# A Deleting Derivations Algorithm for Quantum Nilpotent Algebras at Roots of
Unity

Stéphane Launois Partially supported by EPSRC grant EP/R009279/1. School of
Mathematics, Statistics and Actuarial Science, University of Kent, Canterbury,
Kent, CT2 7FS, UK.  Samuel A. Lopes Partially supported by CMUP, member of
LASI, which is financed by national funds through FCT – Fundação para a
Ciência e a Tecnologia, I.P., under the projects with reference
UIDB/00144/2020 and UIDP/00144/2020. CMUP, Departamento de Matemática,
Faculdade de Ciências, Universidade do Porto, Rua do Campo Alegre s/n,
4169–007 Porto, Portugal.  Alexandra Rogers  School of Mathematics, Statistics
and Actuarial Science, University of Kent, Canterbury, Kent, CT2 7FS, UK.

###### Abstract

This paper extends an algorithm and canonical embedding in [Cau03a] to a large
class of quantum algebras. It applies to iterated Ore extensions over a field
satisfying some suitable assumptions which cover those of Cauchon’s original
setting but also allows for roots of unity. The extended algorithm constructs
a quantum affine space $A^{\prime}$ from the original quantum algebra $A$ via
a series of change of variables within the division ring of fractions
$\mathrm{Frac}(A)$. The canonical embedding takes a completely prime ideal
$P\lhd A$ to a completely prime ideal $Q\lhd A^{\prime}$ such that when $A$ is
a PI algebra, ${\rm PI}\text{-}{\rm deg}(A/P)={\rm PI}\text{-}{\rm
deg}(A^{\prime}/Q)$. When the quantum parameter is a root of unity, combining
our construction with results from [BLR22] allows us to state an explicit
formula for the PI degree of completely prime quotient algebras. This paper
ends with a method to construct a maximum dimensional irreducible
representation of $A/P$ given a suitable irreducible representation of
$A^{\prime}/Q$ when $A$ is PI.

In [Cau03a], Cauchon developed an algorithm for a large class of quantum
algebras originally named CGL extensions after Cauchon, Goodearl and Letzter
who first proved a set of unifying results on the algebras in this class, and
later renamed quantum nilpotent algebras [LLR06]. These are algebras which can
be written in the form of an iterated Ore extension over a field with certain
conditions placed on the defining automorphisms and skew-derivations. In
essence, the algorithm takes a prime quotient of a suitable CGL extension and,
via a series of changes of indeterminates within its division ring of
fractions, obtains a prime quotient of a quantum affine space which shares the
same division ring of fractions as the original algebra. Properties from the
resulting quotiented quantum affine space can then be pulled back to the
original quotient algebra. When the prime ideal is invariant under a rational
action of a torus, the algorithm combined with Goodearl and Letzter’s
$\mathcal{H}$-stratification theory was used to prove the _quantum Gel’fand
Kirillov Conjecture_ on prime quotients of a large class of generic quantum
algebras, such as quantum euclidean spaces, quantum symplectic spaces, quantum
matrices, and quantum Weyl algebras [Cau03a, Théorème 6.1.1]. Cauchon’s
algorithm has also been applied by many authors to gain a better understanding
of the structure of quantum algebras in the generic setting. Such works
include the aforementioned [Cau03b] and [GL98] as well as [Cas14, GY14, GLL19,
GY16, LLN19].

Cauchon’s algorithm cannot be applied to such algebras at roots of unity,
however, as the homomorphism defining the algorithm is not compatible with
roots of unity. Several authors have recreated the main results of [Cau03a]
for quantum algebras at roots of unity satisfying slightly different
conditions ([Hay08] and [LM11]). However, the explicit change of variables in
the original algorithm, that was essential in the applications named above,
was not recreated in either paper. This paper addresses this by extending
Cauchon’s deleting derivations algorithm, including the canonical embedding to
track completely prime ideals, to include the root of unity case by utilising
an adapted version, provided in [Hay08, Section 3], of the homomorphism which
lies at the heart of the procedure.

The main motivation for this paper is the fact that many quantum algebras
become _polynomial identity (PI) algebras_ when taken at roots of unity and in
this case the _PI degree_ is a useful invariant for deducing various
properties of the algebra (see, for example, [BY17], [CPWZ16], and [CPWZ15]).
The PI degree also plays a role in the representation theory of prime affine
PI algebras, giving an upper bound on the dimension of irreducible
representations [BG02, Theorem I.13.5.].

The extended algorithm defined in Section 2 of this paper takes a suitable
iterated Ore extension $A$ and a completely prime ideal
$P\in\mathrm{C.Spec}(A)$ and constructs a quantum affine space $A^{\prime}$
and a canonical embedding
$\psi:\mathrm{C.Spec}(A)\rightarrow\mathrm{C.Spec}(A^{\prime})$ such that
$\mathrm{Frac}(A/P)=\mathrm{Frac}(A^{\prime}/\psi(P))$. Furthermore, if $A$ is
PI then, so too is $A^{\prime}$ and thus ${\rm PI}\text{-}{\rm deg}(A/P)={\rm
PI}\text{-}{\rm deg}(A^{\prime}/\psi(P))$. When $A^{\prime}/\psi(P)$ is itself
a quantum affine space then the PI degree is determined by the rank and values
of the invariant factors of its commutation matrix (see for instance [BLR22,
Lemma 2.4]). With this in mind, we focus on those ideals $P_{w}$ which give
rise to quantum affine spaces $A^{\prime}/\psi(P_{w})$. For such ideals, the
PI degree of $A/P_{w}$ is then computed in Theorem 4.4. We end with a method
to pass certain maximum-dimensional irreducible representations of any
$A^{\prime}/\psi(P)$ back through the deleting derivations algorithm to obtain
maximum irreducible representations of $A/P$ (Section 5) and illustrate our
result by constructing an explicit irreducible representation of maximal
dimension for a completely prime quotient of $U_{q}^{+}(\mathfrak{s0}_{5})$ at
roots of unity.

Acknowledgement: The material presented in this article is part of the PhD
thesis of the third-named author [Rog19]. She would like to thank the School
of Mathematics, Statistics and Actuarial Science at the University of Kent for
their financial support.

##  1 Quantum Nilpotent Algebras at roots of unity

Take $\mathbb{K}$ to be an arbitrary field and $R$ a $\mathbb{K}$-algebra. One
can form an Ore extension $R[x,\sigma,\delta]$ in one indeterminate $x$ by
defining a $\mathbb{K}$-automorphism $\sigma$ and a (left) $\sigma$-derivation
$\delta$ (also called a skew-derivation) such that for any $a,b\in R$ we have
$\delta(ab)=\sigma(a)\delta(b)+\delta(a)b$. These maps define the commutation
rules for $x$ with elements of $R$: given $r\in R$ we have
$xr=\sigma(r)x+\delta(r)$. Ore extensions satisfy a universal mapping property
([GW04, Proposition 2.4]) and thus are unique up to isomorphism ([GW04,
Corollary 2.5]). Iterating this process with $N$ indeterminates gives an
iterated Ore extension
$R[X_{1},\sigma_{1},\delta_{1}]\ldots[X_{N},\sigma_{N},\delta_{N}]$. Note that
this is a noetherian domain by the Skew Hilbert Basis Theorem [BG02, Theorem
I.1.13]. The exact form of $R$ and the properties of the pairs
$(\sigma_{i},\delta_{i})$ are what will define quantum nilpotent algebras at
roots of unity and distinguish them from CGL extensions.

If all the skew-derivations in the iterated Ore extension above are the $0$
map, all the automorphisms act by scalar multiplication on each indeterminate,
i.e. $\sigma_{i}(X_{j})=\lambda_{i,j}X_{j}$ for all $1\leq j<i\leq N$ and some
$\lambda_{i,j}\in\mathbb{K}^{*}$, and $R$ is a field $\mathbb{K}$ then the
iterated Ore extension would describe a quantum affine space, which we can
write as follows:

$\mathbb{K}[X_{1},\sigma_{1}]\ldots[X_{N},\sigma_{N}]=\mathbb{K}_{\Lambda}[X_{1},\ldots,X_{N}]=:\mathcal{O}_{q^{\Lambda}}(\mathbb{K}^{N})$

where $\Lambda:=(\lambda_{i,j})_{i,j}\in M_{N}(\mathbb{K}^{*})$ with
$\lambda_{j,i}=\lambda_{i,j}^{-1}$ for all $1\leq j\leq i\leq N$.

The following terms are needed to define our class of algebras of interest.

######  Definition 1.1.

Let $R$ be a $\mathbb{K}$-algebra and $R[x,\sigma,\delta]$ be an Ore
extension.

  1. 1.

We call the pair $(\sigma,\delta)$, and indeed the Ore extension, _$q$ -skew_
if the automorphism and skew-derivation satisfy the relation
$\delta\circ\sigma=q\sigma\circ\delta$, for some $1\neq q\in\mathbb{K}^{*}$.

  2. 2.

The skew-derivation $\delta$ is _locally nilpotent_ if, for every $0\neq a\in
R$, there exists an integer $n_{a}\geq 0$ such that $\delta^{n_{a}}(a)=0$ and
$\delta^{m}(a)\neq 0$ for any $m<n_{a}$. We define such an $n_{a}$ as the
_$\delta$ -nilpotence index of $a$_.

  3. 3.

[Hay08] We say that the $\sigma$-derivation $\delta$ _extends to a higher
$q$-skew $\sigma$-derivation (h.$q$-s.$\sigma$-d.)_ on $R$ if there is a
sequence $\\{d_{n}\\}_{n=0}^{\infty}$ (denoted simply by $\\{d_{n}\\}$ when it
is obvious which subscript indexes the sequence) of $\mathbb{K}$-linear
operators on $R$ such that:

     * •

$d_{0}$ is the identity;

     * •

$d_{1}=\delta$;

     * •

$d_{n}(rs)=\sum_{i=0}^{n}\sigma^{n-i}d_{i}(r)d_{n-i}(s)$ for all $r,s\in R$
and all $n$,

     * •

$d_{n}\circ\sigma=q^{n}\sigma\circ d_{n}$ for all $n$.

A (h.$q$-s.$\sigma$-d.) is _locally nilpotent_ if, for all $0\neq r\in R$,
there exists an integer $n_{r}\geq 0$ such that $d_{n}(r)=0$ for all $n\geq
n_{r}$, and $d_{m}(r)\neq 0$ for any $m<n_{r}$. In this case we call $n_{r}$
the _$d$ -nilpotence index of $r$_. A h.$q$-s.$\sigma$-d is called _iterative_
if $d_{n}d_{m}=\binom{n+m}{m}_{q}d_{n+m}$ for all $n,m$. Here
$\binom{n+m}{m}_{q}$ is the $q$-Gaussian binomial coefficient: a polynomial in
$q$ over $\mathbb{Z}$ with properties similar to those of regular binomial
coefficients. More precisely, for all $n\geq i\geq 0$, we have
$\binom{n}{i}_{q}:=\frac{(n)!_{q}}{(i)!_{q}(n-i)!_{q}}$, where
$(i)_{q}=\frac{q^{i}-1}{q-1}$ and $(i)!_{q}=(1)_{q}\cdots(i-1)_{q}(i)_{q}$
with the convention that $(0)!_{q}=1$.

The higher skew-derivations are what allowed Haynal to adapt Cauchon’s
_effacement des derivations_ homomorphism to a broader homomorphism which
allows for roots of unity. For practical use, [Hay08, Theorem 2.8] gives a
sufficient condition on a $q$-skew Ore extension $R[x;\sigma,\delta]$ for the
$\sigma$-derivation $\delta$ to extend to a h.$q$-s.$\sigma$-d. for any $1\neq
q\in\mathbb{K}^{*}$. This is used to check that the specific examples we are
interested in do, indeed, satisfy this condition.

The algebras that we study in this article satisfy the following:

######  Hypothesis 1.2.

Let
$A=\mathbb{K}[X_{1}][X_{2};\sigma_{2},\delta_{2}]\ldots[X_{N};\sigma_{N},\delta_{N}]$,
where the $\sigma_{i}$ are $\mathbb{K}$-algebra automorphisms, and the
$\delta_{i}$ are $\sigma_{i}$-derivations on the relevant subalgebras of $A$.
Denote these subalgebras by
$A_{i-1}:=\mathbb{K}[X_{1}][X_{2};\sigma_{2},\delta_{2}]\ldots[X_{i-1};\sigma_{i-1},\delta_{i-1}]$
so that $A_{0}:=\mathbb{K}$ and $A_{N}:=A$. We place the following conditions
on $A$:

  1. H1.2.1:

$\sigma_{i}(X_{l})=\lambda_{i,l}X_{l}$ for all $l<i$ and $2\leq i\leq N$,
where $\lambda_{i,l}\in\mathbb{K}^{*}$.

  2. H1.2.2:

$\Lambda:=(\lambda_{i,j})\in M_{N}(\mathbb{K}^{*})$ is a multiplicatively
antisymmetric matrix. That is, $\lambda_{i,l}=\lambda_{l,i}^{-1}$ for all
$1\leq i,l\leq N$.

  3. H1.2.3:

For $2\leq i\leq N$ there exists some $1\neq q_{i}\in\mathbb{K}^{*}$ such that
$\delta_{i}\circ\sigma_{i}=q_{i}\sigma_{i}\circ\delta_{i}$, i.e.
$(\sigma_{i},\delta_{i})$ is $q_{i}$-skew.

  4. H1.2.4:

For all $2\leq i\leq N$ each $\delta_{i}$ extends to a locally nilpotent,
iterative h.$q_{i}$-s.$\sigma_{i}$-d., $\\{d_{i,n}\\}_{n=0}^{\infty}$, on
$A_{i-1}$, and $\sigma_{l}\circ
d_{i,n}=\lambda_{l,i}^{n}d_{i,n}\circ\sigma_{l}$ on $A_{i-1}$ for all $n\geq
0$ and $i+1\leq l\leq N$.

Algebras satisfying Hypothesis 1.2 are called quantum nilpotent algebras and
include quantum Schubert cell algebras, quantised Weyl algebras, quantised
coordinate rings of affine, symplectic, and euclidean spaces, and quantum
matrices.

##  2 Deleting Derivations Algorithm

The main algorithm in this section will remove each skew-derivation from an
algebra $A$ satisfying Hypothesis 1.2 via an iterative change of
indeterminates within the division ring of fractions of $A$, until we obtain a
quantum affine space. This is developed in much the same way as in [Cau03a,
Section 3.2] but applies the homomorphism defined in [Hay08, Proposition 3.4].
For this application we need the ability to reorder the extensions of $A$.
This is achieved in the following lemma.

######  Lemma 2.1.

Let

$\displaystyle A$
$\displaystyle{}=A_{j-1}[X_{j};\sigma_{j},\delta_{j}][X_{j+1};\sigma_{j+1}]\ldots[X_{N};\sigma_{N}],$
$\displaystyle\hat{A}$
$\displaystyle{}=A_{j-1}[X_{j};\sigma_{j},\delta_{j}][X_{j+1}^{\pm
1};\sigma_{j+1}]\ldots[X_{N}^{\pm 1};\sigma_{N}],$

where the automorphisms and skew-derivations satisfy properties H1.2.1–H1.2.3
and $\delta_{j}\neq 0$.

  1. (I)

Then

$\displaystyle A$
$\displaystyle{}=A_{j-1}[X_{j+1};\sigma^{*}_{j+1}]\ldots[X_{N};\sigma^{*}_{N}][X_{j};\sigma_{j}^{\prime},\delta_{j}^{\prime}],$
$\displaystyle\hat{A}$ $\displaystyle{}=A_{j-1}[X_{j+1}^{\pm
1};\sigma^{*}_{j+1}]\ldots[X_{N}^{\pm
1};\sigma^{*}_{N}][X_{j};\sigma_{j}^{\prime},\delta_{j}^{\prime}],$

where

    1. (i)

$\sigma^{*}_{i}|_{A_{j-1}}=\sigma_{i}|_{A_{j-1}}$ for all $j+1\leq i\leq N$
and $\sigma^{*}_{i}(X_{l})=\sigma_{i}(X_{l})=\lambda_{i,l}X_{l}$ for all
$j+1\leq l<i$;

    2. (ii)

$\sigma_{j}^{\prime}|_{A_{j-1}}=\sigma_{j}$ and
$\delta_{j}^{\prime}|_{A_{j-1}}=\delta_{j}$;

    3. (iii)

$\sigma_{j}^{\prime}(X_{l})=\lambda_{j,l}X_{l}=\lambda_{l,j}^{-1}X_{l}$ and
$\delta_{j}^{\prime}(X_{l})=0$ for all $j+1\leq l\leq N$.

  2. (II)

$(\sigma_{j}^{\prime},\delta_{j}^{\prime})$ is $q_{j}$-skew.

