# Bi-quadratic algebras on 3 generators with PBW: class II.1

V. V. Bavula and A. Al Khabyah

###### Abstract

In [3], an explicit description of bi-quadratic algebras on three generators
with PBW basis was obtained. There are four classes: I-IV. The aim of the
paper is to study algebras that belong to one of the classes: class II.1. In
this class, there are twelve distinct subclasses to consider. The aim of the
paper is to study these algebras. In particular, the centre, the primes
ideals, the automorphism groups and the normal elements are explicitly
described. The Krull, classical Krull, global and Gelfand-Kirillov dimensions
of the algebras and their tensor products are computed.

Key Words: the centre, the prime ideal, the normal element, localization of a
ring, the automorphism group, the Krull dimension, the classical Krull
dimension, the global dimension, the Gelfand-Kirillov dimension, the
$\omega$-weight module, the eigenalgebra.

Mathematics subject classification 2020: 16D60, 16W20, 16S37, 16P20, 16U70,
16S36, 16D25, 16E10, 16S99, 16D60.

${\bf Contents}$

  1. 1.

Introduction.

  2. 2.

The centre of the algebra $A$.

  3. 3.

The centre, the ideal structure and the prime spectrum of the algebra
$A_{x_{1}x_{2}}$.

  4. 4.

The prime spectrum of the algebra $A$.

  5. 5.

The automorphism group of the algebra $A$.

  6. 6.

The Krull, global and Gelfand-Kirillov dimensions of the algebras ${\cal
A}=\bigotimes_{i=1}^{n}A_{i}$ where $A_{i}=A(q_{i},\alpha_{i},\mu_{i})$.

##  1 Introduction

In this paper, $K$ is a field of arbitrary characteristic char$(K)=p$ and
$K^{\times}=K\backslash\\{0\\}$, $\mathbb{F}_{p}=\mathbb{Z}/p\mathbb{Z}$ is
the field with $p$ elements, modules means a left module, $q=q_{1}\in
K\backslash\\{0,1\\}$, $A=A(q,\alpha,\mu)$ is the algebra as in Theorem
1.1.(1) (if it is not stated otherwise), $\mathbb{N}=\\{0,1,\ldots\\}$ is the
set of natural numbers and $\mathbb{N}_{+}=\\{1,2\,\ldots\\}$.

Description of bi-quadratic algebras on 3 generators with PBW basis, an
overview.

Definition, [3]. Given $Q=(q_{1},q_{2},q_{3})\in K^{\times 3},$ a $3\times 3$
matrix with entries in the field $K$, ††margin:  AAabc

$\mathbb{A}=\begin{pmatrix}a&b&c\\\ \alpha&\beta&\gamma\\\ \lambda&\mu&\nu\\\
\end{pmatrix}$ (1)

and $\mathbb{B}=(b_{1},b_{2},b_{3})\in K^{3}$. A bi-quadratic algebra on 3
generators $A=K[x_{1},x_{2},x_{3};Q,\mathbb{A},\mathbb{B}]$ is an algebra
generated by the elements $x_{1},x_{2}$ and $x_{3}$ subject to the defining
relations:
††margin:  DRel1

$x_{2}x_{1}-q_{1}x_{1}x_{2}=ax_{1}+bx_{2}+cx_{3}+b_{1},$ (2) ††margin:  DRel2
$x_{3}x_{1}-q_{2}x_{1}x_{3}=\alpha x_{1}+\beta x_{2}+\gamma x_{3}+b_{2},$ (3)
††margin:  DRel3 $x_{3}x_{2}-q_{3}x_{2}x_{3}=\lambda x_{1}+\mu x_{2}+\nu
x_{3}+b_{3}.$ (4)

Clearly, $A=\sum_{\xi\in\mathbb{N}^{3}}Kx^{\xi}$ where
$\xi=(\xi_{1},\xi_{2},\xi_{3})$ and
$x^{\xi}=x_{1}^{\xi_{1}}x_{2}^{\xi_{2}}x_{3}^{\xi_{3}}$. We say that the
algebra $A$ has PBW basis if $A=\bigoplus_{\xi\in\mathbb{N}^{3}}Kx^{\xi}$.

In [3], an explicit description of bi-quadratic algebras on 3 generators with
PBW basis was obtained. The algebras were partitioned into the following 4
classes:

I: $q_{1}=q_{2}=q_{3}=1$,

II: $q_{1}\neq 1$, $q_{2}=q_{3}=1$,

III: $q_{1}\neq 1$, $q_{2}\neq 1$, $q_{3}=1$,

IV: $q_{1}\neq 1$, $q_{2}\neq 1$, $q_{3}\neq 1$,

and algebras in each subclass were explicitly described. In [3], a
classification (up to isomorphism) of algebras that belong to the first class
($q_{1}=q_{2}=q_{3}=1$) was given. Class II consists of two subclasses: II.1
(Theorem 1.1.(1)) and II.2 (Theorem 1.1.(2)). The aim of the paper is to study
algebras (and their tensor products) that belong to class II.1. The class II.1
consists of twelve subclasses, see below.

Description of bi-quadratic algebras on 3 generators with PBW basis when
$q_{1}\neq 1$, $q_{2}=q_{3}=1$.

######  Theorem 1.1.

[3] Suppose that $q_{1}\neq 1$, $q_{2}=q_{3}=1$ and
$A=K[x_{1},x_{2},x_{3};Q,\mathbb{A},\mathbb{B}]$.

  1. 1.

If $\mu+\alpha\neq 0$ then (up to $G_{3}$):

$x_{2}x_{1}=q_{1}x_{1}x_{2},\;\;x_{3}x_{1}=x_{1}(x_{3}+\alpha)\;\;{\rm
and}\;\;x_{3}x_{2}=x_{2}(x_{3}+\mu)$

where $(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in
K\backslash\\{-1\\}\\}$. The elements $x_{1}$ and $x_{2}$ are regular normal
elements of $A$. The algebra
$A=K[x_{3}][x_{2},x_{1};\sigma,\tau,b=0,\rho=q_{1}]$ is a diskew polynomial
algebra where $\tau(x_{3})=x_{3}-\alpha$ and $\sigma(x_{3})=x_{3}-\mu$.
Furthermore, the algebra $A={\cal D}[x_{2},x_{1};\sigma,\tau,a=h]$ is a GWA
where ${\cal D}=K[x_{3}][h;\tau\sigma]$ is a skew polynomial ring where
$\tau\sigma(x_{3})=x_{3}-\alpha-\mu$, $\sigma(h)=q_{1}h$ and
$\tau(h)=q_{1}^{-1}h$. The element $h$ is a regular normal element of $A$.

    1. (a)

The algebra $A=K[x_{3}][x_{1};\sigma_{1}][x_{2};\sigma_{2}]$ is an iterated
skew polynomial algebra where $\sigma_{1}(x_{3})=x_{3}-\alpha$,
$\sigma_{2}(x_{3})=x_{3}-\mu$ and $\sigma_{2}(x_{1})=q_{1}x_{1}$.

    2. (b)

The algebra homomorphism $A(q_{1},\alpha=1,\mu=0)\rightarrow
A(q_{1}^{-1},\alpha=0,\mu=1)$, $x_{1}\mapsto x_{2}$, $x_{2}\mapsto x_{1}$,
$x_{3}\mapsto x_{3}$ is an algebra isomorphism.

    3. (c)

In particular, the algebra homomorphism $A(q_{1}=-1,\alpha=1,\mu=0)\rightarrow
A(q_{1}=-1,\alpha=0,\mu=1)$, $x_{1}\mapsto x_{2}$, $x_{2}\mapsto x_{1}$,
$x_{3}\mapsto x_{3}$ is an algebra isomorphism.

  2. 2.

If $\mu+\alpha=0$ then (up to $G_{3}$):

$x_{2}x_{1}=q_{1}x_{1}x_{2}+cx_{3}+b_{1},\;\;x_{3}x_{1}=x_{1}(x_{3}+\alpha)\;\;{\rm
and}\;\;x_{3}x_{2}=x_{2}(x_{3}-\alpha)$

where exactly one of the following cases occurs:

    1. (a)

$\alpha=0$ and $c,b_{1}\in\\{0,1\\}$,

    2. (b)

$\alpha=1$ and $(c,b_{1})\in\\{(0,0),(1,1)\\}$,

    3. (c)

$\alpha=1$, $c=0$, $b_{1}=1$ or $\alpha=1$, $c=1$, $b_{1}\in
K\backslash\\{1\\}$.

So, the cases (a)–(c) can be written as $(\alpha,c,b_{1})\in\\{0,1\\}^{3}$ or
$\alpha=1$, $c=1$, $b_{1}\in K\backslash\\{0,1\\}$. The algebra
$A=K[x_{3}][x_{2},x_{1};\sigma,\sigma^{-1},b=cx_{3}+b_{1},\rho=q_{1}]$ is a
diskew polynomial algebra where $\sigma(x_{3})=x_{3}+\alpha$. Furthermore, the
algebra $A=K[x_{3},h][x_{2},x_{1};\sigma,a=h]$ is a classical GWA where
$\sigma(x_{3})=x_{3}+\alpha$ and $\sigma(h)=q_{1}h+cx_{3}+b_{1}$.

Definitions of generalized Weyl algebras and diskew polynomial algebras are
given in [2].

$\bullet$ In this paper, $A=A(q,\alpha,\mu)$ is an algebra that belongs to
class II.1 (see Theorem 1.1.(1)).

Surprisingly, there are 12 different cases/subclasses for the algebras $A$:

$\displaystyle(C1):$ $\displaystyle\text{if $q$ is not a root of unity and
$p=0$},$ $\displaystyle(C2):$ $\displaystyle\text{if $q$ is not a root of
unity, $p>0$
and}\;(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\},$
$\displaystyle(C3):$ $\displaystyle\text{if $q$ is not a root of unity, $p>0$,
$\alpha=1$ and $\mu\not\in\mathbb{F}_{p}$},$ $\displaystyle(C4):$
$\displaystyle\text{if $q$ is a primitive $n$'th root of unity, $p=0$,
$\alpha=1$ and $\mu=0$},$ $\displaystyle(C5):$ $\displaystyle\text{if $q$ is a
primitive $n$'th root of unity, $p=0$, $\alpha=0$ and $\mu=1$},$
$\displaystyle(C6):$ if $q$ is a primitive $n$’th root of unity, $p=0$,
$\alpha=1$ and $\mu=-\frac{\mu_{1}}{\mu_{2}}$ where
$\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ $\displaystyle\text{such that
$\mu_{1}\neq\mu_{2}$ and $(\mu_{1},\mu_{2})=1$},$ $\displaystyle(C7a):$ if $q$
is a primitive $n$’th root of unity, $p=0$, $\alpha=1$ and
$\mu=\frac{\mu_{1}}{\mu_{2}}$ where $\mu_{1},\mu_{2}\in\mathbb{N}_{+}$
$\displaystyle\text{such that $(\mu_{1},\mu_{2})=1$},$ $\displaystyle(C7b):$
$\displaystyle\text{ if $q$ is a primitive $n$'th root of unity, $p=0$,
$\alpha=1$ and $\mu\not\in\mathbb{Q}$},$ $\displaystyle(C8):$
$\displaystyle\text{if $q$ is a primitive $n$'th root of unity, $p>0$,
$\alpha=1$, $\mu=0$ and $p\nmid n$},$ $\displaystyle(C9):$ if $q$ is a
primitive $n$’th root of unity, $p>0$, $\alpha=0$, $\mu=1$ and $p\nmid n$,
$\displaystyle(C10):$ $\displaystyle\text{if $q$ is a primitive $n$'th root of
unity, $p>0$, $\alpha=1$, $\mu\not\in\mathbb{F}_{p}$ and $p\nmid n$},$
$\displaystyle(C11):$ if $q$ is a primitive $n$’th root of unity, $p>0$,
$\alpha=1$, $\mu\in\mathbb{F}_{p}\backslash\\{0,-1\\}=\\{1,\ldots,p-2\\}$ and
$p\nmid n$.

Algebras in these case have different properties and different techniques are
used to study them. The paper is organized as follows.

In Section 2, the centre of the algebra $A$ is described (Theorem 2.3). It is
proven that in the cases (C1)–(C10) the algebra $A$ is a free module over the
centre and an explicit free basis is presented in each case (Theorem 2.4). The
algebra $A$ is a finitely generated module over the centre iff the algebra $A$
belongs to one of the cases C(8)–(C11) (Corollary 2.5).

In Section 3, properties of the localization $A_{x_{1}x_{2}}$ of the algebra
$A$ at the powers of the regular normal element $x_{1}x_{2}$ are considered.
They are used in the classification of the prime ideals of the algebra $A$.
The centre of the algebra $A_{x_{1}x_{2}}$ is described (Theorem 3.2). Ideals
and prime ideals of the algebra $A_{x_{1}x_{2}}$ are classified (Theorem 3.9
and Theorem 3.10). The algebra $A_{x_{1}x_{2}}$ is a free module over its
centre (Theorem 3.4) and an explicit set of free generators is presented (in
each of 12 cases). The elements $x_{1}$ and $x_{2}$ are units of the algebra
$A_{x_{1}x_{2}}$. Let $\omega_{x_{1}}$ and $\omega_{x_{2}}$ be the inner
automorphism of $A_{x_{1}x_{2}}$ that are defined by the elements $x_{1}$ and
$x_{2}$, respectively. Let ${\rm ad}_{x_{3}}$ be the inner derivation of
$A_{x_{1}x_{2}}$ that is determined by the element $x_{3}$. In the triple
$\omega=(\omega_{x_{1}},\omega_{x_{2}},{\rm ad}_{x_{3}})$, the $K$-linear maps
commute. The algebra $A_{x_{1}x_{2}}$ is a generalized weight $\omega$-module
(Proposition 3.5). The $\omega$-eigenalgebra ${\rm
Es}_{A_{x_{1}x_{2}}}(\omega)$ is described (Corollary 3.6). A simplicity
criterion is given for the algebra $A_{x_{1}x_{2}}$ (Theorem 3.8). The algebra
$A_{x_{1}x_{2}}$ is a free left/right ${\rm
Es}_{A_{x_{1}x_{2}}}(\omega)$-module (Corollary 3.6.(2)).

In Section 4, a classification of prime ideals of the algebra $A$ is obtained
(see (15), (18) and Theorem 4.6).

In Section 5, an explicit description the automorphism group ${\rm
Aut}_{K}(A)$ is given (Theorem 5.2), classifications of $\omega$-weight
vectors (Theorem 5.3) and normal elements (Proposition 5.4) of the algebra $A$
are presented. It is shown that every normal element of the algebra $A$ is an
$\omega$-weight vector, and vice versa (Proposition 5.4).

In Section 6, the Krull, classical Krull, global and Gelfand-Kirillov
dimensions of the algebras ${\cal A}=\bigotimes_{i=1}^{n}A_{i}$ are computed
where $A_{i}=A(q_{i},\alpha_{i},\mu_{i})$ (Proposition 6.1.(3) and Theorem
6.2).

##  2 The centre of the algebra $A$

In this section, the centre of the algebra $A$ is described (Theorem 2.3).
There are eleven distinct cases (C1)–(C11) (Theorem 2.3). It is proven that in
the cases (C1)–(C10) the algebra $A$ is a free module over the centre and an
explicit free basis is presented in each case (Theorem 2.4). The algebra $A$
is a finitely generated module over the centre iff the algebra $A$ belongs to
one of the cases C(8)–(C11) (Corollary 2.5).

$\mathbb{N}^{2}$-grading of the algebra $A$. By Theorem 1.1.(1).(a), the
algebra $A$ is an $\mathbb{N}^{2}$-graded algebra,

$A=\bigoplus_{\beta\in\mathbb{N}^{2}}K[x_{3}]x^{\beta}\;\;{\rm
where}\;\;x^{\beta}=x_{1}^{\beta_{1}}x_{2}^{\beta_{2}}\;\;{\rm
and}\;\;\beta=(\beta_{1},\beta_{2}).$ (5)

The product in the algebra $A$ is given by the rule: For all elements $a,b\in
K[x_{3}]$ and $\beta,\gamma\in\mathbb{N}^{2}$,

$ax^{\beta}\cdot
bx^{\gamma}=a\omega^{\beta}(b)q^{\beta_{2}\gamma_{1}}x^{\beta+\gamma}\;\;{\rm
where}\;\;\omega^{\beta}=\omega_{x_{1}}^{\beta_{1}}\omega_{x_{2}}^{\beta_{2}}.$

So, the algebra $A$ is a skew crossed product of the (additive) monoid
$\mathbb{N}^{2}$ with the polynomial algebra $K[x_{3}]$. For all elements
$\beta\in\mathbb{N}^{2}$,

$[x_{3},x^{\beta}]=(\beta_{1}\alpha+\beta_{2}\mu)x^{\beta}$ (6)

where $[a,b]:=ab-ba$ is the commutator of elements $a$ and $b$. So, the direct
sum (5) is a direct sum of eigenspaces for the inner derivation

${\rm ad}_{x_{3}}:A\rightarrow A,\;\;a\mapsto{\rm ad}_{x_{3}}(a):=[x_{3},a]$

of the algebra $A$.

The algebra $K[x_{3}]^{\omega_{x_{1}},\omega_{x_{2}}}$. Given an algebra $R$,
the set of all regular elements of $R$ (i.e. non-zero-divisors) is denoted by
${\cal C}_{R}$. An element $x\in R$ is called a normal element of $R$ if
$Rx=xR$. Let ${\cal N}(R)$ be the set of normal elements of the algebra $R$.
The set ${\cal N}(R)$ is a multiplicative monoid that contains the group of
units $R^{\times}$ and the centre $Z(R)$ of the algebra $R$. Every regular
normal element $x$ of $R$ determines an automorphism $\omega_{x}$ of the
algebra $R$ which is given by the rule:

$xr=\omega_{x}(r)x\;\;{\rm for\;all}\;\;r\in R.$ (7)

If $x\in R^{\times}$ then $\omega_{x}(r)=xrx^{-1}$ for all $r\in R$, i.e.
$\omega_{x}$ is an inner automorphism of $R$ that is determined by the element
$x$. Let $V$ be a $K$-vector space. A $K$-linear map $\phi:V\rightarrow V$ is
called a locally finite  (resp., locally nilpotent) map if for each element
$v\in V$, ${\rm dim}_{K}(K[\phi]v)<\infty$ (resp., $\phi^{i}v$ for all $i\gg
1$). A locally nilpotent map is a locally finite but not vice versa, in
general.

The elements $x_{1}$ and $x_{2}$ of the algebra $A$ are regular normal
elements. Clearly, the automorphisms $\omega_{x_{1}}$ and $\omega_{x_{2}}$ of
the algebra $A$ act on its generators by the rule:

$\displaystyle\omega_{x_{1}}$ $\displaystyle:$ $\displaystyle x_{1}\mapsto
x_{1},\;\;x_{2}\mapsto q^{-1}x_{2},\;\;x_{3}\mapsto x_{3}-\alpha,$
$\displaystyle\omega_{x_{2}}$ $\displaystyle:$ $\displaystyle x_{1}\mapsto
qx_{1},\;\;x_{2}\mapsto x_{2},\;\;x_{3}\mapsto x_{3}-\mu.$

In particular, $\omega_{x_{1}}\omega_{x_{2}}=\omega_{x_{2}}\omega_{x_{1}}$.
The automorphisms $\omega_{x_{1}}$ and $\omega_{x_{2}}$ of the algebra $A$
respect the $\mathbb{N}^{2}$-grading. Also the inner derivation ${\rm
ad}_{x_{3}}$ of the algebra $A$ respects the $\mathbb{N}^{2}$-grading as
follows from the equalities $[x_{3},x_{1}]=\alpha x_{1}$ and
$[x_{3},x_{2}]=\mu x_{2}$. The $K$-linear maps $\omega_{x_{1}}$,
$\omega_{x_{2}}$ and ${\rm ad}_{x_{3}}$ are commuting, locally finite maps
that respect the $\mathbb{N}^{2}$-grading of the algebra $A$.

