_j}{\prod_{\substack{j=2 \\ j \in I}}^{n-1}}q^{-2a_j}				&amp; i=2,\\
\prod_{j=1}^{i-2}q^{-a_j}												&amp; i\in[3,n]\setminus I,\\
\prod_{j=1}^{i-2}q^{-a_j}{\prod_{\substack{j=i \\j \not\in I}}^{n-1}}
q^{2a_j}{\prod_{\substack{j=i \\ j \in I}}^{n-1}}q^{-2a_j}				&amp;i\in I,
\end{dcases}
\\
\mathbb{B}(\underline{a};i)&amp;=
\begin{dcases}
q^{a_1}													&amp;i=2,\\
\prod_{j=1}^{i-2}q^{a_j}
{\prod_{\substack{j=i \\ j \not\in I}}^{n-1}} q^{2a_j}
{\prod_{\substack{j=i \\ j \in I}}^{n-1}} q^{-2a_j}		&amp; i\in [3,n]\setminus I,\\
\prod_{j=1}^{i-2}q^{a_j}									&amp; i \in I.
\end{dcases}
\end{align*}
One can easily verify the defining relations similarly as in Subsection~\ref{sub}.
\begin{theo}
$M_3\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$ is a simple $F_q(n)$-module of dimension $l^{n-1}$.
\end{theo}
\begin{proof}
The calculations are similar to Theorem~\ref{linindep}. Note that for $e(\underline{a}) \neq e(\underline{b})$, if $a_1 \neq b_1$ and $a_2=b_2,\dots,a_{n-1}=b_{n-1}$, then the vectors are eigenvectors corresponding to the operator $x_2y_1$ with distinct eigenvalues.
\end{proof}

\subsection{Classification of simple modules}

Let us first recall the commutation relations and identities for the algebra $F_q(n)$ as mentioned at the beginning of this article. In the root of unity setting, the algebra $F_q(n)$ is a prime affine PI algebra. Proposition~\ref{sim} implies that the $\mathbb{K}$-dimension of each simple $F_q(n)$-module is finite and bounded above by its PI degree $l^{n-1}$. Note that the action of each normal element $\Omega_i$ on a simple $F_q(n)$-module is either zero or invertible.

Let $N$ be a simple $F_q(n)$-module on which each $\Omega_i$ for $i=2,\dots,n-1$ acts invertibly. Then we have the following two cases.

