# Simple Modules and Azumaya Loci over
the PI quantized Weyl Algebras

Sanu Bera1 and Snehashis Mukherjee2
Sanu Bera1 and Snehashis Mukherjee2
School of Mathematical Sciences,
Ramakrishna Mission Vivekananda Educational and Research Institute (rkmveri),
Belur Math, Howrah, Box: 711202, West Bengal, India.
sanubera6575<EMAIL_ADDRESS>
###### Abstract.

In this article, both versions of multiparameter quantum Weyl algebras have
been studied at the roots of unity. The center, PI degree, maximal-dimensional
simple modules, and Azumaya locus have been explicitly computed for such
algebras.

###### Key words and phrases:

Quantum Weyl Algebra, Simple modules, Polynomial Identity algebra

###### 2020 Mathematics Subject Classification:

16D60, 16D70, 16S85

##  1\. Introduction

Let $\mathbb{K}$ be a field and $\mathbb{K}^{*}$ denotes the multiplicative
group of non-zero elements of $\mathbb{K}$ and $n$ is a positive integer. Let
$\Lambda:=\left(\lambda_{ij}\right)_{n\times n}$ be an $n\times n$
multiplicatively antisymmetric matrix over $\mathbb{K}$, that is,
$\lambda_{ii}=1$ and $\lambda_{ij}\lambda_{ji}=1$ for all $1\leq i,j\leq n$
and and let $\underline{q}:=(q_{1},\cdots,q_{n})$ be an $n$-tuple elements of
$\mathbb{K}\setminus\\{0,1\\}$. Given such $\Lambda$ and $\underline{q}$, the
Maltsiniotis multiparameter quantized Weyl algebra
${A_{n}^{\underline{q},\Lambda}}$, called quantum Weyl algebra, is the algebra
generated over the field $\mathbb{K}$ by the variables
$x_{1},y_{1},\cdots,x_{n},y_{n}$ subject to the following relations:

$\displaystyle y_{i}y_{j}$ $\displaystyle=\lambda_{ij}y_{j}y_{i}\ \ \ \ \
\forall\ \ \ 1\leq i<j\leq n,$ $\displaystyle x_{i}x_{j}$
$\displaystyle=q_{i}\lambda_{ij}x_{j}x_{i}\ \ \ \ \ \forall\ \ \ 1\leq i<j\leq
n,$ $\displaystyle x_{i}y_{j}$ $\displaystyle=\lambda_{ij}^{-1}y_{j}x_{i}\ \ \
\ \ \forall\ \ \ 1\leq i<j\leq n,$ $\displaystyle y_{i}x_{j}$
$\displaystyle=q_{i}^{-1}\lambda_{ij}^{-1}x_{j}y_{i}\ \ \ \ \ \forall\ \ \
1\leq i<j\leq n,$ $\displaystyle x_{i}y_{i}-q_{i}y_{i}x_{i}$
$\displaystyle=1+\sum_{k=1}^{i-1}(q_{k}-1)y_{k}x_{k}\ \ \ \ \ \forall\ \ \
1\leq i\leq n.$

The quantum Weyl algebra ${A_{n}^{\underline{q},\Lambda}}$, which arises from
the work of Maltsiniotis [21] on noncommutative differential calculus, has
been extensively studied in [2, 12, 13, 17]. When $n=1$,
${A_{n}^{\underline{q},\Lambda}}$ is a rank one quantized Weyl algebra
$A_{1}^{q}$, whose prime ideals are classified in [13]. The automorphism group
of $A_{1}^{q}$ was determined in [2]. The prime spectrum and the automorphism
group for ${A_{n}^{\underline{q},\Lambda}}$ in the generic case were studied
in [23]. The isomorphism problem for ${A_{n}^{\underline{q},\Lambda}}$ is
solved in [14].

Another family of multiparameter quantized Weyl algebras has been studied in
the literature, including [3], which has more symmetric defining relations
than those of ${{A}_{n}^{\underline{q},\Lambda}}$. The quantized Weyl algebra
of symmetric type is denoted by ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$
and is referred to as an alternative quantum Weyl algebra. The definition of
the algebra ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ in terms of generators
and relations has been mentioned in Section 7.

We note that ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ and
${A_{n}^{\underline{q},\Lambda}}$ are closely related in many aspects [3, 17].
Both algebras have some iterated skew polynomial presentation twisted by
automorphisms and derivations and also have a common localization. The prime
ideals of ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ were classified in
generic case [3]. When $\lambda_{ij}=1$, the algebra
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ is isomorphic to the tensor
product $A_{1}^{q_{1}}\otimes\cdots\otimes A_{1}^{q_{n}}$, whose automorphism
group has been studied in [10] for $q_{i}\neq 1$. The automorphism group and
isomorphism problem for ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ in the
generic case have been studied in [24].

Most of these results concern the generic case when the algebras are not
polynomial identity. In this article, we shall focus on quantized Weyl
algebras when they are PI algebras (equivalently, defining parameters are
roots of unity).

### Assumptions:

In the context of the root of unity setting, throughout this chapter we will
be assuming the following assumption regarding the defining multiparameters
${q_{i}}$ and $\lambda_{ij}$ as follows:

(1) $q_{i}$’s are primitive $l_{i}$-th root of unity for $1\leq i\leq n$. (2)
$\lambda_{ij}$’s are $l_{i}$-th roots of unity for all $1\leq i\leq j\leq n$.
(3) the divisibility condition $l_{1}|l_{2}|\cdots|l_{n}$. ($\star$)

As $q_{i}\in\mathbb{K}\setminus\\{0,1\\}$, it follows that $l_{i}\geq 2$ for
all $1\leq i\leq n$. The assumptions ($\star$ ‣ 1) are satisfied in the
important uniparameter cases:

  * (A)

when $q_{1}=\cdots=q_{n}$ and $\lambda_{ij}=1$ (cf. [5]).

  * (B)

when $q_{i}=q^{2}$, $\lambda_{ij}=q^{-1}$ for $1\leq i<j\leq n$ and
$\operatorname{ord}(q)$ is odd (cf. [8, 9]).

If $q$ is a root of unity, say of order $l$, then the first quantum Weyl
algebra $A_{1}^{q}$ is PI algebra with $\operatorname{PI-deg}(A_{1}^{q})=l$
and hence all irreducible representations of $A_{1}^{q}$ are finite
dimensional and explicitly described up to equivalence in two research
projects directed by E. Letzter [6] and by L. Wang [15] via studying the
matrix solutions $(X,Y)$ of the equation $xy-qyx=1$. To date, there has been
no comprehensive classification of simple modules over the PI quantized Weyl
algebras for $n\geq 2$. Moreover, Chelsea Walton posed a problem in [25,
Problem 2] concerning the explicit classification of irreducible
representations of the quantum Weyl algebras in the uniparameter case (B), up
to equivalence. In this chapter, our objective is to compute the PI degrees,
classify all maximal dimensional simple modules and consequently determine the
Azumaya locus for the multiparameter quantized Weyl algebras
${A_{n}^{\underline{q},\Lambda}}$ and
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ at roots of unity assuming
($\star$ ‣ 1). In particular, these findings provide a partial solution to the
problem stated in [25, Problem 2]. Throughout this chapter $\mathbb{K}$ is an
algebraically closed field of arbitrary characteristics and all modules are
the right modules.

### Arrangement:

The paper is organized as follows. In Section $2$, we discuss some necessary
facts for quantum Weyl algebra and the theory of Polynomial Identity algebras.
Additionally, we present a key technique for calculating the PI degree of
quantum affine space and the derivation erasing process, which remains
independent of characteristics. In Section $3$, we compute the explicit
expression of PI degree for quantum Weyl algebra and some of its prime
factors, under the assumptions ($\star$ ‣ 1). Moving on to Section 4, we
present a classification of all maximal-dimensional simple modules over
quantum Weyl algebra. Consequently, in Sections $5$ and $6$, we compute the
center and the Azumaya locus for quantum Weyl algebra respectively. Finally,
in Section 7, we extend our analysis to alternative quantum Weyl algebra,
obtaining analogous results.

##  2\. Preliminaries

In this section, we begin by revisiting essential results related to the
quantum Weyl algebra. Subsequently, we recall relevant information about the
Polynomial Identity algebra and the process of deleting derivation, enabling
us to comment on the $\mathbb{K}$-dimension of simple modules over both
quantized Weyl algebras at roots of unity. Throughout this paper, $\mathbb{K}$
will denote an algebraically closed field, and all modules under consideration
shall be the right modules.

###  2.1. Torsion and Torsionfree modules

Let $A$ be an algebra and $M$ be a right $A$-module and $S\subset A$ be a
right Ore set. The submodule

$\operatorname{tor}_{S}(M):=\\{m\in M:ms=0\ \text{for some}\ s\in S\\}$

is called the $S$-torsion submodule of $M$. The module $M$ is said to be
$S$-torsion if $\operatorname{tor}_{S}(M)=M$ and $S$-torsionfree if
$\operatorname{tor}_{S}(M)=0$. If Ore set $S$ is generated by $x\in A$, we
simply say that the $S$-torsion/torsionfree module $M$ is
$x$-torsion/torsionfree.

A non zero element $x$ of an algebra ${A}$ is called a normal element if
$x{A}={A}x$. Clearly, if $x$ is a normal element of $A$, then the set
$\\{x^{i}:~{}i\geq 0\\}$ is an Ore set generated by $x$. The next lemma is
obvious.

######  Lemma 2.1.

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

$x_{M}:M\rightarrow M\ \text{given by}\ m\mapsto mx$

is an isomorphism (if $M$ is $x$-torsionfree).

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

###  2.2. Quantum Weyl Algebras

The quantum Weyl algebra ${{A}_{n}^{\underline{q},\Lambda}}$ has an iterated
skew polynomial presentation with respect to the ordering of variables
$y_{1},x_{1},y_{2},x_{2},\cdots,y_{n},x_{n}$ of the form:

$\mathbb{K}[y_{1}][x_{1},\tau_{1},\delta_{1}][y_{2},\sigma_{2}][x_{2},\tau_{2},\delta_{2}]\cdots[y_{n},\sigma_{n}][x_{n},\tau_{n}.\delta_{n}],$

where the $\tau_{j}$ and $\sigma_{j}$ are $\mathbb{K}$-linear automorphisms
and the $\delta_{j}$ are $\mathbb{K}$-linear $\tau_{j}$-derivations such that

$\displaystyle\tau_{j}(y_{i})$ $\displaystyle=q_{i}\lambda_{ij}y_{i},\ i<j$
$\displaystyle\sigma_{j}(y_{i})$ $\displaystyle=\lambda_{ij}^{-1}y_{i},\ i<j$
$\displaystyle\tau_{j}(x_{i})$
$\displaystyle=q_{i}^{-1}\lambda_{ij}^{-1}x_{i},\ i<j$
$\displaystyle\sigma_{j}(x_{i})$ $\displaystyle=\lambda_{ij}x_{i},\ i<j$
$\displaystyle\tau_{j}(y_{j})$ $\displaystyle=q_{j}y_{j},\ \forall~{}j$
$\displaystyle\delta_{j}(x_{i})$ $\displaystyle=\delta_{j}(y_{i})=0,\ i<j$
$\displaystyle\delta_{j}(y_{j})$
$\displaystyle=1+\sum\limits_{i<j}(q_{i}-1)y_{i}x_{i},\ \forall~{}j$

This observation along with the skew Hilbert Basis Theorem (cf. [20, Theorem
2.9]) yields that the algebra ${A_{n}^{\underline{q},\Lambda}}$ is an affine
noetherian domain. Moreover, the family of ordered monomials

$\\{y_{1}^{a_{1}}x_{1}^{b_{1}}\cdots
y_{n}^{a_{n}}x_{n}^{b_{n}}~{}:~{}a_{i},b_{i}\in\mathbb{Z}_{\geq 0}\\}$ (2.1)

is a $\mathbb{K}$-basis of ${A_{n}^{\underline{q},\Lambda}}$. Let us define
$z_{0}:=1$ and

$z_{i}:=x_{i}y_{i}-y_{i}x_{i}\ \ \text{for}\ \ 1\leq i\leq n.$

These elements are going to play a crucial role throughout this chapter. It is
easy to verify that for all $1\leq i\leq n$,

$z_{i}=1+\sum_{j=1}^{i}(q_{j}-1)y_{j}x_{j}=q_{i}^{-1}\left(1+\sum_{j=1}^{i}(q_{j}-1)x_{j}y_{j}\right)=z_{i-1}+(q_{i}-1)y_{i}x_{i}.$

Therefore the last of the listed defining relations for
${A_{n}^{\underline{q},\Lambda}}$ can be rewritten as

$x_{i}y_{i}-q_{i}y_{i}x_{i}=z_{i-1},\ \ \forall\ \ 1\leq i\leq n.$

######  Proposition 2.2.

_([3])_ Direct computations yield the following results for
${A_{n}^{\underline{q},\Lambda}}$.

  1. (1)

For all $1\leq i\leq n$, $z_{i}$ is a normal element of
${A_{n}^{\underline{q},\Lambda}}$. More precisely, we have:

     * (a)

For all $i,j$ with $1\leq i<j\leq n$, $z_{i}x_{j}=x_{j}z_{i}$ and
$z_{i}y_{j}=y_{j}z_{i}$

     * (b)

For all $i,j$ with $1\leq j\leq i\leq n$, $z_{i}x_{j}=q_{j}^{-1}x_{j}z_{i}$
and $z_{i}y_{j}=q_{j}y_{j}z_{i}$

     * (c)

For all $i,j$ with $1\leq i,j\leq n$, $z_{i}z_{j}=z_{j}z_{i}$.

  2. (2)

The following identities hold in the algebra
${A_{n}^{\underline{q},\Lambda}}$, for $k\geq 1$:

     * (a)

$x_{i}^{k}y_{i}=q_{i}^{k}y_{i}x_{i}^{k}+\left(1+q_{i}+\cdots+q_{i}^{k-1}\right)z_{i-1}x_{i}^{k-1}$,

     * (b)

$x_{i}y_{i}^{k}=q_{i}^{k}y_{i}^{k}x_{i}+\left(1+q_{i}+\cdots+q_{i}^{k-1}\right)z_{i-1}y_{i}^{k-1}$.

In Section 7, we have presented all the necessary information and essential
properties regarding the alternative quantum Weyl algebra
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$. It is important to note that
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ and
${{A}_{n}^{\underline{q},\Lambda}}$ share a common localization, which plays a
crucial role in studying ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$.

###  2.3. Polynomial Identity Algebras

In this subsection, we recall some known facts concerning Polynomial Identity
algebra that we shall be applying on (both) quantized Weyl algebras at roots
of unity for further development.

A ring $R$ is said to be a polynomial identity (PI) ring if $R$ satisfies some
monic polynomial $f\in\mathbb{Z}\langle x_{1},\cdots,x_{k}\rangle$ i.e.,
$f(r_{1},\cdots,r_{k})=0$ 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 $x_{1}x_{2}-x_{2}x_{1}$ and therefore
have minimal degree $2$. Let us recall a result which provides a sufficient
condition for a ring to be PI.

######  Proposition 2.3.

_([20, Corollary 13.1.13])_ If $R$ is a ring that is a finitely generated
module over a commutative subring, then $R$ is a PI ring.

