PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS
HEIDI HAYNAL
Abstract. For k a field of arbitrary characteristic, and R a k-algebra, we show that
the PI degree of an iterated skew polynomial ring R[x1; τ1, δ1] · · · [xn; τn, δn] agrees
with the PI degree of R[x1; τ1] · · · [xn; τn] when each (τi, δi) satisfies a qi-skew relation
for qi ∈ k× and extends to a higher qi-skew τi-derivation. We confirm the quantum
Gel’fand-Kirillov conjecture for various quantized coordinate rings, and calculate their
PI degrees. We extend these results to completely prime factor algebras.
1. Introduction
Presented here is a new technique for analyzing skew polynomial rings satisfying a poly-
nomial identity with an eye toward discovering their PI degrees. It combines and extends
the methods of Jøndrup [21] and Cauchon [5], who introduced techniques of “deleting
derivations” in skew polynomial rings, by means of which they showed that some proper-
ties of certain types of iterated skew polynomial ring A = k[x1][x2; τ2, δ2] · · · [xn; τn, δn]
are determined by the corresponding ring A′ = k[x1][x2; τ2] · · · [xn; τn]. Jøndrup’s re-
sults imply that A and A′ have the same PI degree under certain hypotheses, including
characteristic zero for the base field. Cauchon developed an algorithm that gives an
isomorphism between certain localizations of A and A′, but this requires a qi-skew
condition on each (τi, δi) with qi not a root of unity, which usually precludes A from
satisfying a polynomial identity. We relax the restrictions placed on the base field and
its chosen scalars by Jøndrup and Cauchon, respectively, by introducing the notion of
a higher q-skew τ -derivation.
If we “twist” the multiplication in the (commutative) coordinate ring of affine, symplec-
tic, or Euclidean n-space over a field k, we get a (noncommutative) quantized coordinate
ring which has the structure of an iterated skew polynomial ring with coefficients in k.
This structure is also exhibited in the quantized Weyl algebras and in the quantized
coordinate ring of n×n matrices over k. Letting A represent one of these k-algebras, the
quantum Gel’fand-Kirillov conjecture asserts that FractA is isomorphic to the quotient
division ring of a quantum affine space over a purely transcendental extension of k. For
1991 Mathematics Subject Classification. 16R99; 16S36; 81R50; 16P40.
Key words and phrases. noncommutative rings; skew polynomial rings; quantum algebras.
This research will form a part of the author’s PhD dissertation at the University of California at
Santa Barbara.
http://arxiv.org/abs/0704.0846v1
2 HEIDI HAYNAL
more information on the quantum Gel’fand-Kirillov conjecture and proofs of conditions
under which the result holds, see [1] [7] [23] [28] [32] [33]. We will confirm some of these
cases in a new way.
The first section sets up the conventions under which we work, including definitions
and an established result concerning the PI degree of quantum affine space. We assume
that the reader has some familiarity with the subject, so we do not give an exhaustive
collection of definitions. A comprehensive discussion of any unfamiliar terms can be
found in [16] [4] and [27]. In the second section we define higher τ -derivations and
give necessary and sufficient conditions for their existence. Of particular interest are
higher τ -derivations which satisfy a q-skew relation. In the third section we present
a structure theorem for a localization of q-skew polynomial rings. This extends the
work of Cauchon [5], and the calculations are simplified by the presence of higher q-
skew τ -derivations. In the fourth section we deal with the structure of iterated skew
polynomial rings. Sometimes it is advantageous to rearrange the order in which the
indeterminates appear, so we establish a sufficient condition that allows such reordering.
The main theorem there asserts that if A is an iterated q-skew polynomial ring with
certain higher τ -derivations, then there is a finitely generated Ore set T ⊆ A such that
AT−1 is isomorphic to a localization of a much “nicer” iterated skew polynomial ring. In
the fifth section, we use the tools developed in the previous sections to confirm certain
cases of the quantum Gel’fand-Kirillov conjecture and to find the PI degree of some
quantized coordinate rings and quantized Weyl algebras. In the last section, we follow
up with a structure theorem for completely prime factors of iterated skew polynomial
rings. We also present an open question which, if answered positively, would show that
the quantum Gel’fand-Kirillov conjecture holds for certain of the prime factor algebras
we study.
Throughout, k will denote a field of arbitrary characteristic, q ∈ k a nonzero ele-
ment. The following assumptions apply to all skew polynomial rings that we will con-
sider:
• all coefficient rings are k-algebras
• all automorphisms are k-algebra automorphisms
• all skew derivations are k-linear
• in all skew polynomial rings R[x; τ, δ], τ is an automorphism, not just an endo-
morphism.
To say that R[x; τ, δ] is a q-skew polynomial ring means that the auomorphism and skew
derivation satisfy the relation δτ = qτδ. The reader will note that this is opposite to
Cauchon’s conventions, but it matches the presentation in [10] and others. To say that
δ is locally nilpotent means that for every r ∈ R there is an integer nr ≥ 0 such that
δnr(r) = 0, and δp(r) 6= 0 for p < nr. Such nr is called the δ-nilpotence index of r. The
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 3
symbol N refers to the set of positive integers. For a real number m we use the notation
⌊m⌋ in section five to indicate the integer part of m.
Definition 1.1. We say that two rings R and S exhibit PI degree parity when these
two conditions are satisfied:
(1) R is a PI ring if and only if S is a PI ring,
(2) PIdegR = PIdegS.
For a field k and multiplicatively antisymmetric λ ∈ Mn(k), the corresponding mul-
tiparameter quantum affine space is the k-algebra Oλ(kn) with generators x1, . . . , xn
and relations xixj = λijxjxi for all i, j. The corresponding multiparameter quantum
torus is the k-algebra Oλ((k×)n) given by generators x±11 , . . . , x±1n and the same rela-
tions. The multiplicative set generated by x1, . . . , xn in Oλ(kn) is a denominator set,
and Oλ((k×)n) is a localization of Oλ(kn) with respect to this set.
In this paper we’ll show that iterated skew polynomial algebras covering a large class
of standard examples have PI degree parity with Oλ(kn) for an appropriately chosen
λ. To find out what that PI degree may be, we utilize a result of De Concini and
Procesi. In [8, Proposition 7.1], they establish the following formula for calculating the
PI degree of a quantum affine space Oλ(kn). Their assumption of characteristic zero
from [8, Section 4] is not used in this result.
Theorem 1.2. [De Concini - Procesi] Let λ = (λij) be a multiplicatively antisymmetric
n× n matrix over k.
(1) The quantum affine space Oλ(kn) is a PI ring if and only if all the λij are roots of
unity. In this case, there exist a primitive root of unity q ∈ k× and integers aij such
that λij = q
aij for all i, j.
(2) Suppose λij = q
aij for all i, j, where q ∈ k is a primitive ℓth root of unity and the
aij ∈ Z. Let h be the cardinality of the image of the homomorphism
n (aij)−−−−−−→ Zn π−−−−→ (Z/ℓZ)n
where π denotes the canonical epimorphism. Then PI-deg (Oλ(kn)) =
2. Higher q-Skew τ-Derivations
Before the featured definition, a brief discussion of a tool used to study q-skew polyno-
mial rings is needed. Having the q-skew relation δτ = qτδ in place allows us to group
terms of the same degree when we do skew polynomial arithmetic. The means to do
this are provided by the q-Liebnitz rules.
4 HEIDI HAYNAL
Definition 2.1. For an indeterminate t, and integers n ≥ m ≥ 0, we define the following
polynomial functions:
(m)t = t
m−1 + tm−2 + · · ·+ t + 1 (1)
(m)!t = (m)t(m− 1)t · · · (1)t, and (0)!t = 1 (2)(
(n)!t
(m)!t(n−m)!t
The expressions
are called the t-binomial coefficients, or Gaussian polynomials.
The t-binomial coefficients have properties similar to those of the regular binomial
coefficients. Two that will be useful for this work are:(
= 1 for all n ≥ 0 (4)
+ tn−m
for all 0 < m < n
Proofs for these identities may be found in combinatorics texts such as [39]. When we
evaluate the t-binomial coefficients at t = q, we obtain the q-binomial coefficients that
we need for studying q-skew polynomial rings.
As shown in [10, Section 6], the following q-Liebnitz rules hold for any q-skew polynomial
ring R[x; τ, δ]:
δn(rs) =
τn−iδi(r)δn−i(s) for all r, s ∈ R and n = 0, 1, 2, ...
xnr =
τn−iδi(r)xn−i for all r ∈ R and n = 0, 1, 2, ...
Now, taking a cue from the study of Schmidt differential operator rings, for instance
[25], we define a sequence of k-linear maps that allows us to broaden the class of rings
for which we may derive results like those of Jøndrup and Cauchon.
Definition 2.2. A higher q-skew τ -derivation (h.q-s.τ -d.) on a k-algebra R is a sequence
d0, d1, d2, . . . of k-linear operators on R such that
d0 is the identity
dn(rs) =
τn−idi(r)dn−i(s) for all r, s ∈ R and all n
diτ = q
iτdi for all i.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 5
If a sequence of k-linear maps satisfies the first two conditions, we refer to it as a higher
τ -derivation. We abbreviate the sequence {di}∞i=0 usually as just {di}. A h.q-s.τ -d is
locally nilpotent if for all r ∈ R, there exists an integer n ≥ 0 such that di(r) = 0 for all
i ≥ n, and dp(r) 6= 0 for p < n. In this case, n is called the d-nilpotence index of r. A
h.q-s.τ -d is iterative if didj =
di+j for all i, j. This implies that the di commute
with each other. A q-skew τ -derivation δ on R extends to a h.q-s.τ -d. if there is a
h.q-s.τ -d {di} on R with d1 = δ.
For example, consider the k-algebra with two generators x and y, and one relation
xy − qyx = 1, where q ∈ k×. We’ll assume that q 6= 1 and recognize this algebra as
a q-skew polynomial ring k[y][x; τ, δ] with τ(y) = qy and δ(y) = 1, commonly known
as a quantized Weyl algebra and denoted A
1(k). If q is not a root of unity, then the
(i)!q
comprise an iterative higher q-skew τ -derivation that extends δ on k[y]. The prop-
erties of a higher q-skew τ -derivation follow directly from the fact that δ is a q-skew
τ -derivation and the first q-Liebnitz rule. This particular h.q-s.τ -d. is also locally
nilpotent because
yn−i when i ≤ n,
0 when i > n.
Proposition 2.3. Let {di} be a sequence of k-linear maps on a k-algebra R with
d0 = idR, and let R[[x; τ
−1]] be the skew power series ring where τ is a k-linear automor-
phism of R, the coefficients are written on the right of the variable x, and rx = xτ(r)
for all r ∈ R.
(a) Then {di} is a higher τ -derivation on R if and only if the map Ψ : R → R[[x; τ−1]]
given by r 7→
i=0 x
idi(r) is a ring homomorphism.
(b) Extend τ to an automorphism of R[[x; τ−1]] such that τ(x) = xq. Assume that
{di} is a higher τ -derivation. Then the sequence {di} is a h.q-s.τ -d. if and only if this
diagram is commutative:
R[[x; τ−1]]
// R[[x; τ−1]]
6 HEIDI HAYNAL
Proof. (a) Suppose {di} is a higher τ -derivation on R. Consider any r, s ∈ R. It is clear
that Ψ is additive and Ψ(1) = 1. Applying the definition 2.2 gives
Ψ(rs) =
xidi(rs) =
τ i−mdm(r)di−m(s)
Power series multiplication, with rx = xτ(r), gives
Ψ(r)Ψ(s) =
xidi(r)
)( ∞∑
xidi(s)
τ i−mdm(r)di−m(s)
So Ψ preserves products. Therefore, Ψ is a ring homomorphism.
To demonstrate the other implication, suppose Ψ is a ring homomorphism. Then
Ψ(r)Ψ(s) = Ψ(rs) implies that dn(rs) =
i=0 τ
n−idi(r)dn−i(s) for all r, s ∈ R. There-
fore, {di} is a higher τ -derivation.
(b) Suppose that {di} is a h.q-s.τ -d. Then the relations diτ = qiτdi imply that
τΨ(r) =
i=0 x
iqiτdi(r) =
i=0 x
idi(τ(r)) = Ψτ(r), for all r ∈ R.
Now if the diagram is commutative, then comparing the coefficients of
τΨ(r) =
i=0 x
iqiτdi(r) and Ψτ(r) =
i=0 x
idi(τ(r)) for all r ∈ R yields that
diτ = q
iτdi. �
Remark 2.4. If {di} is locally nilpotent on R, we observe that claims analogous to the
proposition can be made for the map Ψ : R → R[x; τ−1].
Proposition 2.5. Let {di} be a h.q-s.τ -d. on a k-algebra R, where τ is an automor-
phism, and let S be a right denominator set in R with τ(S) = S. Then {di} can be
uniquely extended to a h.q-s.τ -d. on RS−1.
Proof. It has been established that τ and d1 extend uniquely to RS
−1 by
τ(rs−1) = τ(r)τ(s)−1 and d1(rs
−1) = d1(r)s
−1 − τ(rs−1)d1(s)s−1 in [10, Lemma 1.3].
Suppose that {di} extends to a h.q-s.τ -d. on RS−1. For r ∈ R and s ∈ S, we apply dn
to the equation r1−1 = (rs−1)(s1−1) to get
dn(r)1
−1 = dn
(rs−1)(s1−1)
τn−jdj(rs
−1)dn−j(s1
= τn(rs−1)dn(s)1
−1 + · · ·+ dn(rs−1)s1−1.
This implies that
dn(rs
−1) =
dn(r)−
τn−jdj(rs
−1)dn−j(s)
So we have uniqueness in case of existence.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 7
To show existence, let Ψ : R → R[[x; τ−1]] be the map defined in Proposition 2.3, and
let φ : R[[x; τ−1]] → RS−1[[x; τ−1]] be the natural map. Consider the composite map
Φ = φΨ : R → RS−1[[x; τ−1]]. For any s ∈ S, the constant term of Φ(s) is a unit.
So we may inductively solve for the coefficients of an inverse for Φ(s) in RS−1[[x; τ−1]].
Details, as in [37, 1.2], are left to the reader. Hence, Φ extends to a ring homomorphism
Φ′ : RS−1 → RS−1[[x; τ−1]] such that Φ′(rs−1) = Φ(r)Φ(s)−1, and we consider the
diagram:
RS−1[[x; τ−1]]
// RS−1[[x; τ−1]]
// RS−1
where τ has been extended to an automorphism of RS−1[[x; τ−1]] as in Proposition 2.3.
Since Φ(r) =
i=0 x
idi(r)1
−1, and {di} is a h.q-s.τ -d. on R, we have
τΦ(r) =
xiqiτdi(r)1
1−1 = Φτ(r)
for all r ∈ R. It follows directly that τΦ′(rs−1) = Φ′τ(rs−1). So, indeed, the diagram is
commutative.
Define a sequence {di} on RS−1 such that di(t) equals the coefficient of xi in Φ′(t) for
all t ∈ RS−1. Then by Proposition 2.3 we conclude that this sequence is a h.q-s.τ -d. on