\setcounter{case}{0}
\begin{case}[$\Omega_1$ acts invertibly on $N$]
Then we can easily verify that the elements
\begin{equation}\label{cev1}
\begin{aligned}
&amp;x_i^{l},\ y_i^{l}, 	&amp;&amp; \text{for $2 \leq i \leq n-1$},\\
&amp;x_1,\ \Omega_i,		&amp;&amp; \forall 1\leq i\leq n,
\end{aligned}
\end{equation}
in $F_q(n)$ commute. Since $N$ has finite $\mathbb{K}$-dimension, there exists a common eigenvector $v$ in $N$ of the operators in~\eqref{cev1}. We define
\[
\begin{aligned}
&amp;vx_i^{l}=\alpha_iv,		&amp;&amp; \forall 2\leq i\leq n,\\
&amp;vy_i^{l}=\beta_iv,		&amp;&amp; \forall 2\leq i\leq n,\\
&amp;v\Omega_i=\zeta_iv,		&amp;&amp; \forall 1\leq i\leq n,\\
&amp;vx_1=\alpha_1v,		&amp;&amp;
\end{aligned}
\]
for some $\alpha_i,\beta_i\in \mathbb{K}$ and for $i=1,\dots,n-1$, $\zeta_i\in \mathbb{K}^*$. Note that since $\Omega_1$ acts invertibly on $N$ and $x_1$ is normal in $F_q(n)$, $\alpha_1 \neq 0$. By Schur's lemma, each of the central elements $x_i^{l}$ and $y_i^{l}$ act as scalars on $N$. Thus, the operators $x_i$ or $y_i$ on $N$ are either invertible or nilpotent. Now set
\[
\mathbb{I}\coloneqq\braces[\big]{i\in [2,n]:\alpha_i= 0} \qquad\text{and}\qquad \mathbb{J}\coloneqq\braces[\big]{i \in [2,n]:\beta_i=0}.
\]
Clearly the set $\mathbb{I}$ (respectively $\mathbb{J}$) consists of indices $i$ in $[2,n]$ for which the operators $x_i$ (respectively $y_i$) on $N$ are nilpotent. Depending on this, the following cases arise:
\begin{subcase}[$\mathbb{I}=\varphi$]
Then $\alpha_i\neq 0$ for all $2\leq i\leq n$ and thus the operators $x_i$ on $N$ are invertible for all $2\leq i\leq n$. Then the vectors
\[
vx_n^{a_{n-1}}\cdots x_2^{a_1}, \quad 0\leq a_i\leq l-1
\]
are nonzero in $N$. Let us set for each $i\in [1,n]$:
\[
\lambda_i\coloneqq\zeta_i,
\]
$\mu_1\coloneqq\alpha_1$ and for each $i \in [2,n]$
\[
\mu_i\coloneqq\text{$l$-th root of $\alpha_i$}.
\]
Thus $\underline{\mu}(\mathbb{I},\mathbb{J})\coloneqq(\mu_1,\dots,\mu_n)\in (\mathbb{K}^{*})^n$ and $\underline{\lambda}(\mathbb{I})\coloneqq(\lambda_1,\dots,\lambda_n)\in (\mathbb{K}^{*})^{n-1}\times \mathbb{K}$. Now define a $\mathbb{K}$-linear map
\[
\phi \colon M\parens[\big]{\underline{\mu}(\mathbb{I},\mathbb{J}),\underline{\lambda}(\mathbb{I})} \to N
\]
by specifying the image of basis vectors:
\[
\phi\parens[\big]{e(a_1,\dots,a_{n-1})}\coloneqq\parens*{\prod_{i=2}^{n}\mu_i^{-a_{i-1}}} vx_n^{a_{n-1}}\cdots x_2^{a_1}.
\]
We can easily check that $\phi$ is a nonzero $F_q(\mathbb{K})$-module homomorphism. Thus by Schur's lemma, $\phi$ is an isomorphism.
\end{subcase}
\begin{subcase}[$\mathbb{I}\neq \varphi$]\label{subcase_1_2}
Then for each $i\in \mathbb{I}$, the operator $x_i$ on $N$ is nilpotent, and hence $\ker (x_i)\coloneqq\braces[\big]{m\in N:mx_i=0}$ is a nonzero subspace of $N$. We now claim that $\cap_{\substack{i\in \mathbb{I}}}\ker (x_i)\neq \{0\}$. Indeed if $i_1,i_2\in \mathbb{I}$, then the nilpotent operator $x_{i_2}$ on $N$ must keep $\ker(x_{i_1})$ invariant because of the commutation relation between $x_{i_1}$ and $x_{i_2}$. Hence $x_{i_2}$ is also nilpotent on the subspace $\ker(x_{i_1})$. That is $\ker(x_{i_1})\cap \ker(x_{i_2})\neq \{0\}$. Continuing this argument we can obtain $\cap_{i\in \mathbb{I}}\ker (x_i)\neq \{0\}$.
\begin{step}
Now each of the commuting operators in~\eqref{cev1} must keep the $\mathbb{K}$-space $\cap_{i\in \mathbb{I}}\ker (x_i)$ invariant due to commutation relations with $x_i$ for $i\in \mathbb{I}$. Therefore we can choose a common eigenvector $w\in \cap_{i\in \mathbb{I}}\ker (x_i)$ of the operators~\eqref{cev1}. We define
\[
\begin{aligned}
&amp; wx_1=\alpha_1w,		&amp;&amp; \\
&amp; wx_i=0,				&amp;&amp; i\in \mathbb{I},\\
&amp; wx^{l}_i=\xi_iw,		&amp;&amp; i\in [2,n]\setminus \mathbb{I},\\
&amp; wy^{l}_i=\eta_iw,		&amp;&amp; i\in [2,n],\\
&amp; w\Omega_i=\zeta_iw,	&amp;&amp; i\in [1,n].
\end{aligned}
\]
Clearly, we can observe that:
\begin{itemize}
\item $\xi_i\in \mathbb{K}^*$ for all $i\in [2,n]\setminus \mathbb{I}$ and $\zeta_i\in \mathbb{K}^*$ for all $i\in [1,n-1]$;
\item $\eta_i=0$ if $i\in \mathbb{J}$ and $\eta_i\in \mathbb{K}^*$ if $i\in [2,n]\setminus \mathbb{J}$.
\end{itemize}
Now for each $i\in \mathbb{I}$, the relation $x_iy_i-y_ix_i=\Omega_{i-1}$ induces the following:
\[
w\Omega_{i-1}=w(x_iy_i-y_ix_i)=-wy_ix_i=q^{-2}w\Omega_i.
\]
Thus we obtain $\zeta_i=q^{2}\zeta_{i-1}$ for each $i\in \mathbb{I}$.
\end{step}
\begin{step}
We now claim that for each $i\in \mathbb{I}$, the vectors $wy_i^{a_i}$, $0\leq a_i\leq l-1$ in $N$ are nonzero. From the description of the set $\mathbb{J}$, if $i\in \mathbb{I}\setminus \mathbb{J}$ then $wy^{l}_i\neq 0$ and we are done. On the other hand if $\mathbb{I}\cap \mathbb{J}\neq \varphi$, then for each $i\in \mathbb{I}\cap \mathbb{J}$, we have $wy^{l}_i=0$. Suppose $k_i$ is the smallest index with $1\leq k_i\leq l$ such that $wy_i^{k_i-1}\neq 0$ and $wy_i^{k_i}=0$. In fact, after simplifying the equality $wy^{k_i}x_i=0$ we obtain
\[
\begin{split}
0	&amp; =wy^{k_i}x_i\\
	&amp; =w\parens[\Bigg]{x_iy_i^{k_i}-\frac{1-q^{-2k_i}}{1-q^{-2}}\Omega_{i-1}y_i^{k_i-1}}\\
	&amp; =-\frac{1-q^{-2k_i}}{1-q^{-2}}w\Omega_{i-1}y_i^{k_i-1}\\
	&amp; =-\frac{1-q^{-2k_i}}{1-q^{-2}}\zeta_{i-1}wy_i^{k_i-1}.
\end{split}
\]
This implies $k_i$ is the smallest index such that $q^{2k_i}=1$ and hence $k_i=l$. Thus we can conclude that for each $i\in \mathbb{I}$, the vectors $wy_i^{a_i}$, $0\leq a_i\leq l-1$ are nonzero in $N$.
\end{step}
\begin{step}
Let us set $\mu_1\coloneqq\alpha_1$ and for each $i\in [2,n]$:
\[
\mu_i\coloneqq
\begin{cases}
\text{$l$-th root of $\xi_i$},	&amp; i\in [2,n]\setminus \mathbb{I},\\
\text{$l$-th root of $\eta_i$}, 	&amp; i\in \mathbb{I}\setminus (\mathbb{I}\cap \mathbb{J}),\\
0,								&amp; i\in \mathbb{I}\cap \mathbb{J},
\end{cases}
\qquad\qquad\text{and}\qquad\gamma_i\coloneqq\zeta_i.
\]
Clearly $\underline{\mu}(\mathbb{I},\mathbb{J})\coloneqq(\mu_1,\dots,\mu_n)\in \mathbb{K}^n$ with $\mu_i=0$ iff $i\in \mathbb{I}\cap \mathbb{J}$ and $\underline{\lambda}(\mathbb{I})\coloneqq(\lambda_1,\dots,\lambda_n)\in (\mathbb{K}^{*})^{n-1}\times \mathbb{K}$ with $\lambda_i=q^2\lambda_{i-1}$ if $i\in \mathbb{I}$. Any $F_q(n)$-module homomorphism must map $\Omega_i$-eigenvectors of $M\parens[\big]{\underline{\mu}(\mathbb{I},\mathbb{J}),\underline{\lambda}(\mathbb{I})}$ to $\Omega_i$-eigenvectors of $N$ with the same eigenvalue. Define a $\mathbb{K}$-linear map
\[
\psi \colon M\parens[\big]{\underline{\mu}(\mathbb{I},\mathbb{J}),\underline{\lambda}(\mathbb{I})} \to N
\]
by setting
\begin{equation*}
\psi\parens[\big]{e(a_1,\dots,a_{n-1})}\coloneqq wf(a_{n-1})\cdots f(a_2),
\end{equation*}
where
\[
f(a_i)=
\begin{cases}
\mu_i^{-a_{i-1}} x_i^{a_{i-1}} 	&amp; i\in [2,n]\setminus \mathbb{I},\\
\mu_i^{-a_{i-1}} y_i^{a_{i-1}} 	&amp; i\in \mathbb{I}\setminus (\mathbb{I}\cap \mathbb{J}),\\
y_i^{a_{i-1}}					&amp; i\in \mathbb{I}\cap \mathbb{J}.
\end{cases}
\]
It can be easily verified that $\psi$ is a nonzero $F_q(n)$-module homomorphism. By Schur's lemma, $\psi$ is an isomorphism.
\end{step}

Thus the above discussions lead us to the main result of this section which provides a classification of simple $F_q(n)$-modules in terms of scalar parameters.