  3. (III)

Suppose that property H1.2.4 is also satisfied. Then $\delta_{j}^{\prime}$
extends to a locally nilpotent, iterative
h.$q_{j}$-s.$\sigma_{j}^{\prime}$-d., $\\{d^{\prime}_{j,n}\\}_{n=0}^{\infty}$,
on $A_{j-1}\langle X_{j+1}^{\pm 1},X_{j+2}^{\pm 1},\ldots,X_{N}^{\pm
1}\rangle$, where the $d_{j,n}^{\prime}$ coincide with the $d_{j,n}$ on
$A_{j-1}$ and, for all $j+1\leq l\leq N$ and $n\geq 1$,
$d_{j,n}^{\prime}(X_{l})=0$. Moreover, $\\{d_{j,n}^{\prime}\\}_{n=0}^{\infty}$
restricts to a h.$q_{j}$-s.$\sigma_{j}^{\prime}$-d. on $A_{j-1}\langle
X_{j+1},X_{j+2},\ldots,X_{N}\rangle$ which is also locally nilpotent and
iterative.

###### Proof.

This is an inductive corollary to [Hay08, Lemma 4.1] and the details are left
to the reader. ∎

In the results that follow, we abuse notation slightly and use the same
notation for maps defined on isomorphic algebras. We do this in the case where
the action of the map on the generators of the algebra does not change, even
though the algebras do. For example, suppose we have two isomorphic iterated
Ore extensions

$\displaystyle\mathbb{K}[X_{1}][X_{2};\sigma_{2},\delta_{2}]\cong\mathbb{K}[x_{1}][x_{2};\bar{\sigma}_{2},\bar{\delta}_{2}],$

where $\sigma_{2}(X_{1})=\lambda X_{1}$ and $\bar{\sigma}_{2}(x_{1})=\lambda
x_{1}$ for the same $\lambda\in\mathbb{K}^{*}$. In this case, we simply denote
$\bar{\sigma}_{2}$ by $\sigma_{2}$ (similarly for $\bar{\delta}_{2}$) and
write $\mathbb{K}[x_{1}][x_{2};\sigma_{2},\delta_{2}]$.

We may also abuse notation in a similar way for restrictions of maps to
isomorphic subalgebras.

We now describe the deleting derivations algorithm for algebras satisfying
Hypothesis 1.2.

###  2.1 The Algorithm

For each $j\in\llbracket 2,N+1\rrbracket$ we define a sequence
$(X_{1}^{(j)},\ldots,X_{N}^{(j)})$ of elements of $F:=\mathrm{Frac}(A)$ and we
set $A^{(j)}:=\mathbb{K}\langle X_{1}^{(j)},\ldots,X_{N}^{(j)}\rangle$ to be
the subalgebra of $F$ generated by these elements. For $j=N+1$ we set
$(X_{1}^{(N+1)},\ldots,X_{N}^{(N+1)}):=(X_{1},\ldots,X_{N})$ so that
$A^{(N+1)}=A$. For some fixed $j\in\llbracket 2,N\rrbracket$ we assume that
the algebra $A^{(j+1)}$ satisfies the following hypothesis, first setting
$(x_{1},\ldots,x_{N}):=(X_{1}^{(j+1)},\ldots,X_{N}^{(j+1)})$ for ease of
notation.

######  Hypothesis 2.2.

  1. H2.2.1:

$A^{(j+1)}\cong\mathbb{K}[X_{1}]\ldots[X_{j};\sigma_{j},\delta_{j}][X_{j+1};\sigma_{j+1}^{(j+1)}]\ldots[X_{N};\sigma_{N}^{(j+1)}]$
by an isomorphism sending $x_{i}$ to $X_{i}$ for all $i\in\llbracket
1,N\rrbracket$.

  2. H2.2.2:

For each $i\in\llbracket j+1,N\rrbracket$, the map $\sigma_{i}^{(j+1)}$ is an
automorphism such that $\sigma_{i}^{(j+1)}(X_{l})=\lambda_{i,l}X_{l}$ for all
$l\in\llbracket 1,i-1\rrbracket$. Furthermore, we have
$\sigma_{i}^{(j+1)}\circ
d_{l,n}=\lambda_{i,l}^{n}d_{l,n}\circ\sigma_{i}^{(j+1)}$ for all
$l\in\llbracket 2,j\rrbracket$ and $n\geq 0$.

This allows us to write

$\displaystyle A^{(j+1)}$ $\displaystyle{}=\mathbb{K}\langle
x_{1},\ldots,x_{N}\rangle$ (1)
$\displaystyle{}=\mathbb{K}[x_{1}]\ldots[x_{j};\sigma_{j},\delta_{j}][x_{j+1};\sigma_{j+1}^{(j+1)}]\ldots[x_{N};\sigma_{N}^{(j+1)}]$
(2)

where, for $i\in\llbracket 2,j\rrbracket$, $\sigma_{i}$ and $\delta_{i}$
satisfy Hypothesis 1.2 and $\delta_{i}$ extends to a locally nilpotent,
iterative h.$q_{i}$-s.$\sigma_{i}$-d., $\\{d_{i,n}\\}_{n=0}^{\infty}$, on
$A_{i-1}$.

We define a new sequence of elements in $F$,
$(y_{1},\ldots,y_{N}):=(X_{1}^{(j)},\ldots,X_{N}^{(j)})$:

$y_{l}=\begin{cases}x_{l}&l\geq j;\\\
\sum_{n=0}^{\infty}q_{j}^{\frac{n(n+1)}{2}}(q_{j}-1)^{-n}d_{j,n}\circ\sigma_{j}^{-n}(x_{l})x_{j}^{-n}&l<j.\end{cases}$
(3)

Note that the sum stated above is finite, since the sequence
$\\{d_{j,n}\\}_{n=0}^{\infty}$ is locally nilpotent. With this we define
$A^{(j)}:=\mathbb{K}\langle y_{1},\ldots,y_{N}\rangle$.

######  Theorem 2.3.

Let $A$ be as in Hypothesis 1.2, with $A^{(j+1)}$ defined as above and
satisfying Hypothesis 2.2. Then we have the following:

  1. (I)

$A^{(j)}\cong\mathbb{K}[X_{1}][X_{2};\sigma_{2},\delta_{2}]\cdots[X_{j-1};\sigma_{j-1},\delta_{j-1}][X_{j};\sigma_{j}^{(j)}]\cdots[X_{N};\sigma_{N}^{(j)}]$
by an isomorphism which sends $y_{l}$ to $X_{l}$ for all $l\in\llbracket
1,N\rrbracket$.

  2. (II)

For all $i\in\llbracket j,N\rrbracket$, the $\sigma_{i}^{(j)}$ are
automorphisms satisfying

    1. (i)

$\sigma_{i}^{(j)}(X_{l})=\lambda_{i,l}X_{l}$ for all $l\in\llbracket
1,i-1\rrbracket$;

    2. (ii)

$\sigma_{i}^{(j)}\circ d_{l,n}=\lambda_{i,l}^{n}d_{l,n}\circ\sigma_{i}^{(j)}$
for all $n\geq 0$ and all $l\in\llbracket 2,j-1\rrbracket$.

  3. (III)

Let $S_{j}:=\\{x_{j}^{n}\mid n\geq 0\\}=\\{y_{j}^{n}\mid n\geq 0\\}$. This is
a multiplicatively closed set of regular elements satisfying the two sided
Ore-condition in $A^{(j+1)}$ and $A^{(j)}$ and, furthermore,
$A^{(j)}S_{j}^{-1}=A^{(j+1)}S_{j}^{-1}$.

###### Proof.

By Hypothesis 2.2, and the discussion thereafter, we can write $A^{(j+1)}$ as

$A^{(j+1)}=\mathbb{K}[x_{1}]\ldots[x_{j};\sigma_{j},\delta_{j}][x_{j+1};\sigma_{j+1}^{(j+1)}]\ldots[x_{N};\sigma_{N}^{(j+1)}].$

Define

$A^{(j+1)}_{j-1}:=\mathbb{K}[x_{1}][x_{2};\sigma_{2},\delta_{2}]\ldots[x_{j-1};\sigma_{j-1},\delta_{j-1}].$

If $\delta_{j}=0$ then $A^{(j)}=A^{(j+1)}$ and the result is trivial. So,
assuming that $\delta_{j}\neq 0$ and applying Lemma 2.1 to
$A^{(j+1)}=A^{(j+1)}_{j-1}[x_{j};\sigma_{j},\delta_{j}][x_{j+1};\sigma_{j+1}^{(j+1)}]\ldots[x_{N};\sigma_{N}^{(j+1)}]$
gives

$A^{(j+1)}=A^{(j+1)}_{j-1}[x_{j+1};\sigma_{j+1}^{(j+1)*}]\ldots[x_{N};\sigma_{N}^{(j+1)*}][x_{j};\sigma^{\prime}_{j},\delta^{\prime}_{j}],$
(4)

where

  1. (a)

$\sigma_{i}^{(j+1)*}|_{A^{(j+1)}_{j-1}}=\sigma_{i}^{(j+1)}|_{A^{(j+1)}_{j-1}}$
for all $i\in\llbracket j+1,N\rrbracket$, and
$\sigma_{i}^{(j+1)*}(x_{l})=\sigma_{i}^{(j+1)}(x_{l})=\lambda_{i,l}x_{l}$ for
all $l\in\llbracket j+1,i-1\rrbracket$;

  2. (b)

$\sigma^{\prime}_{j}|_{A^{(j+1)}_{j-1}}=\sigma_{j}$ and
$\delta^{\prime}_{j}|_{A^{(j+1)}_{j-1}}=\delta_{j}$;

  3. (c)

$\sigma^{\prime}_{j}(x_{l})=\lambda_{j,l}x_{l}=\lambda_{l,j}^{-1}x_{l}$ and
$\delta^{\prime}_{j}(x_{l})=0$ for all $l\in\llbracket j+1,N\rrbracket$.

In particular, $\delta^{\prime}_{j}$ extends to a
h.$q_{j}$-s.$\sigma^{\prime}_{j}$-d., $\\{d^{\prime}_{j,n}\\}_{n=0}^{\infty}$,
on $A^{(j+1)}_{j-1}\langle x_{j+1}^{\pm 1},\ldots,x_{N}^{\pm 1}\rangle$ where

$\displaystyle d^{\prime}_{j,n}|_{A^{(j+1)}_{j-1}}$
$\displaystyle{}=d_{j,n}\quad(\forall\;n\geq 0),$ (5) $\displaystyle
d^{\prime}_{j,n}(x_{l})$ $\displaystyle{}=0\qquad(\forall\;l\in\llbracket
j+1,N\rrbracket\text{ and }n\geq 1).$ (6)

Define

$\widehat{A^{(j+1)}}:=A^{(j+1)}_{j-1}[x_{j+1};\sigma_{j+1}^{(j+1)*}]\ldots[x_{N};\sigma_{N}^{(j+1)*}]$

so that equation (4) becomes

$A^{(j+1)}=\widehat{A^{(j+1)}}[x_{j};\sigma^{\prime}_{j},\delta^{\prime}_{j}].$

Applying [Hay08, Theorem 3.7] to $A^{(j+1)}$ yields the isomorphism,

$\displaystyle f:\widehat{A^{(j+1)}}[x_{j}^{\pm 1};\sigma^{\prime}_{j}]$
$\displaystyle{}~{}\longrightarrow~{}\widehat{A^{(j+1)}}[x_{j};\sigma^{\prime}_{j},\delta^{\prime}_{j}]S_{j}^{-1}$
$\displaystyle\widehat{A^{(j+1)}}~{}\ni~{}a$
$\displaystyle{}~{}\longmapsto~{}f(a)=\sum_{n=0}^{\infty}q_{j}^{\frac{n(n+1)}{2}}(q_{j}-1)^{-n}d^{\prime}_{j,n}\circ(\sigma^{\prime}_{j})^{-n}(a)x_{j}^{-n}$
$\displaystyle x_{j}$ $\displaystyle{}~{}\longmapsto~{}x_{j}.$

Note that $f(x_{l})=x_{l}$ for all $l\in\llbracket j+1,N\rrbracket$ since, by
(6), $d^{\prime}_{j,n}(x_{l})=0$ for all $n\geq 1$. If $l\in\llbracket
1,j-1\rrbracket$, then $x_{l}$ and $(\sigma^{\prime}_{j})^{-n}(x_{l})\in
A^{(j+1)}_{j-1}$. Thus, by (5) and (b) above, we can replace
$\sigma^{\prime}_{j}$ and $d^{\prime}_{j,n}$ in $f(x_{l})$ with $\sigma_{j}$
and $d_{j,n}$ to obtain

$f(x_{l})=\sum_{n=0}^{\infty}q_{j}^{\frac{n(n+1)}{2}}(q_{j}-1)^{-n}d_{j,n}\circ\sigma_{j}^{-n}(x_{l})x_{j}^{-n}=y_{l},$

as defined in (3). Therefore, for any $l\in\llbracket 1,N\rrbracket$ we see
that

$f(x_{l})=\begin{cases}x_{l}&l\geq j;\\\ y_{l}&l<j.\end{cases}$

Hence the isomorphism $f$ takes $x_{l}$ to $y_{l}$ for all $l\in\llbracket
1,N\rrbracket$.

Using [Hay08, Theorem 3.7] we see that restricting $f$ to
$\widehat{A^{(j+1)}}[x_{j};\sigma^{\prime}_{j}]$ gives the deleting derivation
homomorphism as defined in [Hay08, Proposition 3.4] . Therefore

$\operatorname{Im}(f)\cong\widehat{A^{(j+1)}}[x_{j};\sigma^{\prime}_{j}],$ (7)

where $\operatorname{Im}(f)=f(\widehat{A^{(j+1)}}[x_{j};\sigma^{\prime}_{j}])$
is the subalgebra of $\widehat{A^{(j+1)}}[x_{j}^{\pm
1};\sigma^{\prime}_{j},\delta^{\prime}_{j}]$ generated by $x_{j}$ and
$f(\widehat{A^{(j+1)}})$. Since $f(\widehat{A^{(j+1)}})$ is generated by
$\mathbb{K}$ and $\\{y_{l}\mid l\neq j\\}$, and since $x_{j}=y_{j}$, this
simply tells us that

$\operatorname{Im}(f)=\mathbb{K}\langle y_{1},\ldots,y_{N}\rangle=A^{(j)}.$

Using (7) we see that

$A^{(j)}=\operatorname{Im}(f)\cong\mathbb{K}[x_{1}]\ldots[x_{j-1};\sigma_{j-1},\delta_{j-1}][x_{j+1};\sigma_{j+1}^{(j+1)*}]\ldots[x_{N};\sigma_{N}^{(j+1)*}][x_{j};\sigma^{\prime}_{j}],$

and therefore,

$A^{(j)}=\mathbb{K}[y_{1}]\ldots[y_{j-1};\sigma_{j-1},\delta_{j-1}][y_{j+1};\sigma_{j+1}^{(j+1)*}]\ldots[y_{N};\sigma_{N}^{(j+1)*}][y_{j};\sigma^{\prime}_{j}].$
(8)

Finally we apply [Hay08, Proposition 3.6] to conclude that $S_{j}$ is a
multiplicatively closed set of regular elements in both $A^{(j+1)}$ and
$A^{(j)}$, satisfying the two-sided Ore condition, and that

$\displaystyle\operatorname{Im}(f)S_{j}^{-1}$
$\displaystyle{}=\widehat{A^{(j+1)}}[x_{j};\sigma^{\prime}_{j},\delta^{\prime}_{j}]S_{j}^{-1},$
$\displaystyle A^{(j)}S_{j}^{-1}$ $\displaystyle{}=A^{(j+1)}S_{j}^{-1}.$

Thus assertion (III) is proved.