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

######  Proposition 2.4.

The quantum Weyl algebra ${A_{n}^{\underline{q},\Lambda}}$ is a PI algebra if
and only if $q_{i}$’s and $\lambda_{ij}$’s are roots of unity.

###### Proof.

Suppose $q_{i}$ and $\lambda_{ij}$ are roots of unity such that
$l=\operatorname{lcm}(\operatorname{ord}(q_{i}),\operatorname{ord}(\lambda_{ij}):1\leq
i,j\leq n)$. Let $Z$ be the subalgebra of ${A_{n}^{\underline{q},\Lambda}}$
generated by $x_{i}^{l},y_{i}^{l},\ \ \forall\ \ 1\leq i\leq n$. Then from the
defining relations of ${A_{n}^{\underline{q},\Lambda}}$ along with Proposition
2.2, the subalgebra $Z$ is central. Now from the $\mathbb{K}$-basis of
${A_{n}^{\underline{q},\Lambda}}$, one can easily verify that
${A_{n}^{\underline{q},\Lambda}}$ is a finitely generated module over a
central subalgebra $Z$. Hence it follows from Proposition 2.3 that
${A_{n}^{\underline{q},\Lambda}}$ is a PI algebra.

For the converse, just note that the $\mathbb{K}$-subalgebra of
${A_{n}^{\underline{q},\Lambda}}$ generated by $x_{i}$ and $x_{j}$ with
relation $x_{i}x_{j}=q_{i}\lambda_{ij}x_{j}x_{i}$ is not PI if $q_{i}$ or
$\lambda_{ij}$ is not a root of unity (cf. [4, Proposition I.14.2.]). ∎

We now define the PI degree of prime PI-algebras. This definition will suffice
because the algebras covered in this article are all prime. As a consequence
of the Artin-Wedderburn Theorem, any central simple algebra $A$ is isomorphic
to a matrix ring over a central simple division ring. Hence
$\operatorname{dim}_{Z(A)}(A)=n^{2}$ for some natural number $n$. From this,
we define the PI degree of $A$ to be $n$. We now recall one of the fundamental
results from the Polynomial Identity theory.

######  Theorem 2.5 (Posner’s Theorem [20, Theorem 13.6.5]).

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
$\operatorname{dim}_{F}(Q)=(\frac{d}{2})^{2}$.

Note that the $Q$ in Posner’s theorem is PI and, since $Q$ is a central simple
algebra, we can state its PI degree to be $\frac{d}{2}$ by the discussion
above. Furthermore, as a result, [4, I.13.2(6)], $Q$ has the same minimal
degree as $A$, namely $d$. Recognizing that the PI degree can be interpreted
as some measure of how close to being commutative a PI-algebra is and that
this, in turn, is related to its minimal degree, the definition of PI degree
given above can be extended to all prime PI rings in the following way.

######  Definition 2.6.

The PI degree of a prime PI ring $A$ with minimal degree $d$ is
$\operatorname{PI-deg}(A)=\frac{d}{2}$.

######  Remark 2.7.

Based on this definition, for a prime PI ring $R$:

  * (1)

$\operatorname{PI-deg}(S)\leq\operatorname{PI-deg}(R)$ for all subalgebra $S$
of $R$.

  * (2)

$\operatorname{PI-deg}(R/P)\leq\operatorname{PI-deg}(R)$ for all prime ideals
$P\in\operatorname{Spec}(R)$.

As a consequence of Posner’s Theorem, every prime PI ring $R$ has a total ring
of fractions $\operatorname{Fract}(R)$ obtained by inverting all nonzero
central elements of $R$. Now we obtain the following result [4, Corollary
I.13.3]:

######  Corollary 2.8.

Let $R$ be a prime PI ring. If $S$ is a subring of $\operatorname{Fract}(R)$
with $R\subseteq S$, then $S$ is also a prime PI ring and $\operatorname{PI-
deg}(S)=\operatorname{PI-deg}(R)$.

Primitive PI ring exhibits a particularly nice structure, established in
Kaplansky’s Theorem (cf. [20, Theorem 13.3.8]). Now Kaplansky’s Theorem has a
striking consequence in the case of a prime affine PI algebra over an
algebraically closed field. 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. [4, Theorem I.13.5, Lemma III.1.2]):

######  Proposition 2.9.

Let $A$ be a prime affine PI algebra over an algebraically closed field
$\mathbb{K}$, with PI-deg($A$) = $n$ 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 PI-deg($A$)
is attained by some simple $A$-modules.

######  Remark 2.10.

In the roots of unity context, the quantum Weyl algebra is classified as a
prime affine PI algebra. Consequently, according to Proposition 2.9, it is
quite clear that each simple ${A_{n}^{\underline{q},\Lambda}}$-module is
finite dimensional and can have dimension at most $\operatorname{PI-
deg}({A_{n}^{\underline{q},\Lambda}}$). In Section 3, our focus will be on
computing the PI degree of ${A_{n}^{\underline{q},\Lambda}}$.

In the following two subsections, we will focus on deleting derivation process
and PI degree of quantum affine spaces to compute the PI degree of the
concerned algebras.

###  2.4. Derivation Erasing Process

In this discussion we shall recall some results of PI degree parity between
iterated Ore extension and associated quantum polynomial algebra depending on
various assumptions on base ring, existing derivations, and base field.

The main result of [16] states that if $R$ is a noetherian domain which is
also an algebra over a field $\mathbb{K}$ then, under some quantum-like
hypothesis, the iterated Ore extension
$R_{n}=R[x_{1},\sigma_{1},\delta_{1}]\cdots[x_{n},\sigma_{n},\delta_{n}]$ and
$T_{n}=R[x_{1},\sigma_{1}]\cdots[x_{n},\sigma_{n}]$ have the same PI degree.
This result was known earlier only in the case where $R$ is a prime algebra
over a field of $\mathbb{K}$ of characteristic $0$ (cf. [18]). In [19], A.
Leroy and J. Matczuk presented a short and elementary proof of the derivation
erasing process under the hypothesis that the iterated Ore extension is a
prime PI ring. In particular, the following result was shown:

######  Proposition 2.11.

_([19, Theorem 7])_ Suppose that $R=R_{0}$ is a prime PI algebra over a field
$\mathbb{K}$ and $n\geq 1$. Let
$R_{i}:=R_{i-1}[x_{i},\sigma_{i},\delta_{i}],~{}1\leq i\leq n$, be a sequence
of Ore extensions such that each $\sigma_{i}$ is a $\mathbb{K}$-linear
automorphism of $R_{i-1}$ and each $\delta_{i}$ is a $\mathbb{K}$-linear
$\sigma_{i}$-derivation of $R_{i-1}$ such that:

  1. (i)

$\sigma_{i}|_{R_{0}}$ is an automorphism of $R_{0}$ of finite order, for any
$1\leq i\leq n$.

  2. (ii)

$\sigma_{i}(x_{j})=q_{ij}x_{j}$, where $q_{ij}\in\mathbb{K}^{*}$, for any
$1\leq j<i\leq n$.

  3. (iii)

$\delta_{i}$ is a $q_{i}$-skew $\sigma_{i}$-derivation of $R_{i-1}$, where
$1\neq q_{i}\in\mathbb{K}$, for any $1\leq i\leq n$.

Then the following conditions are equivalent:

  1. (1)

$R_{n}$ is a PI algebra.

  2. (2)

$T_{n}=R_{0}[y_{1},\sigma^{\prime}_{1}]\cdots[y_{n},\sigma^{\prime}_{n}]$ is a
PI algebra, where $\sigma^{\prime}_{i}|_{R_{0}}=\sigma_{i}|_{R_{0}}$ and
$\sigma^{\prime}_{i}(y_{j})=q_{ij}y_{j}$, for $1\leq i\leq n$ and $1\leq
j<i\leq n$;

  3. (3)

$q_{ij}$ is a root of unity, for any $1\leq j<i\leq n$;

  4. (4)

$\sigma_{i}$ is an automorphism of finite order of $R_{i-1}$, for any $1\leq
i\leq n$.

Moreover, if one of the above equivalent conditions holds, then the algebras
$R_{n}$ and $T_{n}$ have isomorphic classical rings of quotients and equal PI
degree.

This erasing derivation process does not assume that the derivations are
locally nilpotent or that the prime base ring is of zero characteristic. In
this sense, it gives a generalization of the well-known Cauchon process of
erasing derivations (cf. [11]). Both Theorem $1.2(1)$ and Corollary $(4.7)$ of
[16] are direct consequences of the above proposition.

Most of the quantum algebras can be presented as iterated Ore extension of the
form
$\mathbb{K}[x_{1}][x_{2},\sigma_{2},\delta_{2}]\cdots[x_{n},\sigma_{n},\delta_{n}]$,
where the appearing automorphisms and skew derivations are as in the
proposition (2.11) by taking
$R_{0}=\mathbb{K},\sigma_{1}=id_{\mathbb{K}},\delta_{1}=0$. Thus such algebras
have the same PI degree with quantum affine space
$\mathbb{K}[x_{1}][x_{2},\sigma_{2}]\cdots[x_{n},\sigma_{n}]$.

###  2.5. PI Degree of Quantum Affine Spaces

Let $\mathbf{q}=(q_{ij})$ be a multiplicatively antisymmetric $(n\times
n)$-matrix over $\mathbb{K}$, that is, $q_{ii}=1$ for all $i$ and
$q_{ji}={q_{ij}}^{-1}$ for all $i\neq j$. Given such a matrix, the
multiparameter quantum affine space of degree $n$ is the $\mathbb{K}$-algebra
$\mathcal{O}_{\mathbf{q}}(\mathbb{K}^{n})$ generated by the variables
$x_{1},\cdots,x_{n}$ subject only to the relations

$x_{i}x_{j}=q_{ij}x_{j}x_{i},\ \ \ \forall\ \ \ 1\leq i,j\leq n.$ (2.2)

It is of interest to know when a quantum affine space
$\mathcal{O}_{\mathbf{q}}(\mathbb{K}^{n})$ is a PI ring and what its PI degree
is. Using Kaplansky’s Theorem and Proposition 2.3, one can prove that
$\mathcal{O}_{\mathbf{q}}(\mathbb{K}^{n})$ is a PI ring if and only if all the
$q_{ij}$ are roots of unity. The following result of De Concini and Procesi
provides one of the key techniques for calculating the PI degree of a quantum
affine space independent of characteristic.

######  Proposition 2.12.

_([7, Proposition 7.1])_ Let $\mathbf{q}=\left(q_{ij}\right)$ be an $n\times
n$ multiplicatively antisymmetric matrix over $\mathbb{K}$. Suppose that
$q_{ij}=q^{h_{ij}}$ for all $i,j$, where $q\in\mathbb{K}^{*}$ is a primitive
$m$-th root of unity and the $h_{ij}\in\mathbb{Z}$. Let $h$ be the cardinality
of the image of the homomorphism

$\mathbb{Z}^{n}\xrightarrow{(h_{ij})}\mathbb{Z}^{n}\xrightarrow{\pi}\left(\mathbb{Z}/m\mathbb{Z}\right)^{n},$

where $\pi$ denotes the canonical epimorphism. Then

$\operatorname{PI-
deg}(\mathcal{O}_{\mathbf{q}}(\mathbb{K}^{n}))=\operatorname{PI-
deg}(\mathcal{O}_{\mathbf{q}}\left(\mathbb{(K^{*})}^{n}\right)=\sqrt{h}.$

It is well known that a skew-symmetric matrix over $\mathbb{Z}$ such as our
matrix $H:=(h_{ij})$ 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:

$WHW^{t}=\operatorname{diag}\left(\begin{pmatrix}0&h_{1}\\\
-h_{1}&0\end{pmatrix},\cdots,\begin{pmatrix}0&h_{s}\\\
-h_{s}&0\end{pmatrix},\bf{0}_{n-2s}\right),$

where $\bf{0}_{n-2s}$ is the square matrix of zeros of dimension equals
$\operatorname{dim}(\operatorname{ker}H)$, so that
$2s=\operatorname{rank}(H)=n-\operatorname{dim}(\operatorname{ker}H)$ and
$h_{i}\mid h_{i+1}\in\mathbb{Z}\setminus\\{0\\},~{}~{}\forall\ \ 1\leq i\leq
s-1$. This nonzero $h_{1},h_{1},\cdots,h_{s},h_{s}$ (each occurs twice) are
called the invariant factors of $H$. The following result simplifies the
calculation of $h$ in the statement of proposition (2.12) by the properties of
the integral matrix $H$, namely the dimension of its kernel, along with its
invariant factors and the value of $m$.

######  Lemma 2.13.

_([22, Lemma 5.7])_ Take $1\neq q\in\mathbb{K}^{*}$, a primitive $m$-th root
of unity. Let $H$ be a skew symmetric integral matrix associated to
$\mathbf{q}$ with invariant factors $h_{1},h_{1},\cdots,h_{s},h_{s}$. Then PI
degree of $\mathcal{O}_{\mathbf{q}}(\mathbb{K}^{n})$ is given as

$\operatorname{PI-
deg}(\mathcal{O}_{\mathbf{q}}(\mathbb{K}^{n}))=\operatorname{PI-
deg}(\mathcal{O}_{\mathbf{q}}\left(\mathbb{(K^{*})}^{n}\right)=\prod_{i=1}^{\frac{n-\operatorname{dim}(\operatorname{ker}H)}{2}}\frac{m}{\operatorname{gcd}(h_{i},m)}.$

Let us mention that Haynel applied Corollary $(4.7)$ of [16] and Proposition
(2.12), a result of De Concini and Procesi, to compute PI degree explicitly of
some quantum algebras such as the coordinate ring of odd/even-dimensional
quantum Euclidean space, quantum symplectic space, coordinate ring of quantum
matrices (uniparameter) and quantized Weyl algebras (uniparameter).

In the next section, we will focus on computing the PI degree of
multiparameter quantum Weyl algebras.

##  3\. PI degree for Quantum Weyl Algebras and its prime factors

Let us first recall the assumption ($\star$ ‣ 1) on multiparameters. Hence the
quantum Weyl algebra ${{A}_{n}^{\underline{q},\Lambda}}$ is a PI algebra. This
section aims to calculate an explicit expression of PI degree for quantum Weyl
algebras and some prime factor of it, with the assumption ($\star$ ‣ 1). Here
we will use the derivation erasing process as in Proposition 2.11 and then a
key technique for calculating the PI degree of quantum affine space as in
Proposition 2.12.