RS−1 extending {di} on R. �
Lemma 2.6. Let A be a k-algebra, B ⊆ A a k-subalgebra generated by {b1, b2, . . . }, τ
a k-linear automorphism of A, and {di} a higher τ -derivation on A. If di(bj) ∈ B and
τ(bj) ∈ B, for all i, j ∈ N, then di(B) ⊆ B for all i.
Proof. First, observe that τ(bj) ∈ B for all j implies that τ(B) ⊆ B. Since the di
are k-linear maps, it suffices to check monomials in the bj , using induction on their
length. Suppose, inductively, that for integers m ≥ 1 and 1 ≤ ℓ ≤ m − 1, we have
di(bj1 · · · bjℓ) ∈ B for all i and all j1, . . . , jℓ. Then using the product rule for h.q-s.τ -d.
gives
dn(bj1 · · · bjm) =
τn−1di(bj1 · · · bjm−1)dn−i(bjm) ∈ B
for all n and all j1, . . . , jm, by the induction hypothesis. �
8 HEIDI HAYNAL
Lemma 2.7. Let A be a k-algebra with a set {xj} of generators, τ an automorphism of
A, and {di} a h.q-s.τ -d. on A. If {di} is locally nilpotent for all xj, then {di} is locally
nilpotent on A.
Proof. It suffices to check monomials in the xj because the di are k-linear maps. We
proceed by using induction on the length of such monomials. For a given xn, let i(n)
be its nilpotence index, so di(xn) = 0 for all i ≥ i(n).
Suppose inductively that for n ≥ 2, all integers ℓ with 1 ≤ ℓ ≤ n− 1, and all choices
of j1, . . . , jℓ, there exists an integer m such that di(xj1 · · ·xjl) = 0 for all i ≥ m. For
instance, m = i(j1)+ · · ·+ i(jℓ) will suffice, although the d-nilpotence index of xj1 · · ·xjℓ
may be less than this sum. Then, for p ≥ m+ i(jn), we have
dp(xj1 · · ·xjn) =
τ p−idi(xj1 · · ·xjn−1)dp−i(xjn) = 0,
completing the induction. �
Consider again the quantized Weyl algebra A
1(k). In case q is an ℓ-th root of unity,
the dℓ given in (7) would be undefined due to the occurrence of a zero denominator.
However, realizing A
1(k) as a factor of a quantized Weyl algebra over k[t
±1] allows us to
define a h.q-s.τ -d. on A
1(k) nonetheless. The k[t
±1]-algebra At1(k[t
±1]) has generators
x and y and one relation xy− tyx = 1. This is a t-skew polynomial ring k[t±1][y][x; τ̄ , δ̄]
where τ̄ (y) = ty, τ̄(t) = t, δ̄(y) = 1, and δ̄(t) = 0. Note that
δ̄i(yn) =
(n)!t
(n−i)!t
yn−i when i ≤ n
0 when i > n
implying that δ̄i
k[t±1][y]
⊆ (i)!tk[t±1][y]. So the assignment
d̄i =
(i)!t
defines an iterative, locally nilpotent h.t-s.τ̄ -d. {d̄i} on k[t±1][y]. Now, the relation
xy − tyx = 1 is equivalent to the relation xy − qyx = 1 modulo 〈t − q〉. Hence we
k[t±1]
/〈t− q〉 ∼= Aq1(k).
When q is an ℓth root of unity, we have δ̄ℓ
k[t±1][y]
⊆ 〈t−q〉k[t±1][y]. Nonetheless, the
h.t-s.τ̄ -d. {d̄i} on k[t±1][y] induces a h.q-s.τ -d. {di} on k[y], also iterative and locally
nilpotent, with d1 = δ. Note that even though δ
ℓ = 0 in this algebra, we have di(y
i) = 1
for all i.
This phenomenon is not unique to the quantized Weyl algebras. The conditions that
drive it are codified in the following theorem.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 9
Theorem 2.8. Let R be a k-algebra and R[x; τ, δ] a q-skew polynomial ring where
q ∈ k, q 6= 1. Suppose there exists a torsion-free k[t±1]-algebra R and R[x; τ̄ , δ̄] a t-skew
polynomial ring such that R/〈t− q〉R ∼= R, with τ̄ and δ̄ reducing to τ and δ. Suppose
further that δ̄i(R) ⊆ (i)!tR for all i. Then δ extends to an iterative h.q-s.τ -d. {di} on
R. If δ̄ is locally nilpotent, then so is {di}. If q is not a root of unity, then di = δ
(i)!q
all i. If q is a primitive ℓth root of unity, then di =
(i)!q
for i < ℓ.
Proof. The assumption δ̄i(R) ⊆ (i)!tR for all i implies that the sequence of maps
d̄i =
(i)!t
make up a well-defined iterative h.t-s.τ̄ -d. on R, and also implies that
δ̄ℓ(R) ⊆ 〈t − q〉R because (ℓ)t ≡ (ℓ)q = 0 modulo 〈t − q〉. Since τ̄ and δ̄ reduce to
τ and δ modulo 〈t− q〉, we have an isomorphism R/〈t− q〉[x; τ̄ , δ̄] ∼= R[x; τ, δ] whereby
{d̄i} induces an iterative h.q-s.τ -d. {di} on R. The reduction of the maps from R to R
also implies the remaining results. �
We will find that all of the conditions assumed above are satisfied by the common
quantized coordinate rings and related examples, which will be discussed in a subsequent
section.
3. The τ-Derivation Removing Homomorphism
Following the pattern in [5], let A = R[x; τ, δ], and suppose that δ is locally nilpotent.
Set S = {xn | n ∈ N ∪ {0}} ⊂ A.
Lemma 3.1. The set S is a denominator set in A.
Proof. Clearly, S is a multiplicative set inA. And, since S contains only regular elements
of A, it is left and right reversible. It remains to show that S is an Ore set.
Let a =
i=0 rix
i be an element of A with rn 6= 0. For each ri in the expression of a,
and each mi ≥ 0, we have
xmiri =
τmi−jδj(ri)x
= a′ix+ δ
mi(ri) for some a
i ∈ A.
Since δ is locally nilpotent, we may choose mi to be the δ-nilpotence index of ri to
conclude that xmiri = a
ix for some a
i ∈ A. Set ma = max{mi | 0 ≤ i ≤ n}. Then for
each ri, we have x
mari = ãix, and hence x
maa = ãx for some ã ∈ A.
Now suppose, inductively, that for a given a ∈ A and xp ∈ S we can find elements
xma ∈ S and ā ∈ A such that xmaa = āxp, say ā =
i=0 r̄ix
i. We know that there
10 HEIDI HAYNAL
exists an element xmā such that xmā ā = a′x for some a′ ∈ A. So, xmaa = āxp implies
xmā+maa = a′xp+1, completing the induction.
Hence, for any a ∈ A and s ∈ S, we have Sa∩As 6= ∅. So S is a left Ore set in A. We see
that S is a right Ore set by applying the same argument to Aop = Rop[x; τ−1,−δτ−1]. �
Suppose also that the derivation δ extends to an iterative, locally nilpotent higher q-
skew τ -derivation {di} on R and that q 6= 1. Denote Â = AS−1 = S−1A, the localization
of A with respect to S, and define a map f : R −→ Â by
f(r) =
n(n+1)
2 (q − 1)−ndnτ−n(r)x−n,
noting that {di} is locally nilpotent and that q − 1 is invertible. If q is not a root of
unity and {di} is obtained from a q-skew τ -derivation δ as in (7), the formula for f can
be rewritten as
f(r) =
n(n+1)
(q − 1)−n
(n)!q
δnτ−n(r)x−n.
The rewritten formula matches the one presented in [5, Section 2] when q is replaced by
q−1 to account for the difference between δτ = qτδ (used here) and τδ = qδτ (used in
[5]). We will show that f is a homomorphism and that the the multiplication in imf is
made simpler than that in A by removing the derivation, as seen in the following.
Proposition 3.2. If r ∈ R, then xf(r) = f
x in Â.
Proof. Using the hypothesis that {di} is iterative, we compute that
xf(r) =
n(n+1)
2 (q − 1)−nxdnτ−n(r)x−n
n(n+1)
2 (q − 1)−n
−n(r)x+ d1dnτ
−n(r)
n(n+1)
2 (q − 1)−nq−ndnτ−n+1(r)x−n+1
n(n+1)
2 (q − 1)−n(n+ 1)qdn+1τ−n(r)x−n
n(n+1)
2 (q − 1)−nq−ndnτ−n(τ(r))x−n+1
n(n−1)
2 (q − 1)−n+1(n)qdnτ−n(τ(r))x−n+1
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 11
= τ(r)x
n(n+1)
2 (q − 1)−nq−n + q
n(n−1)
2 (q − 1)−n+1(n)q
−n(τ(r))x−n+1
= τ(r)x
(q − 1)−n
2 + q
2 (qn − 1)
−n(τ(r))x−n+1
= τ(r)x+
(q − 1)−nq
n(n+1)
2 dnτ
−n(τ(r))x−n+1
n(n+1)
2 (q − 1)−ndnτ−n(τ(r))x−n
x = f
which gives the result. �
From Proposition 3.2, it follows by routine induction that
xmf(r) = f
τm(r)
xm ∀m ∈ Z. (8)
This is what we need in order to show that our map is indeed a k-algebra homomor-
phism.
Proposition 3.3. The map f : R −→ Â is a k-algebra homomorphism.
Proof. It is immediate that f is k-linear (τ and {di} are k-linear), and that f(1) = 1.
We’ll show that f is multiplicative. If r, s ∈ R, then using Prop. 3.2,
f(r)f(s) =
i(i+1)
2 (q − 1)−idiτ−i(r)x−if(s)
i(i+1)
2 (q − 1)−idiτ−i(r)f(τ−i(s))x−i
i≥0, j≥0
i(i+1)+j(j+1)
2 (q − 1)−(i+j)diτ−i(r)djτ−(i+j)(s)x−(i+j).
For n ∈ N, the coefficient of x−n in the sum above is
i≥0, j≥0,
i+j=n
i(i+1)+j(j+1)
2 (q − 1)−ndiτ−i(r)djτ−n(s)
(n−p)2+p2+n
2 (q − 1)−ndn−pτ p−n(r)dpτ−n(s)
12 HEIDI HAYNAL
n(n+1)
2 (q − 1)−n
qp(p−n)dn−pτ
pτ−n(r)dpτ
−n(s)
n(n+1)
2 (q − 1)−n
τ pdn−p(τ
−n(r))dp(τ
−n(s))
n(n+1)
2 (q − 1)−ndn(τ−n(r)τ−n(s))
n(n+1)
2 (q − 1)−ndnτ−n(rs),
computed by putting p = j and using the second condition in the Definition 2.2. In
summary, f(r)f(s) =
n=0 q
n(n+1)
2 (q − 1)−ndnτ−n(rs)x−n = f(rs). �
Proposition 3.4. (1) The map f extends uniquely to an algebra homomorphism, also
denoted f , of R[y; τ ] to Â satisfying f(y)=x.
(2) The extended homomorphism is injective.
Proof. (1) This result follows from Proposition 3.2 and the universal property of Ore
extensions.
(2) Let P = pmy
m + · · ·+ p1y + p0 be a nonzero element of R[y; τ ], where each pi ∈ R,
m ≥ 0, pm 6= 0. Then f(P ) = f(pm)xm + · · ·+ f(p1)x+ f(p0). Since
f(pi) =
n(n+1)
2 (q − 1)−ndnτ−n(pi)x−n ∈ AS−1,
we know that there exists an integer l ≥ 0 such that each f(pi)xl is a nonzero element
of A of positive degree l (in x) whenever pi 6= 0. (Because {di} is locally nilpotent,
we may choose an l large enough.) It follows that f(P )xl is a nonzero element of Â of
degree m+ l, hence f(P ) 6= 0. �
Definition 3.5. The algebra homomorphism f : R[y; τ ] −→ Â = AS−1 is called the
derivation removing homomorphism. The image of f , call it A′, is the subalgebra of
Â = AS−1 generated by x and f(R), and is isomorphic (as an algebra) to R[y; τ ] by the
derivation removing homomorphism f .
Observe that A′ contains the multiplicative system S = {xn | n ∈ N ∪ {0}}. Since
equation (8) holds and f(y) = x, the elements of this set are normal in A′. Hence,
S satisfies the (two-sided) Ore condition in A′. The elements of S are regular in A′
because they are regular in Â, and thus:
Proposition 3.6. A′S−1 = AS−1
Proof. We have A′S−1 ⊆ AS−1 because A′ = im(f) ⊆ AS−1. To show the other
inclusion, it suffices to show that R ⊆ A′S−1. (This suffices because A is built up from
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 13
R by x, x2, . . . . So if R ⊆ A′S−1, then AS−1 ⊆ A′S−1.) Consider any r ∈ R and let ℓ
be the d-nilpotence index of r. We show that r ∈ A′S−1 with an induction argument
on ℓ.
If ℓ ≤ 1, then d1(r) = 0, whence f(r) = r ∈ A′ ⊆ AS−1.
If ℓ ≥ 2, we write
f(r) = r +
−n, with rn = q
n(n+1)
2 (q − 1)−ndnτ−n(r) ∈ R.
We’ll show that
n=1 rnx
−n ∈ A′S−1 in order to conclude that r ∈ A′S−1, because
f(r) −
n=1 rnx
−n = r. That is, we need to show that each rn ∈ A′S−1. Suppose,
inductively, that for any element r̃ ∈ R with d-nilpotence index m such that m < ℓ, we
have r̃ ∈ A′S−1.
Note that for n ∈ {1, . . . , ℓ}, we have
dℓ−n(rn) = q
n(n+1)
2 (q − 1)−n
−n(r) = q
n(n+1)
2 (q − 1)−n
q−nℓτ−ndℓ(r) = 0
because dℓ(r) = 0 by hypothesis.
Hence, by the induction hypothesis, each rn ∈ A′S−1 for 1 ≤ n ≤ ℓ− 1. It follows that
r = f(r)−
n=1 rnx
−n also belongs to A′S−1. �
This equality of quotient rings reveals that if A is a PI ring, then
PIdegA = PIdegA′ = PIdegR[y; τ ],
with the second equality arising from the derivation removing homomorphism f . This
recovers the result of Jøndrup [21] without the assumption that k has characteristic
zero. We summarize the results of this section in the following theorem.
Theorem 3.7. Let k be a field, R a k-algebra and A = R[x; τ, δ] a q-skew polynomial
ring in which δ extends to a locally nilpotent, iterative h.q-s.τ -d. {di} on R for some
q ∈ k×, q 6= 1. Let S be the Ore set in A generated by x, and define a map
f : R −→ AS−1 by f(r) =
n=0 q
n(n+1)
2 (q − 1)−ndnτ−n(r)x−n. Then f is a k-algebra
homomorphism, and it extends to an injective homomorphism f : R[y; τ ] −→ AS−1
sending y to x. Furthermore, the extension f : R[y±1; τ ] −→ AS−1 is an isomorphism.
So there is PI degree parity between A and R[y; τ ]. Moreover, if R is a noetherian
domain, then FractA ∼= FractR[y; τ ].
14 HEIDI HAYNAL
4. Main Theorem
In the case where A is an iterated skew polynomial ring, we would like to apply re-
peatedly the method presented above to remove all of the derivations and compare the
resulting Ore localizations. We must first establish some facts about the behavior of
h.q-s.τ -d. when the variables adjoined to the coefficient ring are rearranged, and about
iterated localization. The results of these lemmas will ensure that after the induction
step in the proof of the main theorem we are left with a ring to which the method of
the preceding section applies.
The first parts of the following lemmas hold in a broader class of skew polynomial rings
and also when the q-skew condition is imposed. The final parts assert that h.q-s.τ -d.
are preserved when rearranging of the variables is permissible.
Lemma 4.1. Let S = R[x; τ, δ], A = R[x; τ, δ][y; σ], and Â = R[x; τ, δ][y±1; σ], where
σ(R) = R and σ(x) = λx for some λ ∈ k×.
(1) Then A = R[y; σ′][x; τ ′; δ′], and Â = R[y±1; σ′][x; τ ′; δ′], where σ′ = σ
, τ ′
= τ ,
= δ, τ ′(y) = λ−1y, and δ′(y) = 0.
(2) If (τ, δ) is q-skew, then so is (τ ′, δ′).
(3) Suppose further that δ extends to a h.q-s.τ -d. {di} on R, and that σdi = λidiσ for
all i. Then the τ ′-derivation δ′ extends to a h.q-s.τ ′-d. {d′i} on R[y±1; σ′] such that the
restrictions of the d′i to R coincide with di, and d
i(y) = 0 for all i ≥ 1. Moreover, {d′i}
restricts to a h.q-s.τ ′-d. on R[y; σ′].
(a) If {di} is iterative, then {d′i} is iterative.
(b) If {di} is locally nilpotent, then {d′i} is locally nilpotent.
Proof. (1) Routine details omitted so as not to try the patience of the reader.
(2) Suppose that (τ, δ) is q-skew on R. We’ll check that the two τ ′-derivations τ ′−1δ′τ ′
and qδ′ agree on R[y±1; σ′]. It suffices to check their agreement on a set of generators,
R ∪ {y, y−1}. It is clear that τ ′−1δ′τ ′(r) = qδ′(r) for all r ∈ R. Since δ′(y) = 0, they
agree on {y, y−1} as well. So (τ ′, δ′) is q-skew.
(3) Define a sequence of maps d′i : R[y
±1; σ′] → R[y±1; σ′] by
di(rj)y
Clearly these are k-linear maps, d′i(r) = di(r) for all r ∈ R; also d′i(y) = di(1)y = 0 for
i ≥ 1, and d′0 is the identity on R[y±1; σ′].
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 15
Because δ extends to {di} on R, we get
d1(rj)y
δ(rj)y
j = δ′
for all rj ∈ R. So d′1 = δ′ on R[y±1; σ′].
Now, for integers j,m, n, and elements r, s ∈ R,
(ryj)(sym)
= d′n
rσj(s)yj+m
rσj(s)
τn−idi(r)dn−i(σ
j(s))yj+m
τn−idi(r)y
jσ−jdn−i(σ
j(s))ym
τn−idi(r)y
jλ−j(n−i)dn−i(s)y
(τ ′)n−i
di(r)y
d′n−i(sy
(τ ′)n−id′i(ry
j)d′n−i(sy
So {d′i} satisfies the product rule for a higher τ -derivation on R[y±1; σ′].