The property $y_{l}y_{j}=\lambda_{l,j}y_{j}y_{l}$, along with the fact that
$\lambda_{l,j}=\lambda_{j,l}^{-1}$, allows us to rearrange (8) to obtain

$A^{(j)}=\mathbb{K}[y_{1}]\ldots[y_{j-1};\sigma_{j-1},\delta_{j-1}][y_{j};\sigma^{(j)}_{j}][y_{j+1};\sigma^{(j)}_{j+1}]\ldots[y_{N};\sigma^{(j)}_{N}].$
(9)

Defining

$A^{(j)}_{j-1}:=\mathbb{K}[y_{1}]\ldots[y_{j-1};\sigma_{j-1},\delta_{j-1}]$

we see that $A_{j-1}^{(j)}\cong A_{j-1}$ and there is an isomorphism

$\displaystyle A^{(j)}$
$\displaystyle{}=\mathbb{K}[y_{1}]\ldots[y_{j-1};\sigma_{j-1},\delta_{j-1}][y_{j};\sigma^{(j)}_{j}][y_{j+1};\sigma^{(j)}_{j+1}]\ldots[y_{N};\sigma^{(j)}_{N}]$
$\displaystyle{}\cong\mathbb{K}[X_{1}]\ldots[X_{j-1};\sigma_{j-1},\delta_{j-1}][X_{j};\sigma^{(j)}_{j}][X_{j+1};\sigma^{(j)}_{j+1}]\ldots[X_{N};\sigma^{(j)}_{N}]$
(10)

sending $y_{l}$ to $X_{l}$ for all $l\in\llbracket 1,N\rrbracket$, where the
maps (as defined on suitable subalgebras of $A^{(j)}$) are as follows:

  1. (a′)

$\sigma^{(j)}_{j}=\sigma^{\prime}_{j}|_{A^{(j)}_{j-1}}=\sigma_{j}$;

  2. (b′)

$\sigma_{i}^{(j)}|_{A^{(j)}_{j-1}}=\sigma_{i}^{(j+1)*}|_{A^{(j)}_{j-1}}=\sigma_{i}^{(j+1)}|_{A^{(j)}_{j-1}}$
for all $i\in\llbracket j+1,N\rrbracket$;

  3. (c′)

$\sigma^{(j)}_{i}(y_{l})=\lambda_{i,l}y_{l}$ for all $i\in\llbracket
j+1,N\rrbracket$ and $l\in\llbracket 1,i-1\rrbracket$.

Using the isomorphism in (10) along with the observations (a′)–(c′) above we
can prove assertion (II) for all $i\in\llbracket j,N\rrbracket$: Observation
(a′) proves both parts of assertion (II) when $i=j$, since $\sigma_{j}$
satisfies assertion (II) by definition (see H1.2.1 and H1.2.4). When
$i\in\llbracket j+1,N\rrbracket$, observation (b′) proves (II)(ii), since
$\sigma_{i}^{(j+1)}$ satisfies H2.2.2, and observation (c′) proves (II)(i).

∎

######  Remark 2.4.

If $A$ is an algebra satisfying Hypothesis 1.2 then Hypothesis 2.2 is
satisfied for $j=N+1$. Theorem 2.3 then implies that Hypothesis 2.2 is also
satisfied for all $j\in\llbracket 2,N+1\rrbracket$.

######  Corollary 2.5.

The algebra $A^{\prime}:=A^{(2)}$ is a quantum affine space. More precisely,
by setting $T_{i}:=X_{i}^{(2)}$ for all $i\in\llbracket 1,N\rrbracket$ and
$\Lambda:=(\lambda_{i,j})\in M_{N}(\mathbb{K})$ to be the multiplicatively
antisymmetric matrix, we obtain:

$A^{\prime}=\mathbb{K}_{\Lambda}[T_{1},\ldots,T_{N}].$

######  Remark 2.6.

For all $j\in\llbracket 1,N\rrbracket$, we say that $A^{(j+1)}$ is the algebra
obtained from $A$ after $N-j$ steps of the deleting derivations algorithm.

###  2.2 Ring of fractions

In order to be able to track the completely prime ideals along the deleting
derivations algorithm we need the following two results regarding the division
ring of fractions of the algebras $A^{(j)}$ at each step of the algorithm.
These results were proved in [Cau03a, Subsection 3.3] in the generic setting
and can be applied directly to our setting thanks to the above results.

Let $\Sigma$ be the multiplicatively closed set in $A^{\prime}$ generated by
the elements $T_{1},\ldots,T_{N}$. For $j\in\llbracket 2,N\rrbracket$, define
sets $\Sigma_{j}$:

$\displaystyle\Sigma_{2}$ $\displaystyle{}:=\Sigma,$
$\displaystyle\Sigma_{j+1}$
$\displaystyle{}=A^{(j+1)}\cap\Sigma_{j}\quad\text{ for }j\in\llbracket
2,N\rrbracket.$

######  Proposition 2.7.

For all $j\in\llbracket 2,N+1\rrbracket$ the following are true:

  1. (i)

$\Sigma_{j}$ is a multiplicatively closed set of regular elements in $A^{(j)}$
containing $X_{j-1}^{(j)},\ldots,X_{N}^{(j)}$;

  2. (ii)

$\Sigma_{j}$ satisfies the two-sided Ore condition in $A^{(j)}$;

  3. (iii)

The algebras $A^{(j)}\Sigma_{j}^{-1}\subset\mathrm{Frac}(A)$ are all equal.

###### Proof.

We proceed by induction on $j$. When $j=2$, statements (i) and (ii) are
trivially true from the definition of $\Sigma$ and the fact that the
generators (and monomials in these generators) of a quantum affine space are
regular and normal. Let $j\in\llbracket 2,N\rrbracket$ and suppose statements
(i) and (ii) hold for $j$. We will show they also hold for $j+1$ and that
$A^{(j)}\Sigma_{j}^{-1}=A^{(j+1)}\Sigma_{j+1}^{-1}$.

Recall the notation from the previous section:
$\\{x_{i}\\}_{i}=\\{X_{i}^{(j+1)}\\}_{i}$ and
$\\{y_{i}\\}_{i}=\\{X_{i}^{(j)}\\}_{i}$ where $x_{i}=y_{i}$ for all $i\geq j$.
By the induction hypothesis, $\Sigma_{j}$ is a multiplicatively closed set of
regular elements in $A^{(j)}$ containing $y_{j-1},\ldots,y_{N}$. Therefore
$\Sigma_{j+1}=A^{(j)}\cap\Sigma_{j}$ is a multiplicatively closed set of
regular elements of $A^{(j+1)}$ containing
$y_{j}=x_{j},\;\ldots,\;y_{N}=x_{N}$, thus proving statement (i).

Recall the set $S_{j}=\\{x_{j}^{n}\mid n\in\mathbb{N}\\}=\\{y_{j}^{n}\mid
n\in\mathbb{N}\\}\subset\Sigma_{j}\cap\Sigma_{j+1}$ and use Theorem 2.3(III)
to obtain the inclusions

$\displaystyle A^{(j+1)}\subset A^{(j+1)}S_{j}^{-1}=A^{(j)}S_{j}^{-1}\subset
A^{(j)}\Sigma_{j}^{-1}.$ (11)

Since $\Sigma_{j+1}\subset\Sigma_{j}$ then $\Sigma_{j+1}$ must be invertible
in $A^{(j)}\Sigma_{j}^{-1}$. We use this to show that an element $a\in
A^{(j)}\Sigma_{j}^{-1}$ can be rewritten as an element in
$A^{(j+1)}\Sigma_{j+1}^{-1}$. Write $a=yu^{-1}$, with $y\in A^{(j)}$ and
$u\in\Sigma_{j}$. Since

$\Sigma_{j}\subset A^{(j)}\subset A^{(j)}S_{j}^{-1}=A^{(j+1)}S_{j}^{-1},$

we can write $u=vs_{1}^{-1}$ and $y=xs_{2}^{-1}$, with $v,x\in A^{(j+1)}$ and
$s_{1},s_{2}\in S_{j}$. Then $a$ becomes

$a=xs_{2}^{-1}(vs_{1}^{-1})^{-1}=xs_{2}^{-1}s_{1}v^{-1}=xs_{1}s_{2}^{-1}v^{-1}=xs_{1}(vs_{2})^{-1}.$

Observe that $vs_{2}=(vs_{1}^{-1})s_{1}s_{2}=us_{1}s_{2}\in\Sigma_{j}$ since
$u\in\Sigma_{j}$ and $s_{1},s_{2}\in S_{j}\subset\Sigma_{j}$. Also, $v\in
A^{(j+1)}$ and $s_{2}\in S_{j}\subset A^{(j+1)}$, therefore $vs_{2}\in
A^{(j+1)}\cap\Sigma_{j}=\Sigma_{j+1}$. Similarly, $xs_{1}\in A^{(j+1)}$ so we
can write

$a=bc^{-1}\in A^{(j+1)}\Sigma_{j+1}^{-1},$

where $b=xs_{1}\in A^{(j+1)}$ and $c=vs_{2}\in\Sigma_{j+1}$. From the
inductive hypothesis we know that
$A^{(j)}\Sigma_{j}^{-1}=\Sigma_{j}^{-1}A^{(j)}$, so if $a\in
A^{(j)}\Sigma_{j}^{-1}$ then it must also be true that
$a\in\Sigma_{j}^{-1}A^{(j)}$. We also know from Theorem 2.3(III) that $S_{j}$
is an Ore set in $A^{(j)}$ and $A^{(j+1)}$, thus
$A^{(j)}S_{j}^{-1}=S_{j}^{-1}A^{(j)}$ and
$A^{(j+1)}S_{j}^{-1}=S_{j}^{-1}A^{(j+1)}$. Using these results we can follow a
similar method to before to rewrite $a\in
A^{(j)}\Sigma_{j}^{-1}=\Sigma_{j}^{-1}A^{(j)}$ as

$a=c^{\prime-1}b^{\prime}\in\Sigma_{j+1}^{-1}A^{(j+1)},$

with $c^{\prime}\in\Sigma_{j+1}$ and $b^{\prime}\in A^{(j+1)}$.

If we can prove that $\Sigma_{j+1}$ is a two-sided Ore set in $A^{(j+1)}$ then
the working above implies that $A^{(j)}\Sigma_{j}^{-1}\subseteq
A^{(j+1)}\Sigma_{j+1}^{-1}$. Furthermore, $A^{(j+1)}\subset
A^{(j)}\Sigma_{j}^{-1}$ and $\Sigma_{j+1}\subset\Sigma_{j}$ so we also have
$A^{(j+1)}\Sigma_{j+1}^{-1}\subseteq A^{(j)}\Sigma_{j}^{-1}$. Hence statement
(iii) is true if we can prove that statement (ii) holds.

From the inclusion $A^{(j+1)}\Sigma_{j+1}^{-1}\subseteq
A^{(j)}\Sigma_{j}^{-1}$ we can write any $a=bc^{-1}\in
A^{(j+1)}\Sigma_{j+1}^{-1}$ as $a\in A^{(j)}\Sigma_{j}^{-1}$ and, applying the
above working, we see that there exist $c^{\prime}\in\Sigma_{j+1}$ and
$b^{\prime}\in A^{(j+1)}$ such that
$a=c^{\prime-1}b^{\prime}\in\Sigma_{j+1}^{-1}A^{(j+1)}$. This verifies the
two-sided Ore condition on $\Sigma_{j+1}\subset A^{(j+1)}$ necessary for
proving statement (ii) and, by the comment earlier, statement (iii). ∎

From the above proposition it is clear that:

######  Theorem 2.8.

  1. (i)

There exists a multiplicatively closed set of regular elements $S\subseteq A$
such that $AS^{-1}=A^{\prime}\Sigma^{-1}=\mathbb{K}_{\Lambda}[T_{1}^{\pm
1},\ldots,T_{N}^{\pm 1}]$.

  2. (ii)

$\mathrm{Frac}(A^{(j)})=\mathrm{Frac}(A)$ for all $j\in\llbracket
2,N+1\rrbracket$ and, in particular,
$\mathrm{Frac}(A)=\mathrm{Frac}(A^{\prime})$.

  3. (iii)

$A$ is a PI algebra if and only if the $\lambda_{i,j}$ are roots of unity for
all $i,j\in\llbracket 1,N\rrbracket$ and, in this case, ${\rm PI}\text{-}{\rm
deg}(A)={\rm PI}\text{-}{\rm deg}(A^{\prime})$.

###### Proof.

Let $S=\Sigma_{N+1}$. Proposition 2.7(iii) shows that
$AS^{-1}=A^{\prime}\Sigma^{-1}$ and that all the algebras $A^{(j)}$ have a
common localisation, thus proving (i) and (ii). [CP93, Proposition 7.1] states
that $A^{\prime}$ is PI if and only if the $\lambda_{i,j}$ are roots of unity
for all $1\leq i,j\leq N$ and [Hay08, Corollary 4.7] states the same result
for $A$. Therefore they have the same PI degree since they have equal total
rings of fractions. This proves (iii). ∎

##  3 Deleting Derivations Algorithm on completely prime quotients

In this section we set up a canonical embedding
$\psi:\mathrm{C.Spec}(A)\rightarrow\mathrm{C.Spec}(A^{\prime})$, from the
completely prime spectrum of $A$ into the completely prime spectrum of
$A^{\prime}$, and use this to extend the algorithm of the previous section to
quotient algebras. We also define criteria for some ideal
$Q\in\mathrm{C.Spec}(A^{\prime})$ to lie in the image of $\psi$.

Many of the results of this section are analogues to those found in the
generic setting [Cau03a, Sections 4 and 5] and their proofs follow in almost
the same way, thanks to the results of the previous section. The proofs
concerning membership criteria for the canonical embedding in Subsection 3.2
closely mirror those in the Poisson setting, which can be found in [LL17,
Sections 2.3 and 2.4].

###  3.1 Canonical Embedding and partition of completely prime spectra

####  3.1.1 Embedding

Let $A$ satisfy Hypothesis 1.2 and define the following sets, which we endow
with the induced Zariski topology:

$\displaystyle\mathcal{P}_{j}^{0}(A^{(j)})$
$\displaystyle{}:=\\{P\in\mathrm{C.Spec}(A^{(j)})\mid y_{j}\notin P\\},$
$\displaystyle\mathcal{P}_{j}^{1}(A^{(j)})$
$\displaystyle{}:=\\{P\in\mathrm{C.Spec}(A^{(j)})\mid y_{j}\in P\\},$
$\displaystyle\mathcal{P}_{j}^{0}(A^{(j+1)})$
$\displaystyle{}:=\\{P\in\mathrm{C.Spec}(A^{(j+1)})\mid x_{j}\notin P\\},$
$\displaystyle\mathcal{P}_{j}^{1}(A^{(j+1)})$
$\displaystyle{}:=\\{P\in\mathrm{C.Spec}(A^{(j+1)})\mid x_{j}\in P\\}.$

In the following results, by the term _increasing_ we mean that the
homeomorphism is order-preserving with respect to inclusion of ideals. By _bi-
increasing_ we mean that both the homeomorphism and its inverse are increasing
homeomorphisms.

######  Lemma 3.1.

There is an increasing homeomorphism
$\psi^{0}_{j}:\mathcal{P}^{0}_{j}(A^{(j+1)})\rightarrow\mathcal{P}^{0}_{j}(A^{(j)})$
given by $\psi^{0}_{j}(P):=PS_{j}^{-1}\cap A^{(j)}$. Its inverse is defined by
$(\psi^{0}_{j})^{-1}(Q):=QS_{j}^{-1}\cap A^{(j+1)}$ and is also increasing.
Hence $\psi^{0}_{j}$ is bi-increasing.

###### Proof.

Extension and contraction maps provide bi-increasing inverse homeomorphisms
between $\mathrm{C.Spec}(A^{(j)}S_{j}^{-1})$ and
$\mathcal{P}_{j}^{0}(A^{(j)})$. Similarly, we obtain a bi-increasing
homeomorphism between $\mathcal{P}_{j}^{(0)}(A^{(j+1)})$ and
$\mathrm{C.Spec}(A^{(j+1)}S_{j}^{-1})$. Since
$A^{(j)}S_{j}^{-1}=A^{(j+1)}S_{j}^{-1}$ then their completely prime spectra,
as topological spaces, are equal. Therefore the two bi-increasing
homeomorphisms defined here give rise to the bi-increasing homeomorphism in
the statement of this lemma. ∎

Next we turn our attention to the sets $\mathcal{P}^{1}_{j}(A^{(j)})$ and
$\mathcal{P}^{1}_{j}(A^{(j+1)})$.

######  Lemma 3.2.

There is a surjective algebra homomorphism $g_{j}:A^{(j)}\rightarrow
A^{(j+1)}/\langle x_{j}\rangle$ which takes $y_{i}\mapsto x_{i}+\langle
x_{j}\rangle$, for all $1\leq i\leq N$, where $x_{i}+\langle x_{j}\rangle$ is
the canonical image of $x_{i}$ in $A^{(j+1)}/\langle x_{j}\rangle$.