###  3.1. PI degree for ${{A}_{n}^{\underline{q},\Lambda}}$

Based on the iterated skew polynomial presentation of the quantum Weyl algebra
${{A}_{n}^{\underline{q},\Lambda}}$, we are now ready to use the derivation
erasing independent of characteristic due to by A. Leroy and J. Matczuk [19].
Step 1: (Deleting Derivation) A straightforward calculation shows that the
$\mathbb{K}$-linear maps $\tau_{j}$ and $\delta_{j}$ satisfy the skew relation
$\delta_{j}\tau_{j}=q_{j}\tau_{j}\delta_{j}$ $(q_{j}\neq 1)$. Thus all the
hypotheses of the derivation erasing process as in Proposition 2.11 are
satisfied by the PI algebra ${{A}_{n}^{\underline{q},\Lambda}}$. Hence it
follows that $\operatorname{PI-
deg}{{A}_{n}^{\underline{q},\Lambda}}=\operatorname{PI-
deg}\mathcal{O}_{B}(\mathbb{K}^{2n})$, where the $2n\times 2n$ matrix of
relations $B$ is comprised of $2\times 2$ blocks:

$B_{ii}=\begin{pmatrix}1&q_{i}^{-1}\\\ q_{i}&1\end{pmatrix},\ \
B_{ij}=\begin{pmatrix}\lambda_{ij}&q_{i}^{-1}\lambda_{ij}^{-1}\\\
\lambda_{ij}^{-1}&q_{i}\lambda_{ij}\end{pmatrix}\ i<j,\ \
B_{ji}=\begin{pmatrix}\lambda_{ij}^{-1}&\lambda_{ij}\\\
q_{i}\lambda_{ij}&q_{i}^{-1}\lambda_{ij}^{-1}\end{pmatrix}\ i<j.$

Step 2: (Integral Matrix) Now we form an integral matrix associated with this
$B$ under the assumption ($\star$ ‣ 1). Then the multiplicative group $\Gamma$
generated by all the multiparameters $q_{i}$ and $\lambda_{ij}$ is a cyclic
group of order $l_{n}$ and suppose $q$ be a generator of the group $\Lambda$.
Note that we have a chain of cyclic subgroups

$\langle q_{1}\rangle\subseteq\langle
q_{2}\rangle\subseteq\cdots\subseteq\langle q_{n}\rangle=\langle q\rangle$

due to the divisibility condition $l_{1}|\cdots|l_{n}$. Therefore we can
choose $a_{i}:=\frac{l_{i+1}}{l_{i}}$ such that $\langle q_{i}\rangle=\langle
q_{i+1}^{a_{i}}\rangle$ for all $1\leq i\leq n-1$. Set $a_{n}=1$. So there
exist integers $b_{i},b_{ij}$ such that

$q_{i}=q_{i+1}^{a_{i}b_{i}},\ q_{n}=q^{a_{n}b_{n}}\ \text{and}\
\lambda_{ij}=q_{i}^{b_{ij}}=q_{i+1}^{a_{i}b_{i}b_{ij}}\ \ \forall\ 1\leq i\leq
j\leq n.$

Note that $\operatorname{gcd}(b_{i},l_{i})=1$ for all $1\leq i\leq n$. Define

$s_{i}:=a_{i}\cdots a_{n},\ k_{i}:=b_{i}\cdots b_{n},\ k_{ij}:=b_{ij}\ \
\forall\ 1\leq i\leq j\leq n.$

Thus we can write

$q_{i}=q^{s_{i}k_{i}}\ \text{and}\
\lambda_{ij}=q_{i}^{k_{ij}}=q^{s_{i}k_{i}k_{ij}}\ \ \forall\ 1\leq i\leq j\leq
n.$

Now the powers of this $q$ from the matrix $B$ give a $2n\times 2n$ integer
matrix $B^{{}^{\prime}}$ comprised of $2\times 2$ blocks

$\displaystyle B^{{}^{\prime}}_{ii}=\begin{pmatrix}0&-s_{i}k_{i}\\\
s_{i}k_{i}&0\end{pmatrix},\ \
B^{{}^{\prime}}_{ij}=\begin{pmatrix}s_{i}k_{i}k_{ij}&-s_{i}k_{i}-s_{i}k_{i}k_{ij}\\\
-s_{i}k_{i}k_{ij}&s_{i}k_{i}+s_{i}k_{i}k_{ij}\end{pmatrix}\ i<j,$
$\displaystyle
B^{{}^{\prime}}_{ji}=\begin{pmatrix}-s_{i}k_{i}k_{ij}&s_{i}k_{i}k_{ij}\\\
s_{i}k_{i}+s_{i}k_{i}k_{ij}&-s_{i}k_{i}-s_{i}k_{i}k_{ij}\end{pmatrix}\ i<j.$

Step 3: (Final Step) The PI degree of ${{A}_{n}^{\underline{q},\Lambda}}$ can
be calculated using the integral matrix $B^{\prime}$ from Proposition 2.12.
The cardinality of the image of the homomorphism

$\mathbb{Z}^{2n}\xrightarrow{B^{\prime}}\mathbb{Z}^{2n}\xrightarrow{\pi}\left(\mathbb{Z}/l_{n}\mathbb{Z}\right)^{2n},$
(3.1)

determined by the integral matrix $B^{\prime}$ remains unchanged if we perform
certain elementary reductions on $B^{\prime}$ beforehand. Let us apply the
following sequence of elementary operations on $B^{{}^{\prime}}$ to reduce it
into a simpler form:

  * •

For $i=2,4,\cdots,2n$, replace $\operatorname{row}(i)$ with
$\operatorname{row}(i)+\operatorname{row}(i-1)$.

  * •

For $i=2,4,\cdots,2n$, replace $\operatorname{col}(i)$ with
$\operatorname{col}(i)+\operatorname{col}(i-1)$.

  * •

For $1\leq i<j\leq n$, replace $\operatorname{row}(2j-1)$ by
$\operatorname{row}(2j-1)+k_{ij}\operatorname{row}(2i)$ and replace
$\operatorname{row}(2j)$ with $\operatorname{row}(2j)-\operatorname{row}(2i)$.

  * •

For $1\leq i<j\leq n$, replace $\operatorname{col}(2j-1)$ by
$\operatorname{col}(2j-1)+k_{ij}\operatorname{col}(2i)$ and replace
$\operatorname{col}(2j)$ with $\operatorname{col}(2j)-\operatorname{col}(2i)$.

Then we have a $2n\times 2n$ integral matrix $B^{{}^{\prime\prime}}$ of this
form

$B^{{}^{\prime\prime}}=\operatorname{diag}\left(\begin{pmatrix}0&-s_{1}k_{1}\\\
s_{1}k_{1}&0\end{pmatrix},\cdots,\begin{pmatrix}0&-s_{n}k_{n}\\\
s_{n}k_{n}&0\end{pmatrix}\right).$

Therefore the cardinally of the image of homomorphism (3.1) induced by this
integral matrix $B^{\prime\prime}$ is given by
$\displaystyle\prod\limits_{i=1}^{n}[\operatorname{ord}(s_{i}k_{i})]^{2}$,
where order is taken in the additive group $\mathbb{Z}/{l_{n}\mathbb{Z}}$.
Thus we have

$\operatorname{PI-
deg}{A}_{n}^{\underline{q},\Lambda}=\displaystyle\prod\limits_{i=1}^{n}\operatorname{ord}(s_{i}k_{i})=\displaystyle\prod\limits_{i=1}^{n}\frac{l_{n}}{\operatorname{gcd}(s_{i}k_{i},l_{n})}=\displaystyle\prod\limits_{i=1}^{n}\operatorname{ord}(q_{i})=\displaystyle\prod\limits_{i=1}^{n}l_{i}.$

Finally, we have proved the following

######  Theorem 3.1.

The PI degree of ${A}_{n}^{\underline{q},\Lambda}$ is given by

$\operatorname{PI-
deg}{{A}_{n}^{\underline{q},\Lambda}}=\prod\limits_{i=1}^{n}l_{i},$

under the roots of unity assumption ($\star$ ‣ 1) on the defining
multiparameters.

######  Remark 3.2.

In view of Proposition 2.9, it is worth noting that, there exists a simple
${A}_{n}^{\underline{q},\Lambda}$-module with a $\mathbb{K}$-dimension of
$\prod\limits_{i=1}^{n}l_{i}$. The subsequent Section 4 will provide an
explicit construction and classification of these simple modules.

###  3.2. PI degree for Prime Factors of ${{A}_{n}^{\underline{q},\Lambda}}$

Fix an index $r$ with $1\leq r\leq n$. Then the ideal generated by the normal
element $z_{r}:=x_{r}y_{r}-y_{r}x_{r}$ is a completely prime ideal of
${{A}_{n}^{\underline{q},\Lambda}}$ (see [3, Proposition 4.5]) and hence the
factor algebra ${{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle$ is a
domain. Therefore the algebra ${{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle$ is a prime affine PI algebra under the root of unity assumption
($\star$ ‣ 1). Here, our objective is to explicitly determine the PI degree
for the factor algebra ${{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle$.
Step 1: Let $R_{r}$ denote the $\mathbb{K}$-algebra generated by the variables
$X_{1},Y_{1},\cdots,X_{n},Y_{n}$ subject to the relations:

$\displaystyle Y_{i}Y_{j}$ $\displaystyle=\lambda_{ij}Y_{j}Y_{i}\ \ \ \ \
\forall\ \ \ 1\leq i<j\leq n$ $\displaystyle X_{i}X_{j}$
$\displaystyle=q_{i}\lambda_{ij}X_{j}X_{i}\ \ \ \ \ \forall\ \ \ 1\leq i<j\leq
n$ $\displaystyle X_{i}Y_{j}$ $\displaystyle=\lambda_{ij}^{-1}Y_{j}X_{i}\ \ \
\ \ \forall\ \ \ 1\leq i<j\leq n$ (3.2) $\displaystyle Y_{i}X_{j}$
$\displaystyle=q_{i}^{-1}\lambda_{ij}^{-1}X_{j}Y_{i}\ \ \ \ \ \forall\ \ \
1\leq i<j\leq n$ $\displaystyle X_{r}Y_{r}$ $\displaystyle=Y_{r}X_{r}$
$\displaystyle X_{i}Y_{i}-q_{i}Y_{i}X_{i}$
$\displaystyle=1+\sum_{k=1}^{i-1}(q_{k}-1)Y_{k}X_{k}\ \ \ \ \ \forall\ \ \
1\leq i\neq r\leq n$

The algebra $R_{r}$ is similar to quantum Weyl algebra except for the defining
relation between $x_{r}$ and $y_{r}$. The algebra $R_{r}$ is also a PI algebra
in the root of unity setting. The following result establishes a connection
between the algebras ${{A}_{n}^{\underline{q},\Lambda}}$ and $R_{r}$.

######  Lemma 3.3.

The factor algebra ${{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle$ is
isomorphic to the factor algebra $R_{r}/I_{r}$, where $I_{r}$ is the ideal of
$R_{r}$ generated by $1+\sum\limits_{i=1}^{r}(q_{i}-1)Y_{i}X_{i}$.

###### Proof.

Define a $\mathbb{K}$-linear map
$\Phi:{{A}_{n}^{\underline{q},\Lambda}}\longrightarrow R_{r}/I_{r}$ as
follows:

$x_{i}\mapsto X_{i}+I_{r},\ y_{i}\mapsto Y_{i}+I_{r}.$

We can easily verify that $\Phi$ is an algebra homomorphism with $\langle
z_{r}\rangle\subseteq\operatorname{ker}(\Phi)$. Therefore by the universal
property of quotient algebras, there is a unique $\mathbb{K}$-algebra
homomorphism $\overline{\Phi}:{{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle\longrightarrow R_{r}/I_{r}$ such that

$\overline{\Phi}(x_{i}+\langle z_{r}\rangle)=X_{i}+I_{r},\ \
\overline{\Phi}(y_{i}+\langle z_{r}\rangle)=Y_{i}+I_{r}\ \ \forall\ \ 1\leq
i\leq n.$

On the other hand, define a $\mathbb{K}$-linear map
$\Psi:R_{r}\longrightarrow{{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle$ as follows:

$X_{i}\mapsto x_{i}+\langle z_{r}\rangle,\ Y_{i}\mapsto y_{i}+\langle
z_{r}\rangle.$

We can easily check that $\Psi$ is an algebra homomorphism such that
$I_{r}\subseteq\operatorname{ker}(\Psi)$. Thus by the universal property of
quotient algebras, we have a unique $\mathbb{K}$-algebra homomorphism
$\overline{\Psi}:R_{r}/I_{r}\longrightarrow{{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle$ such that

$\overline{\Psi}(X_{i}+I_{r})=x_{i}+\langle z_{r}\rangle,\ \
\overline{\Psi}(Y_{i}+I_{r})=y_{i}+\langle z_{r}\rangle\ \ \forall\ \ 1\leq
i\leq n.$

Therefore $\overline{\Phi}$ and $\overline{\Psi}$ are inverse to each other
and the lemma follows. ∎

Now based on Lemma 3.3 and Remark 2.7, we can conclude that

$\operatorname{PI-deg}({{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle)=\operatorname{PI-deg}{(R_{r}/I_{r})}\leq\operatorname{PI-
deg}{(R_{r})}.$ (3.3)

Step 2: Our focus now shifts to computing the PI degree for the algebra
$R_{r}$. Similar to the quantum Weyl algebra, the algebra $R_{r}$ has an
iterated skew polynomial presentation with respect to the ordering of
variables $Y_{1},X_{1},\cdots,Y_{n},X_{n}$ of the form:

$\mathbb{K}[Y_{1}][X_{1},\tau_{1},\delta_{1}][Y_{2},\sigma_{2}][X_{2},\tau_{2},\delta_{2}]\cdots[Y_{n},\sigma_{n}][X_{n},\tau_{n}.\delta_{n}]$
(3.4)

where the $\tau_{j}$ and $\sigma_{j}$ are $\mathbb{K}$-linear automorphisms
and the $\delta_{j}$ are $\mathbb{K}$-linear $\tau_{j}$-derivations such that

$\displaystyle\tau_{j}(Y_{i})$ $\displaystyle=q_{i}\lambda_{ij}Y_{i},\ i<j$
$\displaystyle\sigma_{j}(Y_{i})$ $\displaystyle=\lambda_{ij}^{-1}Y_{i},\ i<j$
$\displaystyle\tau_{j}(X_{i})$
$\displaystyle=q_{i}^{-1}\lambda_{ij}^{-1}X_{i},\ i<j$
$\displaystyle\sigma_{j}(X_{i})$ $\displaystyle=\lambda_{ij}X_{i},\ i<j$
$\displaystyle\tau_{j}(Y_{j})$ $\displaystyle=q_{j}Y_{j},\ \forall~{}j\neq r$
$\displaystyle\tau_{r}(Y_{r})$ $\displaystyle=Y_{r}$
$\displaystyle\delta_{j}(Y_{j})$
$\displaystyle=1+\sum\limits_{i<j}(q_{i}-1)Y_{i}X_{i},\ \forall~{}j\neq r$
$\displaystyle\delta_{r}(Y_{r})$ $\displaystyle=0$

We can easily verify that the $\mathbb{K}$-linear maps $\tau_{j}$ and
$\delta_{j}$ satisfy the skew relation
$\delta_{j}\tau_{j}=q_{j}\tau_{j}\delta_{j}$ $(q_{j}\neq 1)$ from this
iterated skew polynomial presentation of $R_{r}$. Hence, all the hypotheses of
the derivation erasing process as in [19, Theorem 7.1] are satisfied by the PI
algebra $R_{r}$. Therefore we obtain $\operatorname{PI-
deg}(R_{r})=\operatorname{PI-deg}\mathcal{O}_{C}(\mathbb{K}^{2n})$, where the
$2n\times 2n$ matrix of relations $C$ is comprised of $2\times 2$ blocks