Furthermore,
τ ′d′i
= τ ′
di(rj)y
τdi(rj)λ
−jyj,
and d′iτ
= d′i
τ(rj)λ
diτ(rj)λ
τdi(rj)λ
−jyj,
giving the q-skew relation d′iτ
′ = qiτ ′d′i on R[y
±1; σ′].
It follows directly from the definition of the maps {di} that their restrictions to the
k-subalgebra R[y; σ′] also exhibit the properties of definition 2.2.
16 HEIDI HAYNAL
If {di} is iterative on R, then d′ℓd′i(rym) = d′ℓ
di(r)y
= dℓdi(r)y
dℓ+i(r)y
d′ℓ+i(ry
m) for all r ∈ R, m ∈ Z, and non-negative integers ℓ, i. Hence, {d′i} is
iterative on R[y±1; σ′].
Suppose that {di} is locally nilpotent on R. By Lemma 2.7 we need only check that
{d′i} is locally nilpotent on R ∪ {y, y−1}, a set of generators for R[y±1; σ′]. This is clear
because d′i(r) = di(r) for all r ∈ R, and d′i(y) = 0 for all i by construction. �
Lemma 4.2. Let
A = R[x1; τ1, δ1][x2; τ2, δ2] · · · [xn; τn, δn][y; σ],
Â = R[x1; τ1, δ1][x2; τ2, δ2] · · · [xn; τn, δn][y±1; σ],
where σ(R) = R, and for all i ∈ {1, . . . , n}, σ(xi) = λixi for some nonzero λi ∈ k. Let
Aj = R[x1; τ1; δ1][x2; τ2, δ2] · · · [xj ; τj, δj ] for j = 1, 2, . . . , n, and A0 = R.
(1) Then
A = R[y; σ∗][x1; τ
1][x2; τ
2] · · · [xn; τ ′n, δ′n],
Â = R[y±1; σ∗][x1; τ
1][x2; τ
2] · · · [xn; τ ′n, δ′n],
where σ∗ = σ
, τ ′i
= τi, δ
= δi, τ
i(y) = λ
i y, and δ
i(y) = 0 for all 1 ≤ i ≤ n
and j ≤ i− 1.
(2) If (τi, δi) is qi-skew for any 1 ≤ i ≤ n, then (τ ′i , δ′i) is also qi-skew.
(3) Suppose that each δi extends to an h.qi-s.τi-d. {di,p}∞p=0, and that σdi,p = λ
i di,pσ
on Ai−1 for all i and p. Then each δ
i extends to a h.qi-s.τ
i -d. {d′i,p}∞p=0 on the algebra
R〈y, y−1, x1, . . . , xi−1〉, where d′i,p coincides with di,p on Aj, for j < i, and d′i,p(y) = 0
for p ≥ 1. Moreover, {d′i,p} restricts to a h.qi-s.τ ′i -d. on R〈y, x1, . . . , xi−1〉.
(a) If {di,p} is iterative for any 1 ≤ i ≤ n, then {d′i,p} is iterative.
(b) If {di,p} is locally nilpotent for any 1 ≤ i ≤ n, then {d′i,p} is locally nilpotent.
Proof. (1) The condition σ(xi) = λixi for all i implies that σ(Ai) = Ai. We will use
induction on n to prove the result.
Lemma 4.1 proves the case n = 1. Suppose the result holds for all m < n, and consider
A = An−1[xn; τn, δn][y; σ]. Application of Lemma 4.1, and then the induction hypothesis,
gives
A = An−1[xn; τn, δn][y; σ]
= An−1[y; σ
′][xn; τ
= R[x1; τ1, δ1] · · · [xn−1; τn−1, δn−1][y; σ′][xn; τ ′n, δ′n]
= R[y; σ∗][x1; τ
1] · · · [xn; τ ′n, δ′n],
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 17
with the desired conditions met by the automorphisms and derivations, completing the
induction. Similarly, Â = R[y±1; σ∗][x1; τ
1] · · · [xn; τ ′n, δ′n].
(2) Consider the two τ ′i -derivations τ
i and qiδ
i on the ring
R[y±1; σ∗][x1; τ
1] · · · [xi−1; τ ′i−1, δ′i−1]
for 1 ≤ i ≤ n. Since (τi, δi) is q-skew, it is clear that these two τ ′i derivations agree on
Ai−1. And since δ
i(y) = 0 for all i = 1, . . . , n, these two τ
i -derivations agree on a full
set of generators of R[y±1; σ∗][x1; τ
1] · · · [xi−1; τ ′i−1, δ′i−1]. Hence, δ′iτ ′i = qiτ ′iδ′i.
(3) Suppose the result holds for the algebra R[x1; τ1, δ1] · · · [xn−1; τn−1, δn−1][y±1; σ].
Then Lemma 4.1 may be applied, with An−1 providing the coefficients, to get
An−1[xn; τn, δn][y
±1; σ] = An−1[y
±1; σ′][xn; τ
where δ′n extends to a h.qn-s.τ
n-d. {d′n,p} on An−1[y±1]. The induction hypothesis gives
the result. �
Definition 4.3. For a k-algebra A and a, b ∈ A, we say that a and b scalar commute
if there is an element α ∈ k× such that ab = αba. We may also say that a and b
α-commute.
In the following two lemmas, we let D denote the division ring of fractions for the
noetherian domain A. When comparing localizations of A, we identify them as subrings
of D.
Lemma 4.4. Let A be a noetherian domain, S ⊆ A \ {0} an Ore set. Let T be an Ore
set in AS−1 \ {0} with S ⊆ T .
(1) Then there exists an Ore set T̃ ⊆ A\{0} with S ⊆ T̃ such that AT̃−1 = (AS−1)T−1.
(2) Suppose A is a k-algebra and S is generated by s1, . . . , sn satisfying sisj = γijsjsi
for all i, j and some γij ∈ k×. Further suppose that T is generated by S ∪ t for some
t ∈ AS−1 that satisfies sit = λitsi for all i and some λi ∈ k×. Then there exist a cyclic
Ore set T̂ ⊆ A \ {0} and an (n + 1)-generator Ore set Ŝ ⊆ A \ {0} such that S ⊆ Ŝ,
and (AS−1)T−1 = AT̂−1 = AŜ−1.
Proof. (1) Consider T ∩A, the subset in T of elements with a denominator of 1. Clearly,
this is a multiplicative set in A which contains S. Set T̃ = T ∩A. Let a ∈ T̃ and α ∈ A.
Then a ∈ T , and since α ∈ AS−1, there exist b′ ∈ T and β ′ ∈ AS−1 such that aβ ′ = αb′.
By [16, 10.2], there exist y ∈ S, and b, β ∈ A such that β ′ = βy−1 and b′ = by−1; hence,
aβy−1 = αby−1 in AS−1. It follows that aβ = αb in A. So T̃ satisfies the right Ore
condition in A, and the left Ore condition by symmetry. By the universal property,
AT̃−1 ∼= (AS−1)T−1. As subrings of D, we have AT̃−1 = (AS−1)T−1.
18 HEIDI HAYNAL
(2) The generating element t has the form t = ā(sm11 s
2 · · · smnn )−1 for some mi ∈ N,
and ā ∈ A. For any si ∈ S, we have
siā(s
2 · · · smnn )−1 = λiā(sm11 sm22 · · · smnn )−1si = µλiāsi(sm11 sm22 · · · smnn )−1,
where µ is a product of powers of the γij. So ā scalar commutes with the genera-
tors of S via the relations siā = µλiāsi. Let Ŝ be the multiplicative set generated by
ā, s1, . . . , sn in A, and T̂ the multiplicative set generated by ās1s2 · · · sn in A. Recall
that (AS−1)T−1 = AT̃−1, where T̃ = T ∩ A from part (1). From the scalar com-
muting relations it follows that any element at̃−1 ∈ AT̃−1 may be written in the form
b(ās1, · · · sn)−m for some m ∈ N ∪ {0}, b ∈ A, or the form cā−ℓn+1s−ℓ11 · · · s−ℓnn , for
ℓj ∈ N ∪ {0}, c ∈ A. So we conclude that Ŝ and T̂ are Ore sets in A and that
(AS−1)T−1 = AT̂−1 = AŜ−1. �
Lemma 4.5. Let A be a noetherian domain, S1 ⊆ A \ {0} an Ore set, and for integers
j = 2, . . . , n let Sj be an Ore set in ((AS
1 ) · · · )S−1j−1 \ {0} with Sj−1 ⊆ Sj.
(1) Then there exists an Ore set T ⊆ A \ {0} such that AT−1 = (((AS−11 )S−12 ) · · · )S−1n .
(2) Suppose A is a k-algebra, S1 is generated by s1, and for j = 2, . . . , n, Sj is generated
by Sj−1 ∪ {sj}, where sisj = γijsjsi for some multiplicatively antisymmetric matrix
(γij) ∈ Mn(k×). Then there are a cyclic Ore set T̂ ⊆ A and an n-generator Ore set
Ŝ ⊆ A such that S1 ⊆ Ŝ, and ((AS−11 )S−12 ) · · ·S−1n = AT̂−1 = AŜ−1.
Proof. (1) The proof proceeds by induction on n. The case n = 1 is covered in the
lemma above. Suppose that for all j ≤ n− 1 there exists an Ore set Tj ⊆ A \ {0} such
that AT−1j = (((AS
2 ) · · · )S−1j . Then the equality
AT−1n−1 = (((AS
2 ) · · · )S−1n−1
identifies an Ore set Tn ⊆ AT−1n−1 \ {0} such that
(AT−1n−1)T
n = (((AS
2 ) · · ·S−1n−1)S−1n .
Furthermore, Lemma 4.4 implies the existence of an Ore set T ⊆ A \ {0} such that
AT−1 = (AT−1n−1)T
n = (((AS
2 ) · · ·S−1n−1)S−1n .
(2) Suppose, inductively, that there exist
(i) a cyclic Ore set T̂n−1 ⊆ A \ {0} generated by s1ā2 · · · ān−1
(ii) an (n− 1)-generator Ore set Ŝn−1 ⊆ A \ {0} with S1 ⊆ Ŝn−1 and generators
s1, ā2, ā3, . . . , ān−1
(iii) the āi scalar commute with s1 and with each other
(iv) ((AS−11 )S
2 ) · · ·S−1n−1 = AT̂−1n−1 = AŜ−1n−1 as subrings of D.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 19
Then sn = ān(s1ā2 · · · ān−1)−r for some ān ∈ A and r ∈ N. Using the relations
sisj = γijsjsi, routine calculations show that the āi scalar commute with the sj ,
and also with each other, for all i, j. Let T̂ be the multiplicative set generated by
s1ā2 · · · ān, and let Ŝ be the multiplicative set generated by s1, ā2, ā3, . . . , ān. Then
((AS−11 )S
2 ) · · ·S−1n = (AT̂−1n−1)S−1n = AT−1 from part (1). Using Lemma 4.4, we con-
clude that T̂ and Ŝ are Ore sets in A and that AT−1 = AT̂−1 = AŜ−1. �
In the proof of the main theorem, we will use without mention the facts gathered here.
For greater details on these statements, see [16, 10X, 10Y] and [10, 1.4].
(1) Given a noetherian ring A and a normal element x ∈ A, the multiplicative set
generated by x is an Ore set.
(2) The multiplicative set generated by a nonempty family of right Ore sets is right
(3) Let A = R[x; τ, δ], and S a right denominator set in R such that τ(S) = S.
Then S is a right denominator set in A and the identity map on AS−1 extends
to an isomorphism of AS−1 onto (RS−1)[x; τ, δ] sending x1−1 to x. Note that
if A is a k-algebra, τ , δ are k-linear, and τ(k×S) = k×S, then the result holds
because S is a denominator set if and only if k×S is a denominator set.
Theorem 4.6. Let R be a k-algebra and noetherian domain,
A = R[x1; τ1, δ1] · · · [xn; τn, δn],
where each τi is a k-linear automorphism of R〈xi, . . . , xi−1〉 such that τi(xj) = λijxj
for all i, j with 1 ≤ j < i ≤ n and some λij ∈ k×, and where each δi is a k-linear τi-
derivation. Assume that there exist elements qi ∈ k× with qi 6= 1 such that δiτi = qiτiδi,
and that δi extends to a locally nilpotent, iterative h.qi-s.τi-d. on R〈xi, . . . , xi−1〉 for
i = 1, . . . , n.
(1) Then there exists an Ore set T ⊆ A generated by n elements of A such that
AT−1 ∼= R[y±11 ; τ1][y±12 ; τ ′2] · · · [y±1n ; τ ′n]
where τ ′i |R = τi and τ ′i(yj) = λijyj for all i, j with 1 ≤ j < i ≤ n
(2) There is PI degree parity between A and R[y1; τ1][y2; τ
2] · · · [yn; τ ′n]. Moreover, these
algebras have isomorphic division rings of fractions.
Proof. (a) Suppose, inductively, that we have
R[x1; τ1, δ1][y
2 ; τ2] · · · [y±1n ; τ ′n] ∼= AS−12
where the restriction of τ ′i to R〈x1〉 coincides with τi, τ ′i(ym) = λimym for 2 ≤ i ≤ n and
1 < m < i, and S2 is an Ore set in A generated by n − 1 elements from A. Then by
20 HEIDI HAYNAL
Lemma 4.2
AS−12
∼= R[y±12 ; τ ′′2 ] · · · [y±1n ; τ ′′n ][x1; τ ′1, δ′1] (9)
where the restrictions of τ ′1 and δ
1 to R coincide with τ1 and δ1, τ
1(yj) = λ
j1 yj, δ
1(yj) =
0, and τ ′′i coincides with the restriction of τi to R〈y2, . . . , yi−1〉 for 2 ≤ i ≤ n. Observe
that by Lemmas 4.2 and 2.7 we also have δ′1τ
1 = q1τ
1, and that δ
1 extends to a
locally nilpotent iterative h.q1-s.τ -d. on R〈y±12 , . . . , y±1n 〉. Then applying the derivation
removing homomorphism to the right hand side of (9) gives an isomorphism
(AS−12 )T
∼= R[y±12 ; τ ′2] · · · [y±1n ; τ ′n][y±11 ; τ ′1]
where T1 ⊆ AS−12 is an Ore set generated by one element of AS−12 . Then Lemma 4.5
and a reordering of variables shows the existence of an Ore set T ⊆ A, generated by n
elements of A, such that AT−1 ∼= R[y±11 ; τ1][y±12 ; τ ′2] · · · [y±1n ; τ ′n].
(2) This follows from part (1). �
Corollary 4.7. Let A = k[x1; τ1, δ1] · · · [xn; τn, δn] with the hypotheses as in Theorem
4.6. Set λ = (λij). Then
(1) A and Oλ(kn) have isomorphic division rings of fractions.
(2) A is a PI-algebra if and only if all the λij are roots of unity, in which case A and
Oλ(kn) have the same PI degree.
In general, identification of the generators for the Ore set T in Theorem 4.6 is very
cumbersome. To illustrate the computations on a fairly short iterated skew polynomial
ring, we consider the multiparameter second quantized Weyl algebra A
2 (k). Here,
Q = (q1, q2) ∈ (k×)2, qi 6= 1 for all i, and Γ = (γij) ∈ M2(k×) with γii = 1 and
γ21 = γ
12 . The algebra A
2 (k) may be presented as an iterated skew polynomial ring
of the form k[y1][x1; τ2, δ2][y2; τ3][x2; τ4, δ4], where the τi are k-linear automorphisms and
the δ2i are k-linear τ2i-derivations such that
τ2(y1) = q1y1, δ2(y1) = 1
τ3(y1) = γ
12 y1
τ3(x1) = γ12x1
τ4(y1) = q1γ12y1, δ4(y1) = 0
τ4(x1) = q
1 γ21x1, δ4(x1) = 0
τ4(y2) = q2y2, δ4(y2) = (q1 − 1)y1x1 + 1.
For greater detail about this algebra, the reader is referred to [1], [23], [12], and [15].
Routine computations show that the pair (τ2, δ2) is a q1-skew derivation and that (τ4, δ4)
is a q2-skew derivation. To show that δ2 and δ4 are locally nilpotent, it suffices to check
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 21
for local nilpotence on a set of generators. Given their definitions, this is accomplished
by verifying their action on powers of y1 and y2:
δi2(y
1 ) =
(n)!q1
(n−i)!q1
yn−i1 i ≤ n
0 i > n
δi4(y
2 ) =
(n)!q2
(n−i)!q2
[δ4(y2)]
iyn−i2 i ≤ n
0 i > n
Using Theorem 2.8 we have a h.q1-s.τ2-d. {d2,i} extending δ2, and a h.q2-s.τ4-d. {d4,i}
extending δ4, both of which are iterative and locally nilpotent. Let S2 ⊆ AQ,Γ2 (k) be the
multiplicative set generated by x2. The derivation removing homomorphism induces an
isomorphism
Φ : k[y1][x1; τ2, δ2][y2; τ3][z
2 ; τ4] −→ A
2 (k)S
whose action on generators is given by
y1 7→ y1
x1 7→ x1
z2 7→ x2
y2 7→ y2 + (q2 − 1)−1
(q1 − 1)y1x1 + 1
x−12 .
For simplicity, label the domain of Φ as BZ−1. Let X1 ⊆ BZ−1 be the Ore set generated
by z2 and x1. Applying the derivation removing homomorphism to BZ
−1 induces an
isomorphism
Ψ : k[y1][z
1 ; τ2][y2; τ3][z
2 ; τ
4] −→ (BZ−1)X−11
whose action on generators is given by
z1 7→ z1
z2 7→ z2
y2 7→ y2
y1 7→ y1 + (q1 − 1)−1x−11 .
The derivation removing homomorphism need not be employed again to achieve the
result. Through iterated localization we find that there is an Ore set T ⊆ AQ,Γ2 (k) such
2 (k)T
−1 ∼= k[y±11 ][x±11 ; τ2][y±12 ; τ3][x±12 ; τ4]
and T is generated by the four elements x2, x1, y2x2(q2 − 1) + y1x1(q1 − 1) + 1, and
y1x1(q1 − 1) + 1. Note that we recover the result of [22, Theorem 5].
22 HEIDI HAYNAL
5. Examples
We will demonstrate how each of the following k-algebras satisfies all the conditions
of Theorem 2.8. Then Corollary 4.7 is applied to obtain an isomorphism of quotient
division rings (thereby confirming the quantum Gel’fand-Kirillov conjecture) and PI
degree parity with a multiparameter quantum affine space. When calculating the PI
degree of a quantum affine space, we encounter an antisymmetric, or skew-symmetric,
integral matrix. As proved in [30, Theorem IV.1], such a matrix is congruent to a matrix
in skew normal form.
Theorem 5.1. [Newman] Let A be a skew-symmetric matrix of rank r which belongs
to Mn(R), where the commutative principal ideal domain R is not of characteristic 2.
Then r = 2s and A is congruent to a matrix in block diagonal form