###### Proof.

By Theorem 2.3 we have

$\displaystyle A^{(j+1)}$ $\displaystyle{}:=\mathbb{K}\langle
x_{1},\ldots,x_{N}\rangle\cong\mathbb{K}[X_{1}]\cdots[X_{j};\sigma_{j},\delta_{j}][X_{j+1};\sigma^{(j+1)}_{j+1}]\cdots[X_{N};\sigma^{(j+1)}_{N}],$
$\displaystyle A^{(j)}$ $\displaystyle{}:=\mathbb{K}\langle
y_{1},\ldots,y_{N}\rangle\cong\mathbb{K}[X_{1}]\cdots[X_{j-1};\sigma_{j-1},\delta_{j-1}][X_{j};\sigma^{(j)}_{j}]\cdots[X_{N};\sigma^{(j)}_{N}].$

Restricting these algebras to $R:=\mathbb{K}\langle
x_{1},\ldots,x_{j-1}\rangle$ and $S:=\mathbb{K}\langle
y_{1},\ldots,y_{j-1}\rangle$ we see that there is an isomorphism $S\rightarrow
R$ sending $y_{i}\mapsto x_{i}$ for all $i\in\llbracket 1,j-1\rrbracket$.
Since $R\subseteq A^{(j+1)}$ we can compose this isomorphism with the natural
surjection $A^{(j+1)}\rightarrow A^{(j+1)}/\langle x_{j}\rangle$ to obtain the
algebra homomorphism $f:S\rightarrow A^{(j+1)}/\langle x_{j}\rangle$, sending
$y_{i}\mapsto x_{i}+\langle x_{j}\rangle$ for all $i\in\llbracket
1,j-1\rrbracket$. Using the the commutation rules for $A^{(j+1)}$ and the fact
that $x_{j}+\langle x_{j}\rangle=0$, we see that $(x_{j}+\langle
x_{j}\rangle)(x_{i}+\langle x_{j}\rangle)=\lambda_{j,i}(x_{i}+\langle
x_{j}\rangle)(x_{j}+\langle x_{j}\rangle)=0$ for all $i\in\llbracket
1,j-1\rrbracket$. The relations on $x_{i}+\langle x_{j}\rangle\in
A^{(j+1)}/\langle x_{j}\rangle$ therefore agree with those on $y_{i}\in
S[y_{j};\sigma^{(j)}_{j}]\cdots[y_{N};\sigma^{(j)}_{N}]=A^{(j)}$, for all
$i\in\llbracket 1,N\rrbracket$. Applying the universal property of Ore
extensions allows us to conclude the construction of the surjective
homomorphism $g_{j}$. ∎

######  Lemma 3.3.

There is an increasing injective map
$\psi_{j}^{1}:\mathcal{P}^{1}_{j}(A^{(j+1)})\rightarrow\mathcal{P}^{1}_{j}(A^{(j)})$
taking $P\mapsto\psi_{j}^{1}(P):=g_{j}^{-1}(P/\langle x_{j}\rangle)$, where
$P/\langle x_{j}\rangle$ denotes the canonical image of $P$ in
$A^{(j+1)}/\langle x_{j}\rangle$, which induces a bi-increasing homeomorphism
between $\mathcal{P}_{j}^{1}(A^{(j+1)})$ and the image
$\psi_{j}^{1}(\mathcal{P}_{j}^{1}(A^{(j+1)}))$.

###### Proof.

Using the First Isomorphism Theorem for algebras, we restrict the map $g_{j}$
from Lemma 3.2 to yield an isomorphism
$g^{\prime}_{j}:A^{(j)}/\ker(g_{j})\xrightarrow{\,\smash{\raisebox{-2.79857pt}{$\scriptstyle\sim$}}\,}A^{(j+1)}/\langle
x_{j}\rangle$. This induces the following bi-increasing homeomorphisms between
sets endowed with the Zariski topology:

$\displaystyle f_{1}:\mathcal{P}_{j}^{1}(A^{(j+1)})$
$\displaystyle{}~{}\longrightarrow~{}\mathrm{C.Spec}(A^{(j+1)}/\langle
x_{j}\rangle),$ $\displaystyle f_{2}:\mathrm{C.Spec}(A^{(j+1)}/\langle
x_{j}\rangle)$
$\displaystyle{}~{}\longrightarrow~{}\mathrm{C.Spec}(A^{(j)}/\ker(g_{j})),$
$\displaystyle f_{3}:\mathrm{C.Spec}(A^{(j)}/\ker(g_{j}))$
$\displaystyle{}~{}\longrightarrow~{}\\{Q\in\mathrm{C.Spec}(A^{(j)})\mid\ker(g_{j})\subseteq
Q\\}.$

The composition of these maps gives a bi-increasing homeomorphism

$\displaystyle f_{3}\circ f_{2}\circ f_{1}:\mathcal{P}_{j}^{1}(A^{(j+1)})$
$\displaystyle{}~{}\longrightarrow~{}\\{Q\in\mathrm{C.Spec}(A^{(j)})\mid\ker(g_{j})\subseteq
Q\\}$ $\displaystyle P$ $\displaystyle{}~{}\longmapsto~{}g_{j}^{-1}(P/\langle
x_{j}\rangle).$

Note that $g_{j}(y_{j})=x_{j}+\langle x_{j}\rangle=0$ so $\langle
y_{j}\rangle\subseteq\ker(g_{j})$, which leads to the inclusion

$\\{Q\in\mathrm{C.Spec}(A^{(j)})\mid\ker(g_{j})\subseteq
Q\\}~{}\subseteq~{}\mathcal{P}_{j}^{1}(A^{(j)}).$

Therefore, from $f_{3}\circ f_{2}\circ f_{1}$, we can define an increasing
injective map

$\displaystyle\psi_{j}^{1}:\mathcal{P}_{j}^{1}(A^{(j+1)})$
$\displaystyle{}~{}\longrightarrow~{}\mathcal{P}_{j}^{1}(A^{(j)})$
$\displaystyle P$ $\displaystyle{}~{}\longmapsto~{}g_{j}^{-1}(P/\langle
x_{j}\rangle),$

which induces a bi-increasing homeomorphism on its image,
$\\{Q\in\mathrm{C.Spec}(A^{(j)})\mid\ker(g_{j})\subseteq Q\\}$. ∎

Using the two previous results we define the map
$\psi_{j}:\mathrm{C.Spec}(A^{(j+1)})\rightarrow\mathrm{C.Spec}(A^{(j)})$
where, for $P\in\mathrm{C.Spec}(A^{(j+1)})$, we set

$\psi_{j}(P):=\begin{cases}\psi_{j}^{0}(P)=PS_{j}^{-1}\cap A^{(j)}&\text{if
}P\in\mathcal{P}_{j}^{0}(A^{(j+1)});\\\ \psi_{j}^{1}(P)=g_{j}^{-1}(P/\langle
x_{j}\rangle)&\text{if }P\in\mathcal{P}_{j}^{1}(A^{(j+1)}).\end{cases}$

The next result follows immediately.

######  Proposition 3.4.

For $j\in\llbracket 2,N\rrbracket$ the map
$\psi_{j}:\mathrm{C.Spec}(A^{(j+1)})\rightarrow\mathrm{C.Spec}(A^{(j)})$ is
injective. For $\epsilon\in\\{0,1\\}$, $\psi_{j}$ induces (by restriction) a
bi-increasing homeomorphism
$\mathcal{P}_{j}^{\epsilon}(A^{(j+1)})\rightarrow\psi_{j}(\mathcal{P}_{j}^{\epsilon}(A^{(j+1)}))$
which is a closed subset of $\mathcal{P}_{j}^{\epsilon}(A^{(j)})$.

Using these maps, we define the canonical embedding:

######  Definition 3.5.

Set $\psi:=\psi_{2}\circ\cdots\circ\psi_{N}$ to be the injective map
$\psi:\mathrm{C.Spec}(A)\longrightarrow\mathrm{C.Spec}(A^{\prime})$. We call
$\psi$ the _canonical embedding_ of $\mathrm{C.Spec}(A)$ into
$\mathrm{C.Spec}(A^{\prime})$.

####  3.1.2 Partition of $\mathrm{C.Spec}(A)$

Let $\mathcal{W}:=\mathbb{P}(\llbracket 1,N\rrbracket)$ denote the power set
of $\llbracket 1,N\rrbracket$ and, for all $w\in\mathcal{W}$, set

$\mathrm{C.Spec}_{w}(A^{\prime}):=\\{Q\in\mathrm{C.Spec}(A^{\prime})\mid
Q\cap\\{T_{1},\ldots T_{N}\\}=\\{T_{i}\mid i\in w\\}\\},$

where $\\{T_{i}\\}_{i=1}^{N}$ are the generators of the quantum affine space
$A^{\prime}$. From [GL98, Section 2.1] we get:

######  Lemma 3.6.

The sets $\\{\mathrm{C.Spec}_{w}(A^{\prime})\\}_{w\in\mathcal{W}}$ provide a
partition of $\mathrm{C.Spec}(A^{\prime})$.

We use $\psi$ to pull this partition back to one on $\mathrm{C.Spec}(A)$. For
each $w\in\mathcal{W}$ we define

$\mathrm{C.Spec}_{w}(A):=\psi^{-1}(\mathrm{C.Spec}_{w}(A^{\prime}))$

and let $\mathcal{W}^{\prime}\subseteq\mathcal{W}$ denote the set of all
$w\in\mathcal{W}$ such that $\mathrm{C.Spec}_{w}(A)\neq\emptyset$. (Note that
$\mathcal{W}^{\prime}$ depends on the expression of the algebra $A$ as an
iterated Ore extension.) We immediately obtain a partition of
$\mathrm{C.Spec}(A)$.

######  Theorem 3.7.

The set $\mathrm{C.Spec}(A)$ has a partition indexed by the family
$\mathcal{W}^{\prime}$ so that,

$\mathrm{C.Spec}(A)=\bigsqcup_{w\in\mathcal{W}^{\prime}}\mathrm{C.Spec}_{w}(A),\quad\text{
where }~{}|\mathcal{W}^{\prime}|\leq|\mathcal{W}|=2^{N}.$

######  Definition 3.8.

We call the partition
$\\{\mathrm{C.Spec}_{w}(A)\\}_{w\in\mathcal{W}^{\prime}}$ the _canonical
partition_ of $\mathrm{C.Spec}(A)$, and we call each
$w\in\mathcal{W}^{\prime}$ a _Cauchon diagram of $A$_.

###  3.2 Properties of the Canonical Embedding

In order to use the deleting derivations algorithm for the purpose of
calculating the PI degree of quotient algebras, we need to be able to test
whether a completely prime ideal of $A^{\prime}$ lies in the image of the
canonical embedding.

######  Lemma 3.9.

Fix some $j\in\llbracket 2,N\rrbracket$ and let
$Q\in\mathrm{C.Spec}(A^{(j)})$. Then,

$Q\in\operatorname{Im}(\psi_{j})\iff[\text{Either }x_{j}=y_{j}\notin Q\text{
or }\ker(g_{j})\subseteq Q].$

###### Proof.

Apply Lemmas 3.1 and 3.3 to the cases $y_{j}\notin Q$ and $y_{j}\in Q$
respectively. ∎

We define injective maps
$f_{j}:\mathrm{C.Spec}(A^{(j+1)})\rightarrow\mathrm{C.Spec}(A^{\prime})$, for
all $j\in\llbracket 1,N\rrbracket$, with
$f_{1}:=\mathrm{id}_{\mathrm{C.Spec}(A^{\prime})}$, the identity on
$\mathrm{C.Spec}(A^{\prime})$ and, for all $j\in\llbracket 2,N\rrbracket$,
$f_{j}:=\psi_{2}\circ\cdots\circ\psi_{j}$, so that $f_{N}=\psi$.

######  Proposition 3.10.

Let $Q\in\mathrm{C.Spec}(A^{\prime})$. The following are equivalent:

  * •

$Q\in\operatorname{Im}(\psi)$.

  * •

For all $j\in\llbracket 2,N\rrbracket$ we have
$Q\in\operatorname{Im}(f_{j-1})$ and either $X_{j}^{(j)}=X_{j}^{(j+1)}\notin
f_{j-1}^{-1}(Q)$ or $\ker(g_{j})\subseteq f_{j-1}^{-1}(Q)$.

###### Proof.

Let $Q\in\mathrm{C.Spec}(A^{\prime})$. Suppose $Q\in\operatorname{Im}(\psi)$.
Then $Q=\psi(P)$ for some $P\in\mathrm{C.Spec}(A)$. Since
$\psi=f_{j-1}\circ\psi_{j}\circ\cdots\psi_{N}$ then $Q=f_{j-1}(P_{j})$ for all
$j\in\llbracket 2,N\rrbracket$, where
$P_{j}=\psi_{j}\circ\cdots\circ\psi_{N}(P)$. Hence
$Q\in\operatorname{Im}(f_{j-1})$ for all $j\in\llbracket 2,N\rrbracket$. From
this we see that $f_{j-1}^{-1}(Q)\in\operatorname{Im}(\psi_{j})$, for all
$j\in\llbracket 2,N\rrbracket$, and we apply Lemma 3.9 to $f_{j-1}^{-1}(Q)$ to
conclude that either $X_{j}^{(j)}=X_{j}^{(j+1)}\notin f_{j-1}^{-1}(Q)$ or
$\ker(g_{j})\subseteq f_{j-1}^{-1}(Q)$.

Now suppose the second statement holds. By Lemma 3.9,
$f_{j-1}^{-1}(Q)\in\operatorname{Im}(\psi_{j})$ for all $j\in\llbracket
2,N\rrbracket$. Let $P^{\prime}:=f_{N-1}^{-1}(Q)$ so that
$P^{\prime}\in\operatorname{Im}(\psi_{N})$. Then $P^{\prime}=\psi_{N}(P)$ for
some $P\in\mathrm{C.Spec}(A)$ and

$Q=f_{N-1}(P^{\prime})=f_{N-1}(\psi_{N}(P))=\psi(P)\in\operatorname{Im}(\psi).$

∎

We finish this subsection by giving a sufficient condition for a completely
prime ideal in $A^{\prime}$ to be in the image of the canonical embedding.
These next two results are not used in this paper, however they are stated
here for the interested reader (proofs can be found in [Rog19, Section 4]).

######  Theorem 3.11.

[Rog19, Theorem 4.17] Let $w\in\mathcal{W}^{\prime}$. Then
$\psi(\mathrm{C.Spec}_{w}(A))$ is a (non-empty) closed subset of
$\mathrm{C.Spec}_{w}(A^{\prime})$ and the map $\psi$ induces (by restriction)
a bi-increasing homeomorphism from $\mathrm{C.Spec}_{w}(A)$ to
$\psi(\mathrm{C.Spec}_{w}(A))$.

######  Proposition 3.12.

[Rog19, Proposition 4.18] Let
$w\in\mathcal{W}^{\prime},\,P\in\mathrm{C.Spec}_{w}(A)$, and
$Q\in\mathrm{C.Spec}_{w}(A^{\prime})$. If $\psi(P)\subseteq Q$ then
$Q\in\operatorname{Im}(\psi)$.

###  3.3 Completely Prime Quotients

In this subsection we extend the algorithm in Section 2 to apply to completely
prime quotient algebras of quantum nilpotent algebras. The results of the
previous sections allow us to construct proofs in a similar way to those found
in [Cau03a, Section 5.3]. This extended algorithm will allow us to pull
certain irreducible representations on $A^{\prime}/Q$ (clarified in Section 5)
back to irreducible representations on $A/P$.

####  3.3.1 $A^{(j)}/P$ and $A^{(j+1)}/\psi_{j}(P)$

We start by extending one step of the algorithm to completely prime quotients.
For some $j\in\llbracket 2,N\rrbracket$, let $P\in\mathrm{C.Spec}(A^{(j+1)})$
and $Q=\psi_{j}(P)\in\mathrm{C.Spec}(A^{(j)})$ be its image under the
canonical embedding. Set

$B^{(j+1)}:=A^{(j+1)}/P,\quad B^{(j)}:=A^{(j)}/Q.$

We denote by $\bar{x}_{1},\ldots,\bar{x}_{N}\in B^{(j+1)}$ and
$\bar{y}_{1},\ldots,\bar{y}_{N}\in B^{(j)}$ the canonical images of the
generators $x_{1},\ldots,x_{N}\in A^{(j+1)}$ in $B^{(j+1)}$ and
$y_{1},\ldots,y_{N}\in A^{(j)}$ in $B^{(j)}$, respectively.

######  Lemma 3.13.

Suppose $\bar{x}_{j}=0$. Then there exists an algebra isomorphism
$B^{(j)}\rightarrow B^{(j+1)}$ sending $\bar{y}_{i}\mapsto\bar{x}_{i}$ for all
$i\in\llbracket 1,N\rrbracket$.

###### Proof.

Since $x_{j}\in P$ then $Q=\psi_{j}^{1}(P)=g_{j}^{-1}(P/\langle
x_{j}\rangle)$. Concatenating $g_{j}$ with the natural surjection
$\pi:A^{(j+1)}/\langle x_{j}\rangle\rightarrow A^{(j+1)}/P$ gives the
following surjective algebra homomorphism:

$\begin{array}[]{ccccc}A^{(j)}&\overset{g_{j}}{\longrightarrow}&A^{(j+1)}/\langle
x_{j}\rangle&\overset{\pi}{\longrightarrow}&A^{(j+1)}/P\\\
y_{i}&\longmapsto&x_{i}+\langle
x_{j}\rangle&\longmapsto&\bar{x}_{i},\end{array}$

The desired isomorphism may then be constructed upon noting that
$\ker(\pi\circ g_{j})=g_{j}^{-1}(P/\langle x_{j}\rangle)=Q$. ∎

######  Lemma 3.14.

Suppose $\bar{x}_{j}\neq 0$ and let $Z_{j}:=\\{\bar{x}_{j}^{n}\mid
n\in\mathbb{N}\\}$. Then the following hold:

  1. (i)

$Z_{j}$ is a multiplicative set of regular elements of $B^{(j+1)}$, which
satisfies the two-sided Ore condition.