$\displaystyle C_{ii}$ $\displaystyle=\begin{pmatrix}1&q_{i}^{-1}\\\
q_{i}&1\end{pmatrix}\ i\neq r$ $\displaystyle C_{ij}$
$\displaystyle=\begin{pmatrix}\lambda_{ij}&q_{i}^{-1}\lambda_{ij}^{-1}\\\
\lambda_{ij}^{-1}&q_{i}\lambda_{ij}\end{pmatrix}\ i<j$ $\displaystyle C_{rr}$
$\displaystyle=\begin{pmatrix}1&1\\\ 1&1\end{pmatrix}$ $\displaystyle C_{ji}$
$\displaystyle=\begin{pmatrix}\lambda_{ij}^{-1}&\lambda_{ij}\\\
q_{i}\lambda_{ij}&q_{i}^{-1}\lambda_{ij}^{-1}\end{pmatrix}\ i<j.$

Now as in the step 2 of Section 3, the integral matrix $C^{\prime}$ associated
with $C$ under the assumption ($\star$ ‣ 1) is comprised of $2\times 2$ blocks

$\displaystyle C^{{}^{\prime}}_{ii}$
$\displaystyle=\begin{pmatrix}0&-s_{i}k_{i}\\\ s_{i}k_{i}&0\end{pmatrix}\
i\neq r$ $\displaystyle C^{{}^{\prime}}_{ij}$
$\displaystyle=\begin{pmatrix}s_{i}k_{i}k_{ij}&-s_{i}k_{i}-s_{i}k_{i}k_{ij}\\\
-s_{i}k_{i}k_{ij}&s_{i}k_{i}+s_{i}k_{i}k_{ij}\end{pmatrix}\ i<j$
$\displaystyle C^{{}^{\prime}}_{rr}$ $\displaystyle=\begin{pmatrix}0&0\\\
0&0\end{pmatrix}$ $\displaystyle C^{{}^{\prime}}_{ji}$
$\displaystyle=\begin{pmatrix}-s_{i}k_{i}k_{ij}&s_{i}k_{i}k_{ij}\\\
s_{i}k_{i}+s_{i}k_{i}k_{ij}&-s_{i}k_{i}-s_{i}k_{i}k_{ij}\end{pmatrix}\ i<j.$

The $\operatorname{PI-deg}R_{r}$ can be computed using the integral matrix
$C^{\prime}$ as stated in Proposition 2.12. However, we can perform certain
elementary reductions on $C^{\prime}$ before using it to compute the
cardinality of the image of the homomorphism

$\mathbb{Z}^{2n}\xrightarrow{C^{\prime}}\mathbb{Z}^{2n}\xrightarrow{\pi}\left(\mathbb{Z}/l_{n}\mathbb{Z}\right)^{2n},$
(3.5)

and this will not change the result. We note that our choice of $s_{i}$ and
$k_{i}$ satisfies $s_{n}k_{n}|\cdots|s_{1}k_{1}$, which we will use in the
following operations. To simplify $C^{\prime}$, we will perform the following
elementary operations:

  * •

For $i=2,4,\cdots,2n$, replace $\operatorname{row}(i)$ with
$\operatorname{row}(i)-\operatorname{row}(i-1)$.

  * •

For $i=2,4,\cdots,2n$, replace $\operatorname{col}(i)$ with
$\operatorname{col}(i)-\operatorname{col}(i-1)$.

  * •

For $1\leq i<j\leq n$ and $i<r$, replace $\operatorname{row}(2j-1)$ by
$\operatorname{row}(2j-1)+k_{ij}\operatorname{row}(2i)$ and replace
$\operatorname{row}(2j)$ with $\operatorname{row}(2j)-\operatorname{row}(2i)$.

  * •

For $1\leq i<j\leq n$ and $i<r$, replace $\operatorname{col}(2j-1)$ by
$\operatorname{col}(2j-1)+k_{ij}\operatorname{col}(2i)$ and replace
$\operatorname{col}(2j)$ with $\operatorname{col}(2j)-\operatorname{col}(2i)$.

  * •

For $1\leq i<j\leq n$ and $r<i$, replace $\operatorname{row}(2j)$ by
$\operatorname{row}(2j)-\operatorname{row}(2i)$ and replace
$\operatorname{col}(2j)$ with $\operatorname{col}(2j)-\operatorname{col}(2i)$.

  * •

For $r+2\leq j\leq n$ and $r\leq i<j$, replace $\operatorname{row}(2i-1)$ with

$\operatorname{row}(2i-1)-\displaystyle\frac{s_{i}k_{i}k_{ij}}{s_{j}k_{j}}\operatorname{row}(2j)$

and replace $\operatorname{col}(2i-1)$ with

$\operatorname{col}(2i-1)-\displaystyle\frac{s_{i}k_{i}k_{ij}}{s_{j}k_{j}}\operatorname{col}(2j).$

  * •

Replace $\operatorname{row}(2r-1)$ by

$\operatorname{row}(2r-1)-\displaystyle\frac{s_{r}k_{r}k_{r(r+1)}}{s_{r+1}k_{r+1}}\operatorname{row}(2r+2)$

and replace $\operatorname{col}(2r-1)$ by

$\operatorname{col}(2r-1)-\displaystyle\frac{s_{r}k_{r}k_{r(r+1)}}{s_{r+1}k_{r+1}}\operatorname{col}(2r+2),$

then replace $\operatorname{row}(2r-1)$ with

$\operatorname{row}(2r-1)-\displaystyle\frac{s_{r}k_{r}}{s_{r+1}k_{r+1}}\operatorname{row}(2r+1)$

and replace $\operatorname{col}(2r-1)$ by

$\operatorname{col}(2r-1)-\displaystyle\frac{s_{r}k_{r}}{s_{r+1}k_{r+1}}\operatorname{col}(2r+1).$

Finally, we obtain a $2n\times 2n$ integer matrix $C^{{}^{\prime\prime}}$ with
the following diagonal form

$C^{{}^{\prime\prime}}=\operatorname{diag}\left(\begin{pmatrix}0&-s_{1}k_{1}\\\
s_{1}k_{1}&0\end{pmatrix},\cdots,\begin{pmatrix}0&0\\\
0&0\end{pmatrix},\cdots,\begin{pmatrix}0&-s_{n}k_{n}\\\
s_{n}k_{n}&0\end{pmatrix}\right).$

Then the cardinality of the image of the homomorphism (3.5), induced by the
matrix $C^{\prime\prime}$, can be expressed as
$\displaystyle\prod\limits_{\begin{subarray}{c}i=1\\\ i\neq
r\end{subarray}}^{n}\left(\operatorname{ord}(s_{i}k_{i})\right)^{2}$, where
order is taken in the additive group $\mathbb{Z}/{l_{n}\mathbb{Z}}$.
Consequently, based on the aforementioned discussions and Proposition 2.12, we
can deduce the following:

$\operatorname{PI-
deg}(R_{r})=\displaystyle\prod\limits_{\begin{subarray}{c}i=1\\\ i\neq
r\end{subarray}}^{n}\operatorname{ord}(s_{i}k_{i})=\displaystyle\prod\limits_{\begin{subarray}{c}i=1\\\
i\neq
r\end{subarray}}^{n}\frac{l_{n}}{\operatorname{gcd}(s_{i}k_{i},l_{n})}=\displaystyle\prod\limits_{\begin{subarray}{c}i=1\\\
i\neq
r\end{subarray}}^{n}\operatorname{ord}(q_{i})=\displaystyle\prod\limits_{\begin{subarray}{c}i=1\\\
i\neq r\end{subarray}}^{n}l_{i}.$

Thus from (3.3), we can conclude that

$\operatorname{PI-deg}({{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle)\leq\displaystyle\prod\limits_{\begin{subarray}{c}i=1\\\ i\neq
r\end{subarray}}^{n}l_{i}.$ (3.6)

Step 3: Finally we aim to establish the equality in (3.6). For simplicity let

$X_{i}:=x_{i}+\langle z_{r}\rangle\ \ \text{and}\ \ Y_{i}:=y_{i}+\langle
z_{r}\rangle$

denote the cosets of $x_{i}$ and $y_{i}$ in
${{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle$ respectively. Suppose
$A(r)$ be the subalgebra of the factor algebra
${{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle$ generated by
$X_{i},Y_{i}$ for all $i\neq r$. Clearly $A(r)$ is the $\mathbb{K}$-algebra
generated by the $(2n-2)$-variables $X_{i},Y_{i}$ for all $1\leq i\neq r\leq
n$ and the following relations:

$\displaystyle Y_{i}Y_{j}$ $\displaystyle=\lambda_{ij}Y_{j}Y_{i}\ \ \ \ \
\forall\ \ \ 1\leq i<j\leq n$ $\displaystyle X_{i}X_{j}$
$\displaystyle=q_{i}\lambda_{ij}X_{j}X_{i}\ \ \ \ \ \forall\ \ \ 1\leq i<j\leq
n$ $\displaystyle X_{i}Y_{j}$ $\displaystyle=\lambda_{ij}^{-1}Y_{j}X_{i}\ \ \
\ \ \forall\ \ \ 1\leq i<j\leq n$ (3.7) $\displaystyle Y_{i}X_{j}$
$\displaystyle=q_{i}^{-1}\lambda_{ij}^{-1}X_{j}Y_{i}\ \ \ \ \ \forall\ \ \
1\leq i<j\leq n$ $\displaystyle X_{i}Y_{i}-q_{i}Y_{i}X_{i}$
$\displaystyle=1+\sum_{k=1}^{i-1}(q_{k}-1)Y_{k}X_{k}\ \ \ \ \ \forall\ \ \
1\leq i\leq r-1$ $\displaystyle X_{r+1}Y_{r+1}$
$\displaystyle=q_{r+1}Y_{r+1}X_{r+1}$ $\displaystyle
X_{i}Y_{i}-q_{i}Y_{i}X_{i}$
$\displaystyle=\sum_{k=r+1}^{i-1}(q_{k}-1)Y_{k}X_{k}\ \ \ \ \ \forall\ \ \
r+2\leq i\leq n$

As a result, based on Remark 2.7, we have:

$\operatorname{PI-deg}({A}(r))\leq\operatorname{PI-
deg}({{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle)$ (3.8)

Moreover, applying the same reasoning as discussed in Step 2 of Subsection
3.2, we can obtain that

$\operatorname{PI-deg}({{A}(r)})=\prod\limits_{\begin{subarray}{c}i=1\\\ i\neq
r\end{subarray}}^{n}l_{i}$ (3.9)

Therefore by combining the relations (3.6), (3.8) and (3.9), we can establish
the following result:

######  Theorem 3.4.

Assuming the root of unity assumption ($\star$ ‣ 1) on the defining
multiparameters, the PI degree of a prime factor of
${A}_{n}^{\underline{q},\Lambda}$ by the normal element $z_{r}$ is given by
$\displaystyle\prod\limits_{\begin{subarray}{c}i=1\\\ i\neq
r\end{subarray}}^{n}l_{i}$.

Question: As the factor algebra ${{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle$ is classified as a prime affine PI algebra, therefore by
Proposition 2.9, there exists a $z_{r}$-torsion simple
${{A}_{n}^{\underline{q},\Lambda}}$-module with $\mathbb{K}$-dimension
$\prod\limits_{\begin{subarray}{c}i=1\\\ i\neq r\end{subarray}}^{n}l_{i}$ for
each $1\leq r\leq n$. It would be interesting to explore the construction of
such simple modules and subsequently classify them. So let us give it a try!

##  4\. Maximal Dimensional Simple ${{A}_{n}^{\underline{q},\Lambda}}$-modules

In view of Theorem 3.4, the following result provides a necessary condition
for an ${A}_{n}^{\underline{q},\Lambda}$-module to attain maximal
$\mathbb{K}$-dimension.

######  Theorem 4.1.

Let $M$ be a maximal $\mathbb{K}$-dimensional (i.e., equal to the
$\operatorname{PI-deg}{{A}_{n}^{\underline{q},\Lambda}}$) simple
${{A}_{n}^{\underline{q},\Lambda}}$-module. Then $M$ is $z_{i}$-torsionfree
for all $1\leq i\leq n$.

###### Proof.

Suppose $M$ be a $z_{r}$-torsion simple
${{A}_{n}^{\underline{q},\Lambda}}$-module for some $1\leq r\leq n$. As
$z_{r}$ is normal element, $M$ becomes a simple module over the factor algebra
${{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle$. Note that the
algebra ${{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle$ is prime
affine PI algebra. Hence by Proposition 2.9 we have

$\operatorname{dim}_{\mathbb{K}}(M)\leq\operatorname{PI-
deg}({{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle).$

Now by Theorem 3.4, it follows that

$\operatorname{dim}_{\mathbb{K}}(M)\leq\operatorname{PI-
deg}({{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle)<\operatorname{PI-
deg}({{A}_{n}^{\underline{q},\Lambda}}).$

Thus we arrive at a contradiction. Hence we conclude that the maximal
$\mathbb{K}$-dimensional simple ${{A}_{n}^{\underline{q},\Lambda}}$-modules
are $z_{i}$-torsionfree for all $1\leq i\leq n$. ∎

In the following subsections, we aim to prove the converse of the above
theorem, by classifying all $z_{i}$-torsionfree simple
${{A}_{n}^{\underline{q},\Lambda}}$-modules. Denote
$[1,n]:=\\{1,2,\cdots,n\\}$ for $n\in\mathbb{N}$.