Let us consider the polynomial

$f_{\mu}(x_{3}):=\prod_{a,b\in\mathbb{F}_{p}}(x_{3}-a-b\mu)=\prod_{b=0}^{p-1}\Big{(}x_{3}^{p}-x_{3}-b(\mu^{p}-\mu)\Big{)}=(x_{3}^{p}-x_{3})^{p}-(\mu^{p}-\mu)^{p-1}(x_{3}^{p}-x_{3}).$
(8)

In more detail,

$\displaystyle f_{\mu}(x_{3})$ $\displaystyle=$
$\displaystyle\prod_{b\in\mathbb{F}_{p}}\prod_{a\in\mathbb{F}_{p}}((x_{3}-b\mu)-a)=\prod_{b\in\mathbb{F}_{p}}((x_{3}-b\mu)^{p}-(x_{3}-b\mu))=\prod_{b\in\mathbb{F}_{p}}\Big{(}x_{3}^{p}-x_{3}-b(\mu^{p}-\mu)\Big{)}$
$\displaystyle=$
$\displaystyle(x_{3}^{p}-x_{3})^{p}-(\mu^{p}-\mu)^{p-1}(x_{3}^{p}-x_{3}).$

Let $G$ be the subgroup of the automorphism group ${\rm Aut}_{K}(K[x_{3}])$
which is generated by the restrictions of the automorphisms $\omega_{x_{1}}$
and $\omega_{x_{2}}$ to the subalgebra $K[x_{3}]$ of $A$. Let
$K[x_{3}]^{G}=K[x_{3}]^{\omega_{x_{1}},\omega_{x_{2}}}=\\{f\in
K[x_{3}]\,|\,\omega_{x_{1}}(f)=f,\omega_{x_{2}}(f)=f\\}$ be the algebra of
$G$-invariants. Proposition 2.1 is a description of the algebra
$K[x_{3}]^{G}$.

######  Proposition 2.1.

$K[x_{3}]^{G}=K[x_{3}]^{\omega_{x_{1}},\omega_{x_{2}}}=\begin{cases}K&\text{if
$p=0$},\\\ K[x_{3}^{p}-x_{3}]&\text{if $p>0$ and
$(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\}$},\\\
K[f_{\mu}]&\text{if $p>0$, $\alpha=1$ and
$\mu\not\in\mathbb{F}_{p}$},\end{cases}$

and
$f_{\mu}(x_{3}):=\prod_{a,b\in\mathbb{F}_{p}}(x_{3}-a-b\mu)=\prod_{b=0}^{p-1}\Big{(}(x_{3}-b\mu)^{p}-(x_{3}-b\mu)\Big{)}=\prod_{a=0}^{p-1}\Big{(}(x_{3}-a)^{p}-\mu^{p-1}(x_{3}-a)\Big{)}$.

Proof. (i) If $p=0$ then $K[x_{3}]^{G}=K$: The condition that $\alpha+\mu\neq
0$ implies that either $\alpha\neq 0$ or $\mu\neq 0$ (or both). If $\alpha\neq
0$ (resp., $\mu\neq 0$) then $K[x_{3}]^{\omega_{x_{1}}}=K$ (resp.,
$K[x_{3}]^{\omega_{x_{2}}}=K$), and the statement (i) follows since
$K\subseteq K[x_{3}]^{G}\subseteq K[x_{3}]^{\omega_{x_{1}}}=K$ (resp.,
$K\subseteq K[x_{3}]^{G}\subseteq K[x_{3}]^{\omega_{x_{2}}}=K$).

(ii) If $p>0$ and
$(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\}$
then $K[x_{3}]^{G}=K[x_{3}^{p}-x_{3}]$: Since
$\mathbb{F}_{p}\alpha+\mathbb{F}_{p}\mu=\mathbb{F}_{p}$ and
$\prod_{i=0}^{p-1}(x_{3}-i)=x_{3}^{p}-x_{3}\in K[x_{3}]^{G}$, the statement
(ii) follows.

(iii) If $p>0$, $\alpha=1$ and $\mu\not\in\mathbb{F}_{p}$ then
$K[x_{3}]^{G}=K[f_{\mu}]$: Since
$f_{\mu}(x_{3}):=\prod_{a,b\in\mathbb{F}_{p}}(x_{3}-a-b\mu)\in K[x_{3}]^{G}$
and the elements $\\{a+b\mu\,|\,a,b\in\mathbb{F}_{p}\\}$ are distinct elements
of the field $K$, we have that $K[x_{3}]^{G}=K[f_{\mu}]$. Indeed, let
$g(x_{3})\in K[x_{3}]^{G}$. We have to show that $g\in K[f_{\mu}]$. We use
induction on the degree $n=\deg(g)$ of the polynomial $g$. The case $n=0$ is
obvious. So, let $n>0$ and we assume that the result holds for all
$G$-invariant polynomials of degree $<n$. By replacing the polynomial
$g(x_{3})$ by $g(x_{3})-g(0)$, we may assume that $g(x_{3})=x_{3}h(x_{3})$ for
some polynomial $h(x_{3})\in K[x_{3}]$. Now, for all numbers $a$ and $b$ such
that $0\leq a,b\leq p-1$,

$g(x_{3})=\omega_{x_{1}}^{a}\omega_{x_{2}}^{b}(g)=(x_{3}-a-b\mu)h(x_{3}-a-b\mu).$

Hence, $g=\prod_{a,b\in\mathbb{F}_{p}}(x_{3}-a-b\mu)\phi=f_{\mu}\phi$ for some
polynomial $\phi=gf_{\mu}^{-1}\in K[x_{3}]^{G}$. Clearly,
$\deg\,\phi=n-\deg\,f_{\mu}<n$. By induction $\phi\in K[f_{\mu}]$, and so
$g\in K[f_{\mu}]$, as required.

Notice that $x_{3}^{p}-x_{3}=\prod_{a\in\mathbb{F}_{p}}(x_{3}-a)$. Hence,
$f_{\mu}=\prod_{b=0}^{p-1}\Big{(}(x_{3}-b\mu)^{p}-(x_{3}-b\mu)\Big{)}$.

Notice that

$\prod_{b\in\mathbb{F}_{p}}(x_{3}-b\mu)=\mu^{p}\prod_{b\in\mathbb{F}_{p}}(\mu^{-1}x_{3}-b)=\mu^{p}\Big{(}(\mu^{-1}x_{3})^{p}-\mu^{-1}x_{3}\Big{)}=x_{3}^{p}-\mu^{p-1}x_{3}.$

Hence,
$f_{\mu}=\prod_{a=0}^{p-1}\Big{(}(x_{3}-a)^{p}-\mu^{p-1}(x_{3}-a)\Big{)}$.
$\Box$

The centralizer $C_{A}(x_{1},x_{2})$ of the elements $x_{1}$ and $x_{2}$ in
the algebra $A(q,\alpha,\mu)$. Let $S$ be a nonempty subset of an algebra $R$,
the set $C_{R}(S):=\\{r\in R\,|\,rs=sr$ for all $s\in S\\}$ is called the
centralizer of $S$ in $R$. The centralizer $C_{R}(S)$ is a subalgebra of $R$.
Proposition 2.2 describes the centralizer $C_{A}(x_{1},x_{2})$.

######  Proposition 2.2.

The centralizer of the elements $x_{1}$ and $x_{2}$ is equal to

  1. 1.

$C_{A}(x_{1},x_{2})=\begin{cases}K[x_{3}]^{G}&\text{if $q$ is not a root of
unity},\\\ K[x_{3}]^{G}[x_{1}^{n},x_{2}^{n}]&\text{if $q$ is a primitive
$n$'th root of unity}.\\\ \end{cases}$

  2. 2.

$C_{A}(x_{1},x_{2})=\begin{cases}K&\text{if $q$ is not a root of unity and
$p=0$},\\\ K[x_{3}^{p}-x_{3}]&\text{if $q$ is not a root of unity, $p>0$
and}\\\
&(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\},\\\
K[f_{\mu}]&\text{if $q$ is not a root of unity, $p>0$, $\alpha=1$ and
$\mu\not\in\mathbb{F}_{p}$},\\\ K[x_{1}^{n},x_{2}^{n}]&\text{if $q$ is a
primitive $n$'th root of unity and $p=0$},\\\
K[x_{3}^{p}-x_{3}][x_{1}^{n},x_{2}^{n}]&\text{if $q$ is a primitive $n$'th
root of unity, $p>0$ and}\\\
&\text{$(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\}$},\\\
K[f_{\mu}][x_{1}^{n},x_{2}^{n}]&\text{if $q$ is a primitive $n$'th root of
unity, $p>0$, $\alpha=1$ and $\mu\not\in\mathbb{F}_{p}$},\\\ \end{cases}$

where the polynomial $f_{\mu}$ is defined in Proposition 2.1. In particular,
the centralizer $C_{A}(x_{1},x_{2})$ is a polynomial algebra in $m$ variables
where $m=0,1,2,3$.

Proof. 1\. The elements $x_{1}$ and $x_{2}$ are regular and normal elements of
the algebra $A$, and so

$C_{A}(x_{1},x_{2})=A^{\omega_{x_{1}},\omega_{x_{2}}}.$

The automorphisms $\omega_{x_{1}}$ and $\omega_{x_{2}}$ of the algebra $A$
respect its $\mathbb{N}^{2}$-grading. Therefore,

$C_{A}(x_{1},x_{2})=\bigoplus_{\beta\in\mathbb{N}^{2}}\Big{(}K[x_{3}]x^{\beta}\Big{)}^{\omega_{x_{1}},\omega_{x_{2}}},$

and the statement follows.

2\. Statement 2 follows from statement 1 and Proposition 2.1. $\Box$

The centre of algebra $A$. For an algebra $R$ and its derivation $\delta$, the
set

$R^{\delta}:=\\{r\in R\,|\,\delta(r)=0\\}={\rm ker}(\delta)$

is called the algebra of $\delta$-constants in $A$. The elements $x_{1}$ and
$x_{2}$ are regular and normal elements of the algebra $A$. Hence,

$Z(A)=A^{\omega_{x_{1}},\omega_{x_{2}},{\rm
ad}_{x_{3}}}=C_{A}(x_{1},x_{2})^{{\rm
ad}_{x_{3}}}=\bigoplus_{\beta\in\mathbb{N}^{2}}\bigg{(}\Big{(}K[x_{3}]x^{\beta}\Big{)}^{\omega_{x_{1}},\omega_{x_{2}}}\bigg{)}^{{\rm
ad}_{x_{3}}}.$ (9)

If $p>0$ and $q$ is a primitive $n$’th root of unity then $p\nmid n$ (since
otherwise $0=q^{n}-1=(q^{\frac{n}{p}}-1)^{p}$ implies $q^{\frac{n}{p}}=1$, a
contradiction). We use this fact in the proof of Theorem 2.3. Theorem 2.3 is
an explicit description of the centre of the algebra $A$. There are eleven
distinct cases (C1)–(C11).

######  Theorem 2.3.

Let
$\Sigma:=K[x_{3}^{p}-x_{3}][x_{1}^{pn},x_{1}^{\xi_{i}n}x_{2}^{in},x_{2}^{pn}]_{i=1,\ldots,p-1}$
where $\xi_{i}$ is a unique natural number such that $0\leq\xi_{i}\leq p-1$
and $\xi_{i}\equiv-i\mu\mod p$. Then

$Z(A)=\begin{cases}K&\text{(C1): if $q$ is not a root of unity and $p=0$},\\\
K[x_{3}^{p}-x_{3}]&\text{(C2): if $q$ is not a root of unity, $p>0$ and}\\\
&(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\},\\\
K[f_{\mu}]&\text{(C3): if $q$ is not a root of unity, $p>0$, $\alpha=1$ and
$\mu\not\in\mathbb{F}_{p}$},\\\ K[x_{2}^{n}]&\text{(C4): if $q$ is a primitive
$n$'th root of unity, $p=0$, $\alpha=1$ and $\mu=0$},\\\
K[x_{1}^{n}]&\text{(C5): if $q$ is a primitive $n$'th root of unity, $p=0$,
$\alpha=0$ and $\mu=1$},\\\ K[x_{1}^{\mu_{1}n}x_{2}^{\mu_{2}n}]&\text{C(6): if
$q$ is a primitive $n$'th root of unity, $p=0$, $\alpha=1$ and
$\mu=-\frac{\mu_{1}}{\mu_{2}}$},\\\ &\text{ where
$\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ such that $\mu_{1}\neq\mu_{2}$ and
$(\mu_{1},\mu_{2})=1$},\\\ K&\text{(C7): if $q$ is a primitive $n$'th root of
unity, $p=0$, $\alpha=1$ and $\mu\not\in\mathbb{Q}_{-}\cup\\{0\\}$},\\\
K[x_{3}^{p}-x_{3}][x_{1}^{pn},x_{2}^{n}]&\text{(C8): if $q$ is a primitive
$n$'th root of unity, $p>0$, $\alpha=1$, $\mu=0$ and $p\nmid n$},\\\
K[x_{3}^{p}-x_{3}][x_{1}^{n},x_{2}^{pn}]&\text{(C9): if $q$ is a primitive
$n$'th root of unity, $p>0$, $\alpha=0$, $\mu=1$ and $p\nmid n$,}\\\
K[f_{\mu}][x_{1}^{pn},x_{2}^{pn}]&\text{(C10): if $q$ is a primitive $n$'th
root of unity, $p>0$, $\alpha=1$, $\mu\not\in\mathbb{F}_{p}$ and $p\nmid
n$},\\\ \Sigma&\text{(C11): if $q$ is a primitive $n$'th root of unity, $p>0$,
$\alpha=1$},\\\
&\text{$\mu\in\mathbb{F}_{p}\backslash\\{0,-1\\}=\\{1,\ldots,p-2\\}$ and
$p\nmid n$,}\\\ \end{cases}$

where $\mathbb{N}_{+}=\mathbb{N}\backslash\\{0\\}$,
$\mathbb{Q}_{-}=\\{q\in\mathbb{Q}\,|\,q<0\\}$ and $\xi_{1}=p-\mu$.

Remarks. 1\. The case (C11) is vacuous if $p=2$ (since
$\mathbb{F}_{2}\backslash\\{0,-1\\}=\emptyset$).

2\. The algebra $\Sigma$ is a commutative subalgebra of $A$ which is generated
by $p+2$ elements that are defined in the theorem.

3\. If $Z(A)\neq K$ then the centre $Z(A)$ uniquely determines the type of the
remaining nine cases in the Theorem 2.3.

4\. The case (C7) consists of two subcases (C7a) and (C7b), see Theorem 3.1.
The ideal structure of the algebra $A$ is different in these two cases as in
the case (C7a) the centre $Z(A_{x_{1}x_{2}})$ of the localized algebra
$A_{x_{1}x_{2}}$ is not equal to $K$ and ideals from the centre
$Z(A_{x_{1}x_{2}})$ generate nontrivial ideals of the algebra $A$.

Proof. The theorem follows from (9), Proposition 2.2.(2) and (6).

In more detail, by Proposition 2.2.(2) we have to consider 6 cases. The first
three cases of Proposition 2.2.(2) are precisely the first three cases of the
theorem (as in these cases $C_{A}(x_{1},x_{2})\subseteq Z(A)$, i.e.
$C_{A}(x_{1},x_{2})=Z(A)$).

The fourth case of Proposition 2.2.(2) ($q$ is a primitive $n$’th root of
unity and $p=0$) is split into 4 subcases, see cases 4-7 of the theorem. The
results follow from the equality $Z(A)=K[x_{1}^{n},x_{2}^{n}]^{{\rm
ad}_{x_{3}}}$ and (6).

For the fifth and sixth cases of Proposition 2.2.(2), we have that $p\nmid n$
(since $q$ is a primitive $n$’th root of unity and $p>0$).

The fifth case of Proposition 2.2.(2) ($q$ is a primitive $n$’th root of
unity, $p>0$ and
$(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\}$)
is split into 3 subcases, see cases 8, 9 and 11 of the theorem. The result
follow from the equality $Z(A)=K[x_{3}^{p}-x_{3}][x_{1}^{n},x_{2}^{n}]^{{\rm
ad}_{x_{3}}}$ and (6).

Finally, the sixth case of Proposition 2.2.(2) ($q$ is a primitive $n$’th root
of unity, $p>0$, $\alpha=1$ and $\mu\not\in\mathbb{F}_{p}$) is the 10’th case
of the theorem. The result follow from the equality
$Z(A)=K[f_{\mu}][x_{1}^{n},x_{2}^{n}]^{{\rm ad}_{x_{3}}}$ and (6). $\Box$

The algebra $A$ is free module over the centre in the cases (C1)–(C10).
Theorem 2.4 shows that the algebra $A$ is free module over the centre in all
cases of Theorem 2.3 but the one when $Z(A)=\Sigma$ (i.e. $q$ is a primitive
$n$’th root of unity, $p>0$, $\alpha=1$,
$\mu\in\mathbb{F}_{p}\backslash\\{0,-1\\}=\\{1,\ldots,p-2\\}$ and $p\nmid n$).
Explicit sets of free generators over $Z(A)$ are presented.

######  Theorem 2.4.

In the cases (C1)–(C10), the algebra $A=\bigoplus_{i\in I,\beta\in{\cal
R}}Z(A)x_{3}^{i}x^{\beta}$ is a free $Z(A)$-module where the pair $(I,{\cal
R})$ of sets $I=I_{A}\subseteq\mathbb{N}$ and ${\cal R}={\cal
R}_{A}\subseteq\mathbb{N}^{2}$ is given below where
$\mathbb{N}_{<i}=\\{0,1,\ldots,i-1\\}$ and
$\mathbb{S}(\mu_{1}n,\mu_{2}n):=\mathbb{N}^{2}\backslash\Big{(}(\mu_{1}n,\mu_{2}n)+\mathbb{N}^{2}\Big{)}$:

$(I,{\cal R})=\begin{cases}(\mathbb{N},\mathbb{N}^{2})&\text{(C1): if $q$ is
not a root of unity and $p=0$},\\\
(\mathbb{N}_{<p},\mathbb{N}^{2})&\text{(C2): if $q$ is not a root of unity,
$p>0$ and}\\\
&(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\},\\\
(\mathbb{N}_{<p^{2}},\mathbb{N}^{2})&\text{(C3): if $q$ is not a root of
unity, $p>0$, $\alpha=1$ and $\mu\not\in\mathbb{F}_{p}$},\\\
(\mathbb{N},\mathbb{N}\times\mathbb{N}_{<n})&\text{(C4): if $q$ is a primitive
$n$'th root of unity, $p=0$, $\alpha=1$ and $\mu=0$},\\\
(\mathbb{N},\mathbb{N}_{<n}\times\mathbb{N})&\text{(C5): if $q$ is a primitive
$n$'th root of unity, $p=0$, $\alpha=0$ and $\mu=1$},\\\
(\mathbb{N},\mathbb{S}(\mu_{1}n,\mu_{2}n))&\text{(C6): if $q$ is a primitive
$n$'th root of unity, $p=0$, $\alpha=1$ and
$\mu=-\frac{\mu_{1}}{\mu_{2}}$},\\\ &\text{ where
$\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ such that $\mu_{1}\neq\mu_{2}$ and
$(\mu_{1},\mu_{2})=1$},\\\ (\mathbb{N},\mathbb{N}^{2})&\text{(C7): if $q$ is a
primitive $n$'th root of unity, $p=0$, $\alpha=1$ and
$\mu\not\in\mathbb{Q}_{-}\cup\\{0\\}$},\\\
(\mathbb{N}_{<p},\mathbb{N}_{<pn}\times\mathbb{N}_{<n})&\text{(C8): if $q$ is
a primitive $n$'th root of unity, $p>0$, $\alpha=1$, $\mu=0$ and $p\nmid
n$},\\\ (\mathbb{N}_{<p},\mathbb{N}_{<n}\times\mathbb{N}_{<pn})&\text{(C9): if
$q$ is a primitive $n$'th root of unity, $p>0$, $\alpha=0$, $\mu=1$ and
$p\nmid n$,}\\\
(\mathbb{N}_{<p^{2}},\mathbb{N}_{<pn}\times\mathbb{N}_{<pn})&\text{(C10): if
$q$ is a primitive $n$'th root of unity, $p>0$, $\alpha=1$,
$\mu\not\in\mathbb{F}_{p}$ and $p\nmid n$}.\\\ \end{cases}$

In the case (C11), $A=\sum_{i\in I,\beta\in{\cal
R}}Z(A)x_{3}^{i}x^{\beta}=\bigoplus_{i\in
I}x_{3}^{i}\Big{(}\sum_{\beta\in{\cal R}}Z(A)x^{\beta}\Big{)}$ where
$I=\mathbb{N}_{<p}$ and ${\cal
R}=\mathbb{N}_{<pn}^{2}\backslash\Big{(}\bigcup_{i=0}^{p-1}((\xi_{i}n,in)+\mathbb{N}_{<pn}^{2})\Big{)}$.