  2. (ii)

There exists an injective algebra homomorphism $\gamma:B^{(j)}\rightarrow
B^{(j+1)}Z_{j}^{-1}$ defined on the generators of $B^{(j)}$ in the following
way:

$\gamma(\bar{y}_{i})=\begin{cases}\bar{x}_{i}&\text{ if }i\geq j;\\\
\sum_{n=0}^{\infty}q_{j}^{\frac{n(n+1)}{2}}(q_{j}-1)^{-n}\lambda_{j,i}^{-n}\overline{d_{j,n}(x_{i})}\bar{x}_{j}^{-n}&\text{
if }i<j,\end{cases}$

where $\overline{d_{j,n}(x_{i})}$ denotes the canonical image of
$d_{j,n}(x_{i})$ in $B^{(j+1)}$.

  3. (iii)

If we identify $B^{(j)}$ with its image $\gamma(B^{(j)})\subseteq
B^{(j+1)}Z_{j}^{-1}$ then $Z_{j}$ is a multiplicative set of regular elements
in $B^{(j)}$ satisfying the two-sided Ore condition. Furthermore,
$B^{(j)}Z_{j}^{-1}=B^{(j+1)}Z_{j}^{-1}$.

###### Proof.

Since $x_{j}\notin P$ then $Q=\psi_{j}^{0}(P)=PS_{j}^{-1}\cap A^{(j)}$, where
$S_{j}=\\{x_{j}^{n}\mid n\in\mathbb{N}\\}$ is the multiplicatively closed set
of regular elements in $A^{(j+1)}$ and $A^{(j)}$ satisfying the two-sided Ore
condition (Theorem 2.3(III)). Denote the subalgebra
$A^{(j+1)}S_{j}^{-1}=A^{(j)}S_{j}^{-1}\subseteq F$ by $\Omega$ and the
completely prime ideal $PS_{j}^{-1}=QS_{j}^{-1}$ by $\Theta\lhd\Omega$. Note
that $\Theta\cap A^{(j+1)}=P$ and $\Theta\cap A^{(j)}=Q$.

We define an injective algebra homomorphism
$\gamma\,^{\prime}:B^{(j+1)}\longrightarrow\Omega/\Theta;~{}a+P\longmapsto
a1^{-1}+\Theta$ and identify $B^{(j+1)}$ with its image. Since
$Z_{j}=\\{\bar{x}_{j}^{n}\mid n\in\mathbb{N}\\}=\\{x_{j}^{n}+P\mid
n\in\mathbb{N}\\}=S_{j}+P$, its image under $\gamma\,^{\prime}$ becomes

$\gamma\,^{\prime}(Z_{j})=Z_{j}1^{-1}+\Theta=(S_{j}+P)1^{-1}+\Theta=S_{j}1^{-1}+\Theta\subseteq
B^{(j+1)}\subseteq\Omega/\Theta.$

Identifying $Z_{j}$ with its image $\gamma\,^{\prime}(Z_{j})$, we observe that
all elements of the set $Z_{j}$ are invertible in $B^{(j+1)}$. We can
therefore write any element $b\in\Omega/\Theta$ as

$\displaystyle b=a_{1}s_{1}^{-1}+\Theta=s_{2}^{-1}a_{2}+\Theta,$ (12)

where $a_{1},a_{2}\in A^{(j+1)}$ and $s_{1},s_{2}\in S_{j}$. Let
$b_{1},b_{2}\in B^{(j+1)}$ and $z_{1},z_{2}\in Z_{j}$ such that

$b_{1}=a_{1}1^{-1}+\Theta,~{}~{}~{}b_{2}=a_{2}1^{-1}+\Theta,~{}~{}~{}z_{1}=s_{1}1^{-1}+\Theta,~{}~{}~{}z_{2}=s_{2}1^{-1}+\Theta.$

Using (12), we see that, for all $b\in\Omega/\Theta$,

$b=b_{1}z_{1}^{-1}=z_{2}^{-1}b_{2}.$

This shows that the set $Z_{j}\subset B^{(j+1)}$ satisfies the two-sided Ore
condition, thus proving property (i) of the lemma. We have also proved the
equality $B^{(j+1)}Z_{j}^{-1}=\Omega/\Theta$, i.e.
$(A^{(j+1)}/P)Z_{j}^{-1}=A^{(j+1)}S_{j}^{-1}/PS_{j}^{-1}$.

For part (ii) we use the fact that $\Theta\cap A^{(j)}=Q$ and
$B^{(j+1)}Z_{j}^{-1}=\Omega/\Theta$ to define an injective homomorphism

$\displaystyle\gamma:B^{(j)}~{}~{}$
$\displaystyle{}~{}\longrightarrow~{}\Omega/\Theta$ $\displaystyle a+Q$
$\displaystyle{}~{}\longmapsto~{}a1^{-1}+\Theta.$

Recall that the generators $y_{i}\in A^{(j)}$ are defined as:

$y_{i}:=\begin{cases}x_{i}&\text{ if }i\geq j;\\\
\sum_{n=0}^{\infty}q_{j}^{\frac{n(n+1)}{2}}(q_{j}-1)^{-n}\lambda_{j,i}^{-n}d_{j,n}(x_{i})x_{j}^{-n}&\text{
if }i<j.\end{cases}$

It is straightforward to check that $\gamma(\bar{y}_{i})=y_{i}1^{-1}+\Theta$
gives the desired results for $i<j$ and $i\geq j$.

Part (iii) is proved in the same way as (i) by identifying $B^{(j)}$ with its
image $\gamma(B^{(j)})$. ∎

Lemmas 3.13 and 3.14 imply:

######  Lemma 3.15.

$\mathrm{Frac}(B^{(j+1)})=\mathrm{Frac}(B^{(j)})$.

####  3.3.2 $A/P$ and $A^{\prime}/\psi(P)$

We continue to extend the algorithm to apply to completely prime quotients
$A/P$. Let $P\in\mathrm{C.Spec}(A)$ and
$Q=\psi(P)\in\mathrm{C.Spec}(A^{\prime})$ and set the following notation:

  * •

$B:=A/P$ and set $\bar{X}_{1},\ldots,\bar{X}_{N}\in B$ to be the canonical
images of $X_{1},\ldots,X_{N}\in A$.

  * •

$B^{\prime}:=A^{\prime}/Q$ and set $t_{1},\ldots,t_{N}\in B^{\prime}$ to be
the canonical images of $T_{1},\ldots,T_{N}\in A^{\prime}$.

  * •

For $j\in\llbracket 2,N+1\rrbracket$, denote by
$P_{j}:=\psi_{j}\circ\cdots\circ\psi_{N}(P)\in\mathrm{C.Spec}(A^{(j)})$ the
image of $P$ after $N-j+1$ steps of the deleting derivations algorithm.

  * •

For each $j\in\llbracket 2,N+1\rrbracket$, define the algebra
$B^{(j)}:=A^{(j)}/P_{j}$ and denote by
$\bar{X}_{1}^{(j)},\ldots,\bar{X}_{N}^{(j)}$ the canonical images of
$X_{1}^{(j)},\ldots,X_{N}^{(j)}$ in $B^{(j)}$. Note: $B^{(N+1)}=B$ with
$(\bar{X}_{1}^{(N+1)},\ldots,\bar{X}_{N}^{(N+1)})=(\bar{X}_{1},\ldots,\bar{X}_{N})$,
and $B^{(2)}=B^{\prime}$ with
$(\bar{X}_{1}^{(2)},\ldots,\bar{X}_{N}^{(2)})=(t_{1},\ldots,t_{N})$.

######  Proposition 3.16.

For each $j\in\llbracket 2,N+1\rrbracket$, $B^{(j)}$ is a subalgebra of
$\mathrm{Frac}(B)$ generated by $\bar{X}_{1}^{(j)},\ldots,\bar{X}_{N}^{(j)}$
and there is an algebra homomorphism,

$\displaystyle f_{j}:~{}~{}A^{(j)}$
$\displaystyle{}~{}\longrightarrow~{}\mathrm{Frac}(B)$ $\displaystyle
X_{i}^{(j)}$ $\displaystyle{}~{}\longmapsto~{}\bar{X}_{i}^{(j)}$

with image $B^{(j)}$ and kernel $P_{j}$.

###### Proof.

Since $B^{(j)}$ is (trivially) a subalgebra of $\text{Frac}(B^{(j)})$
generated by $\bar{X}_{1}^{(j)},\ldots,\bar{X}_{N}^{(j)}$, Lemma 3.15 implies
that $B^{(j)}$ is a subalgebra of $\mathrm{Frac}(B)$ generated by these same
elements. The homomorphism $f_{j}$ is the concatenation of the natural
embedding $B^{(j)}\hookrightarrow\text{Frac}(B)$ with the canonical
surjection, $\pi_{j}:A^{(j)}\rightarrow A^{(j)}/P_{j}=B^{(j)}$. The stated
image and kernel are easily verified. ∎

######  Proposition 3.17.

Let $j\in\llbracket 2,N+1\rrbracket$.

  1. (i)

If $\bar{X}_{j}^{(j+1)}=0$ then $\bar{X}_{i}^{(j)}=\bar{X}_{i}^{(j+1)}$ for
all $i\in\llbracket 1,N\rrbracket$.

  2. (ii)

Suppose $\bar{X}_{j}^{(j+1)}\neq 0$ and set
$(x_{1},\ldots,x_{N}):=(X_{1}^{(j+1)},\ldots,X_{N}^{(j+1)})$. Then the
generators of $B^{(j)}$ are can be obtained as follows:

$\bar{X}_{i}^{(j)}=\begin{cases}\bar{X}_{i}^{(j+1)}&\text{ if }i\geq j;\\\
\sum_{n=0}^{\infty}q_{j}^{\frac{n(n+1)}{2}}(q_{j}-1)^{-n}\lambda_{j,i}^{-n}f_{j+1}\circ
d_{j,n}(x_{i})(\bar{X}_{j}^{(j+1)})^{-n}&\text{ if }i<j,\end{cases}$

where $f_{j+1}$ is the map defined in Proposition 3.16.

  3. (iii)

Suppose $\bar{X}_{j}^{(j+1)}\neq 0$ and let
$Z_{j}=\\{(\bar{X}_{j}^{(j+1)})^{n}\mid
n\in\mathbb{N}\\}=\\{(\bar{X}_{j}^{(j)})^{n}\mid n\in\mathbb{N}\\}$ be a
multiplicatively closed set of regular elements in $B^{(j)}$ and $B^{(j+1)}$.
Then $Z_{j}$ satisfies the two-sided Ore condition in both $B^{(j)}$ and
$B^{(j+1)}$ and we have $B^{(j)}Z_{j}^{-1}=B^{(j+1)}Z_{j}^{-1}$.

###### Proof.

Lemmas 3.13 and 3.15 prove part (i). Part (ii) follows from Lemma 3.14 once
one notes that $f_{j+1}\circ d_{j,n}(x_{i})=\overline{d_{j,n}(x_{i})}\cdot
1^{-1}\in\text{Frac}(B)$. Part (iii) also follows directly from Lemma 3.14. ∎

Let $w\in\mathcal{W}^{\prime}$ with $P\in\mathrm{C.Spec}_{w}(A)$ and
$Q=\psi(P)$. By the definition of $\mathrm{C.Spec}_{w}(A^{\prime})$, we have
that $T_{i}\in Q$ if and only if $i\in w$ or, equivalently, $t_{i}=0$ if and
only if $i\in w$. Let $i\in\bar{w}:=\\{1,\ldots,N\\}\backslash w$ so that
$t_{i}\neq 0$. Then, since $T_{i}$ is normal in $A^{\prime}$ and $Q$ is
completely prime, $t_{i}$ is normal and regular in $B^{\prime}$, hence it is
invertible in $\text{Frac}(B^{\prime})=\text{Frac}(B)$. We denote by $\Sigma$
the multiplicatively closed set of regular elements in $B^{\prime}$ generated
by all $t_{i}$ such that $i\in\bar{w}$. From this set we define, recursively,
the sets $\Sigma_{j}\subset B^{(j)}$ for $j\in\llbracket 2,N+1\rrbracket$ in
the following way:

$\Sigma_{2}:=\Sigma,\quad\Sigma_{j+1}:=B^{(j+1)}\cap\Sigma_{j}.$

The next result extends Proposition 2.7 to quotient algebras.

######  Proposition 3.18.

For each $j\in\llbracket 2,N+1\rrbracket$ the following statements hold:

  1. (i)

$\Sigma_{j}$ is a multiplicatively closed set of regular elements in $B^{(j)}$
which contains, as a subset, $\\{\bar{X}_{i}^{(j)}\mid i\in\llbracket
j-1,N\rrbracket\text{ and }\bar{X}_{i}^{(j)}\neq 0\\}$;

  2. (ii)

$\Sigma_{j}$ satisfies the two-sided Ore condition in $B^{(j)}$;

  3. (iii)

The algebras $B^{(j)}\Sigma_{j}^{-1}$, when considered as subalgebras of
$\mathrm{Frac}(B)$, are all equal.

###### Proof.

We proceed by induction on $j$. When $j=2$, statements (i) and (ii) are
immediately satisfied by the discussion preceding this proposition. Fix
$j\in\llbracket 2,N\rrbracket$ and assume statements (i) and (ii) are true. We
show that these properties are also true when replacing $j$ with $j+1$ and
that $B^{(j)}\Sigma_{j}^{-1}=B^{(j+1)}\Sigma_{j+1}^{-1}$. We consider two
cases: $\bar{X}_{j}^{(j+1)}=0$ and $\bar{X}_{j}^{(j+1)}\neq 0$.

When $\bar{X}_{j}^{(j+1)}=0$ we apply Proposition 3.17(i) to obtain
$\bar{X}_{i}^{(j)}=\bar{X}_{i}^{(j+1)}$ for all $i\in\llbracket
1,N\rrbracket$. Therefore $B^{(j)}=B^{(j+1)}$, and statements (i) and (ii)
follow immediately by the inductive hypothesis and the fact that
$\Sigma_{j+1}\subseteq\Sigma_{j}$.

Now suppose $\bar{X}_{j}^{(j+1)}\neq 0$. Applying Proposition 3.17(ii) gives
$\bar{X}_{i}^{(j)}=\bar{X}_{i}^{(j+1)}$ for all $i\geq j$ and, by the
induction hypothesis, we know that

$\\{\bar{X}_{i}^{(j)}\mid i\in\llbracket
j-1,N\rrbracket,\;\bar{X}_{i}^{(j)}\neq 0\\}~{}\subseteq~{}\Sigma_{j}.$

Therefore,

$\displaystyle\Sigma_{j+1}:=B^{(j+1)}\cap\Sigma_{j}$
$\displaystyle{}~{}\supseteq~{}B^{(j+1)}\cap\\{\bar{X}_{i}^{(j)}\mid
i\in\llbracket j-1,N\rrbracket,\;\bar{X}_{i}^{(j)}\neq 0\\}$
$\displaystyle{}~{}=~{}\\{\bar{X}_{i}^{(j+1)}\mid i\in\llbracket
j,N\rrbracket,\;\bar{X}_{i}^{(j+1)}\neq 0\\}.$

The set $\Sigma_{j}$ is multiplicatively closed, by the induction hypothesis,
which means $\Sigma_{j+1}$ is multiplicatively closed and it contains regular
elements because $P_{j+1}$ is a completely prime ideal. Hence $B^{(j+1)}$ is a
domain in which all nonzero elements are regular. This proves part (i).

Note that $Z_{j}\subset\Sigma_{j+1}$ and $Z_{j}\subset\Sigma_{j}$. Applying
Proposition 3.17(iii), we see that

$B^{(j+1)}\subset B^{(j+1)}Z_{j}^{-1}=B^{(j)}Z_{j}^{-1}\subset
B^{(j)}\Sigma_{j}^{-1}.$

These inclusions mirror those found in (11), and we may apply the rest of the
method used in the proof of Proposition 2.7 to conclude parts (ii) and (iii)
of this proposition. ∎

An immediate consequence of this proposition is that we can now show
equivalence of the total rings of fractions of $A/P$ and $A^{\prime}/\psi(P)$
in the case where $P\in\mathrm{C.Spec}_{w}(A)$, for some
$w\in\mathcal{W}^{\prime}$.