\begin{theo}\label{imp1}
Each $\Omega_i$-torsionfree for all $1\leq i\leq n-1 $ simple $F_q(n)$-module is isomorphic to a simple $F_q(n)$-module $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$ for two subsets $I$ and $J$ of\/ $[1,n]$ with $\underline{\mu}(I,J)\coloneqq(\mu_1,\dots,\mu_n)\in \mathbb{K}^n$ such that $\mu_i=0$ iff $i\in I\cap J$ and $\underline{\lambda}(I)\coloneqq(\lambda_1,\dots,\lambda_n)\in \mathbb{({K}^*)}^{n-1}\times \mathbb{K}$ such that $\lambda_i=q^2\lambda_{i-1}$ if $i\in I$.
\end{theo}
\end{subcase}
\end{case}
\begin{case}
Let $N\Omega_1=0$. Given $\Omega_1=(1-q^{2})y_1x_1=0$ on $N$, where $x_1$ and $y_1$ are normal elements in $F_q(n)$, it follows that they are either $0$ or invertible on $N$. To avoid $N$ becoming a simple module over $F_q(n-1)$, we exclude the possibility that both $x_1$ and $y_1$ are $0$ on $N$. Here the following two cases may arise.
%\setcounter{subcase}{0}
\newsubcaseset
\begin{subcase}[$x_1$ is invertible and $y_1=0$ on $N$]\label{subcase_2_1}
Note that as $\Omega_1=0$ on $N$, $N$ becomes a simple module over $B\coloneqq F_q(n)/\langle \Omega_1 \rangle$ and $x_2$ and $y_2$ are normal in $B$. Since $\Omega_2 \neq 0$ on $N$, both $x_2$ and $y_2$ must be invertible on $N$. Then we can easily verify that the elements
\begin{equation}\label{cev2}
\begin{gathered}
x_i^{l},\ y_i^{l}, \quad \text{for $1 \leq i \leq n-1, \ i\neq 2$},\\
\Omega_i	 \quad \forall 2\leq i\leq n,\\
x_1y_2, \ x_2y_2	
\end{gathered}
\end{equation}
in $F_q(n)$ commute. Since $N$ has finite $\mathbb{K}$-dimension, there exists a common eigenvector $v$ in $N$ of the operators in~\eqref{cev2}. We define
\[
\begin{aligned}
&amp; vx_i^{l}=\alpha_iv,	&amp;&amp; \forall 1\leq i\leq n, \; i\neq 2,\\
&amp; vy_i^{l}=\beta_iv,		&amp;&amp; \forall 2\leq i\leq n, \; i \neq 2,\\
&amp; v\Omega_i=\zeta_iv,	&amp;&amp; \forall 2\leq i\leq n,\\
&amp; vx_1y_2=\nu_1v,		&amp;&amp; \\
&amp; vx_2y_2=\nu_2v,		&amp;&amp;
\end{aligned}
\]
for some $\alpha_i,\beta_i\in \mathbb{K}$, for $1\leq i \leq n$, with $i\neq 2$, $\zeta_i\in \mathbb{K}^*$ for all $i=2,\dots,n-1$ and $\nu_1,\nu_2 \in \mathbb{K}^*$. Note as $\Omega_1=0$ on $N$ and $v\Omega_2=\zeta_2v$, $\nu_2=\frac{\zeta_2}{1-q^2}$. By Schur's lemma, each of the central elements $x_i^{l}$ and $y_i^{l}$ act as scalars on $N$ for $1\leq i \leq n$ with $i\neq 2$. Thus, the operators $x_i$ or $y_i$ on $N$ are either invertible or nilpotent. Now set
\[
\mathbb{I}\coloneqq\braces[\big]{i\in [1,n]\setminus \{2\}:\alpha_i= 0} \qquad \text{and} \qquad \mathbb{J}\coloneqq\braces[\big]{i \in [1,n]\setminus \{2\}:\beta_i=0}.
\]
Clearly the set $\mathbb{I}$ (respectively $\mathbb{J}$) consists of indices $i$ in $[1,n]$ with $i\neq 2$ for which the operators $x_i$ (respectively $y_i$) on $N$ are nilpotent. Depending on this, the following cases arise.

\begin{subsubcase}[$\mathbb{I}=\varphi$]
Then $\alpha_i\neq 0$ for all $1\leq i\leq n$ and $i \neq 2$ and thus the operators $x_i$ on $N$ are invertible for all $1\leq i\leq n$ with $i \neq 2$. Then the vectors
\[
vx_n^{a_{n-1}}\cdots x_3^{a_2}x_1^{a_1}, \quad 0\leq a_i\leq l-1,
\]
are nonzero in $N$. Let us set for each $i\in [1,n-1]$:
\[
\lambda_i\coloneqq\zeta_{i+1},
\]
and for each $i \in [1,n-1]$
\[
\mu_1\coloneqq\text{$l$-th root of $\alpha_1$}
\]
and
\[
\mu_i\coloneqq\text{$l$-th root of $\alpha_{i+1}$}.
\]
Also set $\delta_1\coloneqq\nu_1$.

Thus $\underline{\mu}(\mathbb{I},\mathbb{J})\coloneqq(\mu_1,\dots,\mu_{n-1})\in (\mathbb{K}^{*})^{n-1}$ and $\underline{\lambda}(\mathbb{I})\coloneqq(\lambda_1,\dots,\lambda_{n-1})\in (\mathbb{K}^{*})^{n-2}\times \mathbb{K}$. Now define a $\mathbb{K}$-linear map
\[
\phi \colon M_2\parens[\big]{\underline{\mu}(\mathbb{I},\mathbb{J}),\underline{\lambda}(\mathbb{I}),\delta_1} \to N
\]
by specifying the image of basis vectors:
\[
\phi\parens[\big]{e(a_1,\dots,a_{n-1})}\coloneqq\parens*{\prod_{i=1}^{n-1}\mu_i^{-a_{i}}} vx_n^{a_{n-1}}\cdots x_3^{a_2}x_1^{a_1}.
\]
We can easily check that $\phi$ is a nonzero $F_q(\mathbb{K})$-module homomorphism. Thus by Schur's lemma, $\phi$ is an isomorphism.
\end{subsubcase}
\begin{subsubcase}[$\mathbb{I}\neq \varphi$]%Subcase~\href{\ref{subcase_1_2}}{I.ii}
Then using the similar arguments in Subcase~\ref{subcase_1_2} we can choose a common eigenvector $w\in \cap_{i\in \mathbb{I}}\ker (x_i)$ of the operators~\eqref{cev1}. We define
\[
\begin{aligned}
&amp; wx_i=0,				&amp;&amp; i\in \mathbb{I},\\
&amp; wx^{l}_i=\xi_iw,		&amp;&amp; i\in [1,n]\setminus (\mathbb{I} \cup \{2\}),\\
&amp; wy^{l}_i=\eta_iw,		&amp;&amp; i\in [1,n] \setminus \{2\},\\
&amp; w\Omega_i=\zeta_iw,	&amp;&amp; i\in [2,n],\\
&amp; wx_1y_2=\nu_1w,		&amp;&amp; \\
&amp; wx_2y_2=\nu_2w.		&amp;&amp;
\end{aligned}
\]
Clearly, we can observe that:
\begin{itemize}
\item $\xi_i\in \mathbb{K}^*$ for all $i\in [1,n]\setminus (\mathbb{I} \cup \{2\})$ and $\zeta_i\in \mathbb{K}^*$ for all $i\in [2,n]$;
\item $\eta_i=0$ if $i\in \mathbb{J}$ and $\eta_i\in \mathbb{K}^*$ if $i\in [1,n]\setminus (\mathbb{J}\cup \{2\})$;
\item $\nu_1,\nu_2 \in \mathbb{K}^*$;
\item $\nu_2=\frac{\zeta_2}{1-q^2}$.
\end{itemize}
Now for each $i\in \mathbb{I}$, the relation $x_iy_i-y_ix_i=\Omega_{i-1}$ induces the following
\[
w\Omega_{i-1}=w(x_iy_i-y_ix_i)=-wy_ix_i=q^{-2}w\Omega_i.
\]
Thus we obtain $\zeta_i=q^{2}\zeta_{i-1}$ for each $i\in \mathbb{I}$.

Due to analogous calculations in Subcase~\ref{subcase_1_2} the vectors $wy_i^{a_i},~0\leq a_i\leq l-1$ are nonzero in $N$.