###  4.1. Construction of Simple Modules:

For any two subsets $I$ and $J$ of indices $[1,n]$, let
$\underline{\mu}(I,J):=(\mu_{1},\cdots,\mu_{n})\in\mathbb{K}^{n}$ such that
$\mu_{i}=0$ iff $i\in I\cap J$ and let
$\underline{\gamma}(I):=(\gamma_{1},\cdots,\gamma_{n})\in\mathbb{({K}^{*})}^{n}$
such that $q_{i}\gamma_{i}=\gamma_{i-1}$ if $i\in I$. Given such
$\underline{\mu}(I,J)$ and $\underline{\gamma}(I)$, let
$M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$ be the
$\mathbb{K}$-vector space with basis consisting of

$\\{e(\underline{a})~{}|~{}\underline{a}=(a_{1},\cdots,a_{n}),~{}0\leq
a_{i}\leq l_{i}-1,~{}1\leq i\leq n\\}.$

Let $\epsilon_{i}(\pm 1)$ denote the vector $(0,\cdots,{\pm 1},\cdots,0)$
whose $i$-th coordinate is $\pm 1$ and the remaining are zero. Define the
${{A}_{n}^{\underline{q},\Lambda}}$-module structure on the $\mathbb{K}$-space
$M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$ by the action of
each generator on the basis vectors as follows: for $1\leq i\leq n$,

$\displaystyle e(\underline{a})x_{i}$
$\displaystyle=\begin{cases}\mu_{i}\mathbb{A}(\underline{a};i)e(\underline{a}\oplus\epsilon_{i}(+1)),&\
i\in[1,n]\setminus I\\\
\mu_{i}^{-1}\mathbb{A}(\underline{a};i)\displaystyle\frac{(q_{i}^{-a_{i}}-1)\gamma_{i-1}}{q_{i}-1}~{}e(\underline{a}\oplus\epsilon_{i}(-1)),&\
i\in I\setminus(I\cap J)\\\
\mathbb{A}(\underline{a};i)\displaystyle\frac{(q_{i}^{-a_{i}}-1)\gamma_{i-1}}{q_{i}-1}~{}e(\underline{a}\oplus\epsilon_{i}(-1)),&\
a_{i}\neq 0,\ i\in I\cap J\\\ 0,&a_{i}=0,~{}i\in I\cap J\end{cases}$
$\displaystyle e(\underline{a})y_{i}$
$\displaystyle=\begin{cases}\mu_{i}^{-1}\mathbb{B}(\underline{a};i)\displaystyle\frac{q^{a_{i}}_{i}\gamma_{i}-\gamma_{i-1}}{q_{i}-1}~{}e(\underline{a}\oplus\epsilon_{i}(-1)),&\
i\in[1,n]\setminus I\\\
\mu_{i}\mathbb{B}(\underline{a};i)~{}e(\underline{a}\oplus\epsilon_{i}(+1)),&\
i\in I\setminus(I\cap J)\\\
\mathbb{B}(\underline{a};i)~{}e(\underline{a}\oplus\epsilon_{i}(+1)),&a_{i}\neq
l_{i}-1,\ i\in I\cap J\\\ 0,&a_{i}=l_{i}-1,~{}i\in I\cap J\end{cases}$

where $\gamma_{0}=1$ and $\oplus$ is addition in the additive group
$\bigoplus_{i=1}^{n}\mathbb{Z}/l_{i}\mathbb{Z}$ and the scalars
$\mathbb{A}(\underline{a};i)$ and $\mathbb{B}(\underline{a};i)$ are given by

$\displaystyle\mathbb{A}(\underline{a};i)$
$\displaystyle=\begin{cases}1,&i=1\\\
\displaystyle\left(\prod_{\begin{subarray}{c}s<i\\\ s\notin
I\end{subarray}}(q_{s}\lambda_{si})^{a_{s}}\right)\left(\prod_{\begin{subarray}{c}s<i\\\
s\in I\end{subarray}}(q_{s}\lambda_{si})^{-a_{s}}\right),&2\leq i\leq
n\end{cases}$ $\displaystyle\mathbb{B}(\underline{a};i)$
$\displaystyle=\begin{cases}1,&i=1\\\
\displaystyle\left(\prod_{\begin{subarray}{c}s<i\\\ s\notin
I\end{subarray}}\lambda_{si}^{-a_{s}}\right)\left(\prod_{\begin{subarray}{c}s<i\\\
s\in I\end{subarray}}\lambda_{si}^{a_{s}}\right),&2\leq i\leq n\end{cases}$

In order to establish the well-definedness we need to check that the
$\mathbb{K}$-endomorphisms of
$M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$ defined by the
above rules satisfy the defining relations of
${{A}_{n}^{\underline{q},\Lambda}}$. Indeed we can compute

$\displaystyle e(\underline{a})x_{i}y_{i}$
$\displaystyle=\begin{cases}\mathbb{A}(\underline{a};i)\mathbb{B}(\underline{a}\oplus\epsilon_{i}(+1);i)\displaystyle\frac{q_{i}^{a_{i}+1}\gamma_{i}-\gamma_{i-1}}{q_{i}-1}e(\underline{a}),&i\in[1,n]\setminus
I\\\
\mathbb{A}(\underline{a};i)\mathbb{B}(\underline{a}\oplus\epsilon_{i}(-1);i)\displaystyle\frac{(q_{i}^{-a_{i}}-1)\gamma_{i-1}}{q_{i}-1}e(\underline{a}),&a_{i}\neq
0,i\in I\\\ 0,&a_{i}=0,i\in I\end{cases}$ $\displaystyle
e(\underline{a})y_{i}x_{i}$
$\displaystyle=\begin{cases}\mathbb{A}(\underline{a}\oplus\epsilon_{i}(-1);i)\mathbb{B}(\underline{a};i)\displaystyle\frac{q_{i}^{a_{i}}\gamma_{i}-\gamma_{i-1}}{q_{i}-1}e(\underline{a}),&i\in[1,n]\setminus
I\\\
\mathbb{A}(\underline{a}\oplus\epsilon_{i}(+1);i)\mathbb{B}(\underline{a};i)\displaystyle\frac{(q_{i}^{-(a_{i}+1)}-1)\gamma_{i-1}}{q_{i}-1}e(\underline{a}),&a_{i}\neq
l_{i}-1,i\in I\\\ 0,&a_{i}=l_{i}-1,i\in I\\\ \end{cases}$

so that for each $i\in[1,n]$:

$e(\underline{a})(x_{i}y_{i}-q_{i}y_{i}x_{i})=\displaystyle\left(\prod_{\begin{subarray}{c}s<i\\\
s\notin
I\end{subarray}}q_{s}^{a_{s}}\right)\left(\prod_{\begin{subarray}{c}s<i\\\
s\in I\end{subarray}}q_{s}^{-a_{s}}\right)\gamma_{i-1}e(\underline{a})$

and

$e(\underline{a})\left(1+\sum_{k=1}^{i-1}(q_{k}-1)y_{k}x_{k}\right)=\displaystyle\left(\prod_{\begin{subarray}{c}s<i\\\
s\notin
I\end{subarray}}q_{s}^{a_{s}}\right)\left(\prod_{\begin{subarray}{c}s<i\\\
s\in I\end{subarray}}q_{s}^{-a_{s}}\right)\gamma_{i-1}e(\underline{a}).$

The remaining relations can be easily verified. Therefore the above action on
the $\mathbb{K}$-space
$M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$ is well defined.
Now we have the following result:

######  Theorem 4.2.

The module $M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$ is a
simple ${{A}_{n}^{\underline{q},\Lambda}}$-module of dimension
$\prod\limits_{i=1}^{n}l_{i}$.

###### Proof.

Let $P$ be a nonzero submodule of
$M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$. We claim that $P$
contains a basis vector of the form $e(a_{1},\cdots,a_{n})$. Indeed, any
member $p\in P$ is a finite $\mathbb{K}$-linear combination of such vectors.
i.e.,

$p:=\sum_{\text{finite}}\lambda_{k}~{}e\left(a_{1}^{(k)},\cdots,a_{n}^{(k)}\right)$

for some $\lambda_{k}\in\mathbb{K}$. Suppose there exist two non-zero
coefficients, say, $\lambda_{u},\lambda_{v}$. We can choose the smallest index
$1\leq r\leq n$ such that $a_{r}^{(u)}\neq a_{r}^{(v)}$ in
${\mathbb{Z}}/{l_{r}\mathbb{Z}}$. Then the vectors
$e\left(a_{1}^{(u)},\cdots,a_{n}^{(u)}\right)$ and
$e\left(a_{1}^{(v)},\cdots,a_{n}^{(v)}\right)$ are eigenvectors of $z_{r}$
associated with the eigenvalues

$\Lambda_{u}=\displaystyle\left(\prod_{\begin{subarray}{c}s<r\\\ s\notin
I\end{subarray}}q_{s}^{a_{s}^{(u)}}\right)\left(\prod_{\begin{subarray}{c}s<r\\\
s\in I\end{subarray}}q_{s}^{-a_{s}^{(u)}}\right)\gamma_{r}q_{r}^{a^{(u)}_{r}}\
\ \text{and}\ \
\Lambda_{v}=\displaystyle\left(\prod_{\begin{subarray}{c}s<r\\\ s\notin
I\end{subarray}}q_{s}^{a_{s}^{(v)}}\right)\left(\prod_{\begin{subarray}{c}s<r\\\
s\in I\end{subarray}}q_{s}^{-a_{s}^{(v)}}\right)\gamma_{r}q_{r}^{a^{(v)}_{r}}$

respectively. We claim that $\Lambda_{u}\neq\Lambda_{v}$. Indeed,

$\Lambda_{u}=\Lambda_{v}\implies\gamma_{r}q_{r}^{a^{(u)}_{r}}=\gamma_{r}q_{r}^{a^{(v)}_{r}}\implies
q_{r}^{\left(a_{r}^{(u)}-a_{r}^{(v)}\right)}=1\implies
l_{r}\mid\left(a_{r}^{(u)}-a_{r}^{(v)}\right),$

which is a contradiction.

Now $pz_{r}-\Lambda_{u}p$ 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 $e(a_{1},\cdots,a_{n})$ and hence
$P=M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$ by the actions of
$x_{i}$ and $y_{i}$ for all $1\leq i\leq n$. This completes the proof. ∎

######  Corollary 4.3.

Let $M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$ be a simple
${{A}_{n}^{\underline{q},\Lambda}}$-module. Then

  * (1)

The set $I$ consists of indices $i\in[1,n]$ so that $x_{i}$ act as nilpotent
operator on $M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$.

  * (2)

The set $J$ consists of indices $j\in[1,n]$ so that $y_{j}$ act as nilpotent
operator on $M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$.

  * (3)

The module $M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$ is
$z_{i}$-torsionfree for all $1\leq i\leq n$.

###  4.2. Classification of Simple Modules:

Let us first recall the assumption ($\star$ ‣ 1) as well as the commutation
relations and identities for the algebra ${{A}_{n}^{\underline{q},\Lambda}}$
as mentioned at the beginning of this chapter. In the root of unity setting,
the algebra ${{A}_{n}^{\underline{q},\Lambda}}$ is a prime affine PI algebra.
Proposition 2.9 implies that the $\mathbb{K}$-dimension of each simple
${{A}_{n}^{\underline{q},\Lambda}}$-module is finite and bounded above by its
PI degree $\prod\limits_{i=1}^{n}l_{i}$. Note that the action of each normal
element $z_{i}$ on a simple ${{A}_{n}^{\underline{q},\Lambda}}$-module is
either zero or invertible.

Let $N$ be a simple ${{A}_{n}^{\underline{q},\Lambda}}$-module on which each
$z_{i}$ acts invertibly. We can easily verify that the elements

$x_{i}^{l_{i}},\ y_{i}^{l_{i}},\ z_{i}\ \ \ \forall\ \ \ 1\leq i\leq n$ (4.1)

in ${{A}_{n}^{\underline{q},\Lambda}}$ commute. Since $N$ has finite
$\mathbb{K}$-dimension, there exists a common eigenvector $v$ in $N$ of the
operators in (4.1). We define

$\begin{array}[]{ll}vx_{i}^{l_{i}}=\alpha_{i}v,&\forall\ \ \ 1\leq i\leq n\\\
vy_{i}^{l_{i}}=\beta_{i}v,&\forall\ \ \ 1\leq i\leq n\\\
vz_{i}=\zeta_{i}v,&\forall\ \ \ 1\leq i\leq n\end{array}$

for some $\alpha_{i},\beta_{i}\in\mathbb{K}$ and $\zeta_{i}\in\mathbb{K}^{*}$.
By Schur’s lemma, each of the central elements $x_{i}^{l_{i}}$ and
$y_{i}^{l_{i}}$ act as scalars on $N$. Thus, the operators $x_{i}$ or $y_{i}$
on $N$ are either invertible or nilpotent. Now set

$\mathbb{I}:=\\{i:\alpha_{i}=0\\}\ \ \text{and}\ \
\mathbb{J}:=\\{i:\beta_{i}=0\\}.$

Clearly the set $\mathbb{I}$ (respectively $\mathbb{J}$) consists of indices
$i$ in $[1,n]$ for which the operators $x_{i}$ (respectively $y_{i}$) on $N$
are nilpotent. Depending on this, the following cases arise:
Case I: First consider the case $\mathbb{I}=\varphi$. Then $\alpha_{i}\neq 0$
for all $1\leq i\leq n$ and thus the operators $x_{i}$ on $N$ are invertible
for all $1\leq i\leq n$. Then the vectors

$vx_{n}^{a_{n}}\cdots x_{1}^{a_{1}},\ 0\leq a_{i}\leq l_{i}-1$

are nonzero in $N$. Let us set for each $i\in[1,n]$:

$\gamma_{i}:=\zeta_{i}\ \ \text{and}\ \ \mu_{i}:=l_{i}\text{-th root of}\
\alpha_{i}.$

Thus
$\underline{\mu}(\mathbb{I},\mathbb{J}):=(\mu_{1},\cdots,\mu_{n})\in(\mathbb{K}^{*})^{n}$
and
$\underline{\gamma}(\mathbb{I}):=(\gamma_{1},\cdots,\gamma_{n})\in(\mathbb{K}^{*})^{n}$.
Now define a $\mathbb{K}$-linear map

$\phi:M\left(\underline{\mu}(\mathbb{I},\mathbb{J}),\underline{\gamma}(\mathbb{I})\right)\longrightarrow
N$

by specifying the image of basis vectors:

$\phi\left(e(a_{1},\cdots,a_{n})\right):=\left(\prod\limits_{i=1}^{n}\mu_{i}^{-a_{i}}\right)vx_{n}^{a_{n}}\cdots
x_{1}^{a_{1}}.$

We can easily check that $\phi$ is a nonzero
${{A}_{n}^{\underline{q},\Lambda}}(\mathbb{K})$-module homomorphism. Thus by
Schur’s lemma, $\phi$ is an isomorphism.
Case II: Next consider the case $\mathbb{I}\neq\varphi$. Then for each
$i\in\mathbb{I}$, the operator $x_{i}$ on $N$ is nilpotent, and hence
$\operatorname{ker}(x_{i}):=\\{m\in N:mx_{i}=0\\}$ is a nonzero subspace of
$N$. We now claim that
$\cap_{\begin{subarray}{c}i\in\mathbb{I}\end{subarray}}\operatorname{ker}(x_{i})\neq\\{0\\}$.
Indeed if $i_{1},i_{2}\in\mathbb{I}$, then the nilpotent operator $x_{i_{2}}$
on $N$ must keep $\operatorname{ker}(x_{i_{1}})$ invariant because of the
commutation relation between $x_{i_{1}}$ and $x_{i_{2}}$. Hence $x_{i_{2}}$ is
also nilpotent on the subspace $\operatorname{ker}(x_{i_{1}})$. That is
$\operatorname{ker}(x_{i_{1}})\cap\operatorname{ker}(x_{i_{2}})\neq\\{0\\}$.
Continuing this argument we can obtain
$\cap_{i\in\mathbb{I}}\operatorname{ker}(x_{i})\neq\\{0\\}$.
Step 1: Now each of the commuting operators in (4.1) must keep the
$\mathbb{K}$-space $\cap_{i\in\mathbb{I}}\operatorname{ker}(x_{i})$ invariant
due to commutation relations with $x_{i}$ for $i\in\mathbb{I}$. Therefore we
can choose a common eigenvector
$w\in\cap_{i\in\mathbb{I}}\operatorname{ker}(x_{i})$ of the operators (4.1).
We define

$\begin{array}[]{l}wx_{i}=0,~{}i\in\mathbb{I}\\\
wx^{l_{i}}_{i}=\xi_{i}w,~{}i\in[1,n]\setminus\mathbb{I}\\\
wy^{l_{i}}_{i}=\eta_{i}w,~{}i\in[1,n]\\\
wz_{i}=\zeta_{i}w,~{}i\in[1,n]\end{array}$

Clearly, we can observe that

  * •

$\xi_{i}\in\mathbb{K}^{*}$ for all $i\in[1,n]\setminus\mathbb{I}$ and
$\zeta_{i}\in\mathbb{K}^{*}$ for all $i\in[1,n]$,

  * •

$\eta_{i}=0$ if $i\in\mathbb{J}$ and $\eta_{i}\in\mathbb{K}^{*}$ if
$i\in[1,n]\setminus\mathbb{J}$.