######  Theorem 3.19.

Let $w\in\mathcal{W}^{\prime},\>P\in\mathrm{C.Spec}_{w}(A)$, and
$Q=\psi(P)\in\mathrm{C.Spec}_{w}(A^{\prime})$. Let $\Sigma$ be the
multiplicatively closed set of elements in $A^{\prime}/Q$ which is generated
by all the generators $t_{i}$ such that
$i\in\bar{w}=\\{1,\ldots,N\\}\backslash w$, where $t_{i}$ is the canonical
image of $T_{i}$ in $A^{\prime}/Q$. Then,

  1. (i)

$\Sigma$ is a multiplicatively closed set of regular, normal elements in
$A^{\prime}/Q$.

  2. (ii)

There exists a multiplicatively closed set of regular elements, $\Gamma$, in
$A/P$ which satisfies the two-sided Ore condition in $A/P$ and is such that
$(A/P)\Gamma^{-1}=(A^{\prime}/Q)\Sigma^{-1}$.

  3. (iii)

$\mathrm{Frac}(A/P)=\mathrm{Frac}(A^{\prime}/Q)$. Furthermore, $A/P$ and
$A^{\prime}/Q$ are PI algebras if all $\lambda_{i,j}$ with $i,j\in\bar{w}$ are
roots of unity and, in this case, $\text{PI-deg}(A/P)=\text{PI-
deg}(A^{\prime}/Q)$.

The following ideals are defined for later use as they ensure that at the end
of the deleting derivations algorithm on $A/P$ we obtain a quantum affine
space $A^{\prime}/\psi(P)$.

######  Definition 3.20.

Let $w\in\mathcal{W}^{\prime}$. We call $P_{w}\in\mathrm{C.Spec}_{w}(A)$ a
_Cauchon ideal_ if $\psi(P_{w})=J_{w}:=\langle T_{i}\in A^{\prime}\mid i\in
w\rangle\in\mathrm{C.Spec}_{w}(A^{\prime})$.

######  Remark 3.21.

It can be shown that a rational torus action on $A$ induces a rational torus
action on $A^{\prime}$ and that $\psi$ sends torus-invariant completely prime
ideals in $A$ to torus-invariant completely prime ideals in $A^{\prime}$. This
matches analogous results in the generic setting [Cau03a] and the Poisson
setting [LL17]. Unlike in those settings, this observation does not
immediately yield results allowing us to determine explicitly what
$W^{\prime}$ is for specific algebras $A$. For this reason, we omit these
results from this paper.

##  4 PI degree of completely prime quotients of quantum matrices at roots of
unity

In this section, we consider iterated Ore extensions
$A=\mathbb{K}[X_{1}][X_{2};\sigma_{2},\delta_{2}]\ldots[X_{N};\sigma_{N},\delta_{N}]$
satisfying Hypothesis 1.2 at roots of unity. That is, we assume that all the
parameters $\lambda_{i,j}$ from Hypothesis 1.2 are roots of unity.

In that case, it follows from [Hay08, Theorem 1.2] that the algebra $A$ is PI
and that, with the notation of the previous section:

${\rm PI}\text{-}{\rm deg}(A)={\rm PI}\text{-}{\rm deg}(A^{\prime}).$

Our aim in this section is to provide techniques to compute the PI degree of
completely prime quotients of $A$. In particular, we will focus on the case
where $A$ is a single parameter quantum matrix algebra (at roots of unity).

###  4.1 PI degree of completely prime quotients of $A$

Under our assumptions, $A$ is a PI algebra and this implies that every
completely prime quotient of $A$ is a PI algebra too. Our next aim is to
compute the PI degree and construct irreducible representations of maximal
dimension of quotients by Cauchon ideals.

###  4.2 PI degree of quotients by a Cauchon ideal

In the case where $P_{w}$ is the Cauchon ideal associated to a Cauchon diagram
$w\in\mathcal{W}^{\prime}$, we get that $\psi(P_{w})=\langle T_{i}~{}|~{}i\in
w\rangle$ and so $A^{\prime}/Q$ is a quantum affine space. Namely,
$A^{\prime}/Q=\mathcal{O}_{{\bf\Lambda_{w}}}(\mathbb{K}^{N-|w|})$, where
${\bf\Lambda_{w}}$ is the matrix deduced from ${\bf\Lambda}=(\lambda_{ij})$ by
removing rows and columns indexed by elements of $w$.

In view of Theorem 3.19(iii) and the above discussion, we obtain the following
result.

######  Theorem 4.1.

Let $P_{w}$ be the Cauchon ideal associated to a Cauchon diagram
$w\in\mathcal{W}^{\prime}$. Then ${\rm PI}\text{-}{\rm deg}(A/P_{w})={\rm
PI}\text{-}{\rm deg}(\mathcal{O}_{{\bf\Lambda_{w}}}(\mathbb{K}^{N-|w|}))$.

De Concini and Procesi developed techniques to compute the PI degree of a
quantum affine space. More precisely, they prove the following result.

######  Theorem 4.2.

[CP93, Proposition 7.1] Let $S=(s_{ij})$ be a multiplicatively skew-symmetric
$n\times n$ matrix with coefficients in $\mathbb{K}$. Then:

  1. 1.

If all $s_{ij}$ are roots of unity, then there exists a primitive root of
unity $q\in\mathbb{K}^{\times}$ and integers $a_{ij}$ such that
$s_{ij}=q^{a_{ij}}$ for all $i,j$.

  2. 2.

Assume $s_{ij}=q^{a_{ij}}$ for all $i,j$, where $q$ is a primitive
$\ell^{\mbox{th}}$ root of unity and $a_{ij}\in\mathbb{Z}$ for all $i,j$. Set
$M:=(a_{ij})\in M_{n}(\mathbb{Z})$; this is a skew-symmetric matrix. Then the
PI degree of the corresponding quantum affine space
$\mathcal{O}_{q^{M}}(\mathbb{K}^{n}):=\mathcal{O}_{S}(\mathbb{K}^{n})$ is
$\sqrt{h}$, where $h$ is the cardinality of the image of the homomorphism

${\mathbb{Z}^{n}}$${\mathbb{Z}^{n}}$${(\mathbb{Z}/\ell\mathbb{Z})^{n},}$$\scriptstyle{M}$$\scriptstyle{\pi}$

where $\pi$ denotes the canonical epimorphism.

The above result together with Theorem 4.1 allows us to compute the PI degree
of quotients by Cauchon ideals.

In the following section, we illustrate this in the case when $A$ is a single
parameter quantum matrix algebra.

###  4.3 PI degree of quotients of single parameter quantum matrix algebras by
Cauchon ideals

In this section, we assume that $\lambda_{ij}=q^{m_{ij}}$ for all $i,j$, where
$q$ is a primitive $\ell^{\mbox{th}}$ root of unity and $m_{ij}\in\mathbb{Z}$
for all $i,j$. In this case, we say that $A$ is uniparameter. We set
$M:=(m_{ij})\in M_{N}(\mathbb{K})$, so that $M$ is a skew-symmetric integral
matrix. It is a well-known result that every skew-symmetric integral matrix
$S$ is congruent (in the sense of [New72, Chapter IV]) to its skew-normal
form, that is, to a block diagonal matrix of the form

$S=\left(\begin{smallmatrix}0&h_{1}&&&&&&&\\\ -h_{1}&0&&&&&&&\\\
&&0&h_{2}&&&&&\\\ &&-h_{2}&0&&&&&\\\ &&&&\ddots&&&\\\ &&&&&0&h_{s}&\\\
&&&&&-h_{s}&0&\\\ &&&&&&&\mathbf{0}\end{smallmatrix}\right),$

where $\mathbf{0}$ is a square matrix of zeros of dimension $\dim(\ker(S))$ so
that $2s=N-\dim(\ker(S))$, and
$h_{1},\,h_{1},\,h_{2},\,h_{2},\,\ldots,\,h_{s},\,h_{s}\in\mathbb{Z}\backslash\\{0\\}$
are called the _invariant factors_ of $M$. As they always come in pairs, from
now on we will avoid repetition and list the invariant factors simply as
$h_{1},\,h_{2},\,\ldots,\,h_{s}$. These have the property that $h_{i}|h_{i+1}$
for all $i\in\llbracket 1,s-1\rrbracket$.

######  Theorem 4.3.

Assume $A$ is a uniparameter iterated Ore extension satisfying Hypothesis 1.2
with parameter $q$ being a primitive $\ell^{\mbox{th}}$ root of unity. Let
$w\in\mathcal{W}^{\prime}$ and $P_{w}:=\psi^{-1}(J_{w})$ be the corresponding
Cauchon ideal. Then

${\rm PI}\text{-}{\rm
deg}(A/P_{w})=\prod_{i=1}^{\frac{N-\dim(\ker(M(w)))}{2}}\frac{\ell}{\gcd(h_{i},\ell)},$

where the $h_{i}$ are the invariant factors of the matrix $M(w)$ deduced from
$M$ by removing rows and columns indexed by $w$.

We finish this section by studying the case of single parameter quantum
matrices. That is, we assume that $A=\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$
with $q$ being a primitive $\ell^{\mbox{th}}$ root of unity. Recall that the
quantised coordinate ring of $m\times n$ matrices over $\mathbb{K}$ denoted by
$\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ is the $\mathbb{K}$-algebra generated
by $m\times n$ indeterminates $\\{X_{i,j}\\}_{i,j=1}^{m,n}$ subject to the
following relations: for $(1,1)\leq(i,j)<(s,t)\leq(m,n)$ (in lexicographic
ordering), we have

$\displaystyle X_{i,j}X_{s,t}=\begin{cases}X_{s,t}X_{i,j}&i<s,\,j>t;\\\
qX_{s,t}X_{i,j}&(i=s,\,j<t)\text{ or }(i<s,\,j=t);\\\
X_{s,t}X_{i,j}+(q-q^{-1})X_{i,t}X_{s,j}&i<s,\,j<t.\end{cases}$

Under our assumption that $q$ is a primitive $\ell^{\mbox{th}}$ root of unity,
the algebra $A=\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ is a uniparameter
iterated Ore extension (with the generators added in the lexicographic order)
satisfying Hypothesis 1.2, see [Hay08, Section 5.3] for details.

Let $w$ be a Cauchon diagram and $P_{w}$ the corresponding Cauchon ideal in
$A$. It follows from the above result that to compute the PI degree of
$A/P_{w}$, we need to compute the dimension of $\ker(M(w))$ and the invariant
factors of $M(w))$. This was done in [BCL12] and [BLR22] respectively. Before
stating our main result, we recall necessary results from these two articles.

Given any diagram $w\in\mathcal{W}$, we associate an $m\times n$ diagram
$D(w)$, where $D(w)$ is the $m\times n$ grid whose square in position $(i,j)$
is coloured black if $(i,j)\in w$ and white if $(i,j)\notin w$. We may compute
its _toric permutation_ $\tau=\tau_{w}$, as defined in [BCL12, Section 4.1],
by laying pipes over the squares such that we place a “cross” on each black
square and a “hyperbola” on each white square. We label the sides of the
diagram with the numbers $1,\ldots,m+n$ such that each pair of opposite sides
share the same labels in the same order. The permutation, $\tau_{w}$, may then
be read off this diagram by defining $\tau_{w}(i)$ to be the label (on the
left or top side of $D(w)$) reached by following the pipe starting at label
$i$ (on the right or bottom side of $D(w)$). See Figure 1 for an example of a
diagram with $\tau=(17)(26384)$.

$1$$2$$3$$4$$5$$6$$7$$8$$9$$~{}~{}~{}~{}\longleftrightarrow~{}~{}~{}$ | $1$$2$$3$$1$$2$$3$$4$$5$$6$$7$$8$$4$$5$$6$$7$$8$
---|---
Figure 1: Labelled $3\times 5$ diagram (left) with pipe dream construction
(right) associated to the diagram $\\{(1,2),(1,4),(2,2),(3,1),(3,2),(3,3)\\}$
in $\mathcal{O}_{q}(M_{3,5}(\mathbb{K}))$.

It was proved in [BCL12] that $\dim(\ker(M(w))$ is given by the number $r(w)$
of odd cycles in the disjoint cycle decomposition of its associated toric
permutation $\tau_{w}$. Moreover, [BLR22, Theorem 2.6] shows that all
invariant factors of $M(w)$ are powers of $2$.

Putting all these together, we obtain the following result.

######  Theorem 4.4.

Assume $q$ is a primitive $\ell^{\mbox{th}}$ root of unity with $\ell$ odd.
Let $w$ be a Cauchon diagram for $\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))$ and
$P_{w}$ be the corresponding Cauchon ideal. Then

${\rm PI}\text{-}{\rm
deg}(\mathcal{O}_{q}(M_{m,n}(\mathbb{K}))/P_{w})=\sqrt{\ell^{mn-r(w)}},$

where $r(w)$ is the number of odd cycles in the disjoint cycle decomposition
of the toric permutation associated to $w$.

Note that we did not need an explicit description of the set
$\mathcal{W}^{\prime}$ of Cauchon diagrams to obtain the above result. For
completeness, we note that $\mathcal{W}^{\prime}$ coincides with the set of
$m\times n$ Cauchon-Le diagrams (see [Rog19, Theorem 4.37] for details).

##  5 Irreducible Representations of Maximum Dimension

We take $1\neq q\in\mathbb{K}^{*}$ to be arbitrary, unless stated otherwise.
Let $A$ be an algebra satisfying Hypothesis 1.2 such that
$\lambda_{i,j}=q^{m_{i,j}}$ for some skew-symmetric matrix
$M=(m_{i,j})_{i,j}\in M_{N}(\mathbb{Z})$. Fix $P\in\mathrm{C.Spec}_{w}(A)$,
for some Cauchon diagram $w\in\mathcal{W}^{\prime}$. We make use of the
deleting derivations algorithm to construct an irreducible representation of
$A/P$ given a “suitable” irreducible representation of $A^{\prime}/\psi(P)$.
When $A$ is a PI algebra (which holds if $q$ is a root of unity) and
$\mathbb{K}$ is algebraically closed, it turns out that any irreducible
representation of $A^{\prime}/\psi(P)$ is “suitable”.

Recall the notation set in Section 3.3: $B:=A/P$ with generators
$\bar{X}_{i}:=X_{i}+P$ and $B^{\prime}:=A^{\prime}/\psi(P)$ with generators
$t_{i}:=\bar{T}_{i}$. Set, again, $\Sigma:=\Sigma_{2}\subseteq B^{\prime}$ to
be the (two-sided) Ore set generated by the $t_{i}$, for $i\in\llbracket
1,N\rrbracket\backslash w$ and then define, recursively for all
$j\in\llbracket 2,N\rrbracket$, $\Sigma_{j+1}:=B^{j+1}\cap\Sigma_{j}$, with
$\Gamma:=\Sigma_{N+1}$ being a (two-sided) Ore set in $B$.

######  Proposition 5.1.

Let $A$ be a $\mathbb{K}$-algebra satisfying Hypothesis 1.2,
$P\in\mathrm{C.Spec}_{w}(A)$ for some $w\in\mathcal{W}^{\prime}$ and suppose
that $(\phi^{\prime},V)$ is an irreducible representation of $B^{\prime}$
where, for each $e\in\Sigma$, there exists some $\xi=\xi_{e}\in\mathbb{K}^{*}$
and $\ell=\ell_{e}\in\mathbb{N}_{>1}$ such that
$\phi^{\prime}(e)^{\ell}=\xi\operatorname{Id}_{V}$. Then

  1. 1.

For any $b\in B$, there exists some $b^{\prime}\in B^{\prime}$ and
$e\in\Sigma$ such that

$b=b^{\prime}e^{-1}\in B^{\prime}\Sigma^{-1}$

and $\phi^{\prime}$ induces an irreducible representation of $B$ on $V$ (of
the same dimension), defined by the algebra homomorphism

$\displaystyle\phi:B$
$\displaystyle{}~{}\longrightarrow~{}\mathrm{End}_{\mathbb{K}}(V)$
$\displaystyle b$
$\displaystyle{}~{}\longmapsto~{}\xi^{-1}\phi^{\prime}(b^{\prime})\phi^{\prime}(e)^{\ell-1}.$

  2. 2.

For each $g\in\Gamma$, there exists some $\xi^{\prime}\in\mathbb{K}^{*}$ and
$\ell^{\prime}\in\mathbb{N}_{>1}$ such that
$\phi(g)^{\ell^{\prime}}=\xi^{\prime}\operatorname{Id}_{V}$.

###### Proof.

Let $(\phi^{\prime},V)$ be an irreducible representation of $B^{\prime}$
satisfying the conditions of the proposition. The condition on $\phi^{\prime}$
implies that $\phi^{\prime}(e)^{-1}=\xi_{e}^{-1}\phi^{\prime}(e)^{\ell_{e}-1}$
and induces a representation of $B^{\prime}\Sigma^{-1}$,

$\displaystyle\hat{\phi}:~{}B^{\prime}\Sigma^{-1}$
$\displaystyle{}~{}\longrightarrow~{}\mathrm{End}_{\mathbb{K}}(V)$
$\displaystyle b^{\prime}$
$\displaystyle{}~{}\longmapsto~{}\phi^{\prime}(b^{\prime})$ $\displaystyle
e^{-1}$
$\displaystyle{}~{}\longmapsto~{}\xi_{e}^{-1}\phi^{\prime}(e)^{\ell_{e}-1},$

from which we observe that
$\hat{\phi}(e)^{-1}=\xi_{e}^{-1}\hat{\phi}(e)^{\ell_{e}-1}$, for all
$e\in\Sigma$. The inclusion $B^{\prime}\subseteq B^{\prime}\Sigma^{-1}$
ensures that $(\hat{\phi},V)$ is irreducible.