Let us set for each $i\in [1,n-1]$:
\[
\mu_i\coloneqq
\begin{cases}
\text{$l$-th root of $\xi_1$},		&amp; i=1,\\
\text{$l$-th root of $\xi_{i+1}$},	&amp; i\in [2,n]\setminus \mathbb{I},\\
\text{$l$-th root of $\eta_{i+1}$}, 	&amp; i\in \mathbb{I}\setminus (\mathbb{I}\cap \mathbb{J}),\\
0,									&amp; i\in \mathbb{I}\cap \mathbb{J},
\end{cases}
\qquad\qquad\text{and}\qquad \lambda_i\coloneqq\zeta_{i+1}.
\]
Also set $\delta_1\coloneqq\nu_1$.

Clearly $\underline{\mu}(\mathbb{I},\mathbb{J})\coloneqq(\mu_1,\dots,\mu_{n-1})\in \mathbb{K}^{n-1}$ with $\mu_i=0$ iff $i\in \mathbb{I}\cap \mathbb{J}$ and $\underline{\lambda}(\mathbb{I})\coloneqq(\lambda_1,\dots,\lambda_{n-1})\in (\mathbb{K}^{*})^{n-2}\times \mathbb{K}$ with $\lambda_i=q^2\lambda_{i-1}$ if $i\in \mathbb{I}$. Any $F_q(n)$-module homomorphism must map $\Omega_i$-eigenvectors of $M_2\parens[\big]{\underline{\mu}(\mathbb{I},\mathbb{J}),\underline{\lambda}(\mathbb{I}),\delta_1}$ to $\Omega_i$-eigenvectors of $N$ with the same eigenvalue. Define a $\mathbb{K}$-linear map
\[
\psi \colon M_2\parens[\big]{\underline{\mu}(\mathbb{I},\mathbb{J}),\underline{\lambda}(\mathbb{I}),\delta_1} \to N
\]
by setting
\begin{equation*}
\psi\parens[\big]{e(a_1,\dots,a_{n-1})}\coloneqq wf(a_{n-1})\cdots f(a_1),
\end{equation*}
where
\[
f(a_i)=
\begin{cases}
\mu_1^{-a_1} x_1^{a_1} 			&amp; i=1,\\
\mu_i^{-a_{i}} x_{i+1}^{a_{i}} 	&amp; i\in [2,n]\setminus \mathbb{I},\\
\mu_i^{-a_{i}} y_{i+1}^{a_{i}}	&amp; i\in [2,n] \cap \parens[\big]{\mathbb{I}\setminus (\mathbb{I}\cap \mathbb{J})},\\
y_{i+1}^{a_{i}}					&amp; i\in \mathbb{I}\cap \mathbb{J}.
\end{cases}
\]
It can be easily verified that $\psi$ is a nonzero $F_q(n)$-module homomorphism. By Schur's lemma, $\psi$ is an isomorphism.
\end{subsubcase}
\end{subcase}
\begin{subcase}
Let $x_1$ act as $0$ and $y_1$ act invertibly on $N$. Note that as $\Omega_1=0$ on $N$, $N$ becomes a simple module over $B\coloneqq F_q(n)/\langle \Omega_1 \rangle$ and $x_2$ and $y_2$ are normal in $B$. Since $\Omega_2 \neq 0$ on $N$, both $x_2$ and $y_2$ must be invertible on $N$. Then we can easily verify that the elements
\begin{equation}%\label{cev1}
\begin{gathered}
x_i^{l},\ y_i^{l} \quad \text{for $1 \leq i \leq n-1,\; i\neq 2$},\\
\Omega_i	 \quad \forall 2\leq i\leq n,\\
x_2y_1, \ x_2y_2	
\end{gathered}
\end{equation}
in $F_q(n)$ commute. Since $N$ has finite $\mathbb{K}$-dimension, there exists a common eigenvector $v$ in $N$ of the operators in~\eqref{cev1}. We define
\[
\begin{aligned}
&amp; vx_i^{l}=\alpha_iv, 	&amp;&amp; \forall 1\leq i\leq n, \ i\neq 2,\\
&amp; vy_i^{l}=\beta_iv,		&amp;&amp; \forall 2\leq i\leq n, \ i \neq 2,\\
&amp; v\Omega_i=\zeta_iv, 	&amp;&amp; \forall 2\leq i\leq n,\\
&amp; vx_2y_1=\nu_1v,		&amp;&amp; \\
&amp; vx_2y_2=\nu_2v,		&amp;&amp;
\end{aligned}
\]
for some $\alpha_i,\beta_i\in \mathbb{K}$, for $i=2,\dots,n-1$, $\zeta_i\in \mathbb{K}^*$ and $\nu_1,\nu_2 \in \mathbb{K}^*$. Using similar arguments as in Subcase~\ref{subcase_2_1}, we have $\nu_2=\frac{\zeta_2}{1-q^2}$. By Schur's lemma, each of the central elements $x_i^{l}$ and $y_i^{l}$ act as scalars on $N$ for $1\leq i \leq n$ with $i\neq 2$. Thus, the operators $x_i$ or $y_i$ on $N$ are either invertible or nilpotent. Now set
\[
\mathbb{I}\coloneqq\braces[\big]{i\in [1,n]\setminus \{2\}:\alpha_i= 0} \qquad\text{and}\qquad \mathbb{J}\coloneqq\braces[\big]{i \in [1,n]\setminus \{2\}:\beta_i=0}.
\]
Clearly the set $\mathbb{I}$ is non empty as $1 \in \mathbb{I}$.

Using the similar arguments in Subcase~\ref{subcase_1_2} we can choose a common eigenvector $w\in \cap_{i\in \mathbb{I}}\ker (x_i)$ of the operators~\eqref{cev1}. We define
\[
\begin{aligned}
&amp; wx_i=0,				&amp;&amp; i\in \mathbb{I},\\
&amp; wx^{l}_i=\xi_iw,		&amp;&amp; i\in [1,n]\setminus (\mathbb{I} \cup \{2\}),\\
&amp; wy^{l}_i=\eta_iw,		&amp;&amp; i\in [1,n] \setminus \{2\},\\
&amp; w\Omega_i=\zeta_iw,	&amp;&amp; i\in [2,n],\\
&amp; wx_1y_2=\nu_1w,		&amp;&amp; \\
&amp; wx_2y_2=\nu_2w.		&amp;&amp;
\end{aligned}
\]
Clearly, we can observe that:
\begin{itemize}
\item $\xi_i\in \mathbb{K}^*$ for all $i\in [1,n]\setminus (\mathbb{I} \cup \{2\})$ and $\zeta_i\in \mathbb{K}^*$ for all $i\in [2,n]$;
\item $\eta_i=0$ if $i\in \mathbb{J}$ and $\eta_i\in \mathbb{K}^*$ if $i\in [1,n]\setminus (\mathbb{J}\cup \{2\})$;
\item $\nu_1,\nu_2 \in \mathbb{K}^*$.
\end{itemize}
Now for each $i\in \mathbb{I}$, the relation $x_iy_i-y_ix_i=\Omega_{i-1}$ induces the following
\[
w\Omega_{i-1}=w(x_iy_i-y_ix_i)=-wy_ix_i=q^{-2}w\Omega_i.
\]
Thus we obtain $\zeta_i=q^{2}\zeta_{i-1}$ for each $i\in \mathbb{I}$.