Proof. The theorem follows at once from (5) and Theorem 2.3. $\Box$

######  Corollary 2.5.

The algebra $A$ is a finitely generated module over its centre iff it belongs
to the cases (C8)–(C11) of Theorem 2.3.

Proof. By Theorem 2.3, the algebra that belongs to the case (C11) case is
finitely generated module over its centre. Now, the corollary follows from
Theorem 2.4. $\Box$

The algebra ${\rm Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$. Suppose that the
algebra $A$ belongs to the cases (C1)–(10). Then by Theorem 2.4,

$A=\bigoplus_{i\in I_{A},\beta\in{\cal R}_{A}}Z(A)x_{3}^{i}x^{\beta}.$

So, each element $a\in A$ is a unique sum $a=\sum_{i\in I_{A},\beta\in{\cal
R}_{A}}z_{i\beta}x_{3}^{i}x^{\beta}$ for some elements $z_{i\beta}\in Z(A)$.
For $a\neq 0$, let

${\rm Deg}_{x_{3}}(a):=\max\\{i\in I_{A}\,|\,z_{i\beta}\neq 0\;\;{\rm
for\;some}\;\ \beta\in{\cal R}_{A}\\}.$

For all $z\in Z(A)$, $i\in I_{A}$ and $\beta\in{\cal R}_{A}$,

$\omega_{x_{1}}(zx_{3}^{i}x^{\beta})=q^{-\beta_{2}}z(x_{3}-\alpha)^{i}x^{\beta}\;\;{\rm
and}\;\;\omega_{x_{2}}(zx_{3}^{i}x^{\beta})=q^{\beta_{1}}z(x_{3}-\mu)^{i}x^{\beta}.$
(10)

Clearly, for all elements $a\in A$, ${\rm Deg}_{x_{3}}(\omega_{x_{1}}(a))={\rm
Deg}_{x_{3}}(a)$ and ${\rm Deg}_{x_{3}}(\omega_{x_{2}}(a))={\rm
Deg}_{x_{3}}(a)$.

Recall that $\omega_{x_{1}}$ and $\omega_{x_{2}}$ are commuting automorphisms
of the algebra $A$. The $K$-linear span ${\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$ of all common eigenvectors of the
automorphisms $\omega_{x_{1}}$ and $\omega_{x_{2}}$ is a subalgebra of $A$
which is $\omega_{x_{1}}-$ and $\omega_{x_{2}}$-invariant. The set ${\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$ is a subalgebra of $A$. We denote by
the same symbols the restrictions of the automorphisms $\omega_{x_{1}}$ and
$\omega_{x_{2}}$ to the subalgebra ${\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$.

Corollary 2.6 is a description of the algebra ${\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$ where the algebra $A$ belongs to the
cases (C1)–(C10). It also shows that the algebra $A$ is a free left/right
${\rm Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$-module.

######  Corollary 2.6.

Suppose that the algebra $A$ belongs to one of the cases (C1)–(C10) of Theorem
2.3. Then

  1. 1.

${\rm Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})=\\{a\in A\,|\,{\rm
Deg}_{x_{3}}(a)\leq 0\\}=\bigoplus_{\beta\in{\cal R}_{A}}Z(A)x^{\beta}$, and
for all $z\in Z(A)$ and $\beta\in{\cal R}_{A}$,
$\omega_{x_{1}}(zx^{\beta})=q^{-\beta_{2}}zx^{\beta}$,
$\omega_{x_{2}}(zx^{\beta})=q^{\beta_{1}}zx^{\beta}$, and ${\rm
ad}_{x_{3}}(zx^{\beta})=(\alpha\beta_{1}+\mu\beta_{2})zx^{\beta}$.

  2. 2.

$A=\bigoplus_{i\in I_{A}}x_{3}^{i}{\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})=\bigoplus_{i\in I_{A}}{\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})x_{3}^{i}$ is a direct sum of ${\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$-bimodules $x_{3}^{i}{\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})={\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})x_{3}^{i}$. The algebra $A$ is a free
left/right ${\rm Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$-module.

Proof. 1. Let $E={\rm Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$. The second
equality of statement 1 is obvious. It remains to show that
$E=\bigoplus_{\beta\in{\cal R}_{A}}Z(A)x^{\beta}$. This equality follows from
Theorem 2.4 and (10). In more detail, by (10), the automorphisms
$\omega_{x_{1}}$ and $\omega_{x_{2}}$ respect the $\mathbb{N}^{2}$-grading of
the algebra $A$. Hence, by Theorem 2.4, $E=\bigoplus_{\beta\in{\cal
R}}\Big{(}E\cap\bigoplus_{i\in I}Z(A)x^{i}x^{\beta}\Big{)}$. By (10), for all
elements $z\in Z(A)$ and $\beta\in{\cal R}$,

$\omega_{x_{1}}(zx^{\beta})=q^{-\beta_{2}}zx^{\beta}\;\;{\rm
and}\;\;\omega_{x_{2}}(zx^{\beta})=q^{\beta_{1}}zx^{\beta}.$

If for some polynomial $\phi\in\bigoplus_{i\in I}Kx_{3}^{i}$,

$\omega_{x_{1}}(\phi zx^{\beta})=q^{-\beta_{2}}\phi zx^{\beta}\;\;{\rm
and}\;\;\omega_{x_{2}}(\phi zx^{\beta})=q^{\beta_{1}}\phi zx^{\beta}$

then necessarily $\phi\in K[x_{3}]^{G}\subseteq Z(A)$, and the equality
follows.

2\. Statement 2 follows from statement 1 and Theorem 2.4. $\Box$

Corollary 2.7 is a description of the algebra ${\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$ where the algebra $A$ belongs to the
case (C11). It also shows that the algebra $A$ is a free left/right ${\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$-module.

######  Corollary 2.7.

Suppose that the algebra $A$ belongs to the case(C11) of Theorem 2.3. Let
$\Lambda^{\prime}:=K[x_{3}^{p}-x_{3}][x_{1}^{pn},x_{2}^{pn}]$ and
$I_{A}=\mathbb{N}_{<p}$. Then $\Lambda^{\prime}\subseteq Z(A)$ and

  1. 1.

${\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})=\bigoplus_{\beta\in\mathbb{N}_{<pn}^{2}}\Lambda^{\prime}x^{\beta}$,
and for all $z\in\Lambda^{\prime}$ and $\beta\in\mathbb{N}^{2}_{<pn}$,
$\omega_{x_{1}}(zx^{\beta})=q^{-\beta_{2}}zx^{\beta}$,
$\omega_{x_{2}}(zx^{\beta})=q^{\beta_{1}}zx^{\beta}$, and ${\rm
ad}_{x_{3}}(zx^{\beta})=(\alpha\beta_{1}+\mu\beta_{2})zx^{\beta}$.

  2. 2.

$A=\bigoplus_{i\in I_{A}}x_{3}^{i}{\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})=\bigoplus_{i\in I_{A}}{\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})x_{3}^{i}$ is a direct sum of ${\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$-bimodules $x_{3}^{i}{\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})={\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})x_{3}^{i}$. The algebra $A$ is a free
left/right ${\rm Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$-module.

Proof. 1. The inclusion $\Lambda^{\prime}\subseteq Z(A)$ is obvious. Let
$E={\rm Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})$. Notice that
$A=\bigoplus_{i\in
I_{A},\beta\in\mathbb{N}_{<pn}^{2}}\Lambda^{\prime}x_{3}^{i}x^{\beta}$ and
each direct summand is $\omega_{x_{i}}$-invariant where $i=1,2$. Therefore,
$E=\bigoplus_{i\in
I_{A},\beta\in\mathbb{N}_{<pn}^{2}}\Big{(}E\cap\Lambda^{\prime}x_{3}^{i}x^{\beta}\Big{)}$.
By (10), for all elements $z\in\Lambda^{\prime}$ and $\beta\in{\cal R}$,

$\omega_{x_{1}}(zx^{\beta})=q^{-\beta_{2}}zx^{\beta}\;\;{\rm
and}\;\;\omega_{x_{2}}(zx^{\beta})=q^{\beta_{1}}zx^{\beta}.$

If for some polynomial $\phi\in\bigoplus_{i\in I_{A}}Kx_{3}^{i}$,

$\omega_{x_{1}}(\phi zx^{\beta})=q^{-\beta_{2}}\phi zx^{\beta}\;\;{\rm
and}\;\;\omega_{x_{2}}(\phi zx^{\beta})=q^{\beta_{1}}\phi zx^{\beta}$

then necessarily $\phi\in K[x_{3}]^{G}\subseteq\Lambda^{\prime}$, and the
equality follows.

2\. Statement 2 follows from statement 1 and the equality $A=\bigoplus_{i\in
I_{A},\beta\in\mathbb{N}_{<pn}^{2}}\Lambda^{\prime}x_{3}^{i}x^{\beta}$. $\Box$

##  3 The centre, the ideal structure and the prime spectrum of the algebra
$A_{x_{1}x_{2}}$

The aim of this section is for the algebra $A_{x_{1}x_{2}}$ to describe its
centre (Theorem 3.2) and to classify its ideals (Theorem 3.9) and prime ideals
(Theorem 3.10). We show that the algebra $A_{x_{1}x_{2}}$ is a free module
over its centre (Theorem 3.4) and present an explicit set of free generators
(in each of 12 cases). The algebra $A_{x_{1}x_{2}}$ is a generalized weight
$\omega$-module (Proposition 3.5) where
$\omega=(\omega_{x_{1}},\omega_{x_{2}},{\rm ad}_{x_{3}})$. The
$\omega$-eigenalgebra ${\rm Es}_{A_{x_{1}x_{2}}}(\omega)$ is described
(Corollary 3.6). A simplicity criterion is given for the algebra
$A_{x_{1}x_{2}}$ (Theorem 3.8). The algebra $A_{x_{1}x_{2}}$ is a free
left/right ${\rm Es}_{A_{x_{1}x_{2}}}(\omega)$-module (Corollary 3.6.(2)).

The algebra $A_{x_{1}x_{2}}$ and its $\mathbb{Z}^{2}$-grading. Recall that the
elements $x_{1}$ and $x_{2}$ are regular normal elements of the algebra
$A=A(q,\alpha,\mu)_{x_{1}x_{2}}$. Hence, so is their product $x_{1}x_{2}$. Let
$A_{x_{1}x_{2}}$ be the localization of the algebra $A$ at the powers of the
element $x_{1}x_{2}$. The set S$=\\{x_{1}^{i}x_{2}^{j}\,|\,i,j\geq 0\\}$ is an
Ore set of the domain $A$. Clearly, $A_{x_{1}x_{2}}\simeq S^{-1}A\simeq
AS^{-1}$.

The algebra $A_{x_{1}x_{2}}$ is an $\mathbb{Z}^{2}$-graded algebra,

$A_{x_{1}x_{2}}=\bigoplus_{\beta\in\mathbb{Z}^{2}}K[x_{3}]x^{\beta}\;\;{\rm
where}\;\;x^{\beta}=x_{1}^{\beta_{1}}x_{2}^{\beta_{2}}\;\;{\rm
and}\;\;\beta=(\beta_{1},\beta_{2}).$ (11)

The product in the algebra $A$ is given by the rule: For all elements $a,b\in
K[x_{3}]$ and $\beta,\gamma\in\mathbb{Z}^{2}$,

$ax^{\beta}\cdot
bx^{\gamma}=a\omega^{\beta}(b)q^{\beta_{2}\gamma_{1}}x^{\beta+\gamma}\;\;{\rm
where}\;\;\omega^{\beta}=\omega_{x_{1}}^{\beta_{1}}\omega_{x_{2}}^{\beta_{2}}.$

So, the algebra $A_{x_{1}x_{2}}$ is a skew crossed product of the group
$\mathbb{Z}^{2}$ with the polynomial algebra $K[x_{3}]$. For all elements
$\beta\in\mathbb{Z}^{2}$,

$[x_{3},x^{\beta}]=(\beta_{1}\alpha+\beta_{2}\mu)x^{\beta}.$ (12)

So, the direct sum (11) is a direct sum of eigenspaces for the inner
derivation ${\rm ad}_{x_{3}}$ of the algebra $A_{x_{1}x_{2}}$. The elements
$x_{1}$ and $x_{2}$ of the algebra $A_{x_{1}x_{2}}$ are units. Clearly, the
automorphisms $\omega_{x_{1}}$ and $\omega_{x_{2}}$ of the algebra
$A_{x_{1}x_{2}}$ are the inner automorphisms of the algebra $A_{x_{1}x_{2}}$
that are determined by the units $x_{1}$ and $x_{2}$. The $K$-linear maps
$\omega_{x_{1}}$, $\omega_{x_{2}}$ and ${\rm ad}_{x_{3}}$ are commuting,
locally finite maps that respect the $\mathbb{Z}^{2}$-grading of the algebra
$A_{x_{1}x_{2}}$.

The centralizer $C_{A_{x_{1}x_{2}}}(x_{1},x_{2})$ of the elements $x_{1}$ and
$x_{2}$ in the algebra $A_{x_{1}x_{2}}$. Proposition 3.1 describes the
centralizer $C_{A}(x_{1},x_{2})$.

######  Proposition 3.1.

The centralizer of the elements $x_{1}$ and $x_{2}$ is equal to

  1. 1.

$C_{A_{x_{1}x_{2}}}(x_{1},x_{2})=\begin{cases}K[x_{3}]^{G}&\text{if $q$ is not
a root of unity},\\\ K[x_{3}]^{G}[x_{1}^{\pm n},x_{2}^{\pm n}]&\text{if $q$ is
a primitive $n$'th root of unity}.\\\ \end{cases}$

  2. 2.

$C_{A_{x_{1}x_{2}}}(x_{1},x_{2})=\begin{cases}K&\text{if $q$ is not a root of
unity and $p=0$},\\\ K[x_{3}^{p}-x_{3}]&\text{if $q$ is not a root of unity,
$p>0$ and}\\\
&(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\},\\\
K[f_{\mu}]&\text{if $q$ is not a root of unity, $p>0$, $\alpha=1$ and
$\mu\not\in\mathbb{F}_{p}$},\\\ K[x_{1}^{\pm n},x_{2}^{\pm n}]&\text{if $q$ is
a primitive $n$'th root of unity and $p=0$},\\\ K[x_{3}^{p}-x_{3}][x_{1}^{\pm
n},x_{2}^{\pm n}]&\text{if $q$ is a primitive $n$'th root of unity, $p>0$
and}\\\
&\text{$(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\}$},\\\
K[f_{\mu}][x_{1}^{\pm n},x_{2}^{\pm n}]&\text{if $q$ is a primitive $n$'th
root of unity, $p>0$, $\alpha=1$ and $\mu\not\in\mathbb{F}_{p}$},\\\
\end{cases}$

where the polynomial $f_{\mu}$ is defined in Proposition 2.1.

Proof. 1\. Recall that the commuting automorphisms $\omega_{x_{1}}$ and
$\omega_{x_{2}}$ of the algebra $A_{x_{1}x_{2}}$ respect its
$\mathbb{Z}^{2}$-grading. Therefore,

$C_{A_{x_{1}x_{2}}}(x_{1},x_{2})=A_{x_{1}x_{2}}^{\omega_{x_{1}},\omega_{x_{2}}}=\bigoplus_{\beta\in\mathbb{Z}^{2}}\Big{(}K[x_{3}]x^{\beta}\Big{)}^{\omega_{x_{1}},\omega_{x_{2}}},$

and the statement follows.

2\. Statement 2 follows from statement 1 and Proposition 2.1. $\Box$

The centre of algebra $A_{x_{1}x_{2}}$. Theorem 3.2 is an explicit description
of the centre of the algebra $A_{x_{1}x_{2}}$. Notice that

$Z(A_{x_{1}x_{2}})=A_{x_{1}x_{2}}^{\omega_{x_{1}},\omega_{x_{2}},{\rm
ad}_{x_{3}}}=C_{A_{x_{1}x_{2}}}(x_{1},x_{2})^{{\rm
ad}_{x_{3}}}=\bigoplus_{\beta\in\mathbb{Z}^{2}}\bigg{(}\Big{(}K[x_{3}]x^{\beta}\Big{)}^{\omega_{x_{1}},\omega_{x_{2}}}\bigg{)}^{{\rm
ad}_{x_{3}}}.$ (13)

######  Theorem 3.2.

Let $\Sigma^{\prime}:=K[x_{3}^{p}-x_{3}][x_{1}^{\pm pn},(x_{1}^{-\mu
n}x_{2}^{n})^{\pm 1},x_{2}^{\pm pn}]$. Then

$Z(A_{x_{1}x_{2}})=\begin{cases}K&\text{(C1): if $q$ is not a root of unity
and $p=0$},\\\ K[x_{3}^{p}-x_{3}]&\text{(C2): if $q$ is not a root of unity,
$p>0$ and}\\\
&(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\},\\\
K[f_{\mu}]&\text{(C3): if $q$ is not a root of unity, $p>0$, $\alpha=1$ and
$\mu\not\in\mathbb{F}_{p}$},\\\ K[x_{2}^{\pm n}]&\text{(C4): if $q$ is a
primitive $n$'th root of unity, $p=0$, $\alpha=1$ and $\mu=0$},\\\
K[x_{1}^{\pm n}]&\text{(C5): if $q$ is a primitive $n$'th root of unity,
$p=0$, $\alpha=0$ and $\mu=1$},\\\ K[(x_{1}^{\mu_{1}n}x_{2}^{\mu_{2}n})^{\pm
1}]&\text{(C6): if $q$ is a primitive $n$'th root of unity, $p=0$, $\alpha=1$
and $\mu=-\frac{\mu_{1}}{\mu_{2}}$},\\\ &\text{ where
$\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ such that $\mu_{1}\neq\mu_{2}$ and
$(\mu_{1},\mu_{2})=1$},\\\ K[(x_{1}^{\mu_{1}n}x_{2}^{-\mu_{2}n})^{\pm
1}]&\text{(C7a): if $q$ is a primitive $n$'th root of unity, $p=0$, $\alpha=1$
and $\mu=\frac{\mu_{1}}{\mu_{2}}$},\\\ &\text{ where
$\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ such that $(\mu_{1},\mu_{2})=1$},\\\
K&\text{(C7b): if $q$ is a primitive $n$'th root of unity, $p=0$, $\alpha=1$
and $\mu\not\in\mathbb{Q}$},\\\ K[x_{3}^{p}-x_{3}][x_{1}^{\pm pn},x_{2}^{\pm
n}]&\text{(C8): if $q$ is a primitive $n$'th root of unity, $p>0$, $\alpha=1$,
$\mu=0$ and $p\nmid n$},\\\ K[x_{3}^{p}-x_{3}][x_{1}^{\pm n},x_{2}^{\pm
pn}]&\text{(C9): if $q$ is a primitive $n$'th root of unity, $p>0$,
$\alpha=0$, $\mu=1$ and $p\nmid n$,}\\\ K[f_{\mu}][x_{1}^{\pm pn},x_{2}^{\pm
pn}]&\text{(C10): if $q$ is a primitive $n$'th root of unity, $p>0$,
$\alpha=1$, $\mu\not\in\mathbb{F}_{p}$ and $p\nmid n$},\\\
\Sigma^{\prime}&\text{(C11): if $q$ is a primitive $n$'th root of unity,
$p>0$, $\alpha=1$},\\\
&\text{$\mu\in\mathbb{F}_{p}\backslash\\{0,-1\\}=\\{1,\ldots,p-2\\}$ and
$p\nmid n$.}\\\ \end{cases}$

Remarks. 1\. The case (C7) is split into two sub-cases (C7a) and (C7b).

2\. The (C11) is vacuous if $p=2$ (since
$\mathbb{F}_{2}\backslash\\{0,-1\\}=\emptyset$). The case (C7b) is vacuous if
$K=\mathbb{Q}$.