Now for each $i\in\mathbb{I}$, the relation
$x_{i}y_{i}-q_{i}y_{i}x_{i}=z_{i-1}$ induces the following

$wz_{i-1}=w(x_{i}y_{i}-q_{i}y_{i}x_{i})=-q_{i}wy_{i}x_{i}=q_{i}wz_{i}.$

Thus we obtain $q_{i}\zeta_{i}=\zeta_{i-1}$ for each $i\in\mathbb{I}$.
Step 2: We now claim that for each $i\in\mathbb{I}$, the vectors
$wy_{i}^{a_{i}},~{}0\leq a_{i}\leq l_{i}-1$ in $N$ are nonzero. From the
description of the set $\mathbb{J}$, if $i\in\mathbb{I}\setminus\mathbb{J}$
then $wy^{l_{i}}_{i}\neq 0$ 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 $wy^{l_{i}}_{i}=0$. Suppose $k_{i}$ is
the smallest index with $1\leq k_{i}\leq l_{i}$ such that
$wy_{i}^{k_{i}-1}\neq 0\ \text{and}\ wy_{i}^{k_{i}}=0$. In fact, after
simplifying the equality $wy^{k_{i}}x_{i}=0$ we obtain

$\displaystyle 0=wy^{k_{i}}x_{i}$
$\displaystyle=q_{i}^{-k_{i}}w\left(x_{i}y_{i}^{k_{i}}-\displaystyle\frac{q_{i}^{k_{i}}-1}{q_{i}-1}z_{i-1}y_{i}^{k_{i}-1}\right)$
$\displaystyle=-q_{i}^{-k_{i}}\displaystyle\frac{q_{i}^{k_{i}}-1}{q_{i}-1}wz_{i-1}y_{i}^{k_{i}-1}$
$\displaystyle=-q_{i}^{-k_{i}}\displaystyle\frac{q_{i}^{k_{i}}-1}{q_{i}-1}\zeta_{i-1}wy_{i}^{k_{i}-1}.$

This implies $k_{i}$ is the smallest index such that $q_{i}^{k_{i}}=1$ and
hence $k_{i}=l_{i}$. Thus we can conclude that for each $i\in\mathbb{I}$, the
vectors $wy_{i}^{a_{i}},~{}0\leq a_{i}\leq l_{i}-1$ are nonzero in $N$.
Step 3: Let us set for each $i\in[1,n]$:

$\mu_{i}:=\begin{cases}l_{i}\text{-th root of}\
\xi_{i},&i\in[1,n]\setminus\mathbb{I}\\\ l_{i}\text{-th root of}\
\eta_{i},&i\in\mathbb{I}\setminus(\mathbb{I}\cap\mathbb{J})\\\
0,&i\in\mathbb{I}\cap\mathbb{J}\end{cases}~{}\ \text{and}\ \
~{}\gamma_{i}:=\zeta_{i}.$

Clearly
$\underline{\mu}(\mathbb{I},\mathbb{J}):=(\mu_{1},\cdots,\mu_{n})\in\mathbb{K}^{n}$
with $\mu_{i}=0$ iff $i\in\mathbb{I}\cap\mathbb{J}$ and
$\underline{\gamma}(\mathbb{I}):=(\gamma_{1},\cdots,\gamma_{n})\in(\mathbb{K}^{*})^{n}$
with $q_{i}\gamma_{i}=\gamma_{i-1}$ if $i\in\mathbb{I}$. Any
${{A}_{n}^{\underline{q},\Lambda}}$-module homomorphism must map
$z_{i}$-eigenvectors of
$M\left(\underline{\mu}(\mathbb{I},\mathbb{J}),\underline{\gamma}(\mathbb{I})\right)$
to $z_{i}$-eigenvectors of $N$ with the same eigenvalue. Define a
$\mathbb{K}$-linear map

$\psi:M\left(\underline{\mu}(\mathbb{I},\mathbb{J}),\underline{\gamma}(\mathbb{I})\right)\longrightarrow
N$

by setting

$\psi\left(e(a_{1},\cdots,a_{n})\right):=\displaystyle wf(a_{n})\cdots
f(a_{1}),$

where

$f(a_{i})=\begin{cases}\mu_{i}^{-a_{i}}x_{i}^{a_{i}},&i\in[1,n]\setminus\mathbb{I}\\\
\mu_{i}^{-a_{i}}y_{i}^{a_{i}},&i\in\mathbb{I}\setminus(\mathbb{I}\cap\mathbb{J})\\\
y_{i}^{a_{i}},&i\in\mathbb{I}\cap\mathbb{J}.\end{cases}$

It can be easily verified that $\psi$ is a nonzero
${{A}_{n}^{\underline{q},\Lambda}}$-module homomorphism. By Schur’s lemma,
$\psi$ is an isomorphism.

Thus the above discussions lead us to the main result of this section which
provides a classification of simple
${{A}_{n}^{\underline{q},\Lambda}}$-modules in terms of scalar parameters:

######  Theorem 4.4.

Each $z_{i}$-torsionfree for all $1\leq i\leq n$ simple
${{A}_{n}^{\underline{q},\Lambda}}$-module is isomorphic to a simple
${{A}_{n}^{\underline{q},\Lambda}}$-module
$M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$ for two subsets $I$
and $J$ of $[1,n]$ with
$\underline{\mu}(I,J):=(\mu_{1},\cdots,\mu_{n})\in\mathbb{K}^{n}$ such that
$\mu_{i}=0$ iff $i\in I\cap J$ and
$\underline{\gamma}(I):=(\gamma_{1},\cdots,\gamma_{n})\in\mathbb{({K}^{*})}^{n}$
such that $q_{i}\gamma_{i}=\gamma_{i-1}$ if $i\in I$.

In view of this section together with Theorem 4.1, we have successfully
achieved a comprehensive classification of maximal dimensional simple modules
for ${{A}_{n}^{\underline{q},\Lambda}}$. Specifically, we have proved the
following:

######  Theorem 4.5.

A simple ${{A}_{n}^{\underline{q},\Lambda}}$-module $M$ is $z_{i}$-torsionfree
for all $1\leq i\leq n$ if and only if it is maximal dimensional, i.e.,
$\operatorname{dim}_{\mathbb{K}}(M)=\prod\limits_{i=1}^{n}l_{i}$.

In the next sections, we aim to determine the center and the Azumaya locus for
the quantum Weyl algebra under the roots of unity assumption ($\star$ ‣ 1).

##  5\. The Center for Quantum Weyl Algebras

In this section, we describe the centers of the PI quantized Weyl algebras
under the assumption ($\star$ ‣ 1). Let $Z(A)$ denote the center of the
algebra $A$.

######  Proposition 5.1.

The center $Z({{A}_{n}^{\underline{q},\Lambda}})$ is the subalgebra generated
by $x_{1}^{l_{1}},y_{1}^{l_{1}},\cdots,x_{n}^{l_{n}},y_{n}^{l_{n}}$ and is
isomorphic to a polynomial algebra

$Z({{A}_{n}^{\underline{q},\Lambda}})=\mathbb{K}[x_{1}^{l_{1}},y_{1}^{l_{1}},\cdots,x_{n}^{l_{n}},y_{n}^{l_{n}}].$

###### Proof.

We proceed by induction $n$. The base case $n=1$ is clear from [1, Lemma 2.2].
Fix $n\geq 1$ and let $S$ denote the subalgebra of
${{A}_{n}^{\underline{q},\Lambda}}$ generated by the elements
$x_{1}^{l_{1}},y_{1}^{l_{1}},\cdots,x_{n}^{l_{n}},y_{n}^{l_{n}}$. By
Proposition 2.2 and the defining relations of
${{A}_{n}^{\underline{q},\Lambda}}$ along with the assumptions ($\star$ ‣ 1),
the algebra $S$ is contained in the center of
${{A}_{n}^{\underline{q},\Lambda}}$ and is isomorphic to the polynomial
algebra in variables
$x_{1}^{l_{1}},y_{1}^{l_{1}},\cdots,x_{n}^{l_{n}},y_{n}^{l_{n}}$. Thus it
suffices to show that $Z({{A}_{n}^{\underline{q},\Lambda}})$ is contained in
$S$. Let $w\in Z({{A}_{n}^{\underline{q},\Lambda}})$. By the
$\mathbb{K}$-basis (2.1), we can write

$w=\sum\limits_{a,b}w(a,b)x_{n}^{a}y_{n}^{b},$

where the summation ranges over non-negative integers $a,b$ and each $w(a,b)$
is an element in the subalgebra ${{A}_{n-1}^{\underline{q},\Lambda}}$
generated by the $x_{i},y_{i}$ for all $1\leq i\leq n-1$. The inclusion
$Z({{A}_{n}^{\underline{q},\Lambda}})\subset S$ follows from the following
claim, whose proof is divided into three steps:
Claim: The element $w(a,b)$ belongs to $S$ if $l_{n}$ divides both $a$ and
$b$. Otherwise $w(a,b)=0$.
Step 1: We argue that $x_{n}$ and $y_{n}$ commutes with each $w(a,b)$. First
note that $z_{i}$ commutes with each $w(a,b)$ for all $1\leq i\leq n-1$. This
fact is a consequence of the observations that $w$ is central and $z_{i}$
commutes with $x_{n}$ and $y_{n}$ for all $1\leq i\leq n-1$. Next fix $a,b$
and abbreviate $w(a,b)$ by $w^{\prime}$. Write

$w^{\prime}=\sum\limits_{{\underline{a},\underline{b}}}c_{\underline{a},\underline{b}}x^{\underline{a}}y^{\underline{b}}$

where the sum ranges over pairs of $(n-1)$-tuples
$\underline{a}=(a_{1},\cdots,a_{n-1})$ and
$\underline{b}=(b_{1},\cdots,b_{n-1})$, and
$x^{\underline{a}}y^{\underline{b}}$ abbreviates the element
$x_{1}^{a_{1}}\cdots x_{n-1}^{a_{n-1}}y_{1}^{a_{1}}\cdots y_{n-1}^{a_{n-1}}$.
Since $z_{i}$ commutes with $w^{\prime}$ for all $1\leq i\leq n-1$, we use
Proposition 2.2 to obtain

$\sum\limits_{{\underline{a},\underline{b}}}c_{\underline{a},\underline{b}}x^{\underline{a}}y^{\underline{b}}z_{i}=w^{\prime}z_{i}=z_{i}w^{\prime}=\sum\limits_{{\underline{a},\underline{b}}}\prod\limits_{j=1}^{i}q_{j}^{(b_{j}-a_{j})}c_{\underline{a},\underline{b}}x^{\underline{a}}y^{\underline{b}}z_{i}.$

It follows that if $c_{\underline{a},\underline{b}}\neq 0$ then
$\prod\limits_{j=1}^{i}q_{j}^{(b_{j}-a_{j})}=1$ for all $1\leq i\leq n-1$.
This implies $l_{i}$ divides $(b_{i}-a_{i})$ for all $1\leq i\leq n-1$.
Consequently, using the defining relations of
${{A}_{n}^{\underline{q},\Lambda}}$, we conclude that

$x_{n}w^{\prime}=\sum\limits_{{\underline{a},\underline{b}}}c_{\underline{a},\underline{b}}x_{n}x^{\underline{a}}y^{\underline{b}}=\sum\limits_{{\underline{a},\underline{b}}}c_{\underline{a},\underline{b}}\prod\limits_{i=1}^{n-1}(q_{i}\lambda_{in})^{(b_{i}-a_{i})}x^{\underline{a}}y^{\underline{b}}x_{n}=w^{\prime}x_{n}.$

An analogous argument shows that $y_{n}w^{\prime}=w^{\prime}y_{n}$.
Step 2: We argue that $w(a,b)=0$ if $l_{n}$ does not divide both $a$ and $b$.
We have that

$x_{n}w=\sum\limits_{a,b}w(a,b)x_{n}^{a+1}y_{n}^{b}=\sum\limits_{a\geq 1,b\geq
0}w(a-1,b)x_{n}^{a}y_{n}^{b}$

and

$\begin{array}[]{ll}wx_{n}&=\left(\sum\limits_{a,b}w(a,b)x_{n}^{a}y_{n}^{b}\right)x_{n}\\\
&=\sum\limits_{a,b}q_{n}^{-b}w(a,b)x_{n}^{a+1}y_{n}^{b}-\frac{1-q_{n}^{-b}}{1-q_{n}}z_{n-1}w(a,b)x_{n}^{a}y_{n}^{b-1}\\\
&=\sum\limits_{a\geq 1,b\geq
0}q_{n}^{-b}w(a-1,b)x_{n}^{a}y_{n}^{b}-\sum\limits_{a,b}\frac{1-q_{n}^{-(b+1)}}{1-q_{n}}z_{n-1}w(a,b+1)x_{n}^{a}y_{n}^{b}.\end{array}$

Since $x_{n}w=wx_{n}$, we deduce that

$w(a-1,b)=q_{n}^{-b}w(a-1,b)+\frac{1-q_{n}^{-(b+1)}}{1-q_{n}}z_{n-1}w(a,b+1)$

for $a\geq 1$ and $b\geq 0$. Rearranging this equation, it is straightforward
to verify that this identity gives

$(1-q_{n}^{-b})w(a,b)=\frac{1-q_{n}^{-(b+i)}}{(1-q_{n})^{i}}z_{n-1}^{i}w(a+i,b+i)$

for $a,b,i\geq 0$. Suppose that $l_{n}$ does not divide $b$, so that
$q_{n}^{b}-1\neq 0$. There is a unique $i\in\\{1,\cdots,l_{n}-1\\}$ such that
$l_{n}$ divides $b+i$. For this choice of $i$, we have

$w(a,b)=\frac{(1-q_{n}^{-(b+i)})z_{n-1}^{i}w(a+i,b+i)}{(1-q_{n}^{-b})(1-q_{n})^{i}}=0.$

Analogous argument using the identity $y_{n}w=wy_{n}$ shows that $w(a,b)=0$ if
$l_{n}$ does not divide $a$.
Step 3: Finally, we show that if $l_{n}$ divides both $a$ and $b$, then
$w(a,b)$ belongs to $S$. Since $w(a,b)$ lies in the subalgebra
${{A}_{n-1}^{\underline{q},\Lambda}}$, by induction it suffices to show that
it lies in the center. This follows from the fact $x_{i}$ and $y_{i}$ commute
with $w$ for $1\leq i\leq n-1$ and that $l_{n}$ divides both $a$ and $b$
whenever $w(a,b)$ is non-zero. This completes the proof. ∎

######  Proposition 5.2.

The following identities hold in ${{A}_{n}^{\underline{q},\Lambda}}$, for
$i=1,\cdots,n$ and $z_{0}=1$:

$z_{i}^{l_{i}}=z_{i-1}^{{l_{i}}}+q_{i}^{l_{i}(l_{i}-1)/2}(q_{i}-1)^{l_{i}}y_{i}^{l_{i}}x_{i}^{l_{i}}$

###### Proof.

We claim that the following formula holds in
${{A}_{n}^{\underline{q},\Lambda}}$, for any $k\geq 1$:

$\left((q_{i}-1)y_{i}x_{i}+z_{i-1}\right)^{k}=q_{i}^{k(k-1)/2}(q_{i}-1)^{k}y_{i}^{k}x_{i}^{k}+\sum\limits_{r=1}^{k-1}c_{r}^{(k)}y_{i}^{r}x_{i}^{r}+z_{i-1}^{k}$
(5.1)

where $c_{r}^{(k)}$ are certain element of
${{A}_{n-1}^{\underline{q},\Lambda}}$. This claim can be easily verified by
induction, using the following fact:

  1. (1)

The element $z_{i-1}$ commutes with each of $x_{i}$ and $y_{i}$.