Due to analogous calculations in Subcase~\ref{subcase_1_2} the vectors $wy_i^{a_i},~0\leq a_i\leq l-1$ are nonzero in $N$.

Let us set for each $i\in [1,n-1]$:
\[
\mu_i\coloneqq
\begin{cases}
\text{$l$-th root of $\eta_1$},		&amp; i=1,\\
\text{$l$-th root of $\xi_{i+1}$},	&amp; i\in [2,n]\setminus \mathbb{I},\\
\text{$l$-th root of $eta_{i+1}$}, 	&amp; i\in \mathbb{I}\setminus (\mathbb{I}\cap \mathbb{J}),\\
0,&amp; i\in \mathbb{I}\cap \mathbb{J},
\end{cases}
\qquad\qquad\text{and}\qquad \lambda_i\coloneqq\zeta_{i+1}.
\]
Also set $\delta_1\coloneqq\nu_1$.

Clearly $\underline{\mu}(\mathbb{I},\mathbb{J})\coloneqq(\mu_1,\dots,\mu_{n-1})\in \mathbb{K}^{n-1}$ with $\mu_i=0$ iff $i\in \mathbb{I}\cap \mathbb{J}$ and $\underline{\lambda}(\mathbb{I})\coloneqq(\lambda_1,\dots,\lambda_{n-1})\in (\mathbb{K}^{*})^{n-2}\times \mathbb{K}$ with $\lambda_i=q^2\lambda_{i-1}$ if $i\in \mathbb{I}$. Any $F_q(n)$-module homomorphism must map $\Omega_i$-eigenvectors of $M_3\parens[\big]{\underline{\mu}(\mathbb{I},\mathbb{J}),\underline{\lambda}(\mathbb{I}),\delta_1}$ to $\Omega_i$-eigenvectors of $N$ with the same eigenvalue. Define a $\mathbb{K}$-linear map
\[
\psi \colon M_3\parens[\big]{\underline{\mu}(\mathbb{I},\mathbb{J}),\underline{\lambda}(\mathbb{I}),\delta_1} \to N
\]
by setting
\begin{equation*}
\psi\parens[\big]{e(a_1,\dots,a_{n-1})}\coloneqq wf(a_{n-1})\cdots f(a_1),
\end{equation*}
where
\[
f(a_i)=
\begin{cases}
\mu_1^{-a_1} y_1^{a_1} 			&amp; i=1,\\
\mu_i^{-a_{i}} x_{i+1}^{a_{i}}	&amp; i\in [2,n]\setminus \mathbb{I},\\
\mu_i^{-a_{i}} y_{i+1}^{a_{i}}	&amp; i\in [2,n] \cap \parens[\big]{\mathbb{I}\setminus (\mathbb{I}\cap \mathbb{J})},\\
y_{i+1}^{a_{i}}					&amp; i\in \mathbb{I}\cap \mathbb{J}.
\end{cases}
\]
It can be easily verified that $\psi$ is a nonzero $F_q(n)$-module homomorphism. By Schur's lemma, $\psi$ is an isomorphism.
\end{subcase}
\end{case}

\begin{theo}\label{imp2}
Let $N$ be a $\Omega_1$-torsion and $\Omega_i$-torsionfree for all $2\leq i\leq n-1 $ simple $F_q(n)$-module. Then $N$ is isomorphic to a simple $F_q(n)$-module $M_2\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$ or $M_3\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$ for two subsets $I$ and $J$ of\/ $[1,n]$ with $\underline{\mu}(I,J)\coloneqq(\mu_1,\dots,\mu_{n-1})\in \mathbb{K}^{n-1}$ such that $\mu_{i-1}=0$ iff $i\in I\cap J \cap [3,n]$ and $\underline{\lambda}(I)\coloneqq(\lambda_1,\dots,\lambda_{n-1})\in \mathbb{({K}^*)}^{n-2}\times \mathbb{K}$ such that $\lambda_i=q^2\lambda_{i-1}$ if $i\in I$ and $\delta_1 \in \mathbb{K}^*$.
\end{theo}

In light of this section together with Theorem~\ref{imp1} and Theorem~\ref{imp2}, we have successfully achieved the following:

\begin{theo}\label{main1}
A simple $\Omega_i$-torsionfree for all\/ $2\leq i\leq n-1$ $F_q(n)$-module is maximal dimensional, i.e., $\dime_{\mathbb{K}}(M)=l^{n-1}$.
\end{theo}

\section{PI degree of prime factor algebra of quantum Heisenberg space}\label{section_4}

Our main goal in this section is to give an explicit description of PI degree of prime factor algebra $F_q(n)/\langle \Omega_p\rangle$ for $2 \leq p \leq n$.

\begin{theo}\label{main2}
Under the roots of unity assumption for any $p$ with $2 \leq p \leq n-1$, the factor algebra $F_q(n)/\angles{\Omega_p}$ is a prime PI ring and\/ $\pideg F_q(n)/\angles{\Omega_p} = l^{n-2}$.
\end{theo}
\begin{proof}
Fix an index $p$ such that $2 \leq p \leq n-1$. Then the factor algebra $F_q(n)/\langle \Omega_p\rangle$ is a domain (see~\cite[Lemma~2.6]{aj}). Hence the algebra $F_q(n)/\angles{\Omega_p}$ is a prime PI affine algebra under roots of unity assumption. Let $A_p$ be the $\mathbb{K}$-algebra generated by the variables $X_1,\dots,X_n,Y_1,\dots, Y_n$ subject to the relations:
\begin{gather*}
X_iX_j=q^{-1}X_jX_i \quad\text{and}\quad Y_iY_j=qY_jY_i \quad \forall i&lt;j,\\
X_iY_j=qY_jX_i, \quad \forall i\neq j, \\
X_{p}Y_{p}=q^{2}Y_{p}X_{p},\\
X_iY_i=Y_iX_i+\sum_{k&lt;i}(1-q^{2})Y_kX_k, \quad \forall i \neq p.
\end{gather*}

Note that it is easy to prove that the factor algebra $F_q(n)/\angles{\Omega_p}$ is isomorphic to the factor algebra $A_p/I_p$ where $I_p$ is the ideal of $A_p$ generated by $\sum_{k=1}^{p}(1-q^{2})Y_kX_k$. Hence we conclude that
\begin{equation}\label{noteq}
\pideg \parens[\big]{F_q(n)/\angles{\omega_p}}=\pideg{(A_p/I_p)}\leq \pideg{(A_p)}.
\end{equation}
So now our focus shifts towards the computation of PI-degree of $A_p$. The algebra $A_p$ also has an iterative skew polynomial presentation
\[
\mathbb{K}[Y_1][X_1;\tau_1][Y_2;\sigma_2][X_2;\tau_2,\delta_2]\cdots[Y_n;\sigma_n][X_n;\tau_n,\delta_n]
\]
where the automorphisms $\tau_i,\sigma_i$ and derivations $\delta_i$ are defined by
\[
\begin{aligned}
&amp; \sigma_i(Y_j)=q^{-1}Y_j, \; \sigma_i(X_j)=q^{-1}X_j,	&amp;&amp; j&lt;i,\\
&amp; \tau_i(Y_j)=qY_j, \; \tau_i(X_j)=qX_j, 					&amp;&amp; j&lt;i,\\
&amp; \tau_i(Y_i)=Y_i,										&amp;&amp; \forall i,\\
&amp; \delta_i(X_j)=\delta_i(Y_j)=0,							&amp;&amp; j&lt;i,\\
&amp; \delta_i(Y_i)=\sum _{k &lt; i}(1-q^{2})Y_kX_k, 			&amp;&amp; \forall i \neq p+1,\\
&amp; \delta_{p+1}(Y_{p+1})=0.								&amp;&amp;
\end{aligned}
\]
One checks that the hypothesis of the derivation erasing process independent of characteristic due to by Leroy and Matczuk~\cite[Theorem~7]{lm2} holds for $A_p$, so we obtain $\pideg(A_p)=\pideg\parens[\big]{\mathcal{O}_H(\mathbb{K}^{2n})}$.