3\. If $Z(A_{x_{1}x_{2}})\neq K$ then the centre $Z(A_{x_{1}x_{2}})$ uniquely
determines the type of the remaining nine cases in the Theorem 3.2.

Proof. The theorem follows from (13), Proposition 3.1.(2) and (12).

In more detail, by Proposition 3.1.(2) we have to consider 6 cases. The first
three cases of Proposition 3.1.(2) are precisely the first three cases of the
theorem (as in these cases $C_{A_{x_{1}x_{2}}}(x_{1},x_{2})\subseteq
Z(A_{x_{1}x_{2}})$, i.e. $C_{A_{x_{1}x_{2}}}(x_{1},x_{2})=Z(A_{x_{1}x_{2}})$).

The fourth case of Proposition 3.1.(2) ($q$ is a primitive $n$’th root of
unity and $p=0$) is split into 5 subcases, see the cases (C4)–(C7a) and (C7b)
of the theorem. The results follow from the equality
$Z(A_{x_{1}x_{2}})=K[x_{1}^{\pm n},x_{2}^{\pm n}]^{{\rm ad}_{x_{3}}}$ and
(12).

For the fifth and sixth cases of Proposition 3.1.(2), we have that $p\nmid n$
(since $q$ is a primitive $n$’th root of unity and $p>0$).

The fifth case of Proposition 3.1.(2) ($q$ is a primitive $n$’th root of
unity, $p>0$ and
$(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\}$)
is split into 3 subcases, see cases (C8), (C9) and (C11) of the theorem. The
result follow from the equality
$Z(A_{x_{1}x_{2}})=K[x_{3}^{p}-x_{3}][x_{1}^{\pm n},x_{2}^{\pm n}]^{{\rm
ad}_{x_{3}}}$ and (12).

Finally, the sixth case of Proposition 3.1.(2) ($q$ is a primitive $n$’th root
of unity, $p>0$, $\alpha=1$ and $\mu\not\in\mathbb{F}_{p}$) is the case (C10)
of the theorem. The result follow from the equality
$Z(A_{x_{1}x_{2}})=K[f_{\mu}][x_{1}^{\pm n},x_{2}^{\pm n}]^{{\rm ad}_{x_{3}}}$
and (12). $\Box$

The following obvious lemma is used in several proofs in order to reduce the
number of cases to deal with. In particular, by Lemma 3.3.(1) (resp., Lemma
3.3.(1)), the cases (C4) and (C5) (resp., (C6) and (C7a)) are the same up to
replacing $q$ by $q^{-1}$. Similarly, by Lemma 3.3.(1), the cases (C8) and
(C9) are the same up to replacing $q$ by $q^{-1}$.

######  Lemma 3.3.

  1. 1.

The homomorphism $A_{x_{1}x_{2}}(q,\alpha,\mu)\rightarrow
A_{x_{1}x_{2}}(q^{-1},\mu,\alpha)$, $x_{1}\mapsto x_{2}$, $x_{2}\mapsto x_{1}$
is an isomorphism.

  2. 2.

The homomorphism $A_{x_{1}x_{2}}(q,\alpha,\mu)\rightarrow
A_{x_{1}x_{2}}(q^{-1},-\alpha,\mu)$, $x_{1}\mapsto x_{1}^{-1}$, $x_{2}\mapsto
x_{2}$ is an isomorphism.

Proof. Straightforward. $\Box$

The algebra $A_{x_{1}x_{2}}$ is free module over the centre. Theorem 3.4 shows
that the algebra $A_{x_{1}x_{2}}$ is free module over its centre and explicit
sets of free generators are presented.

######  Theorem 3.4.

The algebra $A_{x_{1}x_{2}}=\bigoplus_{i\in I,\beta\in{\cal
R}}Z(A_{x_{1}x_{2}})x_{3}^{i}x^{\beta}$ is a free $Z(A_{x_{1}x_{2}})$-module
where the pair $(I,{\cal R})$ of sets
$I=I_{A_{x_{1}x_{2}}}\subseteq\mathbb{N}$ and ${\cal R}={\cal
R}_{A_{x_{1}x_{2}}}\subseteq\mathbb{Z}^{2}$ is given below where
$\mathbb{N}_{<i}=\\{0,1,\ldots,i-1\\}$,
$\mathbb{S}^{\prime}(\mu_{1}n,\mu_{2}n):=\mathbb{N}\times\mathbb{Z}\backslash\Big{(}(\mu_{1}n,\mu_{2}n)+\mathbb{N}\times\mathbb{Z}\Big{)}$
and
$\mathbb{S}^{\prime\prime}(-\mu_{1}n,\mu_{2}n):=\mathbb{Z}\times\mathbb{N}\backslash\Big{(}(-\mu_{1}n,\mu_{2}n)+\mathbb{Z}\times\mathbb{N}\Big{)}$:

$(I,{\cal R})=\begin{cases}(\mathbb{N},\mathbb{Z}^{2})&\text{(C1): if $q$ is
not a root of unity and $p=0$},\\\
(\mathbb{N}_{<p},\mathbb{Z}^{2})&\text{(C2): if $q$ is not a root of unity,
$p>0$ and}\\\
&(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\},\\\
(\mathbb{N}_{<p^{2}},\mathbb{Z}^{2})&\text{(C3): if $q$ is not a root of
unity, $p>0$, $\alpha=1$ and $\mu\not\in\mathbb{F}_{p}$},\\\
(\mathbb{N},\mathbb{Z}\times\mathbb{N}_{<n})&\text{(C4): if $q$ is a primitive
$n$'th root of unity, $p=0$, $\alpha=1$ and $\mu=0$},\\\
(\mathbb{N},\mathbb{N}_{<n}\times\mathbb{Z})&\text{(C5): if $q$ is a primitive
$n$'th root of unity, $p=0$, $\alpha=0$ and $\mu=1$},\\\
(\mathbb{N},\mathbb{S}^{\prime}(\mu_{1}n,\mu_{2}n))&\text{(C6): if $q$ is a
primitive $n$'th root of unity, $p=0$, $\alpha=1$ and
$\mu=-\frac{\mu_{1}}{\mu_{2}}$},\\\ &\text{ where
$\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ such that $\mu_{1}\neq\mu_{2}$ and
$(\mu_{1},\mu_{2})=1$},\\\
(\mathbb{N},\mathbb{S}^{\prime\prime}(-\mu_{1}n,\mu_{2}n))&\text{(C7a): if $q$
is a primitive $n$'th root of unity, $p=0$, $\alpha=1$ and
$\mu=\frac{\mu_{1}}{\mu_{2}}$},\\\ &\text{ where
$\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ such that $(\mu_{1},\mu_{2})=1$},\\\
(\mathbb{N},\mathbb{Z}^{2})&\text{(C7b): if $q$ is a primitive $n$'th root of
unity, $p=0$, $\alpha=1$ and $\mu\not\in\mathbb{Q}$},\\\
(\mathbb{N}_{<p},\mathbb{N}_{<pn}\times\mathbb{N}_{<n})&\text{(C8): if $q$ is
a primitive $n$'th root of unity, $p>0$, $\alpha=1$, $\mu=0$ and $p\nmid
n$},\\\ (\mathbb{N}_{<p},\mathbb{N}_{<n}\times\mathbb{N}_{<pn})&\text{C(9): if
$q$ is a primitive $n$'th root of unity, $p>0$, $\alpha=0$, $\mu=1$ and
$p\nmid n$,}\\\
(\mathbb{N}_{<p^{2}},\mathbb{N}_{<pn}\times\mathbb{N}_{<pn})&\text{(C10): if
$q$ is a primitive $n$'th root of unity, $p>0$, $\alpha=1$,
$\mu\not\in\mathbb{F}_{p}$ and $p\nmid n$},\\\
(\mathbb{N}_{<p},\mathbb{N}_{<pn}\times\mathbb{N}_{<n})&\text{(C11): if $q$ is
a primitive $n$'th root of unity, $p>0$, $\alpha=1$},\\\
&\text{$\mu\in\mathbb{F}_{p}\backslash\\{0,-1\\}=\\{1,\ldots,p-2\\}$ and
$p\nmid n$.}\end{cases}$

The algebra $A$ is a finitely generated module over its centre iff it belongs
to the last four cases.

Proof. The theorem follows at once from (11) and Theorem 3.2. $\Box$

Let $V$ be a $K$-vector space and $a_{1},\ldots,a_{m}\in{\rm End}_{K}(V)$ be
commuting $K$-linear maps. A nonzero vector $v\in V$ is called a common
eigenvector of the maps $a_{1},\ldots,a_{m}$ if $a_{i}v=\lambda_{i}v$ for
$i=1,\ldots,m$ and some scalars $\lambda_{i}\in K$. The $m$-tuple
$\lambda=(\lambda_{1},\ldots,\lambda_{m})$ is called a weight for
$a=(a_{1},\ldots,a_{m})$. The set of all weights is denoted by ${\cal
W}(a)={\cal W}(a;V)$. Let ${\cal A}$ be the subalgebra of ${\rm End}_{K}(V)$
which is generated by the elements $a_{1},\ldots,a_{m}$. Then $V$ is a (left)
${\cal A}$-module. In this case, we also say that $V$ is an $a$-module. The
subspace of $V$,

$V_{\lambda}:=V_{\lambda}(a):=\\{v\in V\,|\,a_{i}v=\lambda_{i}v\;\;{\rm
for}\;\;i=1,\ldots,m\\},$

is called the $\lambda$-weight subspace of $V$. Clearly, $V_{\lambda}$ is an
$a$-module and the sum $\sum_{\lambda\in{\cal
W}(a)}V_{\lambda}=\bigoplus_{\lambda\in{\cal W}(a)}V_{\lambda}$ is a direct
sum of $a$-modules. If

$V=\bigoplus_{\lambda\in{\cal W}(a)}V_{\lambda}$

then we say that $V$ is a weight $a$-module. More generally, for each weight
$\lambda\in{\cal W}(a)$, the subspace

$V^{\lambda}:=V^{\lambda}(a):=\\{v\in V\,|\,(a_{i}-\lambda_{i})^{j}v=0\;\;{\rm
for}\;\;j\gg 1\;\;{\rm and}\;\;i=1,\ldots,m\\},$

is called the $\lambda$-generalized weight subspace of $V$. Clearly,
$V_{\lambda}\subseteq V^{\lambda}$ and $V^{\lambda}$ is an $a$-module and the
sum $\sum_{\lambda\in{\cal W}(a)}V^{\lambda}=\bigoplus_{\lambda\in{\cal
W}(a)}V^{\lambda}$ is a direct sum of $a$-modules. If

$V=\bigoplus_{\lambda\in{\cal W}(a)}V^{\lambda}$

then we say that $V$ is a generalized weight $a$-module. If the field $K$ is
an algebraically closed field and the vector space $V$ is a locally finite
$a$-module, i.e. the linear maps $a_{1},\ldots,a_{m}$ are locally finite (i.e.
for all $v\in V$, ${\rm dim}_{K}(K[a_{i}]v)<\infty$ for $i=1,\ldots,m$) then
the $a$-module $V$ is generalized weight module and vice versa. In particular,
every finite dimensional vectors space $V$ is a generalized weight $a$-module.

For all elements $z\in Z(A_{x_{1}x_{2}})$, $i\in I_{A_{x_{1}x_{2}}}$ and
$\beta\in{\cal R}_{A_{x_{1}x_{2}}}$,

$\omega_{x_{1}}(zx_{3}^{i}x^{\beta})=q^{-\beta_{2}}z(x_{3}-\alpha)^{i}x^{\beta}\;\;\omega_{x_{2}}(zx_{3}^{i}x^{\beta})=q^{\beta_{1}}z(x_{3}-\mu)^{i}x^{\beta}\;\;{\rm
and}\;\;{\rm
ad}_{x_{3}}(zx_{3}^{i}x^{\beta})=(\alpha\beta_{1}+\mu\beta_{2})zx_{3}^{i}x^{\beta}.$
(14)

######  Proposition 3.5.

Let $\omega=(\omega_{x_{1}},\omega_{x_{2}},{\rm ad}_{x_{3}})$. Then the
algebra $A_{x_{1}x_{2}}$ is a generalized weight $\omega$-module,
$A_{x_{1}x_{2}}=\bigoplus_{\beta\in{\cal R}}\bigg{(}\bigoplus_{i\in
I}Z(A_{x_{1}x_{2}})x_{3}^{i}x^{\beta}\bigg{)}$ is the direct sum of
generalized weight components where the sets ${\cal R}={\cal
R}_{A_{x_{1}x_{2}}}$ and $I=I_{A_{x_{1}x_{2}}}$ are defined in Theorem 3.4,
${\cal
W}(\omega,A_{x_{1}x_{2}})=\\{(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})\,|\,\beta=(\beta_{1},\beta_{2})\in{\cal
R}\\}$, and for each $\beta\in{\cal R}$ the element $x^{\beta}$ is an
$\omega$-eigenvector with weight
$(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})$.

Proof. By Theorem 3.4, we have the direct sum as in the theorem. It remain to
show that the vector spaces $V_{\beta}:=\bigoplus_{i\in
I}Z(A_{x_{1}x_{2}})x_{3}^{i}x^{\beta}$ are the generalized weight components.
The theorem follows from the Claim.

Claim: The elements of the set ${\cal
W}^{\prime}:=\\{(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})\,|\,\beta=(\beta_{1},\beta_{2})\in{\cal
R}\\}$ are distinct.

In more detail, suppose that the Claim holds. Then, by (14), ${\cal
W}(\omega,A_{x_{1}x_{2}})\subseteq{\cal W}^{\prime}$. The reverse inclusion
follows from the fact that for each $\beta\in{\cal R}$, the element
$x^{\beta}$ is a weight vector with weight
$(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})$, by (14). Hence,
${\cal W}(\omega,A_{x_{1}x_{2}})={\cal W}$. Now, by the Claim and (14), the
vector space $V_{\beta}$ is the generalized weight component that correspond
to the weight $(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})$.

Proof of the Claim. Suppose that for elements $\beta,\beta^{\prime}\in{\cal
R}$,

$(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})=(q^{-\beta_{2}^{\prime}},q^{\beta_{1}^{\prime}},\alpha\beta_{1}^{\prime}+\mu\beta_{2}^{\prime}).$

we have to show that $\beta=\beta^{\prime}$. This is obvious if $q$ is not
root of unity, i.e. if the algebra $A_{x_{1}x_{2}}$ belongs to classes
(C1)–(C3).

So, we assume that $q$ is a primitive $n$’th root of unity, i.e. the algebra
$A_{x_{1}x_{2}}$ belongs to classes (C4)–(C11). Then the equality above is
equivalent to the equalities:

$\displaystyle\beta_{1}^{\prime}$ $\displaystyle\equiv$
$\displaystyle\beta_{1}\mod n,$ $\displaystyle\beta_{2}^{\prime}$
$\displaystyle\equiv$ $\displaystyle\beta_{2}\mod n,$
$\displaystyle(\beta_{1}^{\prime}-\beta_{1})\alpha$ $\displaystyle=$
$\displaystyle(\beta_{2}-\beta_{2}^{\prime})\mu.$

Cases (C4) and (C5): In view of Lemma 3.3.(1), it suffices to consider the
case (C4). By Theorem 3.4, ${\cal R}=\mathbb{Z}\times\mathbb{N}_{n}$. Hence,
$\beta_{2}^{\prime}=\beta_{2}$. Then $\beta_{1}^{\prime}=\beta_{1}$ since
$\alpha=1$.

Cases (C6) and (C7a): In view of Lemma 3.3.(2), it suffices to consider the
case (C6). The third equality,
$\mu_{2}(\beta_{1}^{\prime}-\beta_{1})=-\mu_{1}(\beta_{2}-\beta_{2}^{\prime})$,
implies that $\beta_{1}^{\prime}-\beta_{1}=k\mu_{1}$ and
$\beta_{2}^{\prime}-\beta_{2}=k\mu_{2}$ for some integer $k\in\mathbb{Z}$
(since $(\mu_{1},\mu_{2})=1$). Then
$x^{\beta^{\prime}}=x^{\beta}(x_{1}^{\mu_{1}}x_{2}^{\mu_{2}})^{k}$, and so
$\beta^{\prime}=\beta$ since ${\cal
R}=\mathbb{S}^{\prime}(\mu_{1}n,\mu_{2}n)$, by Theorem 3.4.

Case (C7b): Since $p=0$, $\alpha=1$ and $\mu\not\in\mathbb{Q}$, the third
equality, $\beta_{1}^{\prime}-\beta_{1}=(\beta_{2}-\beta_{2}^{\prime})\mu$,
yields the equality $\beta=\beta^{\prime}$.

Cases (C8) and (C9): In view of Lemma 3.3.(1), it suffices to consider the
case (C8). By Theorem 3.4, ${\cal R}=\mathbb{N}_{<pn}\times\mathbb{N}_{<n}$.
The second equation, $\beta_{2}^{\prime}\equiv\beta_{2}\mod j$, implies the
equality $\beta_{2}^{\prime}=\beta_{2}$. Then the third equality
$\beta_{1}^{\prime}-\beta_{1}=0$ (since $\alpha=1$) implies the equality
$\beta=\beta^{\prime}$.

Case (C10): Since $p>0$, $\alpha=1$ and $\mu\not\in\mathbb{F}_{p}$, the third
equation, $\beta_{1}^{\prime}-\beta_{1}=(\beta_{2}-\beta_{2}^{\prime})\mu$,
yields the equivalences

$\beta_{1}^{\prime}\equiv\beta_{1}\mod p\;\;{\rm
and}\;\;\beta_{2}^{\prime}\equiv\beta_{2}\mod p.$

Then the first two equations yield $\beta_{1}^{\prime}\equiv\beta_{1}\mod pn$
and $\beta_{2}^{\prime}\equiv\beta_{2}\mod pn$ since $(p,n)=1$. Hence,
$\beta=\beta^{\prime}$ since ${\cal
R}=\mathbb{N}_{<pn}\times\mathbb{N}_{<pn}$, by Theorem 3.4.

Case (C11): Recall that $p>0$, $\alpha=1$ and
$\mu\in\mathbb{F}_{p}\backslash\\{0,-1\\}$. By Theorem 3.4, ${\cal
R}=\mathbb{N}_{<pn}\times\mathbb{N}_{<n}$. Then the second equation,
$\beta_{2}^{\prime}\equiv\beta_{2}\mod n$, yields the equality
$\beta_{2}^{\prime}=\beta_{2}$. Then the third equation,
$\beta_{1}^{\prime}-\beta_{1}=0$, yields the equivalence
$\beta_{1}^{\prime}\equiv\beta_{1}\mod p$. Then the first equation,
$\beta_{1}^{\prime}\equiv\beta_{1}\mod n$, implies that
$\beta_{1}^{\prime}\equiv\beta_{1}\mod pn$ since $(p,n)=1$. Hence,
$\beta=\beta^{\prime}$. $\Box$

The $\omega$-eigenalgebra ${\rm Es}_{A_{x_{1}x_{2}}}(\omega)$. By Theorem 3.4,

$A_{x_{1}x_{2}}=\bigoplus_{i\in I_{A_{x_{1}x_{2}}},\beta\in{\cal
R}_{A_{x_{1}x_{2}}}}Z(A_{x_{1}x_{2}})x_{3}^{i}x^{\beta}.$

So, each element $a\in A_{x_{1}x_{2}}$ is a unique sum $a=\sum_{i\in
I_{A_{x_{1}x_{2}}},\beta\in{\cal
R}_{A_{x_{1}x_{2}}}}z_{i\beta}x_{3}^{i}x^{\beta}$ for some elements
$z_{i\beta}\in Z(A_{x_{1}x_{2}})$. For $a\neq 0$, let

${\rm Deg}_{x_{3}}(a):=\max\\{i\in I_{A_{x_{1}x_{2}}}\,|\,z_{i\beta}\neq
0\;\;{\rm for\;some}\;\ \beta\in{\cal R}_{A_{x_{1}x_{2}}}\\}.$

Clearly, the sum

${\rm Es}_{A_{x_{1}x_{2}}}(\omega)=\bigoplus_{\lambda\in{\cal
W}(\omega,A_{x_{1}x_{2}})}A_{x_{1}x_{2},_{\lambda}}$

of all weight $\omega$-submodules of $A_{x_{1}x_{2}}$ is the largest weight
$\omega$-submodule of the algebra $A_{x_{1}x_{2}}$. The space ${\rm
Es}_{A_{x_{1}x_{2}}}(\omega)$ is a subalgebra of $A_{x_{1}x_{2}}$ which is
called the $\omega$-eigenalgebra of $A_{x_{1}x_{2}}$.