−h1 0 0
−h2 0
. . .
0 −hs 0

where hi | hi+1, 1 ≤ i ≤ s− 1.
The same result, in the language of alternating bilinear forms, can be found in [3, Section
5.1].
The matrix S in Theorem 5.1 is clearly equivalent to the more familiar Smith normal
form, diag(h1, h1, h2, h2, . . . , hs, hs, 0, 0, . . . , 0), where the diagonal entries are the in-
variant factors of the matrix A. In the examples that follow, we outline the operations
necessary to obtain the Smith normal form.
Definition 5.2. Let A = k[x1; τ1, δ1] · · · [xn; τn, δn] and A′ = k[x1; τ1] · · · [xn; τn] be
iterated skew polynomial rings. (1) If there exists Q = (q1, . . . , qn) ∈ (k×)n such that
δiτi = qiτiδi for i = 1, . . . , n, then A is called an iterated Q-skew polynomial ring. (2) If
there exist λji ∈ k× such that τj(xi) = λjixi for all i < j, then set λij = λ−1ji and λii = 1
for all i. We call Λ = (λij) ∈ Mn(k×) the matrix of relations for A′.
Lemma 5.3. Let C be a commutative k-algebra, A a C-algebra, B ⊆ A a C-subalgebra
generated by {b1, b2, . . . }. Let τ be a C-algebra automorphism of A, and δ a u-skew
τ -derivation on A for some unit u ∈ C. If τ(bj) ∈ B and δn(bj) ∈ (n)!uB for all j, n,
then δn(B) ⊆ (n)!uB for all n.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 23
Proof. Note that τ(bj) ∈ B for all j implies that τ(B) ⊆ B and hence we have
(j)!uB
⊆ (j)!uB for all j. Suppose that for integers m ≥ 1 and 1 ≤ ℓ ≤ m − 1, we
have δi(bj1 · · · bjℓ) ∈ (i)!uB for all i, and all choices of j1, . . . , jℓ. Then
δn(bj1 · · · bjm) =
τn−iδi(bj1 · · · bj(m−1))δ
n−i(bjm)
(i)!u(n− i)!uB ⊆ (n)!uB
for all n and all j1, . . . , jm by induction. �
For a first family of examples, we take odd-dimensional quantum Euclidean spaces. The
even-dimensional ones will be covered in Example 5.4.
5.1. The coordinate ring of odd-dimensional quantum Euclidean space; Oq(ok2n+1).
For q ∈ k×, assuming q has a (fixed) square root q1/2 ∈ k, the k-algebra Oq(ok2n+1)
may be presented as an iterated skew polynomial ring
k[w][y1; σ1][x1; τ1, δ1] · · · [yn; σn][xn; τn, δn]
with automorphisms σi, τi and derivations δi defined by
σi(w) = q
−1w all i
τi(w) = qw all i
σi(yj) = q
−1yj j < i
σi(xj) = q
−1xj j < i
τi(yj) = qyj i 6= j
τi(xj) = qxj j < i
τi(yi) = yi all i
δi(w) = δi(xj) = δi(yj) = 0 j < i
δi(yi) = (q
1/2 − q3/2)w2 + (1− q2)
yℓxℓ all i.
Quantum Euclidean spaces have been studied since 1990 when they were introduced by
Reshetikhin et al. in [36]. The three-dimensional case has applications to the structure
of space-time at small distances. Musson simplified the original set of relations in [29],
and Oh further simplified them, renaming the generators ω, xi, yi in [31]. Here, we have
made a change to Oh’s variables, yi 7→ qiyi, to obtain the relations in our presentation
of Oq(ok2n+1).
Routine computations show that τ−1i δiτi(yi) = q
−2δi(yi) for all i, and so we conclude
that each (τi, δi) is a q
−2-skew derivation. We may present the analogous k[t±1]-algebra
24 HEIDI HAYNAL
Ot(ok[t±1]2n+1) as an iterated skew polynomial ring with coefficient ring k[t±1] and
generators w, yi, xi for i = 1, . . . , n,
k[t±1][w][y1; σ̄1][x1; τ̄1, δ̄1] · · · [yn; σ̄n][xn; τ̄n, δ̄n]
where the automorphisms and derivations are defined analogously to those of the algebra
Oq(ok2n+1) with t ∈ k[t±1] replacing q ∈ k×. So each (τ̄i, δ̄i) is a t−2-skew derivation. It
is immediate that
Ot(ok[t±1]2n+1)/〈t− q〉 ∼= Oq(ok2n+1)
with each τ̄i and δ̄i reducing to τi and δi respectively.
Let Aj denote the k[t
±1]-subalgebra generated by w, ym, xm for m < j, and yj. To show
that δ̄ij(Aj) ⊆ (i)!t−2Aj, we apply Lemma 5.3 noting that δ̄ij(yj) has been given for i = 1
and is zero for i > 1. So, by Theorem 2.8, each δi in our presentation of Oq(ok2n+1)
extends to an iterative, locally nilpotent h.q−2-s.τi-d. on an appropriate subalgebra.
Then Corollary 4.7 gives
FractOq(ok2n+1) ∼= FractOB(k2n+1),
where the matrix of relations is

1 q q−1 q q−1 · · · q q−1
q−1 1 1 q q−1 · · · q q−1
q 1 1 q q−1 · · · q q−1
q−1 q−1 q−1 1 1 · · · q q−1
q q q 1 1 · · · q q−1
. . .
q−1 q−1 q−1 q−1 q−1 · · · 1 1
q q q q q · · · 1 1

If q ∈ k× is a root of unity, we may assume without loss of generality that it is a
primitive rth root of unity. Then the powers of q from the matrix B become the entries
of a (2n+ 1)× (2n+ 1) integer matrix

0 1 −1 1 −1 · · · 1 −1
−1 0 0 1 −1 · · · 1 −1
1 0 0 1 −1 · · · 1 −1
−1 −1 −1 0 0 · · · 1 −1
1 1 1 0 0 · · · 1 −1
. . .
−1 −1 −1 −1 −1 · · · 0 0
1 1 1 1 1 · · · 0 0

Now, PIdegOq(ok2n+1) can be computed from Theorem 1.2(2) using the matrix B′. The
cardinality of the image will not be changed if we first perform some row reductions on
B′. Letting N = 2n+ 1, n > 2, we manipulate the rows as follows.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 25
• For i = 2, 4, 6, . . . , N − 1, replace row i with row i + row (i+ 1).
• For i = N,N − 2, N − 4, . . . , 5, replace row i with row i − row (i− 2).
• Replace row 5 with row 5 − row 1.
• For i = 2, 4, 6, . . . , N − 5, replace row i with row i − 2row (i+ 5).
• Multiply the even numbered rows, except row 2n− 2, by −1.
The resulting matrix has 2n pivots and one zero row. We put the rows in this order
3, 1, 5, 7, 2, 9, 4, 11, 6, 13, . . . , 2i, 2i+ 7, . . . , N,N − 5, N − 3, N − 1
to place the pivots on the main diagonal and the zero row in the last position. Then we
have a matrix of this form

1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 1 −1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 0 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 0 0 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 0 0 0 4 ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 0 0 0 0 1 1 ∗ ∗ ∗ ∗ ∗
0 0 0 0 0 0 4 ∗ ∗ ∗ ∗ ∗
0 0 0 0 0 0 0
. . . ∗ ∗ ∗ ∗
0 0 0 0 0 0 0 0 1 1 ∗ ∗
0 0 0 0 0 0 0 0 0 4 ∗ ∗
0 0 0 0 0 0 0 0 0 0 2 −2
0 0 0 0 0 0 0 0 0 0 0 0

The diagonal entries of this echelon matrix do not yet reveal the size of its image because
the pivot in row three does not divide all of the (suppressed) entries in its row when
n ≥ 3. So more row reduction is needed.
First replace row 3 with row 3 +
row(4i+ 2).
For n even and j = 5, 7, 9, . . . , 2n− 3, replace row j as follows:
for j = 4p+ 1, p ≥ 1, use row j +
i=p+1
2 · row(4i) + row(2n);
for j = 4p+ 3, p ≥ 1, use row j +
i=p+1
2 · row(4i+ 2).
26 HEIDI HAYNAL
For n odd and j = 5, 7, 9, . . . , 2n− 5, replace row j as follows:
for j = 4p+ 1, p ≥ 1, use row j +
i=p+1
2 · row(4i) + 2 · row(2n);
for j = 4p+ 3, p ≥ 1, use row j +
i=p+1
2 · row(4i+ 2) + row(2n).
Then add row(2n) to row(2n − 3), and add 2·row(2n) to row(2n − 1). For integers
4 ≤ j ≤ 2n − 1, with j 6≡ 2(mod 4), add (−1)jcol 3 to col j. Subtract col(2n + 1)
from col 3; add row 3 to row(2n − 2); and subtract 2·row 3 from row(2n). The result
is an upper echelon matrix in which each pivot divides all the nonzero entries in its
row. So it is trivial to diagonalize by column operations. The Smith normal form for n
odd is diag(1, 1, . . . , 1, 4, 4, . . . , 4, 0) with n+1 ones and n− 1 fours. The Smith normal
form for n even is diag(1, 1, . . . , 1, 2, 2, 4, 4, . . . , 4, 0) with n ones, two twos, and n − 2
fours.
For the cases n = 1, 2, the row-reduced matrices are, respectively,
1 0 0
0 1 −1
0 0 0

1 0 0 1 −1
0 1 −1 1 −1
0 0 2 −2 2
0 0 0 2 −2
0 0 0 0 0

Hence we have, for all n > 0,
PIdegOq(ok2n+1) =
rn, r odd
rn/2⌊
⌋, r even, r /∈ 4Z
rn/2n−1, r ∈ 4Z
5.2. The multiparameter quantized Weyl algebras; AQ,Γn (k). For a fixed n-tuple
Q = (q1, . . . , qn) ∈ (k×)n and Γ = (γij) a multiplicatively antisymmetric n × n matrix
over k, the algebra AQ,Γn (k), studied in [23] and [26], may be presented as an iterated
skew polynomial ring
k[y1][x1; τ1, δ1][y2; σ2][x2; τ2, δ2] · · · [yn; σn][xn; τn, δn]
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 27
where the automorphisms and derivations are defined by
σi(yj) = γjiyj j < i
σi(xj) = γijxj j < i
τi(yj) = qjγjiyj j < i
τi(xj) = q
j γijxj j < i
τi(yi) = qiyi all i
δi(xj) = δi(yj) = 0 j < i
δi(yi) = 1 +
(qℓ − 1)yℓxℓ all i.
Routine computations show that τ−1i δiτi(yi) = qiδi(yi) for all i, and so we conclude
that each (τi, δi) is a qi-skew derivation. We may present the k[t
1 , . . . , t
n ]-algebra
AT,Γn (k[t
1 , . . . , t
n ]) as an iterated skew polynomial ring
k[t±11 , . . . , t
n ][y1][x1; τ̄1, δ̄1][y2; σ̄2][x2; τ̄2, δ̄2] · · · [yn; σ̄n][xn; τ̄n, δ̄n]
where the automorphisms and derivations are defined analogously to those of AQ,Γn (k)
with ti ∈ k[t±11 , . . . , t±1n ] replacing qi ∈ k. So each (τ̄i, δ̄i) is a ti-skew derivation. It is
immediate that
AT,Γn (k[t
1 , . . . , t
n ])/〈t1 − q1, . . . , tn − qn〉 ∼= AQ,Γn (k)
with each τ̄i and δ̄i reducing to τi and δi respectively.
Let Aj denote the k[t
1 , . . . , t
n ]-subalgebra generated by ym, xm for m < j, and yj.
To show that δ̄ij(Aj) ⊆ (i)!tjAj, it suffices to check δ̄ij(yj) by Lemma 5.3. But this is
given by definition for i = 1 and is zero for i > 1. So, by Theorem 2.8, each δi in
our presentation of AQ,Γn (k) extends to an iterative, locally nilpotent h,qi-s.τi-d. on the
appropriate subalgebra. Then Corollary 4.7 gives FractAQ,Γn (k)
∼= FractOΛ(k2n), where
the 2n× 2n matrix of relations Λ is comprised of 2× 2 blocks
Bii =
1 q−1i
, for all i;
Bij =
γji q
i γji
γij qiγij
, for i < j;
Bij =
γji γij
qjγji q
j γij
, for i > j.
If γij and qi are roots of unity for all i, j, then OΛ(k2n) is a PI algebra. Assuming that
γij is an r
ij root of unity and that qi is an r
i root of unity, we let
r = lcm{rij, ri | i, j = 1, . . . , n}.
28 HEIDI HAYNAL
Then there exists a primitive rth root of unity q ∈ k and integers bi, bij such that qi = qbi
and γij = q
bij for i, j = 1, . . . , n. The powers of this q from the matrix Λ give a 2n× 2n
integer matrix Λ′ comprised of 2× 2 blocks
B′ii =
0 −bi
, for all i;
B′ij =
bji bji − bi
bij bij + bi
, for i < j;
B′ij =
bji bij
bj + bji bij − bj
, for i > j.
Then PIdegAQ.Γn (k) can be computed using the matrix Λ
′ in Theorem 1.2 (2).
Consider the single parameter case, denoted Aqn(k), where qi = q for all i, and γij = 1
for i < j, relegating the σi to identity maps. Assuming that q is a primitive r
th root of
unity, then δi(y
i ) = 0 and τi(y
i ) = y
i for all i, implying that y
i is central. The definition
of the τi, along with the q-Liebnitz rule, implies that x
i is central for all i. So the algebra
Aqn(k) is a finitely generated module over the central subring k[y
i , x
1, . . . , y
n]. To
find the PI degree in this case, the integer matrix becomes

0 −1 0 −1 . . . 0 −1
1 0 0 1 0 1
0 0 0 −1 0 −1
1 −1 1 0 0 1
. . .
0 0 0 0 . . . 0 −1
1 −1 1 −1 . . . 1 0

which is seen to have a trivial kernel after these row reductions:
• Replace row 2n with row 2n− row (2n− 2)− row (2n− 3)
• For j = n−1, n−2, . . . , 2, replace row 2j with row 2j−row (2j−2)−row (2j−3)
• Rearrange the rows to order 2, 1, 4, 3, 6, 5 . . . , 2n, 2n− 1.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 29
The resulting matrix has the form

0 −1 ∗
. . .
0 1 0

thus verifying that PIdegAqn(k) = r
5.3. The multiparameter coordinate ring of quantum n×nmatrices; Oλ,p
Mn(k)
The multiparameter coordinate ring of quantum n×n matrices was introduced by Artin,
Schelter, and Tate in [2]. The k-algebra Oλ,p
Mn(k)
is defined by generators xij for
i, j = 1, . . . , n and relations
xℓmxij =
pℓipjmxijxℓm + (λ− 1)pℓiximxlj (ℓ > i, m > j)
λpℓipjmxijxℓm (ℓ > i, m ≤ j)
pjmxijxℓm (ℓ = i, m > j),
where λ ∈ k× and p = (pij) ∈ Mn2(k×) is multiplicatively antisymmetric. It can also
be presented as an iterated skew polynomial ring
k[x11][x12; τ12] · · · [xij ; τij, δij ] · · · [xnn; τnn, δnn]
where each τℓm and δℓm is k-linear and satisfies
τℓm(xij) =
pℓipjmxij when ℓ > i and m 6= j
λpℓipjmxij when ℓ > i and m = j
pjmxij when ℓ = i and m > j,
δℓm(xij) =
(λ− 1)pℓiximxℓj when ℓ > i and m > j
0 otherwise.
Routine computations show τ−1ℓm δℓmτℓm(xij) = λ
−1δℓm(xij) as in [9, Section 5], and so
we conclude that each (τℓm, δℓm) is a λ
−1-skew derivation. We may present the k[t±1]-
algebra Ot,p
Mn(k[t
as an iterated skew polynomial ring with generators xij for
i, j = 1, . . . , n
k[t±1][x11][x12, τ̄12] · · · [xij ; τ̄ij, δ̄ij ] · · · [xnn; τ̄nn, δ̄nn]
30 HEIDI HAYNAL
where the automorphisms and derivations are defined analogously to those of the algebra
Mn(k)
with t ∈ k[t±1] replacing λ ∈ k. So each (τ̄ℓm, δ̄ℓm) is a t−1-skew derivation.
It is immediate that
Mn(k[t
/〈t− λ〉 ∼= Oλ,p
Mn(k)
with each τ̄ℓm and δ̄ℓm reducing to τℓm and δℓm respectively.
Let A−ℓm denote the k[t
±1]-subalgebra generated by the xij with (i, j) < (ℓ,m) in the
lexicographic order. Lemma 5.3 allows us to to verify that δ̄sℓm(A
ℓm) ⊆ (s)!t−1(A−ℓm) by
checking only that δ̄sℓm(xij) is contained in A
ℓm. This is immediate from the formula for
δ̄ℓm given above. Thus, by Theorem 2.8, each δℓm in our presentation of Oλ,p
Mn(k)
extends to an iterative, locally nilpotent h.λ−1-s.τℓm-d. on the appropriate k-subalgebra.
Then Corollary 4.7 gives
FractOλ,p
Mn(k)
) ∼= FractOΛ(kn
where the matrix of relations Λ = (bij) ∈ Mn2(k) is comprised of n× n blocks
Bii =