From Proposition 3.18 and Theorem 3.19 we see that
$B^{\prime}\Sigma^{-1}=B\Gamma^{-1}$, hence each element in $B\Gamma^{-1}$ can
be written in terms of elements in $B^{\prime}\Sigma^{-1}$, and vice versa.
This allows us to view $\hat{\phi}$ as an algebra homomorphism
$\hat{\phi}:B\Gamma^{-1}\rightarrow\mathrm{End}_{\mathbb{K}}(V)$, hence
$(\hat{\phi},V)$ defines an irreducible representation of $B\Gamma^{-1}$.

Every element $b\in B$ may be written as an element $b\cdot 1^{-1}\in
B\Gamma^{-1}$, which we write simply as $b\in B\Gamma^{-1}$. By the equality
of localisations, $b$ may also be written as an element in
$B^{\prime}\Sigma^{-1}$, that is, $b=b^{\prime}e^{-1}\in
B^{\prime}\Sigma^{-1}$. Restricting $\hat{\phi}$ to $B$ gives the following
algebra homomorphism:

$\displaystyle\phi:B$
$\displaystyle{}\longrightarrow\mathrm{End}_{\mathbb{K}}(V)$ $\displaystyle b$
$\displaystyle{}\longmapsto\hat{\phi}\left(b^{\prime}e^{-1}\right)=\xi^{-1}\phi^{\prime}(b^{\prime})\phi^{\prime}(e)^{\ell-1}.$

This defines a (not necessarily irreducible) representation of $B$ on $V$. To
show that this representation is irreducible we note that
$\Gamma\subseteq\Sigma$. Therefore, for all $g\in\Gamma$, there exists some
$\xi^{\prime}\in\mathbb{K}^{*}$ and $\ell^{\prime}\in\mathbb{N}$ such that
$\hat{\phi}(g)^{\ell^{\prime}}=\xi^{\prime}\operatorname{Id}_{V}$, hence

$\phi(g)^{\ell^{\prime}}=\hat{\phi}(g)^{\ell^{\prime}}=\xi^{\prime}\operatorname{Id}_{V}.$

Using the identity $B\Gamma^{-1}=B^{\prime}\Sigma^{-1}$ we may write any
element $b^{\prime}\in B^{\prime}$ as $b^{\prime}=bg^{-1}\in B\Gamma^{-1}$.
Thus

$\phi^{\prime}(b^{\prime})=\hat{\phi}(b^{\prime})=\hat{\phi}\left(bg^{-1}\right)=\xi^{\prime-1}\hat{\phi}(b)\hat{\phi}(g)^{\ell^{\prime}-1}=\phi(\xi^{\prime-1}bg^{\ell^{\prime}-1}),$

which shows that $\phi^{\prime}(B^{\prime})\subseteq\phi(B)$. Therefore, since
$(\phi^{\prime},V)$ is irreducible, so too is $(\phi,V)$. ∎

######  Remark 5.2.

If we take $P=\\{0\\}$ in Proposition 5.1 then we get $B^{\prime}=A^{\prime}$
and $B=A$ and the statement can be stated analogously given the results and
notation of Proposition 2.7 and Theorem 2.8. This means we can construct an
irreducible representation on $A$ provided we have an irreducible
representation on $(\phi^{\prime},V)$ satisfying the condition
$\phi^{\prime}(e)^{\ell_{e}}=\xi_{e}\operatorname{Id}_{V}$ for all elements
$e$ of the Ore set in $A^{\prime}$ generated by $\\{T_{i}\\}_{i=1}^{N}$ and
for some $\ell\in\mathbb{N}_{>1}$.

Restricting the algebra $A$ to the root of unity case and taking $\mathbb{K}$
to be algebraically closed (needed for Schur’s lemma used in the proof), we
see that any irreducible representation on $B^{\prime}$ will satisfy the
conditions of Proposition 5.1.

######  Theorem 5.3.

Take $\mathbb{K}$ to be an algebraically closed field. Let $A$ be a
$\mathbb{K}$-algebra satisfying Hypothesis 1.2 and suppose that the
$\lambda_{i,j}$ in H1.2.1 are of the form $\lambda_{i,j}=q^{m_{i,j}}$ for some
skew-symmetric matrix $M=(m_{i,j})_{i,j}\in M_{N}(\mathbb{Z})$, where
$q\in\mathbb{K}^{*}$ is a primitive $\ell^{\text{th}}$ root of unity with
$\ell\in\mathbb{N}_{>1}$. Then:

  1. 1.

Any irreducible representation $(\phi^{\prime},V)$ of $B^{\prime}$ satisfies
the conditions of Proposition 5.1.

  2. 2.

Any irreducible representation $(\phi^{\prime},V)$ of $B^{\prime}$ induces an
irreducible representation $(\phi,V)$ of $B$ of the same dimension.

###### Proof.

  1. 1.

$A^{\prime}$ is a prime affine PI algebra (Theorem 2.8(iii)), as is
$B^{\prime}$, hence any irreducible representation $(\phi^{\prime},V)$ of
$B^{\prime}$ is finite dimensional. Suppose $P\in\mathrm{C.Spec}_{w}(A)$, for
some $w\in\mathcal{W^{\prime}}$. Then the quotient algebra,
$B^{\prime}=A^{\prime}/\psi(P)$, is generated by all $t_{i}$ such that
$i\in\llbracket 1,N\rrbracket\backslash w$, where $t_{i}:=T_{i}+\psi(P)$.
Hence $t_{i}t_{j}=q^{m_{i,j}}t_{j}t_{i}$, for all $i,j\in\llbracket
1,N\rrbracket\backslash w$. We deduce that $t_{i}^{\ell}\in Z(B^{\prime})$ for
all $i\in\llbracket 1,N\rrbracket\backslash w$, thus, by Schur’s Lemma,
$\phi^{\prime}(t_{i})^{\ell}=\xi_{i}\operatorname{Id}_{V}$ for some
$\xi_{i}\in\mathbb{K}^{*}$. Since $\Sigma$ is defined to be the
multiplicatively closed set generated by $\\{t_{i}\mid i\in\llbracket
1,N\rrbracket\backslash w\\}$, then the conditions of Proposition 5.1 are
satisfied.

  2. 2.

Follows from Proposition 5.1.

∎

######  Remark 5.4.

When $P$ is a Cauchon ideal ($P=P_{w}$ for some $w\in\mathcal{W}^{\prime}$)
then, by definition, $B^{\prime}$ becomes a quantum affine space. If $A$ then
satisfies the conditions of Theorem 5.3 (i.e. we take a quantum nilpotent
algebra at a root of unity, $q$) then

${\rm PI}\text{-}{\rm
deg}(B^{\prime})=\prod_{i=1}^{\frac{N-\dim(\ker(M(w)))}{2}}\frac{\ell}{\gcd(h_{i},\ell)},$

where the $h_{i}$ are the invariant factors of the matrix $M(w)$ deduced from
$M$ by removing rows and columns indexed by $w$ (see Theorem 4.3). We can
construct an irreducible representation of $B^{\prime}$ of maximal dimension
using [BLR22, Proposition 3.4]. This will, by Theorem 5.3, induce an
irreducible representation of $B$ via the method of Proposition 5.1. These
results therefore give a method to construct an explicit maximal dimensional
irreducible representation of $B$ for a given $w\in\mathcal{W}^{\prime}$.

Proposition 5.1 and Theorem 5.3 have been stated for the two extreme algebras
in the DDA. In practice, we would often deal with one step of the DDA at a
time, that is we would deduce an irreducible representation of
$A^{(j+1)}/P_{j+1}$ from an irreducible representation of $A^{(j)}/P_{j}$.

###  5.1 Example: A maximal irreducible representation of
$U_{q}^{+}(\mathfrak{so}_{5})/\langle z^{\prime}\rangle$

To illustrate Remark 5.4 we will construct an irreducible representation of a
quotient of $A:=U_{q}^{+}(\mathfrak{so}_{5})$ by the ideal $\langle
z^{\prime}\rangle$ (defined later in this section). This example will require
applications of the deleting derivations algorithm on both $A$ (Theorem 2.3)
and $A/\langle z^{\prime}\rangle$ (Theorem 3.19), computing the image of
$\langle z^{\prime}\rangle$ by the canonical embedding (Section 3.2), and the
calculation of the PI degree of $A/\langle z^{\prime}\rangle$ (Theorem 4.3).
The full details of these computations are omitted as they are straightforward
but lengthy, however they can be found in [Rog19, Sections 5.3.1 and 7.3.2].

For this example, we take $q$ to be a primitive $\ell^{\text{th}}$ root of
unity with $\ell\notin\\{2,4\\}$.

####  5.1.1 Defining the algebra

$U_{q}^{+}(\mathfrak{so}_{5})$ is the $\mathbb{C}$-algebra generated by two
indeterminates $E_{1},\,E_{2}$ subject to the the following relations:

$\displaystyle
E_{1}^{3}E_{2}-(q^{2}+1+q^{-2})E_{1}^{2}E_{2}E_{1}+(q^{2}+1+q^{-2})E_{1}E_{2}E_{1}^{2}-E_{2}E_{1}^{3}$
$\displaystyle{}=0$ $\displaystyle
E_{2}^{2}E_{1}-(q^{2}+q^{-2})E_{2}E_{1}E_{2}+E_{1}E_{2}^{2}$
$\displaystyle{}=0.$

There is a PBW basis of $U_{q}^{+}(\mathfrak{so}_{5})$ formed by monomials
$E_{1}^{k_{1}}E_{4}^{k_{4}}E_{3}^{k_{3}}E_{2}^{k_{2}}$, where
$k_{1},k_{2},k_{3},k_{4}$ are nonnegative integers and $E_{3},\,E_{4}$ are
certain root vectors. This result can be found, for example, in [Lau06,
Section 2.4]. The same paper also expresses this algebra as an iterated Ore
extension over $\mathbb{C}$ generated by these four indeterminates in the
order $E_{1},\,E_{4},\,E_{3},\,E_{2}$. For easier application of the deleting
derivations algorithm, we relabel the indeterminates:

$X_{1}:=E_{1},\quad X_{2}:=E_{4},\quad X_{3}:=E_{3},\quad X_{4}:=E_{2}.$

The relations between $X_{1},\,X_{2},\,X_{3},\,X_{4}$ in [Lau06] then become

$\displaystyle X_{2}X_{1}$ $\displaystyle=q^{-2}X_{1}X_{2},$ $\displaystyle
X_{3}X_{1}$ $\displaystyle{}=X_{1}X_{3}-(q+q^{-1})X_{2},\quad$ $\displaystyle
X_{3}X_{2}=q^{-2}X_{2}X_{3},$ $\displaystyle X_{4}X_{1}$
$\displaystyle{}=q^{2}X_{1}X_{4}-q^{2}X_{3},\quad$ $\displaystyle
X_{4}X_{2}=X_{2}X_{4}-\frac{q^{2}-1}{q+q^{-1}}X_{3}^{2},\quad~{}~{}X_{4}X_{3}=q^{-2}X_{3}X_{4}.$

This allows us to present $A:=U_{q}^{+}(\mathfrak{so}_{5})$ as the following
iterated Ore extension:

$U_{q}^{+}(\mathfrak{so}_{5})=\mathbb{C}[X_{1}][X_{2};\sigma_{2}][X_{3};\sigma_{3},\delta_{3}][X_{4};\sigma_{4},\delta_{4}]$

where, using the notation $A_{j}:=\mathbb{C}\langle
X_{1},\ldots,X_{j}\rangle\subseteq A$, the automorphisms $\sigma_{i}$ and
skew-derivations $\delta_{i}$ are defined on the generators as:

$\displaystyle\sigma_{2}:A_{1}\longrightarrow A_{1};\quad$ $\displaystyle
X_{1}\longmapsto q^{-2}X_{1},\qquad$
$\displaystyle\sigma_{3}:A_{2}\longrightarrow A_{2};\quad$ $\displaystyle
X_{1}\longmapsto X_{1}\qquad$ $\displaystyle\delta_{3}:A_{2}\longrightarrow
A_{2};\quad$ $\displaystyle X_{1}\longmapsto-(q+q^{-1})X_{2}$ $\displaystyle
X_{2}\longmapsto q^{-2}X_{2},\qquad$ $\displaystyle X_{2}\longmapsto 0,$
$\displaystyle\sigma_{4}:A_{3}\longrightarrow A_{3};\quad$ $\displaystyle
X_{1}\longmapsto q^{2}X_{1}\qquad$
$\displaystyle\delta_{4}:A_{3}\longrightarrow A_{3};\quad$ $\displaystyle
X_{1}\longmapsto-q^{2}X_{3}$ $\displaystyle X_{2}\longmapsto X_{2}\qquad$
$\displaystyle X_{2}\longmapsto-\frac{q^{2}-1}{q+q^{-1}}X_{3}^{2}$
$\displaystyle X_{3}\longmapsto q^{-2}X_{3},\qquad$ $\displaystyle
X_{3}\longmapsto 0.$ (13)

####  5.1.2 Verifying that $U_{q}^{+}(\mathfrak{s0}_{5})$ satisfies Hypothesis
1.2

Routine computations on these maps using [Hay08, Theorem 2.8 and Lemma 5.3]
show that $A$ satisfies all properties of Hypothesis 1.2. In particular:

  1. (H1.2.1)

The $\lambda_{i,j}$’s are as in (13).

  2. (H1.2.2)

$\lambda_{i,j}=q^{m_{i,j}}$, with the $m_{i,j}$ defining the skew-symmetric
matrix

$M:=\left(\begin{smallmatrix}0&2&0&-2\\\ -2&0&2&0\\\ 0&-2&0&2\\\
2&0&-2&0\end{smallmatrix}\right)\in M_{4}(\mathbb{Z}).$

  3. (H1.2.3)

$(\sigma_{3},\delta_{3})$ is $q^{2}$-skew and $(\sigma_{4},\delta_{4})$ is
$q^{4}$-skew. So we set $q_{3}=q^{2}$ and $q_{4}=q^{4}$.

  4. (H1.2.4)

The higher $q_{j}$-skew $\sigma_{j}$-derivations are defined to be:

$\displaystyle d_{j,n}(X_{l})$ $\displaystyle{}=\begin{cases}X_{l}&n=0;\\\
\delta_{j}(X_{l})&n=1;\\\ 0&n>1,\end{cases}$ (14)

for $j\in\\{3,4\\}$ and $l\in\llbracket 1,j-1\rrbracket$.

Since there are two nonzero derivations we apply Theorem 2.3 twice to obtain
$A^{\prime}=\mathbb{C}_{q^{M}}[T_{1},T_{2},T_{3},T_{4}]$, where

$\displaystyle T_{4}$ $\displaystyle{}:=X_{4},$ $\displaystyle T_{3}$
$\displaystyle{}:=X_{3},$ $\displaystyle T_{2}$
$\displaystyle{}:=X_{2}-\frac{q^{4}}{(q^{2}+1)(q+q^{-1})}X_{3}^{2}X_{4}^{-1},$
$\displaystyle T_{1}$
$\displaystyle{}:=X_{1}-\frac{q^{2}(q+q^{-1})}{q^{2}-1}X_{2}X_{3}^{-1}.$ (15)

####  5.1.3 Finding the PI degree of $U_{q}^{+}(\mathfrak{s0}_{5})/\langle
z^{\prime}\rangle$

Consider the ideal $\langle z^{\prime}\rangle\lhd A$, generated by the central
element (see [Lau06, Section 2.4])

$z^{\prime}:=-(q^{2}-q^{-2})(q+q^{-1})X_{2}X_{4}+q^{2}(q^{2}-1)X_{3}^{2}.$

Using localisation theory, Proposition 2.7(iii), and repeated application of
Proposition 3.4 and Lemma 3.1, one can show that $\psi^{-1}(\langle
T_{2}\rangle)=\langle z^{\prime}\rangle$. In particular this proves that
$\langle z^{\prime}\rangle\in\mathrm{C.Spec}_{\\{2\\}}(A)$ and therefore
$\\{2\\}\in\mathcal{W}^{\prime}$. In fact, since $A^{\prime}/\langle
T_{2}\rangle=\mathbb{C}_{q^{M^{\prime}}}[t_{1},t_{3},t_{4}]$, where
$t_{i}:=T_{i}+\langle T_{2}\rangle$ for all $i\in\\{1,3,4\\}$, then $\langle
z^{\prime}\rangle$ is a Cauchon ideal.