Corollary 3.6 is a description of the algebra ${\rm
Es}_{A_{x_{1}x_{2}}}(\omega)$. It also shows that the algebra $A_{x_{1}x_{2}}$
is a free left/right ${\rm Es}_{A_{x_{1}x_{2}}}(\omega)$-module.

######  Corollary 3.6.

  1. 1.

${\rm Es}_{A_{x_{1}x_{2}}}(\omega)=\\{a\in A_{x_{1}x_{2}}\,|\,{\rm
Deg}_{x_{3}}(a)\leq 0\\}=\bigoplus_{\beta\in{\cal
R}_{A_{x_{1}x_{2}}}}Z(A_{x_{1}x_{2}})x^{\beta}$; for all
$\lambda_{\beta}=(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})\in{\cal
W}(\omega,A_{x_{1}x_{2}})=\\{(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})\,|\,\beta=(\beta_{1},\beta_{2})\in{\cal
R}_{A_{x_{1}x_{2}}}\\}$,
$A_{x_{1}x_{2},\lambda_{\beta}}=Z(A_{x_{1}x_{2}})x^{\beta}$; for all $z\in
Z(A_{x_{1}x_{2}})$ and $\beta\in{\cal R}_{A_{x_{1}x_{2}}}$,
$\omega_{x_{1}}(zx^{\beta})=q^{-\beta_{2}}zx^{\beta}$,
$\omega_{x_{2}}(zx^{\beta})=q^{\beta_{1}}zx^{\beta}$ and ${\rm
ad}_{x_{3}}(zx^{\beta})=(\alpha\beta_{1}+\mu\beta_{2})zx^{\beta}$.

  2. 2.

$A_{x_{1}x_{2}}=\bigoplus_{i\in I_{A_{x_{1}x_{2}}}}x_{3}^{i}{\rm
Es}_{A}(\omega)=\bigoplus_{i\in I_{A_{x_{1}x_{2}}}}{\rm
Es}_{A}(\omega)x_{3}^{i}$ is a direct sum of ${\rm
Es}_{A_{x_{1}x_{2}}}(\omega)$-bimodules $x_{3}^{i}{\rm
Es}_{A_{x_{1}x_{2}}}(\omega)={\rm Es}_{A_{x_{1}x_{2}}}(\omega)x_{3}^{i}$. The
algebra $A_{x_{1}x_{2}}$ is a free left/right ${\rm
Es}_{A_{x_{1}x_{2}}}(\omega)$-module.

Proof. 1. Statement 1 follows from Proposition 3.5 and (14). In more detail,
by (14),

$\omega_{x_{1}}(zx^{\beta})=q^{-\beta_{2}}zx^{\beta},\;\;\omega_{x_{2}}(zx^{\beta})=q^{\beta_{1}}zx^{\beta}\;\;{\rm
and}\;\;{\rm
ad}_{x_{3}}(zx^{\beta})=(\alpha\beta_{1}+\mu\beta_{2})zx^{\beta}.$

If for some polynomial $\phi\in K[x_{3}]$,

$\omega_{x_{1}}(\phi zx^{\beta})=q^{-\beta_{2}}\phi
zx^{\beta},\;\;\omega_{x_{2}}(\phi zx^{\beta})=q^{\beta_{1}}\phi
zx^{\beta}\;\;{\rm and}\;\;{\rm ad}_{x_{3}}(\phi
zx^{\beta})=(\alpha\beta_{1}+\mu\beta_{2})\phi zx^{\beta}$

then necessarily $\phi\in K[x_{3}]^{\omega}\subseteq Z(A_{x_{1}x_{2}})$, and
statement 1 follows from Proposition 3.5.

2\. Statement 2 follows from statement 1 and Proposition 3.5. $\Box$

By Corollary 3.6.(1), ${\cal
W}(\omega,A_{x_{1}x_{2}})=\\{\lambda_{\beta}:=(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})\,|\,\beta=(\beta_{1},\beta_{2})\in{\cal
R}_{A_{x_{1}x_{2}}}\\}$ and
$A_{x_{1}x_{2},\lambda_{\beta}}=Z(A_{x_{1}x_{2}})x^{\beta}$. For all
$\beta,\beta^{\prime}\in{\cal R}_{A_{x_{1}x_{2}}}$,

$A_{x_{1}x_{2},\lambda_{\beta}}A_{x_{1}x_{2},\lambda_{\beta^{\prime}}}=Z(A_{x_{1}x_{2}})x^{\beta+\beta^{\prime}}=A_{x_{1}x_{2},\lambda_{\beta+\beta^{\prime}}}\;\;{\rm
and}\;\;A_{x_{1}x_{2},\lambda_{-\beta}}=Z(A_{x_{1}x_{2}})x^{-\beta}.$

Therefore, the set ${\cal R}_{x_{1}x_{2}}$ is an additive abelian group. By
Corollary 2.6 and Corollary 2.7, the set ${\cal R}_{A}$ is an additive
submonoid of the group ${\cal R}_{x_{1}x_{2}}$. Lemma 3.7 is an explicit
description of the group ${\cal R}_{x_{1}x_{2}}$.

For each natural number $n$, let $Z_{n}=\mathbb{Z}/n\mathbb{Z}$.

######  Lemma 3.7.

${\cal R}_{A_{x_{1}x_{2}}}=\begin{cases}\mathbb{Z}^{2}&\text{(C1): if $q$ is
not a root of unity and $p=0$},\\\ \mathbb{Z}^{2}&\text{(C2): if $q$ is not a
root of unity, $p>0$ and}\\\
&(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\},\\\
\mathbb{Z}^{2}&\text{(C3): if $q$ is not a root of unity, $p>0$, $\alpha=1$
and $\mu\not\in\mathbb{F}_{p}$},\\\
\mathbb{Z}\times\mathbb{Z}_{<n}&\text{(C4): if $q$ is a primitive $n$'th root
of unity, $p=0$, $\alpha=1$ and $\mu=0$},\\\
\mathbb{Z}_{<n}\times\mathbb{Z}&\text{(C5): if $q$ is a primitive $n$'th root
of unity, $p=0$, $\alpha=0$ and $\mu=1$},\\\
\mathbb{Z}^{2}/\mathbb{Z}(\mu_{1}n,\mu_{2}n)&\text{(C6): if $q$ is a primitive
$n$'th root of unity, $p=0$, $\alpha=1$ and
$\mu=-\frac{\mu_{1}}{\mu_{2}}$},\\\ &\text{ where
$\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ such that $\mu_{1}\neq\mu_{2}$ and
$(\mu_{1},\mu_{2})=1$},\\\
\mathbb{Z}^{2}/\mathbb{Z}(-\mu_{1}n,\mu_{2}n)&\text{(C7a): if $q$ is a
primitive $n$'th root of unity, $p=0$, $\alpha=1$ and
$\mu=\frac{\mu_{1}}{\mu_{2}}$},\\\ &\text{ where
$\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ such that $(\mu_{1},\mu_{2})=1$},\\\
\mathbb{Z}^{2}&\text{(C7b): if $q$ is a primitive $n$'th root of unity, $p=0$,
$\alpha=1$ and $\mu\not\in\mathbb{Q}$},\\\
\mathbb{Z}_{<pn}\times\mathbb{Z}_{<n}&\text{(C8): if $q$ is a primitive $n$'th
root of unity, $p>0$, $\alpha=1$, $\mu=0$ and $p\nmid n$},\\\
\mathbb{Z}_{<n}\times\mathbb{Z}_{<pn}&\text{C(9): if $q$ is a primitive $n$'th
root of unity, $p>0$, $\alpha=0$, $\mu=1$ and $p\nmid n$,}\\\
\mathbb{Z}_{<pn}\times\mathbb{Z}_{<pn}&\text{(C10): if $q$ is a primitive
$n$'th root of unity, $p>0$, $\alpha=1$, $\mu\not\in\mathbb{F}_{p}$ and
$p\nmid n$},\\\ \mathbb{Z}_{<pn}\times\mathbb{Z}_{<n}&\text{(C11): if $q$ is a
primitive $n$'th root of unity, $p>0$, $\alpha=1$},\\\
&\text{$\mu\in\mathbb{F}_{p}\backslash\\{0,-1\\}=\\{1,\ldots,p-2\\}$ and
$p\nmid n$.}\end{cases}$

Proof. The lemma follows from Theorem 3.4. $\Box$

By Corollary 3.6.(1), ${\cal
W}(\omega,A_{x_{1}x_{2}})=\\{\lambda_{\beta}=(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})\,|\,\beta=(\beta_{1},\beta_{2})\in{\cal
R}_{A_{x_{1}x_{2}}}\\}$, i.e. the map ${\cal
R}_{A_{x_{1}x_{2}}}\rightarrow{\cal W}(\omega,A_{x_{1}x_{2}})$,
$\beta\mapsto\lambda_{\beta}$ is a bijection. By Lemma 3.7, the set ${\cal
W}(\omega,A_{x_{1}x_{2}})$ is an abelian group isomorphic to ${\cal
R}_{A_{x_{1}x_{2}}}$ via the bijection above. So, the group structure on the
set ${\cal W}(\omega,A_{x_{1}x_{2}})$ is given by the rule: for all elements
$\beta,\beta^{\prime}\in{\cal R}_{A_{x_{1}x_{2}}}$,
$\lambda_{\beta}+\lambda_{\beta^{\prime}}:=\lambda_{\beta+\beta^{\prime}}$.

Simplicity criterion for the algebra $A_{x_{1}x_{2}}$.

######  Theorem 3.8.

The algebra $A_{x_{1}x_{2}}$ is a simple algebra iff the algebra $A$ belongs
to the case (C1) and or (C7b) of Theorem 2.3 iff $Z(A_{x_{1}x_{2})}=K$.

Proof. The second ‘iff’ is obvious, see Theorem 3.2.

Let us prove that the first ‘iff’ holds.

$(\Rightarrow)$ The centre of a simple algebra is a field, and so the
implication follows from Theorem 3.2.

$(\Leftarrow)$ Let $J$ be a nonzero ideal of the algebra $A_{x_{1}x_{2}}$. We
have to show that $J=(1)$.

Suppose that the algebra $A$ belongs to the case (C7b). Then the elements
$\\{x^{\beta}\\}_{\beta\in\mathbb{Z}^{2}}$ are eigenvectors of the derivation
${\rm ad}_{x_{3}}$ of the algebra $A_{x_{1}x_{2}}$ with distinct eigenvalues
(since $\alpha=1\in\mathbb{Q}$ and $\mu\not\in\mathbb{Q}$). Using the
$\mathbb{Z}^{2}$-grading of the algebra $A_{x_{1}x_{2}}$, we conclude that
$\varphi x^{\beta}\in J$ for some nonzero element $\varphi\in K[x_{3}]$ and
$\beta\in\mathbb{Z}^{2}$. Hence, $\varphi\in J$ since the element $x^{\beta}$
is a unit. We may assume that the element $\varphi$ has the least possible
degree $d$ in the variable $x_{3}$. If $d=0$ then $\varphi\in K^{\times}$ and
we are done. Suppose that $d>0$, we seek a contradiction. Recall that
$K[x_{3}]^{G}=K$ (Proposition 2.1). The element

$0\neq(1-\omega_{x_{1}})(\varphi(x_{3}))=v(x_{3})-v(x_{3}-1)\in J$

has degree $d-1$, a contradiction.

Suppose that the algebra $A$ belongs to case (C1). Then the elements
$\\{x^{\beta}\\}_{\beta\in\mathbb{Z}^{2}}$ are common eigenvectors of the
commuting inner automorphisms $\omega_{x_{1}}$ and $\omega_{x_{2}}$ of the
algebra $A_{x_{1}x_{2}}$ with distinct weights (since $q$ is not a root of
unity). Furthermore, the algebra
$A=\bigoplus_{\beta\in\mathbb{Z}^{2}}K[x_{3}]x^{\beta}$ is a direct sum of
generalized weight subspaces $K[x_{3}]x^{\beta}$. Hence, $\varphi x^{\beta}\in
J$ for some nonzero element $\varphi\in K[x_{3}]$ and
$\beta\in\mathbb{Z}^{2}$. Hence, $\varphi\in J$ since the element $x^{\beta}$
is a unit. We may assume that the element $\varphi$ has the least possible
degree $d$ in the variable $x_{3}$. If $d=0$ then $\varphi\in K^{\times}$ and
we are done. Suppose that $d>0$, we seek a contradiction. Recall that
$K[x_{3}]^{G}=K$ (Proposition 2.1). Hence, one of the elements

$(1-\omega_{x_{1}})(\varphi(x_{3}))=v(x_{3})-v(x_{3}-\alpha)\in J\;\;{\rm
or}\;\;(1-\omega_{x_{2}})(\varphi(x_{3}))=v(x_{3})-v(x_{3}-\mu)\in J$

is nonzero and has degree strictly smaller than $d$, a contradiction. $\Box$

The ideal structure and the prime spectrum of the algebra $A_{x_{1}x_{2}}$.
For an algebra $R$, we denote by ${\cal I}(R)$ the set of ideals of $R$.

######  Theorem 3.9.

The map

${\cal I}(Z(A_{x_{1}x_{2}}))\rightarrow{\cal
I}(A_{x_{1}x_{2}}),\;\;\mathfrak{p}\mapsto A_{x_{1}x_{2}}\mathfrak{p}$

is a bijection with the inverse $Q\mapsto Z(A_{x_{1}x_{2}})\cap Q$.

Proof. In general, the theorem is true for all simple algebras. In particular,
it is true for simple algebras $A_{x_{1}x_{2}}$, i.e. the algebras that belong
to the cases (C1) and (C7b) (Theorem 3.8).

So, we may assume that the algebra $A_{x_{1}x_{2}}$ does not belong to the
cases (C1) and (C7b). In particular, the centre of the algebra
$A_{x_{1}x_{2}}$ is not the field $K$ (Theorem 3.2). By Lemma 3.3 and the
comments before it, it suffices to consider the cases (C2)–(C4), (C6), (C8),
(C10) and (C11). So, we may assume that the algebra $A_{x_{1}x_{2}}$ belongs
to one of these cases. Let $J$ be a nonzero ideal of the algebra
$A_{x_{1}x_{2}}$. By Theorem 3.4, in order to finish the prove of the theorem
we have to show that

$J=J\cap Z(A_{x_{1}x_{2}}).$

By Theorem 3.4, each element $a\in J$ is a unique sum $a=\sum_{i\in
I,\beta\in{\cal R}}z_{i\beta}x_{3}^{i}x^{\beta}$ where $z_{i\beta}\in
Z(A_{x_{1}x_{2}})$. So, we have to show that all elements $z_{i\beta}$ belong
to the ideal $J$. The ideal $J$ is an $\omega$-module. By Proposition 3.5, all
the elements $\sum_{i\in I}z_{i\beta}x_{3}^{i}x^{\beta}$, where $\beta\in{\cal
R}$, belong to $J$. So, we may assume that $a=\sum_{i\in
I}z_{i\beta}x_{3}^{i}x^{\beta}$ for some $\beta\in{\cal R}$. The element
$x^{\beta}$ is a unit element of the algebra $A_{x_{1}x_{2}}$. Hence, we may
assume that $\beta=0$ (by dividing the element $a$ on the right by
$x^{\beta}$), i.e.

$a=a(x_{3})=\sum_{i=0}^{d}z_{i0}x_{3}^{i}$

for some $d={\rm Deg}_{x_{3}}\geq 0$ such that $z_{d0}\neq 0$. The case when
$D=0$ is obvious. So, we assume that $d>0$ and will use induction on $d$.

(i) Suppose that the algebra $A_{x_{1}x_{2}}$ belongs to one of the cases (C4)
or (C6) (resp.,(C2), (C8) or (C11)).  In each of these cases, $p=0$,
$I=\mathbb{N}$ and $\alpha=1$ (resp., $p>0$, $I=\mathbb{N}_{<p}$, $\alpha=1$
and $d<p$). The element

$J\ni(1-\omega_{x_{1}})(a)=a(x_{3})-a(x_{3}-1)=dz_{d0}x_{3}^{d-1}+\cdots$

has degree $d-1$. By induction $dz_{d0}\in\mathfrak{a}$, and so
$z_{d0}\in\mathfrak{a}$ since the element $d$ is a unit in $A_{x_{1}x_{2}}$ in
all five cases. The element $a-z_{d0}x^{d}\in J$ has degree $<d$. By
induction, all coefficients $z_{i0}$ belong to $J$.

(ii) Suppose that the algebra $A_{x_{1}x_{2}}$ belongs to one of the cases
(C3), or (C10). In each of these cases, $p>0$, $I=\mathbb{N}_{<p^{2}}$,
$\alpha=1$ and $\mu\not\in\mathbb{F}_{p}$. We may assume that $d\geq p$
otherwise we repeat word for word the proof of the statement (i). In this case
the element $a$ is a unique sum $\sum_{j=0}^{p-1}\psi_{j}x_{3}^{j}$ for some
elements $\psi_{j}\in\sum_{i=0}^{p-1}\xi_{ji}(x_{3}^{p}-x_{3})^{i}$ where
$\xi_{ji}\in K[x_{3}]^{G}\subseteq Z(A_{x_{1}x_{2}})$. We have to show that
all the elements $\xi_{ji}$ belong to the ideal $J$. Let

$D^{\prime}(a):=\max\\{j\,|\,\psi_{j}\neq 0\\}.$

Suppose that $D^{\prime}(a)=0$, i.e.
$a=\psi_{0}=\sum_{i=0}^{p-1}\xi_{0i}(x_{3}^{p}-x_{3})^{i}=\sum_{i=0}^{\delta}\xi_{0i}(x_{3}^{p}-x_{3})^{i}$
for some natural number $\delta<p$ such that $\xi_{0\delta}\neq 0$. Clearly,
$\delta\geq 1$ as $d\geq p$. Then $\mu^{p}-\mu\neq 0$ (since
$\mu\not\in\mathbb{F}_{p})$ and

$\displaystyle J$ $\displaystyle\ni$
$\displaystyle(1-\omega_{x_{2}})(a(x_{3}))=a(x_{3})-a(x_{3}-\mu)=\sum_{i=0}^{\delta}\xi_{0i}(x_{3}^{p}-x_{3})^{i}-\sum_{i=0}^{\delta}\xi_{0i}(x_{3}^{p}-x_{3}-(\mu^{p}-\mu))^{i}$
$\displaystyle=$
$\displaystyle\delta(\mu^{p}-\mu)\xi_{0\delta}(x_{3}^{p}-x_{3})^{\delta-1}+\cdots.$

Then by induction on $\delta$, $\xi_{0i}\in J$ for all $i$ such that $1\leq
i\leq\delta$. Now,
$\xi_{00}=v-\sum_{i=1}^{\delta}\xi_{0i}(x_{3}^{p}-x_{3})^{i}\in J$. So, all
the elements $\xi_{0i}$ belong to $J$.

Suppose that $D^{\prime}=D^{\prime}(a)>0$. Now,

$J\ni(1-\omega_{x_{1}})(a(x_{3}))=a(x_{3})-a(x_{3}-1)=\sum_{j=0}^{D^{\prime}}\psi_{j}(x_{3}^{j}-(x_{3}-1)^{j})=D^{\prime}\psi_{D^{\prime}}x_{3}^{D^{\prime}-1}+\cdots.$

By induction on $D^{\prime}$, $\xi_{D^{\prime}i}\in J$. The element
$a^{\prime}:=a-\psi_{D^{\prime}}x_{3}^{D^{\prime}}\in J$ has
$D^{\prime}(a^{\prime})<D^{\prime}(a)=D^{\prime}$. By induction on
$D^{\prime}$, all the coefficients $\xi_{ji}\in J$, as required. This finishes
the proof of statement (ii) and the theorem. $\Box$

Theorem 3.10 is a classification of prime ideals of the algebra
$A_{x_{1}x_{2}}$.