1 p21 p31 · · · pn1
p12 1 p32 · · · pn2
p13 p23 1 · · · pn3
. . .
p1n p2n p3n · · · 1

for all i,
Bij =

λ−1pij pijp21 pijp31 · · · pijpn1
λ−1pijp12 λ
−1pij pijp32 · · · pijpn2
λ−1pijp13 λ
−1pijp23 λ
−1pij · · · pijpn3
. . .
λ−1pijp1n λ
−1pijp2n λ
−1pijp3n · · · λ−1pij

, for i < j,
Bij =

λpij λpijp21 λpijp31 · · · λpijpn1
pijp12 λpij λpijp32 · · · λpijpn2
pijp13 pijp23 λpij · · · λpijpn3
. . .
pijp1n pijp2n pijp3n · · · λpij

, for i > j.
If λ and pij are roots of unity for all i, j, then OΛ(kn
) is a PI algebra. In this case we
may assume that λ is an sth root of unity and that pij is an r
ij root of unity, and let
r = lcm{s, rij | i, j = 1, . . . , n}. Then there exists a primitive rth root of unity q ∈ k
and integers b, bij such that λ = q
b and pij = q
bij . The powers of this q from the matrix
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 31
Λ provide entries for an n2 × n2 integer matrix Λ′ made up of n× n blocks
B′ii =

0 b21 b31 · · · bn1
b12 0 b32 · · · bn2
b13 b23 0 · · · bn3
. . .
b1n b2n b3n · · · 0

for all i,
B′ij =

bij − b bij + b21 bij + b31 · · · bij + bn1
bij + b12 − b bij − b bij + b32 · · · bij + bn2
bij + b13 − b bij + b23 − b bij − b · · · bij + bn3
. . .
bij + b1n − b bij + b2n − b bij + b3n − b · · · bij − b

, for i < j,
B′ij =

bij + b bij + b21 + b bij + b31 + b · · · bij + bn1 + b
bij + b12 bij + b bij + b32 + b · · · bij + bn2 + b
bij + b13 bij + b23 bij + b · · · bij + bn3 + b
. . .
bij + b1n bij + b2n bij + b3n · · · bij + b

, for i > j.
Then PIdegOλ,p
Mn(k)
can be calculated using Λ′ in Theorem 1.2 (2).
The single parameter quantized coordinate ring of n × n matrices, Oq(Mn(k)), is de-
fined over k analogously to Oλ,p(Mn(k)), but with relations that are recovered by setting
λ = q−2 and pij = q for all i > j. When k has characteristic zero and q is a primitive m
root of unity for m odd, Jakobsen and Zhang found in [20] that
PIdegOq(Mn(k)) = m
n(n−1)
2 by using De Concini’s and Procesi’s tool given in Theo-
rem 1.2. This result is reproved in [19] using results of De Concini and Procesi and also
Jøndrup’s work from [21]. Now we can recover PIdegOq(Mn(k) without the assumption
that k has characteristic zero.
In the single parameter case of n × n quantum matrices, the matrix that we use to
calculate the PI degree is

An In In In · · · In
−In An In In · · · In
−In −In An In · · · In
−In −In −In −In · · · An

32 HEIDI HAYNAL
where

0 1 1 1 · · · 1
−1 0 1 1 · · · 1
−1 −1 0 1 · · · 1
−1 −1 −1 · · · −1 0

is n× n and In is the n× n identity matrix.
For any n, the characteristic polynomial of An is the sum of the terms of degree ≡ n
(mod 2) in the binomial expansion of (x+1)n, so in fact χn(x) =
(x+1)n+ 1
(x− 1)n.
But there is also a recursion formula for the characteristic polynomial for n ≥ 3 given
χn(x) = χn−1(x)(x+ 1)− (x− 1)n−1,
which will be useful in the linear algebra that follows.
We will perform the following row reductions on the rows of blocks of Λ′. For ease of
notation, we’ll denote the jth row of blocks as BRj , the interchange of BRi and BRj as
BRi ↔ BRj , and the addition of a multiple of BRi to BRj as MBRi + BRj 7→ BRj ,
where M ∈ Mn(Z).
• BR1 ↔ BRn.
• −InBR1 7→ BR1.
• For i = 2, . . . , n− 1, BR1 +BRi 7→ BRi.
• BRn − AnBR1 7→ BRn.
This yields the matrix

In In In In · · · −An
0 An + In 2In 2In · · · In −An
0 0 An + In 2In · · · In −An
. . .
0 0 0 · · · An + In In −An
0 In −An In −An In −An · · · In + A2n

which can be reduced further by n − 2 block row operations, each of which produces
one zero block in the nth row. We list the first three here along with the resulting (n, n)
block.
• (An + In)BRn − (In − An)BR2 7→ BRn : A3n + 3An
• (An + In)BRn + (In −An)2BR3 7→ BRn : A4n + 6A2n + In
• (An + In)BRn − (In − An)3BR4 7→ BRn : A5n + 10A3n + 5An
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 33
In general, the block row operations that we need to perform in order to obtain a block
upper triangular matrix are:
• For i = 2, . . . , n− 1, (An + In)BRn + (−1)i−1(In − An)i−1BRi 7→ BRn.
These row operations are justified when m is odd because An + In is invertible in
Mn(Z/mZ) in that case, as will be shown below. After applying this step to the i
th row,
the (n, n) block is χi+1(An). So the resulting block upper triangular matrix is

In In In In · · · −An
0 An + In 2In 2In · · · In −An
0 0 An + In 2In · · · In −An
. . .
0 0 0 · · · An + In In −An
0 0 0 0 · · · χn(An)

where χn(An) is the n× n zero matrix. Each block on the diagonal is
An + In =

1 1 1 1 · · · 1
−1 1 1 1 · · · 1
−1 −1 1 1 · · · 1
−1 −1 −1 −1 · · · 1

which can be row reduced just by adding row 1 to rows 2 through n to yield the
matrix 

1 1 1 1 · · · 1
0 2 2 2 · · · 2
0 0 2 2 · · · 2
. . .
0 0 0 0 · · · 2

In particular this shows that An + In is invertible in Mn(Z/mZ) for m odd. Hence
Λ′ can be reduced through row operations to an upper triangular n2 × n2 matrix with
2n− 2 ones, (n− 1)(n− 2) twos, and n zeroes on the diagonal. Assuming that q ∈ k is
a primitive mth root of unity, and recalling Theorem 1.2, the cardinality of the image
in (Z/mZ)n
is mn
2−n if m is odd. Thus we conclude that PIdegOqMn(k) = m
n(n−1)
recovering the result of Jakobsen and Zhang [20] in characteristic zero. By similar
methods, one can show that PIdegOqMn(k) = m
n(n−1)
(n−1)(n−2)
2 when m is even. For
details on this result see [20] or [17].
5.4. The algebra K
n,Γ (k), which generalizes the coordinate rings of even-
dimensional quantum Euclidean space and quantum symplectic space. For
P = (p1, . . . , pn) and Q = (q1, . . . , qn) in (k
×)n with pi 6= qi for all i = 1, . . . , n, and
34 HEIDI HAYNAL
Γ = (γij) ∈ Mn(k×) multiplicatively antisymmetric, the k-algebra KP,Qn,Γ (k) introduced
in [18] is defined by generators xi, yi for i = 1, . . . , n and relations
yiyj = γijyjyi all i, j
xixj = qip
j γijxjxi i < j
xiyj = pjγjiyjxi i < j
xiyj = qjγjiyjxi i > j
xiyi = qiyixi +
(qℓ − pℓ)yℓxℓ all i.
This algebra may be presented in the form of an iterated skew polynomial ring
k[y1][x1; τ1][y2; σ2][x2; τ2, δ2] · · · [yn; σn][xn; τn, δn]
where the automorphisms τi, σi and derivations δi are defined by
σi(yj) = γijyj j < i
σi(xj) = p
i γjixj j < i
τi(yj) = qjγjiyj j < i
τi(xj) = q
j piγijxj j < i
τi(yi) = qiyi all i
δi(xj) = δi(yj) = 0 j < i
δi(yi) =
(qℓ − pℓ)yℓxℓ all i.
Routine computations show that τ−1i δiτi(yi) = qip
i δi(yi) for all i, and so we conclude
that each (τi, δi) is a qip
i -skew derivation. For ease of notation we now shall let
k = k[t±11 , . . . , t
n , u
1 , . . . , u
n ] with T = (t1, . . . , tn) ∈ k and U = (u1, . . . , un) ∈ k.
We may present the k-algebra K
n,Γ (k) as an iterated skew polynomial ring
k[y1][x1; τ̄1][y2; σ̄2][x2; τ̄2, δ̄2] · · · [yn; σ̄n][xn; τ̄n, δ̄n]
where the automorphisms and derivations are defined analogously to those of K
n.Γ (k)
with ti replacing pi and ui replacing qi. Let I ⊆ KT,Un,Γ (k) be the ideal generated by the
2n monomials ti − pi, ui − qi for i = 1, . . . , n. It is immediate that
n,Γ (k)/I
∼= KP,Qn,Γ (k),
with each τ̄i, δ̄i, σ̄i reducing to τi, δi, σi respectively.
Let Aj denote the subalgebra of K
n,Γ (k) generated by ym, xm for m < j and yj. To
show that δ̄ij(Aj) ⊆ (i)!ujt−1j Aj , it suffices to check that δ̄
j(yj) is an element of (i)!ujt−1j
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 35
by Lemma 5.3. This is given for i = 1 by the formula for δ̄j and is zero for i > 1. So,
by Theorem 2.8, each δi in our presentation of K
n,Γ (k) extends to an iterative, locally
nilpotent h.qip
i -s.τi-d. on the appropriate subalgebra. Then Corollary 4.7 gives
FractK
n,Γ (k)
∼= FractOΛ(k2n)
where the 2n× 2n matrix of relations Λ = (Bij) is comprised of 2× 2 blocks
Bii =
1 q−1i
, for all i;
Bij =
γij q
i γji
pjγji qip
j γij
, for i < j;
Bij =
γij p
i γij
qjγji q
j piγij
, for i > j.
If the qi, pi and γi are all roots of unity, then OΛ(k2n) is a PI algebra. Suppose qi is an
rthi root of unity, pi is an s
i root of unity, and γij is an r
ij root of unity for all i, j. Let
r = lcm{ri, si, γij | i, j = 1, . . . , n}. Then there extsis a primitive rth root of unity q ∈ k
and integers bi, ci, bij such that qi = q
bi , pi = q
ci, and γij = q
bij for all i, j. The powers
of q from the matrix Λ provide the entries for an integer matrix Λ′ comprised of 2 × 2
blocks
B′ii =
0 −bi
, for all i;
B′ij =
bij bji − bi
bji + cj bi + bij − cj
, for i < j;
B′ij =
bij bij − ci
bji + bj bij + ci − bj
, for i > j.
Then PIdegK
n,Γ (k) can be calculated using Λ
′ in Theorem 1.2 (2).
The coordinate ring of quantum Euclidean 2n-space over k, Oq(ok2n), is formed by
setting qi = 1, pi = q
−2 for all i, and γij = q
−1 for i < j in the parameters Q, P , and Γ
36 HEIDI HAYNAL
(see [18], Example 2.6). Then the integer matrix, Λ′, is

0 0 −1 1 −1 1 . . . −1 1
0 0 −1 1 −1 1 . . . −1 1
1 1 0 0 −1 1 −1 1
−1 −1 0 0 −1 1 ...
1 1 1 1 0 0
−1 −1 −1 −1 0 0
. . .
1 1 1 1 1 . . . 0 0
−1 −1 −1 −1 −1 . . . 0 0

We perform the following row reductions that preserve the size of the image of the
homomorphism Z2n −→ Z2n given by Λ′:
• For j = 2n, 2n− 1, 2n− 2, . . . , 4, replace row j with row j + row (j − 1)
• Replace row 2 with row 2− row 1
• Replace the (new) row 5 with row 5 + row 1
• For j = 4, 6, 8, . . . , 2n− 4, replace row j with row j + 2row (j + 3)
• For n ≥ 4, rearrange the rows to order 3, 1, 5, 7, 4, 9, 6, 11, . . . , 2i, 2i+ 5, . . . ,
2n− 4, 2n− 2, 2, 2n.
The resulting matrix has the form

0 −1 1
0 2 ∗
0 1 1
. . .
0 1 1
0 0 4
0 −2 2

When n is even, the pivot in the third row does not divide all the entries in its row,
so more elementary row and column operations are needed before it becomes clear that
the matrix can be diagonalized. By a method similar to that used in Example 5.1,
suppressed here in the interest of saving space but listed explicitly in [17], we obtain the
Smith normal form diag(1, 1, . . . , 1, 4, 4, . . . , 4, 0, 0) with n ones and n− 2 fours when n
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 37
is even; and diag(1, 1, . . . , 1, 2, 2, 4, 4, . . . , 4, 0, 0), with n− 1 ones and n− 3 fours when
n is odd. Thus we have
PIdegOq(ok2n) =
rn−1, r odd
rn−1/2⌊
⌋, r even /∈ 4Z
rn−1/2n−2, r ∈ 4Z
. (10)
The low-dimensional cases do not fit the same pattern, but the matrices for the cases
n = 2 and n = 3 are readily transformed to
1 1 0 0
0 0 −1 1
0 0 0 0
0 0 0 0
 and

1 1 0 0 −1 1
0 0 −1 1 −1 1
0 0 0 2 0 0
0 0 0 0 −2 2
0 0 0 0 0 0
0 0 0 0 0 0

respectively. Therefore, formula (10) holds for all n ≥ 2.
As a specific case ofK
n,Γ (k), quantum symplectic spaceOq(sp(k2n)) is formed by setting
qi = q
−2 and pi = 1 for all i, and γij = q for i < j (see [18], Example 2.4). With these
parameters, the 2n× 2n integer matrix Λ′ is

0 2 1 1 1 1 . . . 1 1
−2 0 −1 −1 −1 −1 . . . −1 −1
−1 1 0 2 1 1 1 1
−1 1 −2 0 −1 −1 −1 −1
−1 1 −1 1 0 2 ...
−1 1 −1 1 −2 0
. . .
−1 1 −1 1 −1 1 . . . 0 2
−1 1 −1 1 −1 1 . . . −2 0

We perform the following row reductions that preserve the size of the image of the
homomorphism Z2n −→ Z2n given by Λ′:
• For j = 2n, 2n− 1, . . . , 4, replace row j with row j − row (j − 1)
• Replace row 2 with −(row 2− 2row 3 + row 1)
• For j = 4, 6, 8, . . . , 2n− 2, replace row j with row j + 2row (j + 1)
• For n ≥ 3, order the rows 3, 1, 5, 2, 7, 4, 9, . . . , 2j, 2j + 5, . . . , 2n− 4, 2n, 2n− 2.
38 HEIDI HAYNAL
This yields a matrix whose image is more easily measured:

0 1 1 ∗
. . .
0 0 4
−2 −2

But the pivot in row 2 is problematic because it does not always divide the other
entries in its row. With further elementary row and column operations, full details
of which can be found in [17], we can bring this matrix into Smith normal form
diag(1, 1, . . . , 1, 4, 4, . . . , 4) with n ones and n fours when n is even; or the form
diag(1, 1, . . . , 1, 2, 2, 4, 4, . . . , 4) with n − 1 ones, two twos, and n − 1 fours when n
is odd.
For n = 1, 2, the row reduced matrices are, respectively,
−1 1 0 2
0 1 1 1
0 0 −4 −4
0 0 0 −4
 .
Hence we have, for all n,
PIdegOq(sp(k2n)) =
rn, r odd
rn/2⌊
⌋, r even, r /∈ 4Z
rn/2n, r ∈ 4Z
6. Prime Factor Localizations
In this section we present a structure theorem for completely prime factors of iterated
skew polynomial rings analogous to the main theorem of section four. Applying this
result to the algebras studied in section five, we’d like to strengthen it to the form
of the quantum Gel’fand-Kirillov conjecture. Recall that the assumptions about skew
polynomial rings from section one are still in effect.
Theorem 6.1. Let A = R[x; τ, δ], where R is noetherian and δτ = qτδ for some
q ∈ k×. Assume that δ extends to a locally nilpotent, iterative h.q-s.τ -d., {di}, on R.
Let P ∈ specA be completely prime. Then
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 39
(1) there exists a cyclic Ore set S in A/P such that (A/P )S−1 ∼=
R[y; τ ]/Q
Y −1 for
some completely prime Q ∈ specR[y; τ ] and cyclic Ore set Y ,
(2) FractA/P ∼= FractR[y; τ ]/Q.
Proof. The completely prime ideal P naturally satisfies one of two cases: x ∈ P or
x /∈ P . If x ∈ P , then xA ⊆ P and Ax ⊆ P . So the relation xr = τ(r)x + δ(r)
implies that δ(r) ∈ P for all r ∈ R. Hence, there is a completely prime ideal I ∈ R
such that A/P ∼= R/I ∼= R[y; τ ]/(I + 〈y〉). In this case, we can take S = Y = {1} and
localize. If x /∈ P , then xi /∈ P for all i ∈ N ∪ {0} because A/P is a domain. Letting
S = {1, x, x2, . . . }, which is a known denominator set in A, we have P ∩ S = ∅. Since
extension and contraction provide inverse bijections between the sets specAS−1 and
{I ∈ specA | I ∩ S = ∅}, we know that P e ∈ specAS−1. From Theorem 3.7, we
have AS−1 ∼= R[y±1; τ ], a localization of R[y; τ ]. So there is a completely prime ideal
Q̄⊳R[y±1; τ ] such that AS−1/P e ∼= R[y±1; τ ]/Q̄. Setting Y = {1, y, y2, . . . , }, contraction
to R[y; τ ] gives a completely prime ideal Q, where Q∩ Y = ∅, such that R[y±1; τ ]/Q̄ is
isomorphic to (R[y; τ ]/Q)Y −1. The canonical projection π : AS−1 −→ (A/P )S−1 gives
AS−1/P e ∼= (A/P )S−1. Thus (A/P )S−1 ∼=
R[y; τ ]/Q
Y −1. �
Theorem 6.2. Let R be a noetherian k-algebra, and let
A = R[x1, τ1, δ1] · · · [xn; τn, δn]
be an iterated skew polynomial ring where, for j < i and λij ∈ k×, τi(xj) = λijxj, and δi
is a qi-skew τi-derivation, qi 6= 1, which extends to a locally nilpotent, iterative h.qi-s.τi-d.
{di,p}∞p=0 on R[x1; τ1, δ1] · · · [xi−1; τi−1, δi−1] for all i. Let A′ = R[y1; τ ′1][y2; τ ′2] · · · [yn; τ ′n]
where τ ′i(yj) = λijyj for all i with j < i and the same units λij as above. Let P be a
completely prime ideal in A. Then
(1) there exists a finitely generated Ore set Sn in A/P such that (A/P )S
n is isomorphic
Y −1n for some completely prime ideal Q ⊆ A′ and finitely generated Ore set
(2) FractA/P ∼= FractA′/Q.
Proof. The case n = 1 has been established in Theorem 6.1. Suppose the result holds
for the case n− 1, and let An−1 = R[x1, τ1, δ1] · · · [xn−1; τn−1, δn−1] ⊆ A. Then we have
A = An−1[xn; τn, δn]. If xn ∈ P , then as in Theorem 6.1 there is a completely prime
ideal I ⊆ An−1 such that A/P ∼= An−1/I ∼= An−1[yn; τ ′n]/(I + 〈yn〉). The induction
hypothesis and Lemma 4.2 imply that
An−1[yn; τ
n]/(I + 〈yn〉)
S−1 ∼=
Y −1 for
some finitely generated Ore sets S and Y . Hence there is a finitely generated Ore set
Sn in A such that (A/P )S
∼= (A′/Q)Y −1. If xn /∈ P , let Sn = {1, xn, x2n, . . . } ⊆ A
and Yn = {1, yn, y2n, . . . } ⊆ An−1[yn; τn]. Then from the single-variable result, it follows
that (
An−1[yn; τ
n]/Q̄
Y −1n , (11)
40 HEIDI HAYNAL
for a completely prime ideal Q̄ ⊆ An−1[yn; τ ′n]. From Lemma 4.2, we have
An−1[yn; τ
n] = R[yn; τ
n][x1; τ
1] · · · [xn−1; τ ′n−1, δ′n−1],
which is an iterated skew polynomial ring in n − 1 variables over the coefficient ring
R[yn; τ
n] that satisfies the current assumptions. So, we apply the induction hypothesis
and rearrange variables to obtain
An−1[yn; τn]/Q̄
Y −1n
R[yn; τ
n][y1; τ
1] · · · [yn−1; τ ′n−1]/Q
R[y1; τ
1][y2; τ
2] · · · [yn; τ ′n]/Q
for a completely prime ideal Q ⊆ R[y1; τ ′1][y1; τ ′1] · · · [yn; τ ′n] and a denominator set
Z ⊆ R[y1; τ ′1][y1; τ ′1] · · · [yn; τ ′n]/Q. This, along with isomorphism (11) gives the re-
sult. �
When R is replaced by k, we have the following result.
Corollary 6.3. Let A = k[x1, τ1, δ1] · · · [xn; τn, δn], where τi(xj) = λijxj and δiτi = qiτiδi,
qi 6= 1, for λij , qi ∈ k× and all i with j < i. Assume that each δi extends to a locally
nilpotent, iterative h.qi-s.τi-d. {di,m}∞m=0 on the subalgebra k[x1; τ1, δ1] · · · [xi−1; τi−1, δi−1].
Let P be a completely prime ideal in A and set λii = 1 and λji = λ
ij . Then for
λ = (λij) ∈ Mn(k), and an appropriate completely prime ideal Q ⊆ Oλ(kn), we have
FractA/P ∼= FractOλ(kn)/Q.
We summarize how this applies to the k-algebras of quantized coordinate type.
Corollary 6.4. Let A be any of the examples discussed in sections 5.1 - 5.4, and let P be
a completely prime ideal of A. Then there exist a positive integer N , a multiplicatively
antisymmetric N ×N matrix λ over k, and a completely prime ideal Q ∈ Oλ(kN) such
that FractA/P ∼= FractOλ(kN)/Q.
To complete the question posed by the corollary, one might ask how far the quantum
Gel’fand-Kirillov conjecture extends to prime factor algebras. For instance:
Question 6.5. Find conditions under which we can conclude that for any positive in-
teger n, multiplicatively antisymmetric matrix λ ∈ Mn(k×), and completely prime ideal
Q ∈ specOλ(kn), we have
FractOλ(kn)/Q ∼= FractOp(Km)
for some field extension K ⊇ k, integer m ≤ n, and m×m matrix p over K.
The case n = 1 is trivial. When n = 2 and Q contains x1 or x2, then FractOλ(k2)/Q is
isomorphic either to FractOp(k(y)) where p = (1), or to k itself. In fact, for any n, if Q
is generated by a subset S of {x1, . . . , xn}, then the result holds, with p the submatrix
of λ formed by deleting the ith row and column for xi ∈ S, and K = k. When xi /∈ Q
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 41
for all i, answering the question fully will likely require different methods depending on
the presence of roots of unity among the λij. A positive answer in the generic case has
been provided in the proof of [13, Theorem 2.1]:
Theorem 6.6. [Goodearl - Letzter] Let k be a field, λ = (λij) a multiplicatively anti-
symmetric n× n matrix over k×, and Λ the subgroup of k× generated by the λij. If Λ
is torsionfree, then all of the prime ideals Q of Oλ(kn) are completely prime.
In their proof, they showed that FractOλ(kn)/Q ∼= FractOp(Km), and identified K as
the quotient field of a commutative domain embedded in the center of Oλ((k×)n)/Q′,
where Q′ is the prime ideal in Oλ((k×)n) induced by localization.
Quantum affine space is included in a class called quantum solvable algebras by A. N. Panov.
The main theorem of [34, Section 3], states that when the group generated by the λij is
torsionfree, then FractOλ(kn)/Q is isomorphic to the quotient division ring of a quan-
tum torus. The main theorem of [35, Section 3], allows roots of unity and states that
when Q satisfies the extra condition of being stable under a certain set of derivations,
then FractOλ(kn)/Q is isomorphic to the quotient division ring of a quantum torus.
Cauchon’s work may also be specialized to apply to quantum affine space when the
group generated by the λij is torsionfree. The result of [5, Theorem 6.1.1], indicates
that FractOλ(kn)/Q is isomorphic to FractOp(Km) which specializes to this result.
But the division ring of real quaternions provides an example showing that Question
6.5 needs to have some conditions imposed. Note that
H ∼= Oλ(R3)/Q, where λ =
1 −1 −1
−1 1 −1
−1 −1 1
 , andQ = 〈x21 + 1, x22 + 1, x23 + 1〉.
Therefore, we cannot obtain the desired isomorphism of quotient division rings in this
case, illustrating the necessity of an extra condition such as the one imposed by Panov
in [35].
acknowledgments
The author thanks her dissertation advisor, Ken Goodearl, for his direction that was so
freely given in many inspiring discussions.
References
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42 HEIDI HAYNAL
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http://arxiv.org/abs/math/9902133
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 43
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Department of Mathematics, University of California, Santa Barbara, California
93106
E-mail address : heidi@softerhardware.com
	1. Introduction
	2. Higher q-Skew -Derivations
	3. The -Derivation Removing Homomorphism
	4. Main Theorem
	5. Examples
	5.1. The coordinate ring of odd-dimensional quantum Euclidean space; Oq (o k2n+1)
	5.2. The multiparameter quantized Weyl algebras; AnQ, (k)
	5.3. The multiparameter coordinate ring of quantum n n matrices; O, bold0mu mumu pppppp(to.Mn(k))to.
	5.4. The algebra Kn, P, Q (k), which generalizes the coordinate rings of even-dimensional quantum Euclidean space and quantum symplectic space
	6. Prime Factor Localizations
	acknowledgments
	References

  For k a field of arbitrary characteristic, and R a k-algebra, we show that
the PI degree of an iterated skew polynomial ring
R[x_1;\tau_1,\delta_1]...b[x_n;\tau_n,\delta_n] agrees with the PI degree of
R[x_1;\tau_1]...b[x_n;\tau_n] when each (\tau_i,\delta_i) satisfies a q_i-skew
relation for q_i \in k^{\times} and extends to a higher q_i-skew
\tau_i-derivation. We confirm the quantum Gel'fand-Kirillov conjecture for
various quantized coordinate rings, and calculate their PI degrees. We extend
these results to completely prime factor algebras.