By Theorem 3.19,

${\rm PI}\text{-}{\rm deg}(U_{q}^{+}(\mathfrak{s0}_{5})/\langle
z^{\prime}\rangle)={\rm PI}\text{-}{\rm deg}(A^{\prime}/\langle
T_{2}\rangle)={\rm PI}\text{-}{\rm
deg}(\mathbb{C}_{q^{M^{\prime}}}[t_{1},t_{3},t_{4}])$

where $M^{\prime}=\left(\begin{smallmatrix}0&0&-2\\\ 0&0&2\\\
2&-2&0\end{smallmatrix}\right)$ is obtained from $M$ by deleting the second
row and second column. It is easily verified that the skew normal form of
$M^{\prime}$ is $S=\left(\begin{smallmatrix}0&2&0\\\ -2&0&0\\\
0&0&0\end{smallmatrix}\right)$, hence $M^{\prime}$ has a kernel of dimension
$1$ and one pair of invariant factors: $h_{1}=2$. Applying Theorem 4.3 with
$P_{\\{2\\}}=\langle z^{\prime}\rangle$ gives:

$\text{PI-deg}(A/\langle
z^{\prime}\rangle)=\prod_{i=1}^{\frac{3-1}{2}}\frac{\ell}{\gcd(h_{i},\ell)}=\frac{\ell}{\gcd(2,\ell)}=\begin{cases}\ell&\ell\text{
is odd};\\\ \ell/2&\ell>4\text{ is even}.\end{cases}$

####  5.1.4 Constructing an irreducible representation of
$U_{q}^{+}(\mathfrak{so}_{5})/\langle z^{\prime}\rangle$ of maximum dimension

Recall that
$B^{\prime}:=\mathbb{C}_{q^{M^{\prime}}}[t_{1},t_{3},t_{4}],~{}B:=U_{q}^{+}(\mathfrak{so}_{5})/\langle
z^{\prime}\rangle$ and let $\Sigma\subseteq B^{\prime}$ be the
multiplicatively closed set generated by $\\{t_{1},t_{3},t_{4}\\}$. The
quantum affine space associated to $S$ is
$D:=\mathbb{C}_{q^{S}}[x_{1},y_{1},z_{1}]$. By [BLR22, Proposition 3.4(ii)]
there is an $\ell$-dimensional $\mathbb{C}$-vector space $V$,
$\lambda,\xi\in\mathbb{C}^{*}$ and an algebra homomorphism
$\varphi:D\rightarrow\mathrm{End}_{\mathbb{C}}(V)$ whose image on the
generators of $D$, upon fixing a basis $v_{1},\ldots,v_{\ell}\in V$, may be
presented as matrices in the following way:

$\varphi(x_{1})=\left(\begin{smallmatrix}\lambda&&&&\\\ &\lambda q^{2}&&&\\\
&&\lambda q^{4}&&\\\ &&&\ddots&\\\ &&&&\lambda
q^{(\ell-1)2}\end{smallmatrix}\right),\qquad\varphi(y_{1})=\left(\begin{smallmatrix}0&0&\ldots&0&1\\\
1&0&\ldots&0&0\\\ 0&1&\ldots&0&0\\\ \vdots&&\ddots&\vdots&\vdots\\\
0&\ldots&\ldots&1&0\end{smallmatrix}\right),\qquad\varphi(z_{1})=\xi\operatorname{Id}_{V}.$

The pair $(\varphi,V)$ defines an irreducible representation of $D$.

To define an irreducible representation of
$\mathbb{C}_{q^{M^{\prime}}}[t_{1},t_{3},t_{4}]$ we apply [BLR22, Proposition
3.4(iii)] using $E^{-1}=\left(\begin{smallmatrix}1&0&1\\\ -1&0&0\\\
0&-1&0\end{smallmatrix}\right)$, where $E\in M_{3}(\mathbb{Z})$ is the
invertible matrix satisfying $EM^{\prime}E^{T}=S$. The resulting algebra
homomorphism
$\phi^{\prime}:\mathbb{C}_{q^{M^{\prime}}}[t_{1},t_{3},t_{4}]\rightarrow\mathrm{End}_{\mathbb{C}}(V)$,
is defined on $t_{1},\,t_{3},\,t_{4}$ as

$\displaystyle\phi^{\prime}(t_{1})$
$\displaystyle{}=\varphi(x_{1}^{1}y_{1}^{0}z_{1}^{1})=\varphi(x_{1})\varphi(z_{1})=\xi\varphi(x_{1}),$
$\displaystyle\phi^{\prime}(t_{3})$
$\displaystyle{}=\varphi(x_{1}^{-1}y_{1}^{0}z_{1}^{0})=\varphi(x_{1})^{-1}=\lambda^{-\ell}\varphi(x_{1}^{\ell-1}),$
$\displaystyle\phi^{\prime}(t_{4})$
$\displaystyle{}=\varphi(x_{1}^{0}y_{1}^{-1}z_{1}^{0})=\varphi(y_{1})^{-1}=\varphi(y_{1}^{\ell-1}),$
(16)

and it defines an irreducible representation of
$\mathbb{C}_{q^{M^{\prime}}}[t_{1},t_{3},t_{4}]$ on $V$. Using the definition
of $\varphi$, we see that

$\displaystyle\phi^{\prime}(t_{1})^{\ell}$
$\displaystyle{}=\xi^{\ell}\varphi(x_{1})^{\ell}=(\xi\lambda)^{\ell}\operatorname{Id}_{V},$
$\displaystyle\phi^{\prime}(t_{3})^{\ell}$
$\displaystyle{}=\lambda^{-\ell^{2}}\varphi(x_{1})^{\ell^{2}-\ell}=\lambda^{-\ell^{2}}\lambda^{\ell^{2}-\ell}\operatorname{Id}_{V}=\lambda^{-\ell}\operatorname{Id}_{V},$
$\displaystyle\phi^{\prime}(t_{4})^{\ell}$
$\displaystyle{}=\varphi(y_{1})^{\ell^{2}-\ell}=\operatorname{Id}_{V},$

hence the conditions of Proposition 5.1 are satisfied. Applying this
proposition allows us to define an irreducible representation of $B$ once we
know how to write the generators $\bar{X}_{1},\ldots,\bar{X}_{4}\in B$ in
terms of the generators $t_{1},t_{3},t_{4}\in B^{\prime}$. Using (15) we see
that

$\displaystyle t_{4}:=\bar{X}_{4},\quad t_{3}=\bar{X}_{3},\quad
t_{2}=\bar{X}_{2}-\frac{q^{4}}{(q^{2}+1)(q+q^{-1})}\bar{X}_{3}^{2}\bar{X}_{4}^{-1},\quad
t_{1}=\bar{X}_{1}-\frac{q^{2}(q+q^{-1})}{q^{2}-1}\bar{X}_{2}\bar{X}_{3}^{-1}.$

Rearranging these identities, and noting that $t_{2}=0$, allows us to write

$\bar{X}_{4}=t_{4},\quad\bar{X}_{3}=t_{3},\quad\bar{X}_{2}=\frac{q^{4}}{(q^{2}+1)(q+q^{-1})}t_{3}^{2}t_{4}^{-1},\quad\bar{X}_{1}=t_{1}+\frac{q^{4}}{q^{4}-1}t_{3}t_{4}^{-1}.$

Applying Proposition 5.1 and using (5.1.4), we deduce that there is an algebra
homomorphism $\phi:B\rightarrow\mathrm{End}_{\mathbb{C}}(V)$ defined on
generators as

$\displaystyle\phi(\bar{X}_{4})$
$\displaystyle{}=\phi^{\prime}(t_{4})=\varphi(y_{1})^{-1},$
$\displaystyle\phi(\bar{X}_{3})$
$\displaystyle{}=\phi^{\prime}(t_{3})=\varphi(x_{1})^{-1},$
$\displaystyle\phi(\bar{X}_{2})$
$\displaystyle{}=\phi^{\prime}\left(\frac{q^{4}}{(q^{2}+1)(q+q^{-1})}t_{3}^{2}t_{4}^{-1}\right)$
$\displaystyle\phi(\bar{X}_{1})$
$\displaystyle{}=\phi^{\prime}\left(t_{1}+\frac{q^{4}}{q^{4}-1}t_{3}t_{4}^{-1}\right)$
$\displaystyle{}=\frac{q^{4}}{(q^{2}+1)(q+q^{-1})}\varphi(x_{1})^{-2}\varphi(y_{1}),$
$\displaystyle{}=\xi\varphi(x_{1})+\frac{q^{4}}{q^{4}-1}\varphi(x_{1})^{-1}\varphi(y_{1}).$

which defines an irreducible representation $(\phi,V)$ of $B$. Substituting in
the matrices for $\varphi(x_{1})$ and $\varphi(y_{1})$ and taking, for
example, $\ell=5$ gives the following explicit form of this irreducible
representation:

$\displaystyle\phi(\bar{X}_{4})$
$\displaystyle{}=\left(\begin{matrix}0&1&0&0&0\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr 0&0&1&0&0\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&0&0&1&0\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr 0&0&0&0&1\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr 1&0&0&0&0\end{matrix}\right),\quad$
$\displaystyle\phi(\bar{X}_{3})$
$\displaystyle{}=\left(\begin{matrix}{\lambda}^{-1}&0&0&0&0\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr 0&\lambda^{-1}q^{3}&0&0&0\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr 0&0&\lambda^{-1}q&0&0\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&0&0&\lambda^{-1}q^{4}&0\\\ \vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&0&0&0&\lambda^{-1}q^{2}\end{matrix}\right),$
$\displaystyle\phi(\bar{X}_{2})$ $\displaystyle{}=\left(\begin{matrix}\vskip
6.0pt plus 2.0pt minus 2.0pt\cr 0&0&0&0&\frac{1}{(q^{2}+1)^{2}\lambda^{2}}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\frac{q}{(q^{2}+1)^{2}\lambda^{2}}&0&0&0&0\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr 0&\frac{q^{2}}{(q^{2}+1)^{2}\lambda^{2}}&0&0&0\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr 0&0&\frac{q^{3}}{(q^{2}+1)^{2}\lambda^{2}}&0&0\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&0&0&\frac{q^{4}}{(q^{2}+1)^{2}\lambda^{2}}&0\end{matrix}\right),$
$\displaystyle\phi(\bar{X}_{1})$
$\displaystyle{}=\left(\begin{matrix}\xi\lambda&0&0&0&\frac{q^{4}}{(q^{4}-1)\lambda}\\\
\vskip 6.0pt plus 2.0pt minus
2.0pt\cr\frac{q^{2}}{(q^{4}-1)\lambda}&\xi\lambda q^{2}&0&0&0\\\ \vskip 6.0pt
plus 2.0pt minus 2.0pt\cr 0&\frac{1}{(q^{4}-1)\lambda}&\xi\lambda q^{4}&0&0\\\
\vskip 6.0pt plus 2.0pt minus 2.0pt\cr
0&0&\frac{q^{3}}{(q^{4}-1)\lambda}&\xi\lambda q&0\\\ \vskip 6.0pt plus 2.0pt
minus 2.0pt\cr 0&0&0&\frac{q}{(q^{4}-1)\lambda}&\xi\lambda
q^{3}\end{matrix}\right).$

## References

  * [BCL12] J. P. Bell, K. Casteels, and S. Launois.  Enumeration of $\mathcal{H}$-strata in quantum matrices with respect to dimension.  Journal of Combinatorial Theory. Series A, 119(1):83–98, 2012.
  * [BLR22] J. Bell, S. Launois, and A. Rogers.  PI degree and irreducible representations of quantum determinantal rings and their associated quantum schubert varieties.  arXiv.2212.03799v2
  * [BG02] K. A. Brown and K. R. Goodearl.  Lectures on algebraic quantum groups.  Advanced Courses in Mathematics. CRM Barcelona. Birkhäuser Verlag, Basel, 2002.
  * [BY17] K. A. Brown and M. T. Yakimov.  Azumaya loci and discriminant ideals of PI algebras.  Advances in Mathematics, 340: 1219–1255, 2018.
  * [Cas14] K. Casteels.  Quantum matrices by paths.  Algebra & Number Theory, 8(8):1857–1912, 2014.
  * [Cau03a] G. Cauchon.  Effacement des dérivations et spectres premiers des algèbres quantiques.  Journal of Algebra, 260(2):476–518, 2003.
  * [Cau03b] G. Cauchon.  Spectre premier de $O_{q}(M_{n}(k))$: image canonique et séparation normale.  Journal of Algebra, 260(2):519–569, 2003.
  * [CPWZ15] S. Ceken, J. H. Palmieri, Y.-H. Wang, and J. J. Zhang.  The discriminant controls automorphism groups of noncommutative algebras.  Advances in Mathematics, 269:551–584, 2015.
  * [CPWZ16] S. Ceken, J. H. Palmieri, Y.-H. Wang, and J. J. Zhang.  The discriminant criterion and automorphism groups of quantized algebras.  Advances in Mathematics, 286:754–801, 2016.
  * [CP93] C. De Concini and C. Procesi.  Quantum groups.  In $D$-modules, representation theory, and quantum groups (Venice, 1992), volume 1565 of Lecture Notes in Math., pages 31–140. Springer, Berlin, 1993.
  * [GY14] J. Geiger and M. Yakimov.  Quantum Schubert cells via representation theory and ring theory.  Michigan Mathematical Journal, 63: 125–157, 2014.
  * [GLL19] K. R. Goodearl, S. Launois, and T. H. Lenagan.  Tauvel’s height formula for quantum nilpotent algebras.  Communications in Algebra, 47(10): 4194–4209, 2019.
  * [GL98] K. R. Goodearl and E. S. Letzter.  Prime and primitive spectra of multiparameter quantum affine spaces.  In Trends in ring theory (Miskolc, 1996), volume 22 of CMS Conf. Proc., pages 39–58. American Mathematical Society, Providence, RI, 1998.
  * [GW04] K. R. Goodearl and R. B. Warfield, Jr.  An introduction to noncommutative Noetherian rings, volume 61 of London Mathematical Society Student Texts.  Cambridge University Press, Cambridge, second edition, 2004.
  * [GY16] K. R. Goodearl and M. T. Yakimov.  From quantum Ore extensions to quantum tori via noncommutative UFDs.  Advances in Mathematics, 300: 672–716, 2016.
  * [Hay08] H. Haynal.  PI degree parity in $q$-skew polynomial rings.  Journal of Algebra, 319(10):4199–4221, 2008.
  * [KLR04] A. C. Kelly, T. H. Lenagan, and L. Rigal.  Ring theoretic properties of quantum Grassmannians.  Journal of Algebra and its Applications, 3(1):9–30, 2004.
  * [Lau06] S. Launois.  On the automorphism groups of $q$-enveloping algebras of nilpotent Lie algebras.  In Proceedings From Lie Algebras to Quantum Groups, 2006.
  * [LL17] S. Launois, C. Lecoutre.  Poisson deleting derivations algorithm and Poisson spectrum  Communications in Algebra, 45(3):1294–1313, 2017.
  * [LLN19] S. Launois, T. H. Lenagan, and B. Nolan.  Total positivity is a quantum phenomenon: the grassmannian case.  To appear in Memoirs of the American Mathematical Society, 2019.
  * [LLR06] S. Launois, T. H. Lenagan, and L. Rigal.  Quantum unique factorisation domains.  Journal of the London Mathematical Society. Second Series, 74(2):321–340, 2006.
  * [LR06] T. H. Lenagan and L. Rigal.  Quantum graded algebras with a straightening law and the AS-Cohen-Macaulay property for quantum determinantal rings and quantum Grassmannians.  Journal of Algebra, 301(2):670–702, 2006.
  * [LR08] T. H. Lenagan and L. Rigal.  Quantum analogues of Schubert varieties in the Grassmannian.  Glasgow Mathematical Journal, 50(1):55–70, 2008.
  * [LM11] A. Leroy and J. Matczuk.  On $q$-skew iterated Ore extensions satisfying a polynomial identity.  Journal of Algebra and its Applications, 10(4):771–781, 2011.
  * [MR01] J. C. McConnell and J. C. Robson.  Noncommutative Noetherian rings, volume 30 of Graduate Studies in Mathematics.  American Mathematical Society, Providence, RI, revised edition, 2001.  With the cooperation of L. W. Small.
  * [New72] M. Newman.  Integral matrices.  Academic Press, New York-London, 1972.  Pure and Applied Mathematics, Vol. 45.
  * [Pos06] A. Postnikov.  Total positivity, Grassmannians, and networks.  arXiv Mathematics e-prints, page math/0609764, Sep 2006.
  * [Rog19] A. Rogers.  Irreducible Representations of Quantum Nilpotent Algebras at Roots of Unity, and Their Completely Prime Quotients.  PhD thesis, University of Kent, 2019.