######  Theorem 3.10.

The map

${\rm Spec}(Z(A_{x_{1}x_{2}}))\rightarrow{\rm
Spec}(A_{x_{1}x_{2}}),\;\;\mathfrak{p}\mapsto A_{x_{1}x_{2}}\mathfrak{p}$

is a bijection with the inverse $Q\mapsto Z(A_{x_{1}x_{2}})\cap Q$. In
particular, every nonzero prime ideal of $A_{x_{1}x_{2}}$ meets the centre.

Proof. The theorem follows at once from Theorem 3.9. $\Box$

##  4 The prime spectrum of the algebra $A$

The aim of the section is to classify the prime ideals of the algebra $A$ (see
(15), (18) and Theorem 4.6).

For an algebra $R$, let ${\rm Spec}\,(R)$ be the set of its prime ideals. The
set $({\rm Spec}\,(R),\subseteq)$ is a partially ordered set (poset) with
respect to inclusion of prime ideals. A prime ideal $\mathfrak{p}$ of an
algebra $R$ is called a _completely prime ideal_ if $R/\mathfrak{p}$ is a
domain. We denote by ${\rm Spec}_{c}(R)$ the set of completely prime ideals of
$R$; it is called the _completely prime spectrum_ of $R$. The annihilator of a
simple $R$-module is a prime ideal of $R$. That kind of prime ideals are
called primitive. The set of all primitive ideal of $R$ is denoted by ${\rm
Prim}(R)$. By the very definition, the poset $({\rm Spec}\,(R),\subseteq)$ is
determined by the containment information about prime ideals of $R$. In
general, it is difficult to say whether one prime ideal contains or does not
contain another prime ideal. Proposition 4.1.(3) is a general result about
containments of primes. It splits the prime spectrum of a ring into two
disjoint subsets such that elements of one are not contained in elements of
the other.

Each element $r\in R$ determines two maps from $R$ to $R$, $r\cdot:x\mapsto
rx$ and $\cdot r:x\mapsto xr$ where $x\in R.$ An element $a\in R$ is called a
_normal element_ if $Ra=aR$.

######  Proposition 4.1.

([1].) Let $R$ be a Noetherian ring and $s$ be an element of $R$ such that
$S_{s}:=\\{s^{i}\,|\,i\in\mathbb{N}\\}$ is a left denominator set of the ring
$R$ and $(s^{i})=(s)^{i}$ for all $i\geqslant 1$ (e.g., $s$ is a normal
element such that ${\rm ker}(\cdot s_{R})\subseteq{\rm ker}(s_{R}\cdot)$).
Then ${\rm Spec}\,(R)={\rm Spec}(R,s)\,\sqcup\,{\rm Spec}_{s}(R)$ where ${\rm
Spec}(R,s):=\\{\mathfrak{p}\in{\rm Spec}\,(R)\,|\,s\in\mathfrak{p}\\}$, ${\rm
Spec}_{s}(R)=\\{\mathfrak{q}\in{\rm Spec}\,(R)\,|\,s\notin\mathfrak{q}\\}$ and

  1. 1.

the map ${\rm Spec}\,(R,s)\rightarrow{\rm
Spec}\,(R/(s)),\,\,\mathfrak{p}\mapsto\mathfrak{p}/(s)$, is a bijection with
the inverse $\mathfrak{q}\mapsto\pi^{-1}(\mathfrak{q})$ where
$\pi:R\rightarrow R/(s),r\mapsto r+(s),$

  2. 2.

the map ${\rm Spec}_{s}(R)\rightarrow{\rm
Spec}\,(R_{s}),\,\,\mathfrak{p}\mapsto S_{s}^{-1}\mathfrak{p},$ is a bijection
with the inverse $\mathfrak{q}\mapsto\sigma^{-1}(\mathfrak{q})$ where
$\sigma:R\rightarrow R_{s}:=S_{s}^{-1}R,\,r\mapsto\frac{r}{1}$.

  3. 3.

For all $\mathfrak{p}\in{\rm Spec}\,(R,s)$ and $\mathfrak{q}\in{\rm
Spec}_{s}(R),\mathfrak{p}\not\subseteq\mathfrak{q}.$

The prime spectra and simple modules for the algebras $A/(x_{1})$ and
$A/(x_{2})$.

$\displaystyle A/(x_{1})$ $\displaystyle=$ $\displaystyle K\langle
x_{2},x_{3}\,|\,x_{2}x_{3}=(x_{3}-\mu)x_{2}\rangle\simeq\begin{cases}\mathbb{A}&\text{if
}\mu\neq 0,\\\ K[x_{2},x_{3}]&\text{if }\mu=0.\\\ \end{cases}$ $\displaystyle
A/(x_{2})$ $\displaystyle=$ $\displaystyle K\langle
x_{1},x_{3}\,|\,x_{1}x_{3}=(x_{3}-\alpha)x_{1}\rangle\simeq\begin{cases}\mathbb{A}&\text{if
}\alpha\neq 0,\\\ K[x_{1},x_{3}]&\text{if }\alpha=0.\\\ \end{cases}$

The algebras $A/(x_{1})$ and $A/(x_{2})$ are subalgebras of $A$. The prime
ideals and simple modules for the algebra $\mathbb{A}$ are classified in [4].

For each subset $\mathfrak{a}$ of $A$,
$V(\mathfrak{a}):=\\{\mathfrak{p}\in{\rm
Spec}(A)\,|\,\mathfrak{a}\subseteq\mathfrak{p}\\}$. The element $x_{1}x_{2}\in
A$ is a regular normal element. Therefore, $A\subseteq A_{x_{1}x_{2}}$. By
Proposition 4.1,

${\rm Spec}(A)=\bigg{(}V(x_{1})\cup V(x_{2})\bigg{)}\coprod{\rm
Spec}_{x_{1}x_{2}}(A)$ (15)

since ${\rm Spec}(A,x_{1}x_{2})=V(x_{1}x_{2})=V(x_{1})\cup V(x_{2})$ as the
elements $x_{1}$ and $x_{2}$ are normal, and the map

${\rm Spec}_{x_{1}x_{2}}(A)\rightarrow{\rm Spec}(A_{x_{1}x_{2}}),\;\;P\mapsto
P_{x_{1}x_{2}}$ (16)

is a bijection with the inverse $Q\mapsto A\cap Q$. By Theorem 3.10, the map

${\rm Spec}(Z(A_{x_{1}x_{2}}))\rightarrow{\rm
Spec}(A_{x_{1}x_{2}}),\;\;\mathfrak{p}\mapsto A_{x_{1}x_{2}}\mathfrak{p}$

is a bijection with the inverse $Q\mapsto Z(A_{x_{1}x_{2}})\cap Q$. Therefore,
the map

${\rm Spec}(A_{x_{1}x_{2}})\rightarrow{\rm
Spec}_{x_{1}x_{2}}(A),\;\;\mathfrak{q}\mapsto A\cap\mathfrak{q}A_{x_{1}x_{2}}$
(17)

is a bijection with the inverse $P\mapsto Z(A_{x_{1}x_{2}})\cap
P_{x_{1}x_{2}}$. Since the elements $x_{1}$ and $x_{2}$ are regular and normal
elements of the algebra $A$ we have the equalities

$V(x_{1})={\rm Spec}(A/(x_{2}))\;\;{\rm and}\;\;V(x_{2})={\rm
Spec}(A/(x_{1})).$ (18)

The prime spectrum of $A$: Cases (C1) and (C7b).

######  Theorem 4.2.

The following statements are equivalent:

  1. 1.

The algebra $A$ belongs to one of the cases (C1) or (C7b).

  2. 2.

${\rm Spec}(A)=\\{0\\}\cup V(x_{1})\cup V(x_{2})=\\{0\\}\cup{\rm
Spec}(A/(x_{1}))\cup{\rm Spec}(A/(x_{2}))$.

  3. 3.

${\rm Spec}_{x_{1}x_{2}}(A)=\\{0\\}$.

  4. 4.

${\rm Spec}(A_{x_{1}x_{2}})=\\{0\\}$.

  5. 5.

The algebra $A_{x_{1}x_{2}}$ is a simple algebra.

Proof. By (15), ${\rm Spec}(A)=\\{0\\}\cup V(x_{1})\cup V(x_{2})$ iff ${\rm
Spec}_{x_{1}x_{2}}(A)=\\{0\\}$ iff ${\rm Spec}(A_{x_{1}x_{2}})=\\{0\\}$ iff
the algebra $A_{x_{1}x_{2}}$ is a simple algebra iff the algebra $A$ belongs
to one of the cases (C1) or (C7b), by Theorem 3.8. $\Box$

The set $T:=T_{A}:=Z(A)\cap\\{x^{\beta}\,|\,\beta\in\mathbb{N}^{2}\\}$ is an
Ore set of the algebra $A$ that belongs to the centre of $A$. Lemma 4.3
describes generators of the monoid $T$.

######  Lemma 4.3.

Let $M_{\Sigma}:=\langle
x_{1}^{pn},x_{1}^{\xi_{i}n}x_{2}^{in},x_{2}^{pn}\rangle_{i=1,\ldots,p-1}$ be
the multiplicative submonoid of $\Sigma$ (which is generated by the elements
in the brackets) where $\xi_{i}$ is a unique natural number such that
$0\leq\xi_{i}\leq p-1$ and $\xi_{i}\equiv-i\mu\mod p$ (see Theorem 2.3) and
$T=Z(A)\cap\\{x^{\beta}\,|\,\beta\in\mathbb{N}^{2}\\}$. Then

$T=\begin{cases}\\{1\\}&\text{(C1): if $q$ is not a root of unity and
$p=0$},\\\ \\{1\\}&\text{(C2): if $q$ is not a root of unity, $p>0$ and}\\\
&(\alpha,\mu)\in\\{(1,\mu^{\prime}),(0,1)\,|\,\mu^{\prime}\in\mathbb{F}_{p}\backslash\\{-1\\}\\},\\\
\\{1\\}&\text{(C3): if $q$ is not a root of unity, $p>0$, $\alpha=1$ and
$\mu\not\in\mathbb{F}_{p}$},\\\ \langle x_{2}^{n}\rangle&\text{(C4): if $q$ is
a primitive $n$'th root of unity, $p=0$, $\alpha=1$ and $\mu=0$},\\\ \langle
x_{1}^{n}\rangle&\text{(C5): if $q$ is a primitive $n$'th root of unity,
$p=0$, $\alpha=0$ and $\mu=1$},\\\ \langle
x_{1}^{\mu_{1}n}x_{2}\rangle&\text{(C6): if $q$ is a primitive $n$'th root of
unity, $p=0$, $\alpha=1$ and $\mu=-\frac{\mu_{1}}{\mu_{2}}$},\\\ &\text{ where
$\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ such that $\mu_{1}\neq\mu_{2}$ and
$(\mu_{1},\mu_{2})=1$},\\\ \\{1\\}&\text{(C7a): if $q$ is a primitive $n$'th
root of unity, $p=0$, $\alpha=1$ and $\mu=\frac{\mu_{1}}{\mu_{2}}$},\\\
&\text{ where $\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ such that
$(\mu_{1},\mu_{2})=1$},\\\ \\{1\\}&\text{(C7b): if $q$ is a primitive $n$'th
root of unity, $p=0$, $\alpha=1$ and $\mu\not\in\mathbb{Q}$},\\\ \langle
x_{1}^{pn},x_{2}^{n}\rangle&\text{(C8): if $q$ is a primitive $n$'th root of
unity, $p>0$, $\alpha=1$, $\mu=0$ and $p\nmid n$},\\\ \langle
x_{1}^{n},x_{2}^{pn}\rangle&\text{(C9): if $q$ is a primitive $n$'th root of
unity, $p>0$, $\alpha=0$, $\mu=1$ and $p\nmid n$,}\\\ \langle
x_{1}^{pn},x_{2}^{pn}\rangle&\text{(C10): if $q$ is a primitive $n$'th root of
unity, $p>0$, $\alpha=1$, $\mu\not\in\mathbb{F}_{p}$ and $p\nmid n$},\\\
M_{\Sigma}&\text{(C11): if $q$ is a primitive $n$'th root of unity, $p>0$,
$\alpha=1$},\\\
&\text{$\mu\in\mathbb{F}_{p}\backslash\\{0,-1\\}=\\{1,\ldots,p-2\\}$ and
$p\nmid n$.}\\\ \end{cases}$

Proof. The lemma follows from the description of the centre of the algebra
$A_{x_{1}x_{2}}$ (Theorem 3.2. $\Box$

######  Proposition 4.4.

Let $T=Z(A)\cap\\{x^{\beta}\,|\,\beta\in\mathbb{N}^{2}\\}$. Then the following
statements are equivalent:

  1. 1.

The algebra $A$ is not of type (C7a).

  2. 2.

$Z(A_{x_{1}x_{2}})=T^{-1}Z(A)$.

Proof. Suppose that the algebra $A$ is of type (C7a). Then $T=\\{1\\}$, and so
$T^{-1}Z(A)=Z(A)=K\neq Z(A_{x_{1}x_{2}})$, by Theorem 2.3 and Theorem 3.2.

Suppose that the algebra $A$ is not of type (C7a). We have to show that
$Z(A_{x_{1}x_{2}})=T^{-1}Z(A)$. This follows straightaway from the description
of the set $T$ (Lemma 4.3) and the descriptions of the centres of the algebras
$A$ (Theorem 2.3) and $A_{x_{1}x_{2}}$ (Theorem 3.2). $\Box$

######  Lemma 4.5.

Let $T=Z(A)\cap\\{x^{\beta}\,|\,\beta\in\mathbb{N}^{2}\\}$. Then the following
statements are equivalent:

  1. 1.

The algebra $A$ is of type (C6) or (C8)–(C11).

  2. 2.

$A_{x_{1}x_{2}}=T^{-1}A$.

  3. 3.

$Z(A_{x_{1}x_{2}})=T^{-1}Z(A)$.

Remark. Lemma 4.5 states that if the algebra $A$ is of type (C6) or (C8)–(C11)
then the localization $A_{x_{1}x_{2}}$ of the algebra $A$ at the Ore set that
is generated by a normal element $x_{1}x_{2}$ is in fact the localization of
the algebra $A$ at an Ore set that is generated by central elements. Also the
centre of the algebra $A_{x_{1}x_{2}}$ is the localization of the centre of
the algebra $A$ at an Ore set that is generated by central elements. In
general, both these statements are badly wrong (as the algebras $A$ of type
(C7a) show: $T_{A}=\\{1\\}$,
$Z(A_{x_{1}x_{2}})=K[(x_{1}^{\mu_{1}n}x_{2}^{-\mu_{2}n})^{\pm 1}]\neq
K=Z(A)=T^{-1}_{A}Z(A)$ and $A_{x_{1}x_{2}}\neq A=T^{-1}_{A}A$).

Proof. The lemma follows from Lemma 4.3 and Proposition 4.4. $\Box$

Theorem 4.6 is an explicit description of the set ${\rm
Spec}_{x_{1}x_{2}}(A)$.

######  Theorem 4.6.

  1. 1.

Suppose that the algebra $A$ is not of type (C7a) or (C11). Then the map

${\rm Spec}(Z(A))\backslash V(T_{A})\rightarrow{\rm
Spec}_{x_{1}x_{2}}(A),\;\;\mathfrak{p}\mapsto\mathfrak{p}A=\bigoplus_{i\in
I_{A},\beta\in{\cal R}_{A}}\mathfrak{p}x_{3}^{i}x^{\beta}$

is a bijection with the inverse $P\mapsto P\cap Z(A)$ where the set $T_{A}$ is
defined in Lemma 4.3 and the sets $I_{A}$ and ${\cal R}_{A}$ are defined in
Theorem 2.4.

  2. 2.

Suppose that the algebra $A$ is of type (C7a) or (C11). Then the map

${\rm Spec}(Z(A_{x_{1}x_{2}}))\rightarrow{\rm
Spec}_{x_{1}x_{2}}(A),\;\;\mathfrak{q}\mapsto A\cap\mathfrak{q}A_{x_{1}x_{2}}$

is a bijection with the inverse $P\mapsto P_{x_{1}x_{2}}\cap
Z(A_{x_{1}x_{2}})$.

Proof. 1. Let $T=T_{A}$. By Proposition 4.4, $Z(A_{x_{1}x_{2}})=T^{-1}Z(A)$,
and so ${\rm Spec}(Z(A_{x_{1}x_{2}}))={\rm Spec}(Z(A))\backslash V(T)$. Now,
by (17), the map

${\rm Spec}(Z(A))\backslash V(T)\rightarrow{\rm
Spec}_{x_{1}x_{2}}(A),\;\;\mathfrak{p}\mapsto A\cap\mathfrak{p}A_{x_{1}x_{2}}$

is a bijection with the inverse $P\mapsto P_{x_{1}x_{2}}\cap
Z(A_{x_{1}x_{2}})$. By Proposition 4.4 and Theorem 2.4,

$A\cap\mathfrak{p}A_{x_{1}x_{2}}=A\cap\mathfrak{p}T^{-1}A=\bigoplus_{i\in
I_{A},\beta\in{\cal
R}_{A}}\Big{(}Z(A)\cap\mathfrak{p}T^{-1}Z(A)\Big{)}x_{3}^{i}x^{\beta}=\bigoplus_{i\in
I_{A},\beta\in{\cal R}_{A}}\mathfrak{p}x_{3}^{i}x^{\beta},$

$\mathfrak{p}A_{x_{1}x_{2}}\cap
Z(A_{x_{1}x_{2}})=\mathfrak{p}A_{x_{1}x_{2}}\cap
T^{-1}Z(A)=\mathfrak{p}T^{-1}Z(A)=T^{-1}\mathfrak{p}$, and statement 1
follows.

2\. Statement 2 is (17). $\Box$

In contrast to Theorem 3.10 not every nonzero prime ideal of the set ${\rm
Spec}_{x_{1}x_{2}}(A)$ meets the centre of the algebra $A$.

######  Corollary 4.7.

The following statements are equivalent:

  1. 1.

For all nonzero prime ideals $P\in{\rm Spec}_{x_{1}x_{2}}(A)$, $P\cap Z(A)\neq
0$.

  2. 2.

The algebra $A$ is not of the type (C7a).

Proof. By Proposition 4.4, if the algebra $A$ is not of the type (C7a) then
$P\cap Z(A)\neq 0$ for all nonzero prime ideals $P\in{\rm
Spec}_{A_{x_{1}x_{2}}}(A)$ (since the algebra $A$ is an essential left/right
$A$-submodule of $A_{x_{1}x_{2}}$, $Z(A_{x_{1}x_{2}})=T_{A}^{-1}Z(A)$ and
Theorem 3.10).

If the algebra $A$ is of the type (C7a) then $Z(A_{x_{1}x_{2}})\neq K=Z(A)$,
and so $P\cap Z(A)=P\cap K=0$ for all nonzero prime ideals $P\in{\rm
Spec}_{x_{1}x_{2}}(A)$. $\Box$

######  Corollary 4.8.

Suppose that the algebra $A$ belongs to the case (C7a). Then for all nonzero
prime ideals $P\in{\rm Spec}_{x_{1}x_{2}}(A)$, $P\cap Z(A)=0$.

Proof. See the proof of Corollary 4.7. $\Box$

For a ring $R$ and its prime ideal $P$, ${\rm cht}(P):={\rm Kdim}(R/P)$ is
called the co-height of $P$.

The algebra $A$ is an $\omega$-submodule of the algebra $A_{x_{1}x_{2}}$.

######  Lemma 4.9.

Every prime ideal $P\in{\rm Spec}_{x_{1}x_{2}}(A)$ is an $\omega$-submodule of
the algebra $A$.

Proof. Since $P\in{\rm Spec}_{x_{1}x_{2}}(A)$, $x_{i}\not\in P$ for $i=1,2$.
Then it follows from the inclusions:

$\displaystyle P\supseteq(x_{i})P$ $\displaystyle=$
$\displaystyle(x_{i}P)=(\omega_{x_{i}}(P)x_{i})=\omega_{x_{i}}(P)(x_{i}),$
$\displaystyle P\supseteq P(x_{i})$ $\displaystyle=$
$\displaystyle(Px_{i})=(x_{i}\omega_{x_{i}}^{-1}(P))=(x_{i})\omega_{x_{i}}^{-1}(P),$

that $\omega_{x_{i}}^{\pm 1}(P)\subseteq P$, and so $\omega_{x_{i}}(P)=P$.
Clealry, ${\rm ad}_{x_{3}}(P)\subseteq P$, and the lemma follows. $\Box$

######  Corollary 4.10.

For all nonzero prime ideals $P\in{\rm Spec}(A)$, $P\cap{\rm
Es}_{A}(\omega)\neq 0$.