Introduction
Presented here is a new technique for analyzing skew polynomial rings satisfying a poly-
nomial identity with an eye toward discovering their PI degrees. It combines and extends
the methods of Jøndrup [21] and Cauchon [5], who introduced techniques of “deleting
derivations” in skew polynomial rings, by means of which they showed that some proper-
ties of certain types of iterated skew polynomial ring A = k[x1][x2; τ2, δ2] · · · [xn; τn, δn]
are determined by the corresponding ring A′ = k[x1][x2; τ2] · · · [xn; τn]. Jøndrup’s re-
sults imply that A and A′ have the same PI degree under certain hypotheses, including
characteristic zero for the base field. Cauchon developed an algorithm that gives an
isomorphism between certain localizations of A and A′, but this requires a qi-skew
condition on each (τi, δi) with qi not a root of unity, which usually precludes A from
satisfying a polynomial identity. We relax the restrictions placed on the base field and
its chosen scalars by Jøndrup and Cauchon, respectively, by introducing the notion of
a higher q-skew τ -derivation.
If we “twist” the multiplication in the (commutative) coordinate ring of affine, symplec-
tic, or Euclidean n-space over a field k, we get a (noncommutative) quantized coordinate
ring which has the structure of an iterated skew polynomial ring with coefficients in k.
This structure is also exhibited in the quantized Weyl algebras and in the quantized
coordinate ring of n×n matrices over k. Letting A represent one of these k-algebras, the
quantum Gel’fand-Kirillov conjecture asserts that FractA is isomorphic to the quotient
division ring of a quantum affine space over a purely transcendental extension of k. For
1991 Mathematics Subject Classification. 16R99; 16S36; 81R50; 16P40.
Key words and phrases. noncommutative rings; skew polynomial rings; quantum algebras.
This research will form a part of the author’s PhD dissertation at the University of California at
Santa Barbara.
http://arxiv.org/abs/0704.0846v1
2 HEIDI HAYNAL
more information on the quantum Gel’fand-Kirillov conjecture and proofs of conditions
under which the result holds, see [1] [7] [23] [28] [32] [33]. We will confirm some of these
cases in a new way.
The first section sets up the conventions under which we work, including definitions
and an established result concerning the PI degree of quantum affine space. We assume
that the reader has some familiarity with the subject, so we do not give an exhaustive
collection of definitions. A comprehensive discussion of any unfamiliar terms can be
found in [16] [4] and [27]. In the second section we define higher τ -derivations and
give necessary and sufficient conditions for their existence. Of particular interest are
higher τ -derivations which satisfy a q-skew relation. In the third section we present
a structure theorem for a localization of q-skew polynomial rings. This extends the
work of Cauchon [5], and the calculations are simplified by the presence of higher q-
skew τ -derivations. In the fourth section we deal with the structure of iterated skew
polynomial rings. Sometimes it is advantageous to rearrange the order in which the
indeterminates appear, so we establish a sufficient condition that allows such reordering.
The main theorem there asserts that if A is an iterated q-skew polynomial ring with
certain higher τ -derivations, then there is a finitely generated Ore set T ⊆ A such that
AT−1 is isomorphic to a localization of a much “nicer” iterated skew polynomial ring. In
the fifth section, we use the tools developed in the previous sections to confirm certain
cases of the quantum Gel’fand-Kirillov conjecture and to find the PI degree of some
quantized coordinate rings and quantized Weyl algebras. In the last section, we follow
up with a structure theorem for completely prime factors of iterated skew polynomial
rings. We also present an open question which, if answered positively, would show that
the quantum Gel’fand-Kirillov conjecture holds for certain of the prime factor algebras
we study.
Throughout, k will denote a field of arbitrary characteristic, q ∈ k a nonzero ele-
ment. The following assumptions apply to all skew polynomial rings that we will con-
sider:
• all coefficient rings are k-algebras
• all automorphisms are k-algebra automorphisms
• all skew derivations are k-linear
• in all skew polynomial rings R[x; τ, δ], τ is an automorphism, not just an endo-
morphism.
To say that R[x; τ, δ] is a q-skew polynomial ring means that the auomorphism and skew
derivation satisfy the relation δτ = qτδ. The reader will note that this is opposite to
Cauchon’s conventions, but it matches the presentation in [10] and others. To say that
δ is locally nilpotent means that for every r ∈ R there is an integer nr ≥ 0 such that
δnr(r) = 0, and δp(r) 6= 0 for p < nr. Such nr is called the δ-nilpotence index of r. The
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 3
symbol N refers to the set of positive integers. For a real number m we use the notation
⌊m⌋ in section five to indicate the integer part of m.
Definition 1.1. We say that two rings R and S exhibit PI degree parity when these
two conditions are satisfied:
(1) R is a PI ring if and only if S is a PI ring,
(2) PIdegR = PIdegS.
For a field k and multiplicatively antisymmetric λ ∈ Mn(k), the corresponding mul-
tiparameter quantum affine space is the k-algebra Oλ(kn) with generators x1, . . . , xn
and relations xixj = λijxjxi for all i, j. The corresponding multiparameter quantum
torus is the k-algebra Oλ((k×)n) given by generators x±11 , . . . , x±1n and the same rela-
tions. The multiplicative set generated by x1, . . . , xn in Oλ(kn) is a denominator set,
and Oλ((k×)n) is a localization of Oλ(kn) with respect to this set.
In this paper we’ll show that iterated skew polynomial algebras covering a large class
of standard examples have PI degree parity with Oλ(kn) for an appropriately chosen
λ. To find out what that PI degree may be, we utilize a result of De Concini and
Procesi. In [8, Proposition 7.1], they establish the following formula for calculating the
PI degree of a quantum affine space Oλ(kn). Their assumption of characteristic zero
from [8, Section 4] is not used in this result.
Theorem 1.2. [De Concini - Procesi] Let λ = (λij) be a multiplicatively antisymmetric
n× n matrix over k.
(1) The quantum affine space Oλ(kn) is a PI ring if and only if all the λij are roots of
unity. In this case, there exist a primitive root of unity q ∈ k× and integers aij such
that λij = q
aij for all i, j.
(2) Suppose λij = q
aij for all i, j, where q ∈ k is a primitive ℓth root of unity and the
aij ∈ Z. Let h be the cardinality of the image of the homomorphism
n (aij)−−−−−−→ Zn π−−−−→ (Z/ℓZ)n
where π denotes the canonical epimorphism. Then PI-deg (Oλ(kn)) =
2. Higher q-Skew τ-Derivations
Before the featured definition, a brief discussion of a tool used to study q-skew polyno-
mial rings is needed. Having the q-skew relation δτ = qτδ in place allows us to group
terms of the same degree when we do skew polynomial arithmetic. The means to do
this are provided by the q-Liebnitz rules.
4 HEIDI HAYNAL
Definition 2.1. For an indeterminate t, and integers n ≥ m ≥ 0, we define the following
polynomial functions:
(m)t = t
m−1 + tm−2 + · · ·+ t + 1 (1)
(m)!t = (m)t(m− 1)t · · · (1)t, and (0)!t = 1 (2)(
(n)!t
(m)!t(n−m)!t
The expressions
are called the t-binomial coefficients, or Gaussian polynomials.
The t-binomial coefficients have properties similar to those of the regular binomial
coefficients. Two that will be useful for this work are:(
= 1 for all n ≥ 0 (4)
+ tn−m
for all 0 < m < n
Proofs for these identities may be found in combinatorics texts such as [39]. When we
evaluate the t-binomial coefficients at t = q, we obtain the q-binomial coefficients that
we need for studying q-skew polynomial rings.
As shown in [10, Section 6], the following q-Liebnitz rules hold for any q-skew polynomial
ring R[x; τ, δ]:
δn(rs) =
τn−iδi(r)δn−i(s) for all r, s ∈ R and n = 0, 1, 2, ...
xnr =
τn−iδi(r)xn−i for all r ∈ R and n = 0, 1, 2, ...
Now, taking a cue from the study of Schmidt differential operator rings, for instance
[25], we define a sequence of k-linear maps that allows us to broaden the class of rings
for which we may derive results like those of Jøndrup and Cauchon.
Definition 2.2. A higher q-skew τ -derivation (h.q-s.τ -d.) on a k-algebra R is a sequence
d0, d1, d2, . . . of k-linear operators on R such that
d0 is the identity
dn(rs) =
τn−idi(r)dn−i(s) for all r, s ∈ R and all n
diτ = q
iτdi for all i.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 5
If a sequence of k-linear maps satisfies the first two conditions, we refer to it as a higher
τ -derivation. We abbreviate the sequence {di}∞i=0 usually as just {di}. A h.q-s.τ -d is
locally nilpotent if for all r ∈ R, there exists an integer n ≥ 0 such that di(r) = 0 for all
i ≥ n, and dp(r) 6= 0 for p < n. In this case, n is called the d-nilpotence index of r. A
h.q-s.τ -d is iterative if didj =
di+j for all i, j. This implies that the di commute
with each other. A q-skew τ -derivation δ on R extends to a h.q-s.τ -d. if there is a
h.q-s.τ -d {di} on R with d1 = δ.
For example, consider the k-algebra with two generators x and y, and one relation
xy − qyx = 1, where q ∈ k×. We’ll assume that q 6= 1 and recognize this algebra as
a q-skew polynomial ring k[y][x; τ, δ] with τ(y) = qy and δ(y) = 1, commonly known
as a quantized Weyl algebra and denoted A
1(k). If q is not a root of unity, then the
(i)!q
comprise an iterative higher q-skew τ -derivation that extends δ on k[y]. The prop-
erties of a higher q-skew τ -derivation follow directly from the fact that δ is a q-skew
τ -derivation and the first q-Liebnitz rule. This particular h.q-s.τ -d. is also locally
nilpotent because
yn−i when i ≤ n,
0 when i > n.
Proposition 2.3. Let {di} be a sequence of k-linear maps on a k-algebra R with
d0 = idR, and let R[[x; τ
−1]] be the skew power series ring where τ is a k-linear automor-
phism of R, the coefficients are written on the right of the variable x, and rx = xτ(r)
for all r ∈ R.
(a) Then {di} is a higher τ -derivation on R if and only if the map Ψ : R → R[[x; τ−1]]
given by r 7→
i=0 x
idi(r) is a ring homomorphism.
(b) Extend τ to an automorphism of R[[x; τ−1]] such that τ(x) = xq. Assume that
{di} is a higher τ -derivation. Then the sequence {di} is a h.q-s.τ -d. if and only if this
diagram is commutative:
R[[x; τ−1]]
// R[[x; τ−1]]
6 HEIDI HAYNAL
Proof. (a) Suppose {di} is a higher τ -derivation on R. Consider any r, s ∈ R. It is clear
that Ψ is additive and Ψ(1) = 1. Applying the definition 2.2 gives
Ψ(rs) =
xidi(rs) =
τ i−mdm(r)di−m(s)
Power series multiplication, with rx = xτ(r), gives
Ψ(r)Ψ(s) =
xidi(r)
)( ∞∑
xidi(s)
τ i−mdm(r)di−m(s)
So Ψ preserves products. Therefore, Ψ is a ring homomorphism.
To demonstrate the other implication, suppose Ψ is a ring homomorphism. Then
Ψ(r)Ψ(s) = Ψ(rs) implies that dn(rs) =
i=0 τ
n−idi(r)dn−i(s) for all r, s ∈ R. There-
fore, {di} is a higher τ -derivation.
(b) Suppose that {di} is a h.q-s.τ -d. Then the relations diτ = qiτdi imply that
τΨ(r) =
i=0 x
iqiτdi(r) =
i=0 x
idi(τ(r)) = Ψτ(r), for all r ∈ R.
Now if the diagram is commutative, then comparing the coefficients of
τΨ(r) =
i=0 x
iqiτdi(r) and Ψτ(r) =
i=0 x
idi(τ(r)) for all r ∈ R yields that
diτ = q
iτdi. �
Remark 2.4. If {di} is locally nilpotent on R, we observe that claims analogous to the
proposition can be made for the map Ψ : R → R[x; τ−1].
Proposition 2.5. Let {di} be a h.q-s.τ -d. on a k-algebra R, where τ is an automor-
phism, and let S be a right denominator set in R with τ(S) = S. Then {di} can be
uniquely extended to a h.q-s.τ -d. on RS−1.
Proof. It has been established that τ and d1 extend uniquely to RS
−1 by
τ(rs−1) = τ(r)τ(s)−1 and d1(rs
−1) = d1(r)s
−1 − τ(rs−1)d1(s)s−1 in [10, Lemma 1.3].
Suppose that {di} extends to a h.q-s.τ -d. on RS−1. For r ∈ R and s ∈ S, we apply dn
to the equation r1−1 = (rs−1)(s1−1) to get
dn(r)1
−1 = dn
(rs−1)(s1−1)
τn−jdj(rs
−1)dn−j(s1
= τn(rs−1)dn(s)1
−1 + · · ·+ dn(rs−1)s1−1.
This implies that
dn(rs
−1) =
dn(r)−
τn−jdj(rs
−1)dn−j(s)
So we have uniqueness in case of existence.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 7
To show existence, let Ψ : R → R[[x; τ−1]] be the map defined in Proposition 2.3, and
let φ : R[[x; τ−1]] → RS−1[[x; τ−1]] be the natural map. Consider the composite map
Φ = φΨ : R → RS−1[[x; τ−1]]. For any s ∈ S, the constant term of Φ(s) is a unit.
So we may inductively solve for the coefficients of an inverse for Φ(s) in RS−1[[x; τ−1]].
Details, as in [37, 1.2], are left to the reader. Hence, Φ extends to a ring homomorphism
Φ′ : RS−1 → RS−1[[x; τ−1]] such that Φ′(rs−1) = Φ(r)Φ(s)−1, and we consider the
diagram:
RS−1[[x; τ−1]]
// RS−1[[x; τ−1]]
// RS−1
where τ has been extended to an automorphism of RS−1[[x; τ−1]] as in Proposition 2.3.
Since Φ(r) =
i=0 x
idi(r)1
−1, and {di} is a h.q-s.τ -d. on R, we have
τΦ(r) =
xiqiτdi(r)1
1−1 = Φτ(r)
for all r ∈ R. It follows directly that τΦ′(rs−1) = Φ′τ(rs−1). So, indeed, the diagram is
commutative.
Define a sequence {di} on RS−1 such that di(t) equals the coefficient of xi in Φ′(t) for
all t ∈ RS−1. Then by Proposition 2.3 we conclude that this sequence is a h.q-s.τ -d. on
RS−1 extending {di} on R. �
Lemma 2.6. Let A be a k-algebra, B ⊆ A a k-subalgebra generated by {b1, b2, . . . }, τ
a k-linear automorphism of A, and {di} a higher τ -derivation on A. If di(bj) ∈ B and
τ(bj) ∈ B, for all i, j ∈ N, then di(B) ⊆ B for all i.
Proof. First, observe that τ(bj) ∈ B for all j implies that τ(B) ⊆ B. Since the di
are k-linear maps, it suffices to check monomials in the bj , using induction on their
length. Suppose, inductively, that for integers m ≥ 1 and 1 ≤ ℓ ≤ m − 1, we have
di(bj1 · · · bjℓ) ∈ B for all i and all j1, . . . , jℓ. Then using the product rule for h.q-s.τ -d.
gives
dn(bj1 · · · bjm) =
τn−1di(bj1 · · · bjm−1)dn−i(bjm) ∈ B
for all n and all j1, . . . , jm, by the induction hypothesis. �
8 HEIDI HAYNAL
Lemma 2.7. Let A be a k-algebra with a set {xj} of generators, τ an automorphism of
A, and {di} a h.q-s.τ -d. on A. If {di} is locally nilpotent for all xj, then {di} is locally
nilpotent on A.
Proof. It suffices to check monomials in the xj because the di are k-linear maps. We
proceed by using induction on the length of such monomials. For a given xn, let i(n)
be its nilpotence index, so di(xn) = 0 for all i ≥ i(n).
Suppose inductively that for n ≥ 2, all integers ℓ with 1 ≤ ℓ ≤ n− 1, and all choices
of j1, . . . , jℓ, there exists an integer m such that di(xj1 · · ·xjl) = 0 for all i ≥ m. For
instance, m = i(j1)+ · · ·+ i(jℓ) will suffice, although the d-nilpotence index of xj1 · · ·xjℓ
may be less than this sum. Then, for p ≥ m+ i(jn), we have
dp(xj1 · · ·xjn) =
τ p−idi(xj1 · · ·xjn−1)dp−i(xjn) = 0,
completing the induction. �
Consider again the quantized Weyl algebra A
1(k). In case q is an ℓ-th root of unity,
the dℓ given in (7) would be undefined due to the occurrence of a zero denominator.
However, realizing A
1(k) as a factor of a quantized Weyl algebra over k[t
±1] allows us to
define a h.q-s.τ -d. on A
1(k) nonetheless. The k[t
±1]-algebra At1(k[t
±1]) has generators
x and y and one relation xy− tyx = 1. This is a t-skew polynomial ring k[t±1][y][x; τ̄ , δ̄]
where τ̄ (y) = ty, τ̄(t) = t, δ̄(y) = 1, and δ̄(t) = 0. Note that
δ̄i(yn) =
(n)!t
(n−i)!t
yn−i when i ≤ n
0 when i > n
implying that δ̄i
k[t±1][y]
⊆ (i)!tk[t±1][y]. So the assignment
d̄i =
(i)!t
defines an iterative, locally nilpotent h.t-s.τ̄ -d. {d̄i} on k[t±1][y]. Now, the relation
xy − tyx = 1 is equivalent to the relation xy − qyx = 1 modulo 〈t − q〉. Hence we
k[t±1]
/〈t− q〉 ∼= Aq1(k).
When q is an ℓth root of unity, we have δ̄ℓ
k[t±1][y]
⊆ 〈t−q〉k[t±1][y]. Nonetheless, the
h.t-s.τ̄ -d. {d̄i} on k[t±1][y] induces a h.q-s.τ -d. {di} on k[y], also iterative and locally
nilpotent, with d1 = δ. Note that even though δ
ℓ = 0 in this algebra, we have di(y
i) = 1
for all i.
This phenomenon is not unique to the quantized Weyl algebras. The conditions that
drive it are codified in the following theorem.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 9
Theorem 2.8. Let R be a k-algebra and R[x; τ, δ] a q-skew polynomial ring where
q ∈ k, q 6= 1. Suppose there exists a torsion-free k[t±1]-algebra R and R[x; τ̄ , δ̄] a t-skew
polynomial ring such that R/〈t− q〉R ∼= R, with τ̄ and δ̄ reducing to τ and δ. Suppose