  2. (2)

For any $r\geq 1$, we have
$(y_{i}^{r}x_{i}^{r})(y_{i}x_{i})=q_{i}^{r}y_{i}^{r+1}x_{i}^{r+1}+\displaystyle\frac{q_{i}^{r}-1}{q_{i}-1}z_{i-1}y_{i}^{r}x_{i}^{r}$.

In particular the above identity (5.1) for $k=l_{i}$ becomes

$z_{i}^{l_{i}}=q_{i}^{l_{i}(l_{i}-1)/2}(q_{i}-1)^{l_{i}}y_{i}^{l_{i}}x_{i}^{l_{i}}+\sum\limits_{r=1}^{{l_{i}}-1}c_{r}^{({l_{i}})}y_{i}^{r}x_{i}^{r}+z_{i-1}^{l_{i}}.$

This implies

$\sum\limits_{r=1}^{{l_{i}}-1}c_{r}^{({l_{i}})}y_{i}^{r}x_{i}^{r}=z_{i}^{l_{i}}-q_{i}^{l_{i}(l_{i}-1)/2}(q_{i}-1)^{l_{i}}y_{i}^{l_{i}}x_{i}^{l_{i}}-z_{i-1}^{l_{i}}.$

The right-hand side is central, so it follows that the left-hand side is also
central. By Proposition 5.1, we conclude that $c_{r}^{(l_{i})}=0$ for $1\leq
r\leq l_{i}-1$. Thus we obtain

$z_{i}^{l_{i}}=z_{i-1}^{l_{i}}+q_{i}^{l_{i}(l_{i}-1)/2}(q_{i}-1)^{l_{i}}y_{i}^{l_{i}}x_{i}^{l_{i}},\
\ 1\leq i\leq n.$

∎

Let $S$ be the multiplicative set generated by the elements $z_{i}$ for $1\leq
i\leq n$. Since each $z_{i}$ is a normal element, $S$ forms an Ore set. We
denote the localization of ${{A}_{n}^{\underline{q},\Lambda}}$ with respect to
the Ore set $S$ as
${{B}_{n}^{\underline{q},\Lambda}}:={{A}_{n}^{\underline{q},\Lambda}}[S^{-1}]$.
Then the center of ${{B}_{n}^{\underline{q},\Lambda}}$ is obtained by
localizing $Z({{A}_{n}^{\underline{q},\Lambda}})$ with respect to the
multiplicative set $S_{c}$ generated by the elements $z_{i}^{l_{i}}$ for
$1\leq i\leq n$. Therefore

$Z({{B}_{n}^{\underline{q},\Lambda}})=\mathbb{K}[x_{1}^{l_{1}},y_{1}^{l_{1}},\cdots,x_{n}^{l_{n}},y_{n}^{l_{n}}][S_{c}^{-1}].$

Now it follows from Corollary 2.8 that the algebra
${{B}_{n}^{\underline{q},\Lambda}}$ is a PI algebra with $\operatorname{PI-
deg}({{B}_{n}^{\underline{q},\Lambda}})=\operatorname{PI-
deg}({{A}_{n}^{\underline{q},\Lambda}})$.

##  6\. The Azumaya Locus for Quantum Weyl Algebras

Assume the assumption ($\star$ ‣ 1) on the defining multiparameters of
${{A}_{n}^{\underline{q},\Lambda}}$. Then the algebra
${{A}_{n}^{\underline{q},\Lambda}}$ is prime affine PI algebra satisfying the
hypothesis (H) with $\operatorname{PI-
deg}{{A}_{n}^{\underline{q},\Lambda}}=\prod\limits_{i=1}^{n}l_{i}$. This
allows us to define the Azumaya locus of ${{A}_{n}^{\underline{q},\Lambda}}$,
which parameterizes the simple ${{A}_{n}^{\underline{q},\Lambda}}$-modules of
maximal dimension. As $\mathbb{K}$ is algebraically closed, it is clear from
Proposition 5.1, the maximal ideals of $Z({{A}_{n}^{\underline{q},\Lambda}})$
are of the form

$\mathfrak{m}:=\langle x_{i}^{l_{i}}-\alpha_{i},y_{i}^{l_{i}}-\beta_{i}:1\leq
i\leq n\rangle$

for some scalars $\alpha_{i},\beta_{i}\in\mathbb{K}$. Suppose

$\chi_{\mathfrak{m}}:Z({{A}_{n}^{\underline{q},\Lambda}})\rightarrow\mathbb{K}$

is the central character corresponding to $\mathfrak{m}\in
Z({{A}_{n}^{\underline{q},\Lambda}})$. Then
$\chi_{\mathfrak{m}}(x_{i}^{l_{i}})=\alpha_{i}$ and
$\chi_{\mathfrak{m}}(y_{i}^{l_{i}})=\beta_{i}$. As each $z_{i}^{l_{i}}$ is
central, therefore using Proposition 5.2, we can write

$\chi_{\mathfrak{m}}(z_{i}^{l_{i}})=\chi_{\mathfrak{m}}(z_{i-1}^{l_{i}})+q_{i}^{l_{i}(l_{i}-1)/2}(q_{i}-1)^{l_{i}}\beta_{i}\alpha_{i}\
\ \text{for}\ \ 1\leq i\leq n.$

By Theorem 4.5, a simple ${{A}_{n}^{\underline{q},\Lambda}}$-module $N$ is
maximal dimensional if and only if the action of each $z_{i}$ on $N$ is
invertible. Therefore a simple ${{A}_{n}^{\underline{q},\Lambda}}$-module is
maximal dimensional if and only if it is a simple module over
${{B}_{n}^{\underline{q},\Lambda}}$. Consequently, the Azumaya locus of
${{A}_{n}^{\underline{q},\Lambda}}$ is given by

$\displaystyle\mathcal{AL}\left({{A}_{n}^{\underline{q},\Lambda}}\right)$
$\displaystyle=\\{\mathfrak{m}\in\operatorname{Maxspec}Z({{A}_{n}^{\underline{q},\Lambda}}):\chi_{\mathfrak{m}}(z_{i}^{l_{i}})\neq
0,~{}\forall~{}1\leq i\leq n\\}$
$\displaystyle=\operatorname{Maxspec}Z({{B}_{n}^{\underline{q},\Lambda}}).$

Thus we have established the following

######  Theorem 6.1.

The Azumaya locus of ${{A}_{n}^{\underline{q},\Lambda}}$ is
$\operatorname{Maxspec}Z({{B}_{n}^{\underline{q},\Lambda}})$. Moreover,
${{B}_{n}^{\underline{q},\Lambda}}$ is Azumaya algebra over
$Z({{B}_{n}^{\underline{q},\Lambda}})$.

##  7\. Alternative Quantum Weyl Algebras

Another family of multiparameter quantized Weyl algebras has been studied in
the literature, including [3]. This family exhibits more symmetric defining
relations compared to those of ${{A}_{n}^{\underline{q},\Lambda}}$. Given such
$\Lambda$ and $\underline{q}$, the multiparameter quantized Weyl algebra
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ of symmetric type due to M.
Akhavizadegan and D. Jordan, called as alternative quantum Weyl algebra, is
the algebra generated over the field $\mathbb{K}$ by the variables
$x_{1},y_{1},\cdots,x_{n},y_{n}$ subject to the following relations:

$\displaystyle y_{i}y_{j}=\lambda_{ij}y_{j}y_{i}\ \ \ \ \ \forall\ \ \ 1\leq
i<j\leq n,$ $\displaystyle x_{i}x_{j}=\lambda_{ij}x_{j}x_{i}\ \ \ \ \ \forall\
\ \ 1\leq i<j\leq n,$ $\displaystyle x_{i}y_{j}=\lambda_{ij}^{-1}y_{j}x_{i}\ \
\ \ \ \forall\ \ \ 1\leq i<j\leq n,$ $\displaystyle
y_{i}x_{j}=\lambda_{ij}^{-1}x_{j}y_{i}\ \ \ \ \ \forall\ \ \ 1\leq i<j\leq n,$
$\displaystyle x_{i}y_{i}-q_{i}y_{i}x_{i}=1\ \ \ \ \ \forall\ \ \ 1\leq i\leq
n.$

The algebra ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ has an iterated skew
polynomial presentation with respect to the ordering of indices
$y_{1},x_{1},\cdots,y_{n},x_{n}$ of the form:

$\mathbb{K}[y_{1}][x_{1},\tau_{1},\delta_{1}][y_{2},\sigma_{2}][x_{2},\tau_{2},\delta_{2}]\cdots[y_{n},\sigma_{n}][x_{n},\tau_{n},\delta_{n}]$

where the $\tau_{j}$ and $\sigma_{j}$ are $\mathbb{K}$-linear automorphisms
and the $\delta_{j}$ are $\mathbb{K}$-linear $\tau_{j}$-derivations such that

$\displaystyle\tau_{j}(y_{i})$ $\displaystyle=\lambda_{ij}y_{i},\ i<j$
$\displaystyle\sigma_{j}(y_{i})$ $\displaystyle=\lambda_{ij}^{-1}y_{i},\ i<j$
$\displaystyle\tau_{j}(x_{i})$ $\displaystyle=\lambda_{ij}^{-1}x_{i},\ i<j$
$\displaystyle\sigma_{j}(x_{i})$ $\displaystyle=\lambda_{ij}x_{i},\ i<j$
$\displaystyle\tau_{j}(y_{j})$ $\displaystyle=q_{j}y_{j},\ \forall~{}j$
$\displaystyle\delta_{j}(x_{i})$ $\displaystyle=\delta_{j}(y_{i})=0,\ i<j$
$\displaystyle\delta_{j}(y_{j})$ $\displaystyle=1,\ \forall~{}j$

Using the skew Hilbert basis theorem, we can conclude that the algebra
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ is an affine noetherian domain.
Moreover, the family

$\\{y_{1}^{a_{1}}x_{1}^{b_{1}}\cdots
y_{n}^{a_{n}}x_{n}^{b_{n}}~{}:~{}a_{i},b_{i}\in\mathbb{Z}_{\geq 0}\\}$

is a $\mathbb{K}$-basis of ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$.
Similar to the quantum Weyl algebra, in the context of the alternative quantum
Weyl algebra, we define $z_{0}=1$ and

$z_{i}=x_{i}y_{i}-y_{i}x_{i}\ \ \forall\ \ 1\leq i\leq n.$

These elements will also play a crucial role in this section. It can be easily
verified that for all $1\leq i\leq n$,

${z}_{i}=1+(q_{i}-1)y_{i}x_{i}=q_{i}^{-1}\left(1+(q_{i}-1)x_{i}y_{i}\right).$

The following proposition is well known (cf. [3]) for the algebra
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ and can be proved using the
defining relations.

######  Proposition 7.1.

Direct computations yield the following results for
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$.

  1. (1)

For all $1\leq i\leq n$, $z_{i}$ is a normal element of
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$. More precisely, we have:

     * (a)

For all $i,j,~{}~{}1\leq i\neq j\leq n$, $z_{i}x_{j}=x_{j}z_{i}$ and
$z_{i}y_{j}=y_{j}z_{i}$

     * (b)

For all $i,~{}~{}1\leq i\leq n$, $z_{i}x_{i}=q_{i}^{-1}x_{i}z_{i}$ and
$z_{i}y_{i}=q_{i}y_{i}z_{i}$

     * (c)

For all $i,j$ with $1\leq i,j\leq n$, $z_{i}z_{j}=z_{j}z_{i}$.

  2. (2)

For $k\geq 1$, the following identities hold in the algebra
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$

     * (a)

$x_{i}^{k}y_{i}=q_{i}^{k}y_{i}x_{i}^{k}+\left(1+q_{i}+\cdots+q_{i}^{k-1}\right)x_{i}^{k-1}$,

     * (b)

$x_{i}y_{i}^{k}=q_{i}^{k}y_{i}^{k}x_{i}+\left(1+q_{i}+\cdots+q_{i}^{k-1}\right)y_{i}^{k-1}$.

In this section, our objective is to establish analogous results for
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ in parallel to those obtained for
${{A}_{n}^{\underline{q},\Lambda}}$. To begin, we derive the following
necessary and sufficient conditions for the alternative quantum Weyl algebra
to be a PI algebra.

######  Proposition 7.2.

The alternative quantum Weyl algebra
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ is a PI algebra if and only if the
parameters $q_{i}$ and $\lambda_{ij}$ are roots of unity.

###### Proof.

The argument is parallel to Proposition 2.4, based on the defining relations
of ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ and the identities in
Proposition 7.1. ∎

We denote by ${\mathcal{B}_{n}^{\underline{q},\Lambda}}$ the localization of
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ with respect to the Ore set
generated by the normal elements $z_{i},\ 1\leq i\leq n$. Now we recall a
result from [3] which states that the localizations of both versions of
quantum Weyl algebras are isomorphic.

######  Theorem 7.3.

_([3, 1.7])_ The map
$\theta:{\mathcal{B}_{n}^{\underline{q},\Lambda}}\longrightarrow{{B}_{n}^{\underline{q},\Lambda}}$
defined by

$\theta(y_{i})=y_{i},\theta(x_{i})=z_{i-1}^{-1}x_{i}$

with $z_{0}=1$ is an isomorphism of $\mathbb{K}$-algebras.

The above isomorphism will play a crucial role to obtain the PI degree,
maximal dimensional simple module, center, and the Azumaya locus for
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$.

###  7.1. PI degree for ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$

Here we wish to compute the PI degree of
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$, taking into account the roots of
unity assumption ($\star$ ‣ 1) concerning the defining multiparameters
${q_{i}}$ and $\lambda_{ij}$. In the roots of unity setting, it follows from
Corollary 2.8 that the algebras ${B}_{n}^{\underline{q},\Lambda}$ and
$\mathcal{B}_{n}^{\underline{q},\Lambda}$ are PI algebras and

$\operatorname{PI-deg}({{A}_{n}^{\underline{q},\Lambda}})=\operatorname{PI-
deg}({B}_{n}^{\underline{q},\Lambda}),\ \operatorname{PI-
deg}({\mathcal{A}_{n}^{\underline{q},\Lambda}})=\operatorname{PI-
deg}(\mathcal{B}_{n}^{\underline{q},\Lambda}).$

Therefore using the isomorphism in Theorem 7.3 we obtain

$\operatorname{PI-deg}({A}_{n}^{\underline{q},\Lambda})=\operatorname{PI-
deg}(\mathcal{A}_{n}^{\underline{q},\Lambda}).$

Consequently, based on Theorem 3.1, we have now established that

######  Theorem 7.4.

The PI degree of ${A}_{n}^{\underline{q},\Lambda}$ is equal to the PI degree
of ${\mathcal{B}_{n}^{\underline{q},\Lambda}}$, and both are given by

$\operatorname{PI-
deg}{\mathcal{A}_{n}^{\underline{q},\Lambda}}=\operatorname{PI-
deg}{\mathcal{B}_{n}^{\underline{q},\Lambda}}=\prod\limits_{i=1}^{n}l_{i}.$
(7.1)

######  Remark 7.5.

It is worth noting that there exists a simple
$\mathcal{A}_{n}^{\underline{q},\Lambda}$-module with $\mathbb{K}$-dimension
$\prod\limits_{i=1}^{n}l_{i}$. Subsection 7.3 will present an explicit
construction and classification of these simple modules.