Now the $2n \times 2n$ matrix $H$ is of the form
\begin{equation*}
\begin{pmatrix}
0 &amp; 0 &amp; 1 &amp; -1 &amp;1 &amp; -1 	&amp;\cdots &amp; 1 &amp; -1 &amp; 1 &amp; -1 &amp;\cdots &amp; 1 &amp; -1\\
0 &amp; 0 &amp; 1 &amp; -1 &amp;1 &amp; -1 	&amp;\cdots &amp; 1 &amp; -1 &amp; 1 &amp; -1 &amp;\cdots &amp; 1 &amp; -1\\
-1 &amp; -1 &amp; 0 &amp; 0 &amp;1 &amp; -1 	&amp;\cdots &amp; 1 &amp; -1 &amp; 1 &amp; -1 &amp;\cdots &amp; 1 &amp; -1\\
1 &amp; 1 &amp; 0 &amp; 0 &amp;1 &amp; -1 	&amp;\cdots &amp; 1 &amp; -1 &amp; 1 &amp; -1 &amp;\cdots &amp; 1 &amp; -1\\
\vdots &amp;\vdots &amp;\vdots &amp;\vdots &amp;\vdots &amp;\vdots &amp;\ddots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;\vdots&amp;\vdots\\
-1 &amp; -1 &amp; -1 &amp; -1 &amp; -1 &amp; -1 	&amp;\cdots &amp; 0 &amp; -2 &amp; 1 &amp; -1 	&amp;\cdots &amp; 1 &amp; -1\\
1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 		&amp;\cdots &amp; 2 &amp; 0 &amp; 1 &amp; -1 	&amp;\cdots &amp; 1 &amp; -1\\
-1 &amp; -1 &amp; -1 &amp; -1 &amp; -1 &amp; -1 	&amp;\cdots &amp; -1 &amp; -1 &amp; 0 &amp; 0 	&amp;\cdots &amp; 1 &amp; -1\\
1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 		&amp;\cdots &amp; 1 &amp; 1 &amp; 0 &amp; 0 		&amp;\cdots &amp; 1 &amp; -1\\
\vdots &amp;\vdots &amp;\vdots &amp;\vdots &amp;\vdots &amp;\vdots &amp;\ddots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\vdots&amp;\ddots&amp;\vdots&amp;\vdots\\
-1 &amp; -1 &amp; -1 &amp; -1 &amp; -1 &amp; -1 	&amp;\cdots &amp; -1 &amp; -1 &amp; -1 &amp; -1 	&amp;\cdots &amp; 0 &amp; 0\\
1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 &amp; 1 		&amp;\cdots &amp; 1 &amp; 1 &amp; 1 &amp; 1 		&amp;\cdots &amp; 0 &amp; 0
\end{pmatrix}
\end{equation*}
where:
\begin{itemize}
\item the $(2p-1)$-th row is $(-1, -1, -1, -1,\dots, 0, -2, 1, -1,\dots, 1, -1)$;
\item the $2p$-th row is $(1, 1, 1, 1,\dots, 2, 0,1,-1,\dots, 1, 1)$;
\item the $(2p+1)$-th row is $(-1, -1, -1, -1, -1, -1,\dots, -1, -1, 0, 0,\dots, 1,-1)$;
\item the $(2p+2)$-th row is $(1, 1, 1, 1, 1, 1,\dots, 1, 1, 0, 0,\dots, 1,-1)$.
\end{itemize}
Now we perform the following elementary row operations:
\begin{itemize}
\item Replace row $1$ with row $1-\text{row } 2$.
\item Replace row $(2p-1)$ with $\text{row } (2p-1)-\text{row } 2p$.
\item Replace row $(2p+1)$ with $\text{row } (2p+1)-\text{row } (2p+2)$.
\item Replace row $(2n-1)$ with $\text{row } (2n-1)+\text{row } 2n$.
\end{itemize}
The resulting matrix has two zero rows: row $1$ and row $(2n-1)$ and two identical rows: row $(2p-1)$ and row $(2p+1)$, which are both equal to $(-2,-2,\dots,-2,-2,0,0,\dots,0)$ where the first $2p$ many coordinates are $-2$.

Since the rank of a skew-symmetric matrix is even, we have $\ran H \leq 2n-4$. So from Corollary~\ref{PI}, we have $\pideg(A_p)=\pideg\parens[\big]{\mathcal{O}_H(\mathbb{K}^{2n})} \leq l^{n-2}$. Hence $\pideg F_q(n)/\angles{\Omega_p} \leq l^{n-2}$.

Now we can establish the equality of the above by constructing a simple $F_q(n)/\angles{\Omega_p}$-module with $\mathbb{K}$-dimension $l^{n-2}$.

Let $(\underline{\alpha}, \underline{\lambda},\underline{\gamma})\coloneqq(\alpha_1,\dots,\alpha_n,\lambda_1,\dots,\lambda_n, \gamma_1,\gamma_2,\gamma_3)\in \mathbb{K}^{2n+3}$ be such that all the scalar except $\alpha_{p+1}, \lambda_p$ are non zero. Given such $(\underline{\alpha}, \underline{\lambda},\underline{\gamma})$, let $M(\underline{\alpha},\underline{\lambda},\underline{\gamma})$ be the $\mathbb{K}$-vector space with basis consisting of
\[
\braces[\big]{e(\underline{a}) : \underline{a}=(a_1,\dots,a_n),\ 0\leq a_i \leq m-1,\ 1\leq i\leq n \ \ \text{with $i \neq 1$},\ p+1,\  a_1=a_{p+1}=0}.
\]
Let $e_{\pm i}$ denote the vector $(0,\dots,{\pm 1},\dots,0)$ whose $i$-th coordinate is $\pm 1$ and the remaining are zero. Define the $F_q(n)$-module structure on the $\mathbb{K}$-space $M(\underline{\alpha},\underline{\lambda},\underline{\gamma})$ by the action of each generator on the basis vectors as follows:
\begin{align*}
&amp; e(\underline{a})X_1 =\gamma_1 \prod_{\substack{j=2\\ j \neq i+1}}^{n}q^{a_j}e(\underline{a}),\\
&amp; e(\underline{a})Y_1 =\frac{\gamma_1^{-1}\lambda_1}{1-q^{2}} \prod_{\substack{j=2\\ j \neq i+1}}^{n}q^{a_j} e(\underline{a}),\\
&amp; e(\underline{a})X_k =\alpha_k\prod_{\substack{j=2\\ j \neq p+1}}^{k-1}q^{-a_j}~e(\underline{a}+e_k) \quad \text{for $k \neq p+1$},\\
&amp; e(\underline{a})Y_k =\alpha_k^{-1}\prod_{\substack{j=2\\ j \neq p+1}}^{k-1}q^{a_j}{\prod_{\substack{j=k+1\\ j \neq p+1}}^{n}}q^{2a_j}\ \frac{\lambda_k-q^{2a_k}\lambda_{k-1}}{1-q^{2}}e(\underline{a}+e_{-k}) \quad \text{for $k \neq p+1$},\\
&amp; e(\underline{a})X_{p+1} =\alpha_p \prod_{j=2}^{p-1}q^{-a_j}\ q^{-(a_p+1)}\ \gamma_2 \ (q^{2}-1)\ \lambda_{p-1}^{-1}\ e(\underline{a}+e_p),\\
&amp; e(\underline{a})Y_{p+1} =\alpha_p^{-1}\prod_{j=p+2}^{n}q^{a_j}{\prod_{\substack{j=2 \\ j \neq p, \ p+1}}^{n}}q^{a_j} q^{(a_p-1)} \ \gamma_3 \ e(\underline{a}+e_{-p}).
\end{align*}
It can be easily verified this is $F_q(n)$-module with $M(\underline{\alpha},\underline{\lambda},\underline{\gamma})\Omega_p=0$. Hence $M(\underline{\alpha},\underline{\lambda},\underline{\gamma})$ is a simple $F_q(n)/\angles{\Omega_p}$-module with $\mathbb{K}$-dimension $m^{n-2}$. Thus
\[
\pideg F_q(n)/\angles{\Omega_p}=l^{n-2}.\qedhere
\]
\end{proof}