Proof. The corollary holds for all nonzero prime ideals of the algebra $A$
that contains the element $x_{1}x_{2}\in{\rm Es}_{A}(\omega)$. By (15), we may
assume that $0\neq P\in{\rm Spec}_{x_{1}x_{2}}(A)$.

If the algebra $A$ is not the type (C7a) then the corollary follows from
Corollary 4.7.

Suppose that the algebra is of the type (C7a). Since
$Z(A_{x_{1}x_{2}})=K[x_{1}^{\mu_{1}n}x_{2}^{-\mu_{2}n})^{\pm 1}]$ (Theorem
3.2), the corollary follows from Theorem 4.6.(2). $\Box$

##  5 The automorphism group of the algebra $A$

The aim of the section is to describe the automorphism group ${\rm
Aut}_{K}(A)$ (Theorem 5.2), to classify $\omega$-weight vectors (Theorem 5.3)
and normal elements (Proposition 5.4) of the algebra $A$. Furthermore, we will
show that every normal element of the algebra $A$ is an $\omega$-weight
vector, and vice versa (Proposition 5.4).

The automorphism group ${\rm Aut}_{K}(A)$. Proposition 5.1 is the key fact in
finding an explicit description for the automorphism group of the algebra $A$
(Theorem 5.2).

######  Proposition 5.1.

  1. 1.

The ideals $(x_{1})$ and $(x_{2})$ of the algebra $A$ has co-height 2. The
ideals $(x_{1})$ and $(x_{2})$ of the algebra $A$ are the only ideals that
have co-height 2 iff the algebra $A$ is of type (C1)–(C7).

  2. 2.

If the algebra $A$ is of type (C8) (resp., (C9)) then the ideal $(x_{1})$
(resp., $(x_{2}))$ is the only height 1 prime ideal $P$ of $A$ such that the
algebra $A/P$ is commutative.

  3. 3.

Suppose that the algebra $A$ is of type (C10) or (C11). Then the ideals
$(x_{1})$ and $(x_{2})$ of the algebra $A$ is the only pair of prime ideals of
the algebra $A$ of height 1, $P$ and $Q$, the factor algebra $A/(P+Q)$ is
commutative domain.

Proof. 1. Clearly, ${\rm cht}(x_{i})={\rm Kdim}(A/(x_{i}))=2$. The second part
of the corrollary follows from the classification of prime ideals in the set
${\rm Spec}_{x_{1}x_{2}}(A)$ (Theorem 4.6) and the descriptions of the centres
of the algebras $A$ and $A_{x_{1}x_{2}}$ (Theorem 2.3 and Theorem 3.2).

2\. In the view of the $(x_{1},x_{2})$-symmetry, it suffices to consider the
case (C8). The algebra $A/(x_{1})=K[x_{2},x_{3}]$ is obviously a commutative
domain and the algebra $A/(x_{2})$ is obviously not (since
$x_{1}x_{3}=(x_{3}-1)x_{1}$).

Suppose that $P\in{\rm Spec}_{x_{1}x_{2}}(A)$. By Theorem 2.4,
$A=\bigoplus_{i\in I,\beta\in{\cal R}}Z(A)x_{3}^{i}x^{\beta}$. Then the
algebra

$A/(P)=\bigoplus_{i\in I,\beta\in{\cal
R}}\Big{(}Z(A)/(P)\Big{)}x_{3}^{i}x^{\beta}$

is not commutative, by Theorem 2.4, and the result follows (from the
classification of prime ideals of the algebra $A$).

3\. The algebra $A/(x_{1},x_{2})=K[x_{3}]$ is obviously a commutative domain.

Suppose that $P,Q\in{\rm Spec}_{x_{1}x_{2}}(A)$. By Theorem 3.4,
$A_{x_{1}x_{2}}=\bigoplus_{i\in I,\beta\in{\cal
R}}Z(A_{x_{1}x_{2}})x_{3}^{i}x^{\beta}$. Then the algebra

$\Big{(}A/(P,Q)\Big{)}_{x_{1}x_{2}}=A_{x_{1}x_{2}}/(P,Q)=\bigoplus_{i\in
I,\beta\in{\cal R}}\Big{(}Z(A_{x_{1}x_{2}})/(P,Q)\Big{)}x_{3}^{i}x^{\beta}$

is not commutative, by Theorem 3.4. Hence, the algebra $A/(P,Q)$ is also not a
commutative domain.

It remains to consider the case when the pair of ideals is $(x_{i})$, $P$
where $P\in{\rm Spec}_{x_{1}x_{2}}(A)$. Suppose that the algebra $A$ is of
type (C10). Then, by Theorem 4.6.(1), $P=\mathfrak{p}A$ for some prime ideal
$\mathfrak{p}\in{\rm Spec}Z(A))\backslash V(T_{A})$. By Theorem 2.4, the
algebra

$A/(x_{i},P)=\bigg{(}\bigoplus_{i\in I,\beta\in{\cal
R}}\Big{(}Z(A)/\mathfrak{p}\Big{)}x_{3}^{i}x^{\beta}\bigg{)}/(x_{i})$

is not commutative.

Suppose that the algebra $A$ is of type (C11) and the algebra $A/(x_{i},P)$ is
a commutative domain. We seek a contradiction. The factor algebras

$\displaystyle A/(x_{1})$ $\displaystyle=$ $\displaystyle K\langle
x_{2},x_{3}\,|\,x_{2}x_{3}=(x_{3}-\mu)x_{2}\rangle\simeq\mathbb{A},$
$\displaystyle A/(x_{2})$ $\displaystyle=$ $\displaystyle K\langle
x_{1},x_{3}\,|\,x_{1}x_{3}=(x_{3}-\alpha)x_{1}\rangle\simeq\mathbb{A},$

are both isomorphic to the skew polynomial ring
$\mathbb{A}=K[x_{3}][x_{1},\;\sigma]$ where $\sigma(x_{3})=x_{3}-1$. By the
classification of prime ideals of the algebra $\mathbb{A}$, every prime ideal
$\mathfrak{q}$ of the algebra $\mathbb{A}$ such that $\mathbb{A}/\mathfrak{q}$
is a commutative algebra contains the element $x_{1}$. Therefore,
$x_{2}\in(x_{1},P)$ (resp., $x_{1}\in(x_{2},P$)). In a view of the
$(x_{1},x_{2})$-symmetry, let us consider the first case. Recall that
$A_{x_{1}x_{2}}=T^{-1}_{A}A$ (Lemma 4.5) where $T_{A}=M_{\Sigma}$ (Lemma 4.3).
Then, by Theorem 4.6.(2),

$x_{2}=x_{1}a+(x_{1}^{n}x_{2}^{n})^{-s}(bx_{1}^{pn}+\sum_{i=1}^{p-1}b_{i}x_{1}^{\xi_{i}n}x_{2}^{in}+cx_{2}^{pn})$

for some elements $a,b,c,b_{i}\in A$ and a natural number $s$. By multiplying
the equality above on the left by the element $(x_{1}^{n}x_{2}^{n})^{s}$ and
using the $\mathbb{N}^{2}$-grading of the algebra $A$ we get a contradiction.
$\Box$

######  Theorem 5.2.

Let $A=A(q,\alpha,\mu)$ and $C:=C_{A}(x_{1},x_{2})$, see Proposition 2.2. Then

  1. 1.

Suppose that $q^{2}\neq 1$. Then

${\rm
Aut}_{K}(A)=\\{\sigma_{\lambda}\,|\,\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})\in
K^{\times}\times K^{\times}\times C\\}\simeq K^{\times}\times K^{\times}\times
C,\;\;\sigma_{\lambda}\mapsto\lambda$

where $\sigma_{\lambda}:x_{1}\mapsto\lambda_{1}x_{1}$,
$x_{2}\mapsto\lambda_{2}x_{2}$, $x_{3}\mapsto x_{3}+\lambda_{3}$.

  2. 2.

Suppose that $q^{2}=1$ ($q\neq 1$). Then

${\rm
Aut}_{K}(A)=\\{\sigma_{\lambda}\tau^{i}\,|\,\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})\in
K^{\times}\times K^{\times}\times C,i=0,1\\}\simeq\Big{(}K^{\times}\times
K^{\times}\times C\Big{)}\rtimes\mathbb{Z}_{2},$

a semidirect product of groups where
$\sigma_{\lambda}:x_{1}\mapsto\lambda_{1}x_{1}$,
$x_{2}\mapsto\lambda_{2}x_{2}$, $x_{3}\mapsto x_{3}+\lambda_{3}$ and
$\tau:x_{1}\mapsto x_{2}$, $x_{2}\mapsto x_{1}$, $x_{3}\mapsto x_{3}$;
$\tau^{2}=1$,
$\tau\sigma_{(\lambda_{1},\lambda_{2},\lambda_{3})}=\sigma_{(\lambda_{2},\lambda_{1},\tau(\lambda_{3}))}\tau$
and $\mathbb{Z}_{2}=\mathbb{Z}/2\mathbb{Z}\simeq\langle\tau\rangle$.

Proof. Suppose that $q^{2}\neq 1$ (resp., $q^{2}=1$). Let
$\Gamma=\\{\sigma_{\lambda}\,|\,\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})\in
K^{\times}\times K^{\times}\times C\\}$. Using the defining relations of the
algebra $A$, we see that $\Gamma$ (resp.,
$\Gamma^{\prime}:=\\{\sigma_{\lambda}\tau^{i}\,|\,\lambda=(\lambda_{1},\lambda_{2},\lambda_{3})\in
K^{\times}\times K^{\times}\times C,i=0,1\\}$) is a subgroup of ${\rm
Aut}_{K}(A)$ which is isomorphic to the group $K^{\times}\times
K^{\times}\times C$ via $\sigma_{\lambda}\mapsto\lambda$ (resp.,
$\Gamma\rtimes\mathbb{Z}_{2}$).

Given an automorphism $\sigma$ of the algebra $A$. We have to show that
$\sigma\in\Gamma$ (resp., $\sigma\in\Gamma^{\prime}$). The theorem follows
from the following Claim.

Claim: The group ${\rm Aut}_{K}(A)$ acts on the set
$\\{K^{\times}x_{1},K^{\times}x_{2}\\}$ by the rule
$(\sigma,S)\mapsto\sigma(S)$ where
$S\in\\{K^{\times}x_{1},K^{\times}x_{2}\\}$.

Suppose that the Claim holds. Then we have two options either
$\sigma(x_{1})=\lambda_{1}x_{1}$, $\sigma(x_{2})=\lambda_{2}x_{2}$ or
$\sigma(x_{1})=\lambda_{1}x_{2}$, $\sigma(x_{2})=\lambda_{2}x_{1}$ for some
elements $\lambda_{1},\lambda_{2}\in K^{\times}$. The second option is only
possible in the case when $q^{2}=1$. This follows from the relation
$x_{2}x_{1}=qx_{1}x_{2}$. In the second case, by replacing the automorphism
$\sigma$ by $\sigma\tau$ we may assume that in both cases
$\sigma(x_{1})=\lambda_{1}x_{1}$ and $\sigma(x_{2})=\lambda_{2}x_{2}$.
Furthermore, by replacing $\sigma$ by the automorphism
$\sigma\sigma^{-1}_{(\lambda_{1},\lambda_{2},0)}$, we may assume that
$\lambda_{1}=\lambda_{2}=1$. Let $x_{3}^{\prime}=\sigma(x_{3})$. Then

$\displaystyle[x_{3}^{\prime}-x_{3},x_{1}]$ $\displaystyle=$
$\displaystyle\sigma([x_{3},x_{1}])-[x_{3},x_{1}]=\sigma(\alpha x_{1})-\alpha
x_{1}=\alpha x_{1}-\alpha x_{1}=0,$
$\displaystyle\,[x_{3}^{\prime}-x_{3},x_{2}]$ $\displaystyle=$
$\displaystyle\sigma([x_{3},x_{2}])-[x_{3},x_{2}]=\sigma(\mu x_{2})-\mu
x_{2}=\mu x_{2}-\mu x_{2}=0,$

and so $c:=x_{3}^{\prime}-x_{3}\in C$, i.e.
$\sigma=\sigma_{(1,1,c)}\in\Gamma$, as required.

Proof of the Claim. Suppose that the algebra $A$ is of type (C1)–(C7) (resp.,
(C10) or (C11)). By Proposition 5.1.(1) (resp., Proposition 5.1.(3)), the
group ${\rm Aut}_{K}(A)$ acts on the set $\\{(x_{1}),(x_{2})\\}$. Since
$K^{\times}$ is the group of units of the algebra $A$ the Claim follows.

Suppose that the algebra $A$ is of type (C8) or (C9). In the view of
$(x_{1},x_{2})$-symmetry, it suffices to consider only the case (C8), i.e. $q$
is a primitive $n$’th root of unity, $p>0$, $\alpha=1$, $\mu=0$ and $p\nmid
n$. Clearly, the algebra $A=K[x_{2},x_{3}][x_{1};\sigma]$ is a skew polynomial
ring where $\sigma(x_{2})=q^{-1}x_{2}$ and $\sigma(x_{3})=x_{3}-1$. By
Proposition 5.1.(2), the ideal $(x_{1})$ is the only height 1 prime ideal $P$
of the algebra $A$ such that $A/P$ is a commutative domain. Hence,
$\sigma((x_{1}))=(x_{1})$, and so $\sigma(x_{1})=\lambda_{1}x_{1}$ for some
unit $\lambda_{1}$ of the algebra $A$, i.e. $\lambda_{1}\in K^{\times}$.

By Theorem 2.4, $A=\bigoplus_{i\in I,\beta\in{\cal R}}Z(A)x_{3}^{i}x^{\beta}$
where $I=\mathbb{N}_{<p}$ and ${\cal
R}=\mathbb{N}_{<pn}\times\mathbb{N}_{<n}$. Notice that

$\bigoplus_{\beta_{1}\in
N_{<pn},\beta_{2}\in\mathbb{N}_{<n}}Z(A)x_{1}^{\beta_{1}}x_{2}^{\beta_{2}}$

is the direct sum of the eigenspaces of the automorphism $\omega_{x_{1}}$
where the set of eigenvalues is $\\{q^{-\beta_{2}}\,|\,\beta_{2}\in
N_{<n}\\}$. In particular,

$\sigma^{\pm 1}\Bigg{(}\bigoplus_{\beta_{1}\in
N_{<pn},\beta_{2}\in\mathbb{N}_{<n}}Z(A)x_{1}^{\beta_{1}}x_{2}\Bigg{)}=\bigoplus_{\beta_{1}\in
N_{<pn},\beta_{2}\in\mathbb{N}_{<n}}Z(A)x_{1}^{\beta_{1}}x_{2}.$

Hence, $\sigma(x_{2})=\lambda_{2}x_{2}$ for some element $\lambda_{2}\in
K^{\times}$, and the Claim holds. $\Box$

Classification of normal and $\omega$-weight elements of the algebra $A$.
Theorem 5.3 is a classification of $\omega$-weight elements of the algebra
$A$. It is also a classification of normal elements of the algebra $A$ as we
will show that every normal element of $A$ is an $\omega$-weight element and
vice versa (Proposition 5.4).

######  Theorem 5.3.

  1. 1.

Suppose that the algebra $A$ belongs to one of the cases (C1)–(C6), (C7b) or
(C8)–(C10). Then ${\cal
W}_{A}(\omega)=\\{(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})\,|\,\beta\in{\cal
R}_{A}\\}$ and for each weight
$\lambda=(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})$,
$A_{\lambda}=Z(A)x^{\beta}$ where the set ${\cal R}_{A}$ is defined in Theorem
2.4.

  2. 2.

Suppose that the algebra $A$ belongs to the case (C7a). Then ${\cal
W}_{A}(\omega)=\\{(q^{-\beta_{2}},q^{\beta_{1}},\beta_{1}+\mu\beta_{2}+kn)\,|\,\beta\in\mathbb{N}^{2}_{<n},k\in\mathbb{N}\\}$
and for each weight
$\lambda=(q^{-\beta_{2}},q^{\beta_{1}},\beta_{1}+\mu\beta_{2}+kn)\in{\cal
W}_{A}(\omega)$,

$A_{\lambda}=\bigoplus_{i=0}^{k}Kx_{1}^{\beta_{1}+in}x_{2}^{\beta_{2}+(k-i)n}\;\;{\rm
and}\;\;{\rm dim}_{K}(A_{\lambda})=k+1<\infty.$

In particular, the weight spaces are finite dimensional.

  3. 3.

Suppose that the algebra $A$ belongs to the case (C11). Then ${\cal
W}_{A}(\omega)={\cal
W}_{A_{x_{1}x_{2}}}(\omega)=\\{(q^{-\beta_{2}},q^{\beta_{1}},\beta_{1}+\mu\beta_{2})\,|\,\beta\in{\cal
R}_{x_{1}x_{2}}=\mathbb{N}_{<pn}\times\mathbb{N}_{<n}\\}$ and for each weight
$\lambda_{\beta}=(q^{-\beta_{2}},q^{\beta_{1}},\beta_{1}+\mu\beta_{2})\in{\cal
W}_{A}(\omega)$,

$A_{\lambda_{\beta}}=\bigoplus_{i=0}^{p-1}\Lambda^{\prime}x^{\theta(\beta,i)}$

where $\Lambda^{\prime}=K[x_{3}^{p}-x_{3}][x_{1}^{pn},x_{2}^{pn}]$ and
$\theta(\beta,i)$ is a unique element of the set $\mathbb{N}_{<pn}^{2}$ such
that $\theta(\beta,i)\equiv\beta+in(-\mu,1)\mod pn$.

Proof. 1. By Corollary 2.6.(1), ${\rm Es}_{A}(\omega)={\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})=\bigoplus_{\beta\in{\cal
R}_{A}}Z(A)x^{\beta}$, and for all $z\in Z(A)$ and $\beta\in{\cal R}_{A}$,
$\omega_{x_{1}}(zx^{\beta})=q^{-\beta_{2}}zx^{\beta}$,
$\omega_{x_{2}}(zx^{\beta})=q^{\beta_{1}}zx^{\beta}$, and ${\rm
ad}_{x_{3}}(zx^{\beta})=(\alpha\beta_{1}+\mu\beta_{2})zx^{\beta}$. Statement 1
follows from the Claim.

Claim. The map ${\cal R}_{A}\rightarrow{\cal W}_{A}(\omega)$,
$\beta\mapsto\lambda_{\beta}=(q^{-\beta_{2}},q^{\beta_{1}},\alpha\beta_{1}+\mu\beta_{2})$
is an injection.

Suppose that the Claim holds. Then necessarily the map is well defined and a
bijection since ${\rm Es}_{A}(\omega)=\bigoplus_{\beta\in{\cal
R}_{A}}Z(A)x^{\beta}$, and this direct sum is the direct sum of
$\omega$-eigenspaces.

Proof of the Claim.

Suppose that the algebra $A$ is of type (C1)–(C3). The the Claim is true since
the element $q$ is not a root of unity.

Suppose that the algebra $A$ is of type (C4) or (C5). In view of Theorem
1.1.(1b), it suffices to consider the case (C4). Recall that in this case, $q$
is a primitive $n$’th root of unity, $p=0$, $\alpha=1$, $\mu=0$ and ${\cal
R}_{A}=\mathbb{N}\times\mathbb{N}_{n}$ (Theorem 2.4). Then
$\lambda_{\beta}=(q^{-\beta_{2}},q^{\beta_{1}},\beta_{1})=\lambda_{\beta^{\prime}}=(q^{-\beta_{2}^{\prime}},q^{\beta_{1}^{\prime}},\beta_{1}^{\prime})$
implies that $\beta_{2}=\beta_{2}^{\prime}$ and
$\beta_{1}=\beta_{1}^{\prime}$, and so
$\lambda_{\beta}=\lambda_{\beta^{\prime}}$.