###  7.2. PI degree for Prime Factors of
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$

Fix an index $r$ with $1\leq r\leq n$. Then the ideal generated by the normal
element $z_{r}:=x_{r}y_{r}-y_{r}x_{r}$ is a completely prime ideal of
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ (see [3, Proposition 4.5]) and
hence the factor algebra ${\mathcal{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle$ is a domain. Note that the algebra
${\mathcal{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle$ is a prime
affine PI algebra under the root of unity assumption ($\star$ ‣ 1). Here we
wish to compute the PI degree of the factor algebra
${\mathcal{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle$ explicitly.
Step 1: First we consider an algebra that is similar to Step 1 in Section 3.2.
Let $\mathcal{R}_{r}$ denote the $\mathbb{K}$-algebra generated by the
variables $X_{1},Y_{1},\cdots,X_{n},Y_{n}$ subject to the relations:

$\displaystyle Y_{i}Y_{j}$ $\displaystyle=\lambda_{ij}Y_{j}Y_{i}\ \ \ \ \
\forall\ \ \ 1\leq i<j\leq n$ $\displaystyle X_{i}X_{j}$
$\displaystyle=\lambda_{ij}X_{j}X_{i}\ \ \ \ \ \forall\ \ \ 1\leq i<j\leq n$
$\displaystyle X_{i}Y_{j}$ $\displaystyle=\lambda_{ij}^{-1}Y_{j}X_{i}\ \ \ \ \
\forall\ \ \ 1\leq i<j\leq n$ (7.2) $\displaystyle Y_{i}X_{j}$
$\displaystyle=\lambda_{ij}^{-1}X_{j}Y_{i}\ \ \ \ \ \forall\ \ \ 1\leq i<j\leq
n$ $\displaystyle X_{r}Y_{r}$ $\displaystyle=Y_{r}X_{r}$ $\displaystyle
X_{i}Y_{i}$ $\displaystyle-q_{i}Y_{i}X_{i}=1\ \ \ \ \ \forall\ \ \ 1\leq i\neq
r\leq n$

In the root of unity setting ($\star$ ‣ 1), the algebra $\mathcal{R}_{r}$
becomes a PI algebra. The following result provides a connection between the
algebras ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ and $\mathcal{R}_{r}$.

######  Lemma 7.6.

The factor algebra ${\mathcal{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle$ is isomorphic to the factor algebra
$\mathcal{R}_{r}/\mathcal{I}_{r}$, where $\mathcal{I}_{r}$ is the ideal of
$\mathcal{R}_{r}$ generated by $1+(q_{r}-1)Y_{r}X_{r}$.

The proof of this lemma follows a similar approach as that of Lemma 3.3. As a
result, based on Remark 2.7, we can conclude that

$\operatorname{PI-deg}({\mathcal{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle)=\operatorname{PI-
deg}{(\mathcal{R}_{r}/\mathcal{I}_{r})}\leq\operatorname{PI-
deg}{(\mathcal{R}_{r})}.$ (7.3)

Step 2: Our objective now is to determine the PI degree of the algebra
$\mathcal{R}_{r}$. By following the same reasoning outlined in Step 2 of
Section 3.2, we can establish that

$\operatorname{PI-
deg}(\mathcal{R}_{r})=\prod\limits_{\begin{subarray}{c}i=1\\\ i\neq
r\end{subarray}}^{n}l_{i}.$

Consequently, using (7.3), we obtain the inequality

$\operatorname{PI-deg}({\mathcal{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle)\leq\displaystyle\prod\limits_{\begin{subarray}{c}i=1\\\ i\neq
r\end{subarray}}^{n}l_{i}.$ (7.4)

Step 3: Finally we aim to establish the equality in (7.4). It is important to
note that the alternative quantum Weyl algebra
${\mathcal{A}_{n-1}^{\underline{q}(r),\Lambda(r)}}$ can be viewed as a
subalgebra of the factor algebra
${\mathcal{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle$. Here
$\underline{q}(r)$ is the $(n-1)$-tuple elements of
$\mathbb{K}\setminus\\{0,1\\}$ obtained from $\underline{q}$ by removing the
$r$-th element and $\Lambda(r)$ is the $(n-1)\times(n-1)$ multiplicatively
antisymmetric matrix obtained by simultaneously removing the $r$-th row and
column from $\Lambda$. Thus from Remark 2.7 we have:

$\operatorname{PI-
deg}({\mathcal{A}_{n-1}^{\underline{q}(r),\Lambda(r)}})\leq\operatorname{PI-
deg}({\mathcal{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle)$ (7.5)

As a consequence of (7.1), we can establish that

$\operatorname{PI-
deg}({\mathcal{A}_{n-1}^{\underline{q}(r),\Lambda(r)}})=\displaystyle\prod\limits_{\begin{subarray}{c}i=1\\\
i\neq r\end{subarray}}^{n}l_{i}.$ (7.6)

Consequently, we have established the following

######  Theorem 7.7.

Assuming the root of unity assumption ($\star$ ‣ 1) on the defining
multiparameters, the PI degree of a prime factor of
$\mathcal{A}_{n}^{\underline{q},\Lambda}$ by the normal element $z_{r}$ is
given by $\displaystyle\prod\limits_{\begin{subarray}{c}i=1\\\ i\neq
r\end{subarray}}^{n}l_{i}.$

Question: In view of Proposition 2.9, there exists a $z_{r}$-torsion simple
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$-module with $\mathbb{K}$-dimension
$\prod\limits_{\begin{subarray}{c}i=1\\\ i\neq r\end{subarray}}^{n}l_{i}$ for
each $1\leq r\leq n$. It would be interesting to explore the construction of
such simple modules and subsequently classify them. So let us give it a try!

###  7.3. Classification of Maximal Dimensional Simple Modules

Here we classify all maximal dimensional simple
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$-modules. As a direct consequence
of Theorem 7.3, we can derive the following result, establishing a significant
connection between the simple modules of both versions of quantum Weyl
algebras.

######  Proposition 7.8.

There is a a one-to-one correspondence between the simple $z_{i}$-torsionfree
${{A}_{n}^{\underline{q},\Lambda}}$-modules and the simple $z_{i}$-torsionfree
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$-modules.

In light of this proposition along with Theorem 4.4, we can classify simple
$z_{i}$-torsionfree $\mathcal{A}_{n}^{\underline{q},\Lambda}$-modules. In
particular, each $z_{i}$-torsionfree for all $1\leq i\leq n$ simple
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$-module is isomorphic to a simple
${{A}_{n}^{\underline{q},\Lambda}}$-module
$M\left(\underline{\mu}(I,J),\underline{\gamma}(I)\right)$. Moreover, the
$\mathbb{K}$-dimension of each simple $z_{i}$-torsionfree
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$-module is equal to
$\operatorname{PI-deg}{\mathcal{A}_{n}^{\underline{q},\Lambda}}$ which is
maximal.

Next, we aim to prove that each maximal dimensional (i.e., equal to the
$\operatorname{PI-deg}{\mathcal{A}_{n}^{\underline{q},\Lambda}}$) simple
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$-module is $z_{i}$-torsionfree for
all $1\leq i\leq n$. If possible let $M$ be a maximal dimensional simple
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$-module which is $z_{r}$-torsion
for some $1\leq r\leq n$. As $z_{r}$ is a normal element then $M$ becomes a
simple module over the factor algebra
${\mathcal{A}_{n}^{\underline{q},\Lambda}}/\langle z_{r}\rangle$. Hence by
Proposition 2.9 along with Theorem 7.7, we obtain

$\operatorname{dim}_{\mathbb{K}}(M)\leq\operatorname{PI-
deg}({\mathcal{A}_{n}^{\underline{q},\Lambda}}/\langle
z_{r}\rangle)<\operatorname{PI-
deg}({\mathcal{A}_{n}^{\underline{q},\Lambda}}).$

This is a contradiction.

Consequently, we have achieved a comprehensive classification of maximal
dimensional simple ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$-modules. In
particular, we have established the following.

######  Theorem 7.9.

A simple ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$-module $M$ is
$z_{i}$-torsionfree for all $1\leq i\leq n$ if and only if it is maximal
dimensional, i.e.,
$\operatorname{dim}_{\mathbb{K}}(M)=\prod\limits_{i=1}^{n}l_{i}$.

###  7.4. The Center and Azumaya Locus for
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$

In this subsection, we describe $Z({\mathcal{A}_{n}^{\underline{q},\Lambda}})$
and determine the Azumaya locus. By the isomorphism $\theta$ in Theorem 7.3,
we have

$Z({\mathcal{B}_{n}^{\underline{q},\Lambda}})=\theta^{-1}(Z({{B}_{n}^{\underline{q},\Lambda}}))=\theta^{-1}(\mathbb{K}[x_{1}^{l_{1}},y_{1}^{l_{1}},\cdots,x_{n}^{l_{n}},y_{n}^{l_{n}}][S_{c}^{-1}]).$
(7.7)

From the definition of $\theta$, it follows that

$\theta^{-1}(x_{i}^{l_{i}})=(z_{1}\cdots z_{i-1}x_{i})^{l_{i}},\
\theta^{-1}(y_{i}^{l_{i}})=y_{i}^{l_{i}},\
\theta^{-1}(z_{i}^{l_{i}})=(z_{1}\cdots z_{i-1}z_{i})^{l_{i}}$

and $\mathcal{S}_{c}:=\theta^{-1}(S_{c})$ is also a multiplicative set in
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ contained in the multiplicative
set $T_{c}$ generated by the $z_{i}^{l_{i}},\ 1\leq i\leq n$. Now applying
$\theta^{-1}$ in the identity of Proposition 5.2, we obtain the following
identity for the central element $z_{i}^{l_{i}}$ in
$\mathcal{A}_{n}^{\underline{q},\Lambda}$

$z_{i}^{l_{i}}=1+q_{i}^{l_{i}(l_{i}-1)/2}(q_{i}-1)^{l_{i}}y_{i}^{l_{i}}x_{i}^{l_{i}},\
\ 1\leq i\leq n$ (7.8)

Therefore from (7.7),

$\displaystyle Z({\mathcal{B}_{n}^{\underline{q},\Lambda}})$
$\displaystyle=\mathbb{K}[\theta^{-1}(x_{i}^{l_{i}}),\theta^{-1}(y_{i}^{l_{i}}):1\leq
i\leq n][\mathcal{S}_{c}^{-1}]$
$\displaystyle=\mathbb{K}[x_{i}^{l_{i}},y_{i}^{l_{i}}:1\leq i\leq
n][T_{c}^{-1}].$

Thus the center of $\mathcal{A}_{n}^{\underline{q},\Lambda}$ is

$Z(\mathcal{A}_{n}^{\underline{q},\Lambda})=Z({\mathcal{B}_{n}^{\underline{q},\Lambda}})\cap\mathcal{A}_{n}^{\underline{q},\Lambda}=\mathbb{K}[x_{i}^{l_{i}},y_{i}^{l_{i}}:1\leq
i\leq n].$

As $\mathbb{K}$ is algebraically closed, the maximal ideals of
$Z({\mathcal{A}_{n}^{\underline{q},\Lambda}})$ are of the form

$\mathfrak{m}:=\langle x_{i}^{l_{i}}-\alpha_{i},y_{i}^{l_{i}}-\beta_{i}:1\leq
i\leq n\rangle$

for some scalars $\alpha_{i},\beta_{i}\in\mathbb{K}$. Suppose

$\chi_{\mathfrak{m}}:Z({\mathcal{A}_{n}^{\underline{q},\Lambda}})\rightarrow\mathbb{K}$

is the central character corresponding to $\mathfrak{m}\in
Z({\mathcal{A}_{n}^{\underline{q},\Lambda}})$. Then
$\chi_{\mathfrak{m}}(x_{i}^{l_{i}})=\alpha_{i}$ and
$\chi_{\mathfrak{m}}(y_{i}^{l_{i}})=\beta_{i}$. Hence from (7.8), we can write

$\chi_{\mathfrak{m}}(z_{i}^{l_{i}})=1+q_{i}^{l_{i}(l_{i}-1)/2}(q_{i}-1)^{l_{i}}\beta_{i}\alpha_{i}\
\ \text{for}\ \ 1\leq i\leq n.$

By Theorem 7.9, a simple ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$-module
$N$ is maximal dimensional if and only if the action of each $z_{i}$ on $N$ is
invertible. Consequently, the Azumaya locus of
${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ is given by

$\displaystyle\mathcal{AL}\left({\mathcal{A}_{n}^{\underline{q},\Lambda}}\right)$
$\displaystyle=\\{\mathfrak{m}\in\operatorname{Maxspec}Z({\mathcal{A}_{n}^{\underline{q},\Lambda}}):\chi_{\mathfrak{m}}(z_{i}^{l_{i}})\neq
0,~{}\forall~{}1\leq i\leq n\\}$
$\displaystyle=\operatorname{Maxspec}Z({\mathcal{B}_{n}^{\underline{q},\Lambda}}).$

Thus we have established the following

######  Theorem 7.10.

The center of $\mathcal{A}_{n}^{\underline{q},\Lambda}$ is
$\mathbb{K}[x_{i}^{l_{i}},y_{i}^{l_{i}}:1\leq i\leq n]$ and the Azumaya locus
of ${\mathcal{A}_{n}^{\underline{q},\Lambda}}$ is
$\operatorname{Maxspec}Z({\mathcal{B}_{n}^{\underline{q},\Lambda}})$.
Moreover, ${\mathcal{B}_{n}^{\underline{q},\Lambda}}$ is Azumaya algebra over
$Z({\mathcal{B}_{n}^{\underline{q},\Lambda}})$.

Finally, it is worth mentioning that the Azumaya property of the algebra
${\mathcal{B}_{n}^{\underline{q},\Lambda}}$ can also be established by
utilizing the isomorphism presented in Theorem 7.3 along with Theorem 6.1.

##  8\. Concluding Remarks

In conclusion, this chapter provides significant contributions to the study of
multiparameter quantized algebras at roots of unity. We have achieved an
explicit expression for the PI degree and the center, as well as a
comprehensive classification of the maximal dimensional simple modules.
Consequently, we have obtained the Azumaya locus of the multiparameter
quantized algebras at roots of unity.

It is worth noting that recent studies by Brown and Yakimov [5] have shed
light on the connection between the Azumaya locus $\mathcal{AL}(R)$ and the
discriminant ideal $D_{l}$ of a prime PI algebra $R$. Specifically, in their
main theorem [5, Main Theorem], they established that the zero set of $D_{l}$
is the complement of the Azumaya locus of $R$, in the case when $l$ is the
square of the PI degree of $R$. As an application, the classification of the
Azumaya locus of the multiparameter quantized algebras at roots of unity was
obtained in [5, Theorem 6.2] by determining its discriminant ideal.

Thus, in particular, this chapter provides an alternative approach to
determine the Azumaya locus of the multiparameter quantized Weyl algebras
without explicitly finding its discriminant ideal.

## Acknowledgements

The authors are grateful to the National Board of Higher Mathematics,
Department of Atomic Energy, Government of India for providing financial
support to carry out this research.

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