Hence combining Theorem~\ref{main1} and Theorem~\ref{main2} we can characterize the maximal dimensional simple modules over $F_q(n)$.

\begin{theo}\label{main}
Let $N$ be a simple $F_q(n)$-module. Then $N$ is maximal dimensional if and only if $N$ is $\Omega_i$-torsionfree for $i=2,\dots,n-1$. Moreover $N$ is isomorphic to one of the following modules
\begin{enumerate}
\item $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$ for two subsets $I$ and $J$ of\/ $[1,n]$ with $\underline{\mu}(I,J)\coloneqq(\mu_1,\dots,\mu_n)\in \mathbb{K}^n$ such that $\mu_i=0$ iff $i\in I\cap J$ and $\underline{\lambda}(I)\coloneqq(\lambda_1,\dots,\lambda_n)\in \mathbb{({K}^*)}^{n-1}\times \mathbb{K}$ such that $\lambda_i=q^2\lambda_{i-1}$ if $i\in I$.
\item $M_2\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$ for two subsets $I$ and $J$ of\/ $[1,n]$ with $\underline{\mu}(I,J)\coloneqq(\mu_1,\dots,\mu_{n-1})\in \mathbb{K}^{n-1}$ such that $\mu_i=0$ iff $i\in I\cap J \cap [3,n]$ and $\underline{\lambda}(I)\coloneqq(\lambda_1,\dots,\lambda_{n-1})\in \mathbb{({K}^*)}^{n-2}\times \mathbb{K}$ such that $\lambda_i=q^2\lambda_{i-1}$ if $i\in I$ and $\delta_1 \in \mathbb{K}^*$.
\item $M_3\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$ for two subsets $I$ and $J$ of\/ $[1,n]$ with $\underline{\mu}(I,J)\coloneqq(\mu_1,\dots,\mu_{n-1})\in \mathbb{K}^{n-1}$ such that $\mu_i=0$ iff $i\in I\cap J \cap [3,n]$ and $\underline{\lambda}(I)\coloneqq(\lambda_1,\dots,\lambda_{n-1})\in \mathbb{({K}^*)}^{n-2}\times \mathbb{K}$ such that $\lambda_i=q^2\lambda_{i-1}$ if $i\in I$ and $\delta_1 \in \mathbb{K}^*$.
\end{enumerate}
\end{theo}

\section{The Azumaya Locus for Quantum Heisenberg Algebras}\label{azuqwa}\label{section_5}

We define the Azumaya locus of ${F_q(n)}$ over ${Z\parens[\big]{F_q(n)}}$ to be the set
\[
\mathcal{AL}\parens[\big]{F_q(n)}\coloneqq\braces[\big]{\mathfrak{m}\in \mspect Z\parens[\big]{F_q(n)}:\ \exists V\ \text{simple module},\ V\mathfrak{m}=0,\ \dime_{\mathbb{K}}(V)=l^{n-1}}.
\]
It is clear from Theorem~\ref{cen}, the center $Z\parens[\big]{F_q(n)}$ of $F_q(n)$ are generated by
\[
x_1^ay_1^b\ \ \text{where}\ \ a+b=l, \ x_i^l,\ y_i^l\ \ \text{for}\ \ i=2,\dots,n.
\]
Consider the homomorphism $\Psi \colon \mathbb{K}[z_1,\dots,z_{2n+l-1}] \to Z\parens[\big]{F_q(n)}$ defined by $\Psi(z_i)=x_1^{l-i}y_1^{i}$ for $i=1,\dots,l-1$, $\Psi(z_{l-1+i})=x_i^l$ for $i=1,3,\dots,2n-1$ and $\Psi(z_{l-1+i})=y_i^l$ for $i=2,4,\dots,2n$. Then $\ker\Psi$ will be a homogeneous ideal which will be generated by all possible $2 \times 2$ cofactors of the determinant
\[
\begin{vmatrix}
z_1 &amp; z_2 &amp; z_3 &amp; \cdots &amp; z_l \\
z_2 &amp; z_3 &amp; z_4 &amp; \cdots &amp; z_{l+1} \\
\end{vmatrix}.
\]
Since $\mathbb{K}$ is algebraically closed, the maximal ideals of $Z\parens[\big]{F_q(n)}$ are in one-one correspondence with points $(\gamma_1,\dots,\gamma_{2n+l-1})\in \mathbb{K}^{2n+l-1}$ satisfying the condition that each $2\times 2$ cofactor of the determinant
\[
\begin{vmatrix}
\gamma_1 &amp; \gamma_2 &amp; \gamma_3 &amp; \cdots &amp; \gamma_l \\
\gamma_2 &amp; \gamma_3 &amp; \gamma_4 &amp; \cdots &amp; \gamma_{l+1} \\
\end{vmatrix}
\]
is $0$. Indeed each maximal ideal of the center is of the form $\angles[\big]{z_i-\gamma_i \big\vert i=1,\dots,2n+l-1}/I$ satisfying the above mentioned condition. Suppose
\[
\psi_{\mathfrak{m}} \colon Z\parens[\big]{F_q(n)} \to \mathbb{K}
\]
is the algebra homomorphism corresponding to $\mathfrak{m}\in Z\parens[\big]{F_q(n)}$ such that $\psi_{\mathfrak{m}}(x_i^l)=\gamma_{l+2i-2}$ and $\psi_{\mathfrak{m}}(y_i^{l})=\gamma_{l+2i-1}$. As each $\Omega_i^{l}$ is central, therefore using Lemma~\ref{vimp}, we can write
\begin{equation}\label{cenid}
\psi_{\mathfrak{m}}(\Omega_i^{l})=(1-q^2)^{l}\sum_{p\leq i}\gamma_{l+2p-2}\gamma_{l+2p-1}\ \ \text{for}\ \ 1\leq i\leq n.
\end{equation}
In light of the classification of maximal dimensional simple ${F_q(n)}$-modules, we have the following theorem.