Suppose that the algebra $A$ is of type (C6). In this case, $q$ is a primitive
$n$’th root of unity, $p=0$, $\alpha=1$ and $\mu=-\frac{\mu_{1}}{\mu_{2}}$
where $\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ such that $\mu_{1}\neq\mu_{2}$ and
$(\mu_{1},\mu_{2})=1$, and ${\cal
R}_{A}=\mathbb{S}(\mu_{1}n,\mu_{2}n)=\mathbb{N}^{2}\backslash\Big{(}(\mu_{1}n,\mu_{2}n)+\mathbb{N}^{2}\Big{)}$
(Theorem 2.4). Then
$\lambda_{\beta}=(q^{-\beta_{2}},q^{\beta_{1}},\beta_{1}+\mu\beta_{2})=\lambda_{\beta^{\prime}}=(q^{-\beta_{2}^{\prime}},q^{\beta_{1}^{\prime}},\beta_{1}^{\prime}+\mu\beta_{2}^{\prime})$
implies the equality
$\mu_{2}(\beta_{1}^{\prime}-\beta_{1})=\mu_{1}(\beta_{2}^{\prime}-\beta_{2})$,
and so $\beta^{\prime}=\beta+k(\mu_{1},\mu_{2})$ for some $k\in\mathbb{N}$
since $(\mu_{1},\mu_{2})=1$, i.e. $\beta^{\prime}=\beta$ (since
$\beta,\beta^{\prime}\in\mathbb{S}(\mu_{1}n,\mu_{2}n)$), as required.

Suppose that the algebra $A$ is of type (C7b). In this case, $\alpha=1$,
$\mu\not\in\mathbb{Q}$ and ${\cal R}_{A}=\mathbb{N}^{2}$ (Theorem 2.4). Then
$\lambda_{\beta}=(q^{-\beta_{2}},q^{\beta_{1}},\beta_{1}+\mu\beta_{2})=\lambda_{\beta^{\prime}}=(q^{-\beta_{2}^{\prime}},q^{\beta_{1}^{\prime}},\beta_{1}^{\prime}+\mu\beta_{2}^{\prime})$
implies $\beta=\beta^{\prime}$ since $\mu\not\in\mathbb{Q}$ (use the equality
of the last coordinates).

Suppose that the algebra $A$ is of type (C8) or (C9). In view of the obvious
symmetry, it suffices to consider the type (C8). In this case, $q$ is a
primitive $n$’th root of unity, $p>0$, $\alpha=1$, $\mu=0$, $p\nmid n$ and
${\cal R}_{A}=\mathbb{N}_{<pn}\times\mathbb{N}_{<n}$ (Theorem 2.4). Then
$\lambda_{\beta}=(q^{-\beta_{2}},q^{\beta_{1}},\beta_{1})=\lambda_{\beta^{\prime}}=(q^{-\beta_{2}^{\prime}},q^{\beta_{1}^{\prime}},\beta_{1}^{\prime})$
implies $\beta=\beta^{\prime}$ by comparing the first and the last coordinates
of the vectors $\lambda_{\beta}$ and $\lambda_{\beta^{\prime}}$.

Suppose that the algebra $A$ is of type (C10). In this case, $q$ is a
primitive $n$’th root of unity, $p>0$, $\alpha=1$, $\mu\not\in\mathbb{F}_{p}$,
$p\nmid n$ and ${\cal R}_{A}=\mathbb{N}_{<pn}\times\mathbb{N}_{<pn}$ (Theorem
2.4). Then
$\lambda_{\beta}=(q^{-\beta_{2}},q^{\beta_{1}},\beta_{1}+\mu\beta_{2})=\lambda_{\beta^{\prime}}=(q^{-\beta_{2}^{\prime}},q^{\beta_{1}^{\prime}},\beta_{1}^{\prime}+\mu\beta_{2}^{\prime})$
implies $\beta=\beta^{\prime}$ since $\mu\not\in\mathbb{F}_{p}$ (use the
equality of the last coordinates).

2\. By Corollary 2.6.(1), ${\rm Es}_{A}(\omega)={\rm
Es}_{A}(\omega_{x_{1}},\omega_{x_{2}})=\bigoplus_{\beta\in{\cal
R}_{A}}Z(A)x^{\beta}$, and for all $z\in Z(A)$ and $\beta\in{\cal R}_{A}$,
$\omega_{x_{1}}(zx^{\beta})=q^{-\beta_{2}}zx^{\beta}$,
$\omega_{x_{2}}(zx^{\beta})=q^{\beta_{1}}zx^{\beta}$, and ${\rm
ad}_{x_{3}}(zx^{\beta})=(\alpha\beta_{1}+\mu\beta_{2})zx^{\beta}$. Recall that
in the case (C7a), $q$ is a primitive $n$’th root of unity, $p=0$, $\alpha=1$,
$\mu=\frac{\mu_{1}}{\mu_{2}}$, where $\mu_{1},\mu_{2}\in\mathbb{N}_{+}$ such
that $(\mu_{1},\mu_{2})=1$, $Z(A)=K$ (Theorem 2.3) and ${\cal
R}_{A}=\mathbb{N}^{2}$ (Theorem 2.4). Then
$\lambda_{\beta}=(q^{-\beta_{2}},q^{\beta_{1}},\beta_{1}+\mu\beta_{2})=\lambda_{\beta^{\prime}}=(q^{-\beta_{2}^{\prime}},q^{\beta_{1}^{\prime}},\beta_{1}^{\prime}+\mu\beta_{2}^{\prime})$
iff $\beta_{i}=\gamma_{i}+l_{i}n$,
$\beta_{i}^{\prime}=\gamma_{i}+l_{i}^{\prime}n$ for unique $\gamma_{i}\in
N_{<p}$ and for some $l_{i},l_{i}^{\prime}\in N$ where $i=1,2$, and
$\gamma_{1}+\mu\gamma_{2}+(l_{1}+l_{2})n=\gamma_{1}+\mu\gamma_{2}+(l_{1}^{\prime}+l_{2}^{\prime})n$.
It follows that ${\cal
W}_{A}(\omega)=\\{(q^{-\gamma_{2}},q^{\gamma_{1}},\gamma_{1}+\mu\gamma_{2}+kn)\,|\,(\gamma_{1}\gamma_{2})\in\mathbb{N}^{2}_{<n},k\in\mathbb{N}\\}$
and for each weight
$\lambda=(q^{-\gamma_{2}},q^{\gamma_{1}},\gamma_{1}+\mu\gamma_{2}+kn)\in{\cal
W}_{A}(\omega)$,

$A_{\lambda}=\bigoplus_{i=0}^{k}Kx_{1}^{\gamma_{1}+in}x_{2}^{\gamma_{2}+(k-i)n}\;\;{\rm
and}\;\;{\rm dim}_{K}(A_{\lambda})=k+1<\infty.$

3\. By Corollary 3.6, ${\cal
W}_{A_{x_{1}x_{2}}}(\omega)=\\{(q^{-\beta_{2}},q^{\beta_{1}},\beta_{1}+\mu\beta_{2})\,|\,\beta\in{\cal
R}_{x_{1}x_{2}}=\mathbb{N}_{<pn}\times\mathbb{N}_{<n}\\}$ and for each weight
$\lambda_{\beta}=(q^{-\beta_{2}},q^{\beta_{1}},\beta_{1}+\mu\beta_{2})\in{\cal
W}_{A_{x_{1}x_{2}}}(\omega)$,
$A_{x_{1}x_{2},\lambda_{\beta}}=Z(A_{x_{1}x_{2}})x^{\beta}$. Notice that ${\rm
Es}_{A}(\omega)\subseteq A\cap{\rm Es}_{A_{x_{1}x_{2}}}(\omega)$. Hence, to
prove statement 3, it suffices to show that

$A\cap
A_{x_{1}x_{2},\lambda_{\beta}}=\bigoplus_{i=0}^{p-1}\Lambda^{\prime}x^{\theta(\beta,i)}$

(since then ${\cal W}_{A}(\omega)={\cal W}_{A_{x_{1}x_{2}}}(\omega)$ and for
each element $\lambda_{\beta}\in{\cal W}_{A}(\omega)$,
$A_{\lambda_{\beta}}=A\cap A_{x_{1}x_{2},\lambda_{\beta}}$).

$\displaystyle A\cap A_{x_{1}x_{2},\lambda_{\beta}}$ $\displaystyle=$
$\displaystyle A\cap Z(A_{x_{1}x_{2}})x^{\beta}=A\cap
K[x_{3}^{p}-x_{3}][x_{1}^{\pm pn},(x_{1}^{-\mu n}x_{2}^{n})^{\pm 1},x_{2}^{\pm
pn}]x^{\beta}$ $\displaystyle=$ $\displaystyle
A\cap\bigoplus_{i=0}^{p-1}K[x_{3}^{p}-x_{3}][x_{1}^{\pm pn},x_{2}^{\pm
pn}](x_{1}^{-\mu n}x_{2}^{n})^{i}x^{\beta}$ $\displaystyle=$ $\displaystyle
A\cap\bigoplus_{i=0}^{p-1}K[x_{3}^{p}-x_{3}][x_{1}^{\pm pn},x_{2}^{\pm
pn}]x^{\beta+in(-\mu,i)}$ $\displaystyle=$
$\displaystyle\bigoplus_{i=0}^{p-1}\Lambda^{\prime}x^{\theta(\beta,i)}$

where $\theta(\beta,i)$ is a unique element of the set $\mathbb{N}_{<pn}^{2}$
such that $\theta(\beta,i)\equiv\beta+in(-\mu,1)\mod pn$. $\Box$

######  Proposition 5.4.

Every normal element of the algebra $A$ is an $\omega$-weight vector, and vice
versa.

Proof. $(\Rightarrow)$ Let
$a=\sum_{i\in\mathbb{N},\beta\in\mathbb{N}^{2}}\lambda_{i\beta}x_{3}^{i}x^{\beta}$
be a nonzero normal element of the algebra $A$ where $\lambda_{i\beta}\in K$.
Then $ab=\omega_{a}(b)b$ for all elements $b\in A$ where $\omega_{a}$ is the
automorphism of the algebra $A$ that os determined by the normal nd regular
element $A$. In particular, for $i=1,2$,

$\omega_{a}(x_{i})a=ax_{i}=x_{i}\omega_{x_{i}}^{-1}(a),$

and so $\omega_{x_{i}}^{-1}(a)=c_{i}a$ where
$c_{i}=x_{i}^{-1}\omega_{a}(x_{i})\in A_{x_{i}}$. Let
$(\alpha_{1},\alpha_{2}):=(\alpha,\mu)$. On the other hand,
$\omega_{x_{i}}^{-1}(a)=\sum_{i\in\mathbb{N},\beta\in\mathbb{N}^{2}}\lambda_{i\beta}(x_{3}+\alpha_{i})^{i}\rho_{i,\beta}x^{\beta}$
where

$\rho_{i,\beta}:=\begin{cases}q^{\beta_{2}}&\text{if }i=1,\\\
q^{-\beta_{1}}&\text{if }i=2.\\\ \end{cases}$

Using the $\mathbb{N}^{2}$-grading of the algebra $A$ we see that $c_{i}\in
K^{\times}$.

Similarly,
$\omega_{a}(x_{3})a=ax_{3}=\sum_{i\in\mathbb{N},\beta\in\mathbb{N}^{2}}(x_{3}-(\alpha\beta_{1}+\mu\beta_{2}))\lambda_{i\beta}x_{3}^{i}x^{\beta}$.
Using the $\mathbb{N}^{2}$-grading of the algebra $A$ we see that
$\alpha\beta_{1}+\mu\beta_{2}$ is a constant, say $\nu$, for all
$i\in\mathbb{N}$ and $\beta\in\mathbb{N}^{2}$ such that $\lambda_{i\beta}\neq
0$. So, ${\rm ad}_{x_{3}}(a)=\nu a$. Therefore, the element $a$ is an
$\omega$-weight vector.

$(\Leftarrow)$ Every $\omega$-weight vector of the algebra $A$ is a normal
element: If $a$ is an $\omega$-eigenvector with eigenvalue $\lambda_{\beta}$
then $x_{1}a=q^{-\beta_{2}}ax_{1}$, $x_{2}a=q^{-\beta_{1}}ax_{2}$ and
$ax_{3}=(x_{3}-\alpha\beta_{1}-\mu\beta_{2})a$. $\Box$

##  6 The Krull, global and Gelfand-Kirillov dimensions of the algebras ${\cal
A}=\bigotimes_{i=1}^{n}A_{i}$ where $A_{i}=A(q_{i},\alpha_{i},\mu_{i})$

The aim of this section is to compute the Krull, classical Krull, global and
Gelfand-Kirillov dimensions of the algebras ${\cal
A}=\bigotimes_{i=1}^{n}A_{i}$ where $A_{i}=A(q_{i},\alpha_{i},\mu_{i})$. The
centre of the algebra ${\cal A}$ is described.

The algebras ${\cal A}=\bigotimes_{i=1}^{n}A_{i}$ where
$A_{i}=A(q_{i},\alpha_{i},\mu_{i})$.

######  Proposition 6.1.

Let ${\cal A}=\bigotimes_{i=1}^{n}A_{i}$ be a tensor product of algebras
$A_{i}=A(q_{i},\alpha_{i},\mu_{i})=K\langle x_{i1},x_{i2},x_{i3}\rangle$ in
Theorem 1.1.(1).

  1. 1.

The algebra ${\cal A}$ is an iterated skew polynomial algebra,

${\cal
A}=K[x_{11}][x_{12};\sigma_{12}][x_{13};\sigma_{13}]\cdots[x_{n1}][x_{n2};\sigma_{n2}][x_{n3};\sigma_{n3}].$

  2. 2.

The algebra ${\cal A}$ is a Noetherian domain with PBW basis, ${\cal
A}=\bigoplus_{\beta\in\mathbb{N}^{3n}}Kx^{\beta}$.

  3. 3.

The Gelfand-Kirillov dimension of the algebra ${\cal A}$ is $3n$.

  4. 4.

$Z({\cal A})=\bigotimes_{i=1}^{n}Z(A_{i})$.

Proof. 1\. Statement 1 follows at once from Theorem 1.1.(1,a).

2\. By statement 1, the algebra ${\cal A}$ is a Noetherian domain with PBW
basis (since ${\cal A}=\otimes_{i=1}^{n}A_{i}$).

3\. Statement 3 follows at once from statement 2.

4\. This statement is obvious (as the centre of a tensor product of algebras
is the tensor product of their centres). $\Box$

$\mathbb{N}^{2n}$-grading of the algebra ${\cal A}=\bigotimes_{i=1}^{n}A_{i}$.
By (5), the algebra ${\cal A}$ is an $\mathbb{N}^{2n}$-graded algebra,

${\cal A}=\bigoplus_{\beta\in\mathbb{N}^{2n}}P_{n}x^{\beta}\;\;{\rm
where}\;\;P_{n}=K[x_{13},\ldots,x_{n3}]\;\;{\rm
and}\;\;x^{\beta}=x_{11}^{\beta_{1}}x_{12}^{\beta_{2}}\cdots
x_{n1}^{\beta_{2n-1}}x_{n2}^{\beta_{2n}}.$ (19)

The $\mathbb{N}^{2n}$-grading on the algebra ${\cal A}$ is the tensor product
of $\mathbb{N}^{2}$-grading of the tensor multiples $A_{i}$ of the algebra
${\cal A}$. The algebra $P_{n}$ is a polynomial algebra in $n$ variables.

The product in the algebra $A$ is given by the rule: For all elements $a,b\in
P_{n}$ and $\beta,\gamma\in\mathbb{N}^{2n}$,

$ax^{\beta}\cdot
bx^{\gamma}=a\omega^{\beta}(b)q^{[\beta,\gamma]}x^{\beta+\gamma}\;\;{\rm
where}\;\;\omega^{\beta}=\prod_{i=1}^{n}\omega_{x_{i1}}^{\beta_{2i-1}}\omega_{x_{i2}}^{\beta_{2i}}\;\;{\rm
and}\;\;q^{[\beta,\gamma]}=\prod_{i=1}^{n}q_{i}^{\beta_{2i}\gamma_{2i-1}}.$

So, the algebra ${\cal A}$ is a skew crossed product of the (additive) monoid
$\mathbb{N}^{2n}$ with the polynomial algebra $P_{n}$.

The global, Krull and classical Krull dimensions of algebras ${\cal A}$. For
an algebra $A$, we denote by ${\cal K}(A)$ and cl.Kdim $(A)$ the Krull and the
classical Krull dimension of $A$, respectively.

######  Theorem 6.2.

Let ${\cal A}=\bigotimes_{i=1}^{n}A_{i}$ be a tensor product of algebras
$A_{i}=A(q_{i},\alpha_{i},\mu_{i})=K\langle x_{i1},x_{i2},x_{i3}\rangle$ in
Theorem 1.1.(1).

  1. 1.

The global dimensional of ${\cal A}$ is $3n$.

  2. 2.

The Krull dimensional of ${\cal A}$ is $3n$.

  3. 3.

The classical Krull dimensional of ${\cal A}$ is $3n$.

Proof. 1\. By Proposition 6.1.(1), the algebra ${\cal A}$ is an iterated skew
polynomial algebra. Now, the result follows from [5, Theorem 7.5.3.(iii)].

2\. By Proposition 6.1.(1), the algebra ${\cal A}$ is an iterated skew
polynomial algebra. Now, the result follows from [5, Proposition 6.5.4.(i)].

3\. (i) cl.Kdim $({\cal A})\geq 3n$: By Theorem 1.1.(1), there is a strictly
ascending chain of primes in the algebra ${\cal A}$:

$0\subset\mathfrak{p}_{1}\subset\cdots\subset\mathfrak{p}_{3n}=(x_{11},x_{12},x_{13},\ldots,x_{n1},x_{n2},x_{n3})$

where for each natural number $k=1,\ldots,n-1$,

$\mathfrak{p}_{3k+i}=\begin{cases}(x_{11},x_{12},x_{13},\ldots,x_{k1},x_{k2},x_{k3})&\text{if
}i=0,\\\ (x_{11},x_{12},x_{13},\ldots,x_{k1},x_{k2},x_{k3},x_{k+1,1})&\text{if
}i=1,\\\
(x_{11},x_{12},x_{13},\ldots,x_{k1},x_{k2},x_{k3},x_{k+1,1},x_{k+1,2})&\text{if
}i=2,\\\ \end{cases}$

and the statement (i) follows.

(ii) cl.Kdim $({\cal A})\leq 3n$: The algebra ${\cal A}$ is a Noetherian
algebra (Proposition 6.1.(2)), hence cl.Kdim $({\cal A})\leq{\cal K}(A)=3n$,
by [5, Lemma 6.4.5].

By the statements (i) and (ii), cl.Kdim $({\cal A})=3n$. $\Box$

Licence. For the purpose of open access, the author has applied a Creative
Commons Attribution (CC BY) licence to any Author Accepted Manuscript version
arising from this submission.

Disclosure statement. No potential conflict of interest was reported by the
author.

Data availability statement. Data sharing not applicable – no new data
generated.

## References

  * [1] V. V. Bavula and T. Lu, The prime spectrum and simple modules over the quantum spatial ageing algebra, _Algebr. Represent. Theory_ 19 (2016), 1109–1133.
  * [2] V. V. Bavula, Generalized Weyl algebras and diskew polynomial rings, J. Algebra Appl., 19 (2020) no. 10, 2050194, 43pp.
  * [3] V. V. Bavula, Description of bi-quadratic algebras on 3 generators with PBW basis, J. Algebra, 631 (2023) 695–730. arXiv:2112.11105.
  * [4] V. V. Bavula, Classifications of prime ideals and simple modules of the Weyl algebra $A_{1}$ in prime characteristic, Tokyo J. Math., 46 (2023), no. 1, 1–31.
  * [5] J. C. McConnell and J. C. Robson, Noncommutative Noetherian rings, Wiley, Chichester, 1987.

V. V. Bavula | Ali Al Khabyah
---|---
Department of Pure Mathematics | Department of Mathematics
University of Sheffield | Faculty of Science
Hicks Building | Jazan University
Sheffield S3 7RH | Jazan 45142
UK | Saudi Arabia
email<EMAIL_ADDRESS>| email<EMAIL_ADDRESS>