\begin{theo}\label{azu}
The Azumaya locus of ${F_q(n)}$ is given by
\[
\mathcal{AL}\parens[\big]{F_q(n)} = \braces[\big]{\mathfrak{m}\in \mspect Z\parens[\big]{F_q(n)} : \psi_{\mathfrak{m}}(\Omega_i^{l})\neq 0, \ \forall 2\leq i\leq n-1}.
\]
\end{theo}
\begin{proof}
Suppose $\mathfrak{m}\in\mathcal{AL}\parens[\big]{F_q(n)}$. Then by definition there exists a simple module $V$ over $F_q(n)$ with $\dime(V)=\pideg\parens[\big]{F_q(n)}$ and $V\mathfrak{m}=0$. Now by Theorem~\ref{main}, $V$ is maximal dimensional i.e.\ having dimension equal to $\pideg\parens[\big]{F_q(n)}$ if and only if the action of each $\Omega_i$ on $V$ is invertible for each $i=2,\dots,n-1$. As each $\Omega_i^{l}$ is central, therefore by Schur's lemma we can conclude that $\psi_{\mathfrak{m}}(\Omega_i^{l})\neq 0$, $\forall 2\leq i\leq n-1$.

Conversely suppose $\mathfrak{m}=\langle z_i-\gamma_i: 1\leq i\leq 2n+l-1\rangle/I\in \mspect Z\parens[\big]{F_q(n)}$ for some $\gamma_i\in\mathbb{K}^{2n+l-1}$ is such that $\psi_{\mathfrak{m}}(\Omega_i^{l})\neq 0,~\forall 2\leq i\leq n-1$. Then the following two cases arise.

\setcounter{case}{0}
\newcaseset
\begin{case}\label{theo_26_case_1}
Let $\psi_{\mathfrak{m}}(\Omega_1^l)\neq 0$. Define
\[
I\coloneqq\braces[\big]{i\in[1,n]:\gamma_{l+2i-2}=0} \qquad\text{and}\qquad J\coloneqq\braces[\big]{j\in[1,n]:\gamma_{l+2i-1}=0}.
\]
Also for all $i\in[1,n]$, we take
\begin{align*}
\mu_{i}=&amp;
\begin{cases}
\gamma_{l+2i-2}^{\frac{1}{l}}	&amp; \text{when $i \notin I$},\\
\gamma_{l+2i-1}^{\frac{1}{l}} 	&amp; \text{when $i\in I\setminus J$},\\
0								&amp; \text{when $i\in I\cap J$},
\end{cases}
\qquad\qquad\text{and}\qquad\qquad
\lambda_{i}=
\begin{cases}
\parens[\big]{\psi_{\mathfrak{m}}(\Omega_{i}^{l})}^{\frac{1}{l}}	&amp;\text{when $i\notin I$},\\
q^{2}\lambda_{i-1}												&amp;\text{when $i\in I$}.
\end{cases}
\end{align*}
Then by Subsection~\ref{sub}, $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$ is a maximal dimensional simple $F_q(n)$-module where $\underline{\mu}(I,J)\coloneqq(\mu_1,\dots,\mu_{n-1})\in \mathbb{K}^{n-1}$ and $\underline{\lambda}(I)\coloneqq(\lambda_1,\dots,\lambda_{n-1})\in \mathbb{({K}^*)}^{n-1}$.
When $i\notin I$ i.e., $\gamma_{l+2i-2}\neq0$, from the equality~\eqref{cenid} we have
\[
\gamma_{l+2i-1}=(1-q^2)^{-l}\parens[\big]{\psi_{\mathfrak{m}}(\Omega_i^{l})-\psi_{\mathfrak{m}}(\Omega_{i-1}^{l})}\gamma_{l+2i-2}^{-1}.
\]
Thus it follows from Corollary~\ref{anncoro} that $\mathfrak{m}$ annihilates the simple module $M_1\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I)}$.
\end{case}

\begin{case}
Let $\psi_{\mathfrak{m}}(\Omega_1^l)= 0$ i.e.\ $\gamma_{l}\gamma_{l+1}=0$. As $\psi_{\mathfrak{m}}(\Omega_2^l) \neq 0$, we have $\gamma_{l+2}\gamma_{l+3}\neq 0$. Define
\[
I\coloneqq\braces[\big]{i\in[1,n]\setminus\{2\}:\gamma_{l+2i-2}=0} \qquad\text{and}\qquad J\coloneqq\braces[\big]{j\in[1,n]\setminus\{2\}:\gamma_{l+2i-1}=0}.
\]

If $\gamma_{l}\neq 0$, then for $i \in [1,n-1]$
\begin{align*}
\mu_{i}=&amp;
\begin{cases}
\gamma_l^{\frac{1}{l}}			&amp; i=1,\\
\gamma_{l+2i}^{\frac{1}{l}}		&amp; \text{when $i \notin I$},\\
\gamma_{l+2i+1}^{\frac{1}{l}}	&amp; \text{when $i\in I\setminus J$},\\
0								&amp; \text{when $i\in I\cap J$},
\end{cases}
\qquad\qquad\text{and}\qquad\qquad
\lambda_{i}=
\begin{cases}
\parens[\big]{\psi_{\mathfrak{m}}(\Omega_{i+1}^{l})}^{\frac{1}{l}},	&amp;\text{when $i\notin I$}\\
q^{2}\lambda_{i-1},												&amp;\text{when $i\in I$}.
\end{cases}
\end{align*}
Also $\delta_1=(\gamma_l\gamma_{l+3})^{\frac{1}{l}}$. Then as in Case~\ref{theo_26_case_1} we can show that $\mathfrak{m}$ annihilates the simple module $M_2\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$.

If $\gamma_l=0$, then define for $i \in [1,n-1]$
\begin{align*}
\mu_{i}=&amp;
\begin{cases}
\gamma_{l+1}^{\frac{1}{l}}		&amp;i=1,\\
\gamma_{l+2i}^{\frac{1}{l}}		&amp;\text{when $i \notin I$},\\
\gamma_{l+2i+1}^{\frac{1}{l}}	&amp;\text{when $i\in I\setminus J$},\\
0								&amp;\text{when $i\in I\cap J$},
\end{cases}
\qquad\qquad\text{and}\qquad\qquad
\lambda_{i}=
\begin{cases}
\parens[\big]{\psi_{\mathfrak{m}}(\Omega_{i+1}^{l})}^{\frac{1}{l}}	&amp;\text{when $i\notin I$},\\
q^{2}\lambda_{i-1}													&amp;\text{when $i\in I$}.
\end{cases}
\end{align*}
Also $\delta_1=(\gamma_{l+2}\gamma_{l+1})^{\frac{1}{l}}$. Then as in Case~\ref{theo_26_case_1} we can show that $\mathfrak{m}$ annihilates the simple module $M_3\parens[\big]{\underline{\mu}(I,J),\underline{\lambda}(I),\delta_1}$.\qedhere
\end{case}
\let\qed\relax
\end{proof}

\section*{Declaration of interests}

The author does not work for, advise, own shares in, or receive funds from any organization that could benefit from this article, and has declared no affiliations other than their research organizations.

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