further that δ̄i(R) ⊆ (i)!tR for all i. Then δ extends to an iterative h.q-s.τ -d. {di} on
R. If δ̄ is locally nilpotent, then so is {di}. If q is not a root of unity, then di = δ
(i)!q
all i. If q is a primitive ℓth root of unity, then di =
(i)!q
for i < ℓ.
Proof. The assumption δ̄i(R) ⊆ (i)!tR for all i implies that the sequence of maps
d̄i =
(i)!t
make up a well-defined iterative h.t-s.τ̄ -d. on R, and also implies that
δ̄ℓ(R) ⊆ 〈t − q〉R because (ℓ)t ≡ (ℓ)q = 0 modulo 〈t − q〉. Since τ̄ and δ̄ reduce to
τ and δ modulo 〈t− q〉, we have an isomorphism R/〈t− q〉[x; τ̄ , δ̄] ∼= R[x; τ, δ] whereby
{d̄i} induces an iterative h.q-s.τ -d. {di} on R. The reduction of the maps from R to R
also implies the remaining results. �
We will find that all of the conditions assumed above are satisfied by the common
quantized coordinate rings and related examples, which will be discussed in a subsequent
section.
3. The τ-Derivation Removing Homomorphism
Following the pattern in [5], let A = R[x; τ, δ], and suppose that δ is locally nilpotent.
Set S = {xn | n ∈ N ∪ {0}} ⊂ A.
Lemma 3.1. The set S is a denominator set in A.
Proof. Clearly, S is a multiplicative set inA. And, since S contains only regular elements
of A, it is left and right reversible. It remains to show that S is an Ore set.
Let a =
i=0 rix
i be an element of A with rn 6= 0. For each ri in the expression of a,
and each mi ≥ 0, we have
xmiri =
τmi−jδj(ri)x
= a′ix+ δ
mi(ri) for some a
i ∈ A.
Since δ is locally nilpotent, we may choose mi to be the δ-nilpotence index of ri to
conclude that xmiri = a
ix for some a
i ∈ A. Set ma = max{mi | 0 ≤ i ≤ n}. Then for
each ri, we have x
mari = ãix, and hence x
maa = ãx for some ã ∈ A.
Now suppose, inductively, that for a given a ∈ A and xp ∈ S we can find elements
xma ∈ S and ā ∈ A such that xmaa = āxp, say ā =
i=0 r̄ix
i. We know that there
10 HEIDI HAYNAL
exists an element xmā such that xmā ā = a′x for some a′ ∈ A. So, xmaa = āxp implies
xmā+maa = a′xp+1, completing the induction.
Hence, for any a ∈ A and s ∈ S, we have Sa∩As 6= ∅. So S is a left Ore set in A. We see
that S is a right Ore set by applying the same argument to Aop = Rop[x; τ−1,−δτ−1]. �
Suppose also that the derivation δ extends to an iterative, locally nilpotent higher q-
skew τ -derivation {di} on R and that q 6= 1. Denote Â = AS−1 = S−1A, the localization
of A with respect to S, and define a map f : R −→ Â by
f(r) =
n(n+1)
2 (q − 1)−ndnτ−n(r)x−n,
noting that {di} is locally nilpotent and that q − 1 is invertible. If q is not a root of
unity and {di} is obtained from a q-skew τ -derivation δ as in (7), the formula for f can
be rewritten as
f(r) =
n(n+1)
(q − 1)−n
(n)!q
δnτ−n(r)x−n.
The rewritten formula matches the one presented in [5, Section 2] when q is replaced by
q−1 to account for the difference between δτ = qτδ (used here) and τδ = qδτ (used in
[5]). We will show that f is a homomorphism and that the the multiplication in imf is
made simpler than that in A by removing the derivation, as seen in the following.
Proposition 3.2. If r ∈ R, then xf(r) = f
x in Â.
Proof. Using the hypothesis that {di} is iterative, we compute that
xf(r) =
n(n+1)
2 (q − 1)−nxdnτ−n(r)x−n
n(n+1)
2 (q − 1)−n
−n(r)x+ d1dnτ
−n(r)
n(n+1)
2 (q − 1)−nq−ndnτ−n+1(r)x−n+1
n(n+1)
2 (q − 1)−n(n+ 1)qdn+1τ−n(r)x−n
n(n+1)
2 (q − 1)−nq−ndnτ−n(τ(r))x−n+1
n(n−1)
2 (q − 1)−n+1(n)qdnτ−n(τ(r))x−n+1
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 11
= τ(r)x
n(n+1)
2 (q − 1)−nq−n + q
n(n−1)
2 (q − 1)−n+1(n)q
−n(τ(r))x−n+1
= τ(r)x
(q − 1)−n
2 + q
2 (qn − 1)
−n(τ(r))x−n+1
= τ(r)x+
(q − 1)−nq
n(n+1)
2 dnτ
−n(τ(r))x−n+1
n(n+1)
2 (q − 1)−ndnτ−n(τ(r))x−n
x = f
which gives the result. �
From Proposition 3.2, it follows by routine induction that
xmf(r) = f
τm(r)
xm ∀m ∈ Z. (8)
This is what we need in order to show that our map is indeed a k-algebra homomor-
phism.
Proposition 3.3. The map f : R −→ Â is a k-algebra homomorphism.
Proof. It is immediate that f is k-linear (τ and {di} are k-linear), and that f(1) = 1.
We’ll show that f is multiplicative. If r, s ∈ R, then using Prop. 3.2,
f(r)f(s) =
i(i+1)
2 (q − 1)−idiτ−i(r)x−if(s)
i(i+1)
2 (q − 1)−idiτ−i(r)f(τ−i(s))x−i
i≥0, j≥0
i(i+1)+j(j+1)
2 (q − 1)−(i+j)diτ−i(r)djτ−(i+j)(s)x−(i+j).
For n ∈ N, the coefficient of x−n in the sum above is
i≥0, j≥0,
i+j=n
i(i+1)+j(j+1)
2 (q − 1)−ndiτ−i(r)djτ−n(s)
(n−p)2+p2+n
2 (q − 1)−ndn−pτ p−n(r)dpτ−n(s)
12 HEIDI HAYNAL
n(n+1)
2 (q − 1)−n
qp(p−n)dn−pτ
pτ−n(r)dpτ
−n(s)
n(n+1)
2 (q − 1)−n
τ pdn−p(τ
−n(r))dp(τ
−n(s))
n(n+1)
2 (q − 1)−ndn(τ−n(r)τ−n(s))
n(n+1)
2 (q − 1)−ndnτ−n(rs),
computed by putting p = j and using the second condition in the Definition 2.2. In
summary, f(r)f(s) =
n=0 q
n(n+1)
2 (q − 1)−ndnτ−n(rs)x−n = f(rs). �
Proposition 3.4. (1) The map f extends uniquely to an algebra homomorphism, also
denoted f , of R[y; τ ] to Â satisfying f(y)=x.
(2) The extended homomorphism is injective.
Proof. (1) This result follows from Proposition 3.2 and the universal property of Ore
extensions.
(2) Let P = pmy
m + · · ·+ p1y + p0 be a nonzero element of R[y; τ ], where each pi ∈ R,
m ≥ 0, pm 6= 0. Then f(P ) = f(pm)xm + · · ·+ f(p1)x+ f(p0). Since
f(pi) =
n(n+1)
2 (q − 1)−ndnτ−n(pi)x−n ∈ AS−1,
we know that there exists an integer l ≥ 0 such that each f(pi)xl is a nonzero element
of A of positive degree l (in x) whenever pi 6= 0. (Because {di} is locally nilpotent,
we may choose an l large enough.) It follows that f(P )xl is a nonzero element of Â of
degree m+ l, hence f(P ) 6= 0. �
Definition 3.5. The algebra homomorphism f : R[y; τ ] −→ Â = AS−1 is called the
derivation removing homomorphism. The image of f , call it A′, is the subalgebra of
Â = AS−1 generated by x and f(R), and is isomorphic (as an algebra) to R[y; τ ] by the
derivation removing homomorphism f .
Observe that A′ contains the multiplicative system S = {xn | n ∈ N ∪ {0}}. Since
equation (8) holds and f(y) = x, the elements of this set are normal in A′. Hence,
S satisfies the (two-sided) Ore condition in A′. The elements of S are regular in A′
because they are regular in Â, and thus:
Proposition 3.6. A′S−1 = AS−1
Proof. We have A′S−1 ⊆ AS−1 because A′ = im(f) ⊆ AS−1. To show the other
inclusion, it suffices to show that R ⊆ A′S−1. (This suffices because A is built up from
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 13
R by x, x2, . . . . So if R ⊆ A′S−1, then AS−1 ⊆ A′S−1.) Consider any r ∈ R and let ℓ
be the d-nilpotence index of r. We show that r ∈ A′S−1 with an induction argument
on ℓ.
If ℓ ≤ 1, then d1(r) = 0, whence f(r) = r ∈ A′ ⊆ AS−1.
If ℓ ≥ 2, we write
f(r) = r +
−n, with rn = q
n(n+1)
2 (q − 1)−ndnτ−n(r) ∈ R.
We’ll show that
n=1 rnx
−n ∈ A′S−1 in order to conclude that r ∈ A′S−1, because
f(r) −
n=1 rnx
−n = r. That is, we need to show that each rn ∈ A′S−1. Suppose,
inductively, that for any element r̃ ∈ R with d-nilpotence index m such that m < ℓ, we
have r̃ ∈ A′S−1.
Note that for n ∈ {1, . . . , ℓ}, we have
dℓ−n(rn) = q
n(n+1)
2 (q − 1)−n
−n(r) = q
n(n+1)
2 (q − 1)−n
q−nℓτ−ndℓ(r) = 0
because dℓ(r) = 0 by hypothesis.
Hence, by the induction hypothesis, each rn ∈ A′S−1 for 1 ≤ n ≤ ℓ− 1. It follows that
r = f(r)−
n=1 rnx
−n also belongs to A′S−1. �
This equality of quotient rings reveals that if A is a PI ring, then
PIdegA = PIdegA′ = PIdegR[y; τ ],
with the second equality arising from the derivation removing homomorphism f . This
recovers the result of Jøndrup [21] without the assumption that k has characteristic
zero. We summarize the results of this section in the following theorem.
Theorem 3.7. Let k be a field, R a k-algebra and A = R[x; τ, δ] a q-skew polynomial
ring in which δ extends to a locally nilpotent, iterative h.q-s.τ -d. {di} on R for some
q ∈ k×, q 6= 1. Let S be the Ore set in A generated by x, and define a map
f : R −→ AS−1 by f(r) =
n=0 q
n(n+1)
2 (q − 1)−ndnτ−n(r)x−n. Then f is a k-algebra
homomorphism, and it extends to an injective homomorphism f : R[y; τ ] −→ AS−1
sending y to x. Furthermore, the extension f : R[y±1; τ ] −→ AS−1 is an isomorphism.
So there is PI degree parity between A and R[y; τ ]. Moreover, if R is a noetherian
domain, then FractA ∼= FractR[y; τ ].
14 HEIDI HAYNAL
4. Main Theorem
In the case where A is an iterated skew polynomial ring, we would like to apply re-
peatedly the method presented above to remove all of the derivations and compare the
resulting Ore localizations. We must first establish some facts about the behavior of
h.q-s.τ -d. when the variables adjoined to the coefficient ring are rearranged, and about
iterated localization. The results of these lemmas will ensure that after the induction
step in the proof of the main theorem we are left with a ring to which the method of
the preceding section applies.
The first parts of the following lemmas hold in a broader class of skew polynomial rings
and also when the q-skew condition is imposed. The final parts assert that h.q-s.τ -d.
are preserved when rearranging of the variables is permissible.
Lemma 4.1. Let S = R[x; τ, δ], A = R[x; τ, δ][y; σ], and Â = R[x; τ, δ][y±1; σ], where
σ(R) = R and σ(x) = λx for some λ ∈ k×.
(1) Then A = R[y; σ′][x; τ ′; δ′], and Â = R[y±1; σ′][x; τ ′; δ′], where σ′ = σ
, τ ′
= τ ,
= δ, τ ′(y) = λ−1y, and δ′(y) = 0.
(2) If (τ, δ) is q-skew, then so is (τ ′, δ′).
(3) Suppose further that δ extends to a h.q-s.τ -d. {di} on R, and that σdi = λidiσ for
all i. Then the τ ′-derivation δ′ extends to a h.q-s.τ ′-d. {d′i} on R[y±1; σ′] such that the
restrictions of the d′i to R coincide with di, and d
i(y) = 0 for all i ≥ 1. Moreover, {d′i}
restricts to a h.q-s.τ ′-d. on R[y; σ′].
(a) If {di} is iterative, then {d′i} is iterative.
(b) If {di} is locally nilpotent, then {d′i} is locally nilpotent.
Proof. (1) Routine details omitted so as not to try the patience of the reader.
(2) Suppose that (τ, δ) is q-skew on R. We’ll check that the two τ ′-derivations τ ′−1δ′τ ′
and qδ′ agree on R[y±1; σ′]. It suffices to check their agreement on a set of generators,
R ∪ {y, y−1}. It is clear that τ ′−1δ′τ ′(r) = qδ′(r) for all r ∈ R. Since δ′(y) = 0, they
agree on {y, y−1} as well. So (τ ′, δ′) is q-skew.
(3) Define a sequence of maps d′i : R[y
±1; σ′] → R[y±1; σ′] by
di(rj)y
Clearly these are k-linear maps, d′i(r) = di(r) for all r ∈ R; also d′i(y) = di(1)y = 0 for
i ≥ 1, and d′0 is the identity on R[y±1; σ′].
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 15
Because δ extends to {di} on R, we get
d1(rj)y
δ(rj)y
j = δ′
for all rj ∈ R. So d′1 = δ′ on R[y±1; σ′].
Now, for integers j,m, n, and elements r, s ∈ R,
(ryj)(sym)
= d′n
rσj(s)yj+m
rσj(s)
τn−idi(r)dn−i(σ
j(s))yj+m
τn−idi(r)y
jσ−jdn−i(σ
j(s))ym
τn−idi(r)y
jλ−j(n−i)dn−i(s)y
(τ ′)n−i
di(r)y
d′n−i(sy
(τ ′)n−id′i(ry
j)d′n−i(sy
So {d′i} satisfies the product rule for a higher τ -derivation on R[y±1; σ′].
Furthermore,
τ ′d′i
= τ ′
di(rj)y
τdi(rj)λ
−jyj,
and d′iτ
= d′i
τ(rj)λ
diτ(rj)λ
τdi(rj)λ
−jyj,
giving the q-skew relation d′iτ
′ = qiτ ′d′i on R[y
±1; σ′].
It follows directly from the definition of the maps {di} that their restrictions to the
k-subalgebra R[y; σ′] also exhibit the properties of definition 2.2.
16 HEIDI HAYNAL
If {di} is iterative on R, then d′ℓd′i(rym) = d′ℓ
di(r)y
= dℓdi(r)y
dℓ+i(r)y
d′ℓ+i(ry
m) for all r ∈ R, m ∈ Z, and non-negative integers ℓ, i. Hence, {d′i} is
iterative on R[y±1; σ′].
Suppose that {di} is locally nilpotent on R. By Lemma 2.7 we need only check that
{d′i} is locally nilpotent on R ∪ {y, y−1}, a set of generators for R[y±1; σ′]. This is clear
because d′i(r) = di(r) for all r ∈ R, and d′i(y) = 0 for all i by construction. �
Lemma 4.2. Let
A = R[x1; τ1, δ1][x2; τ2, δ2] · · · [xn; τn, δn][y; σ],
Â = R[x1; τ1, δ1][x2; τ2, δ2] · · · [xn; τn, δn][y±1; σ],
where σ(R) = R, and for all i ∈ {1, . . . , n}, σ(xi) = λixi for some nonzero λi ∈ k. Let
Aj = R[x1; τ1; δ1][x2; τ2, δ2] · · · [xj ; τj, δj ] for j = 1, 2, . . . , n, and A0 = R.
(1) Then
A = R[y; σ∗][x1; τ
1][x2; τ
2] · · · [xn; τ ′n, δ′n],
Â = R[y±1; σ∗][x1; τ
1][x2; τ
2] · · · [xn; τ ′n, δ′n],
where σ∗ = σ
, τ ′i
= τi, δ
= δi, τ
i(y) = λ
i y, and δ
i(y) = 0 for all 1 ≤ i ≤ n
and j ≤ i− 1.
(2) If (τi, δi) is qi-skew for any 1 ≤ i ≤ n, then (τ ′i , δ′i) is also qi-skew.
(3) Suppose that each δi extends to an h.qi-s.τi-d. {di,p}∞p=0, and that σdi,p = λ
i di,pσ
on Ai−1 for all i and p. Then each δ
i extends to a h.qi-s.τ
i -d. {d′i,p}∞p=0 on the algebra
R〈y, y−1, x1, . . . , xi−1〉, where d′i,p coincides with di,p on Aj, for j < i, and d′i,p(y) = 0
for p ≥ 1. Moreover, {d′i,p} restricts to a h.qi-s.τ ′i -d. on R〈y, x1, . . . , xi−1〉.
(a) If {di,p} is iterative for any 1 ≤ i ≤ n, then {d′i,p} is iterative.
(b) If {di,p} is locally nilpotent for any 1 ≤ i ≤ n, then {d′i,p} is locally nilpotent.
Proof. (1) The condition σ(xi) = λixi for all i implies that σ(Ai) = Ai. We will use
induction on n to prove the result.
Lemma 4.1 proves the case n = 1. Suppose the result holds for all m < n, and consider
A = An−1[xn; τn, δn][y; σ]. Application of Lemma 4.1, and then the induction hypothesis,
gives
A = An−1[xn; τn, δn][y; σ]
= An−1[y; σ
′][xn; τ
= R[x1; τ1, δ1] · · · [xn−1; τn−1, δn−1][y; σ′][xn; τ ′n, δ′n]
= R[y; σ∗][x1; τ
1] · · · [xn; τ ′n, δ′n],
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 17
with the desired conditions met by the automorphisms and derivations, completing the
induction. Similarly, Â = R[y±1; σ∗][x1; τ
1] · · · [xn; τ ′n, δ′n].
(2) Consider the two τ ′i -derivations τ
i and qiδ
i on the ring
R[y±1; σ∗][x1; τ
1] · · · [xi−1; τ ′i−1, δ′i−1]
for 1 ≤ i ≤ n. Since (τi, δi) is q-skew, it is clear that these two τ ′i derivations agree on
Ai−1. And since δ
i(y) = 0 for all i = 1, . . . , n, these two τ
i -derivations agree on a full
set of generators of R[y±1; σ∗][x1; τ
1] · · · [xi−1; τ ′i−1, δ′i−1]. Hence, δ′iτ ′i = qiτ ′iδ′i.
(3) Suppose the result holds for the algebra R[x1; τ1, δ1] · · · [xn−1; τn−1, δn−1][y±1; σ].
Then Lemma 4.1 may be applied, with An−1 providing the coefficients, to get
An−1[xn; τn, δn][y
±1; σ] = An−1[y
±1; σ′][xn; τ
where δ′n extends to a h.qn-s.τ
n-d. {d′n,p} on An−1[y±1]. The induction hypothesis gives
the result. �
Definition 4.3. For a k-algebra A and a, b ∈ A, we say that a and b scalar commute
if there is an element α ∈ k× such that ab = αba. We may also say that a and b
α-commute.
In the following two lemmas, we let D denote the division ring of fractions for the
noetherian domain A. When comparing localizations of A, we identify them as subrings
of D.
Lemma 4.4. Let A be a noetherian domain, S ⊆ A \ {0} an Ore set. Let T be an Ore
set in AS−1 \ {0} with S ⊆ T .
(1) Then there exists an Ore set T̃ ⊆ A\{0} with S ⊆ T̃ such that AT̃−1 = (AS−1)T−1.
(2) Suppose A is a k-algebra and S is generated by s1, . . . , sn satisfying sisj = γijsjsi
for all i, j and some γij ∈ k×. Further suppose that T is generated by S ∪ t for some
t ∈ AS−1 that satisfies sit = λitsi for all i and some λi ∈ k×. Then there exist a cyclic
Ore set T̂ ⊆ A \ {0} and an (n + 1)-generator Ore set Ŝ ⊆ A \ {0} such that S ⊆ Ŝ,
and (AS−1)T−1 = AT̂−1 = AŜ−1.
Proof. (1) Consider T ∩A, the subset in T of elements with a denominator of 1. Clearly,
this is a multiplicative set in A which contains S. Set T̃ = T ∩A. Let a ∈ T̃ and α ∈ A.
Then a ∈ T , and since α ∈ AS−1, there exist b′ ∈ T and β ′ ∈ AS−1 such that aβ ′ = αb′.
By [16, 10.2], there exist y ∈ S, and b, β ∈ A such that β ′ = βy−1 and b′ = by−1; hence,
aβy−1 = αby−1 in AS−1. It follows that aβ = αb in A. So T̃ satisfies the right Ore
condition in A, and the left Ore condition by symmetry. By the universal property,
AT̃−1 ∼= (AS−1)T−1. As subrings of D, we have AT̃−1 = (AS−1)T−1.
18 HEIDI HAYNAL
(2) The generating element t has the form t = ā(sm11 s
2 · · · smnn )−1 for some mi ∈ N,
and ā ∈ A. For any si ∈ S, we have
siā(s
2 · · · smnn )−1 = λiā(sm11 sm22 · · · smnn )−1si = µλiāsi(sm11 sm22 · · · smnn )−1,
where µ is a product of powers of the γij. So ā scalar commutes with the genera-
tors of S via the relations siā = µλiāsi. Let Ŝ be the multiplicative set generated by
ā, s1, . . . , sn in A, and T̂ the multiplicative set generated by ās1s2 · · · sn in A. Recall
that (AS−1)T−1 = AT̃−1, where T̃ = T ∩ A from part (1). From the scalar com-
muting relations it follows that any element at̃−1 ∈ AT̃−1 may be written in the form
b(ās1, · · · sn)−m for some m ∈ N ∪ {0}, b ∈ A, or the form cā−ℓn+1s−ℓ11 · · · s−ℓnn , for
ℓj ∈ N ∪ {0}, c ∈ A. So we conclude that Ŝ and T̂ are Ore sets in A and that
(AS−1)T−1 = AT̂−1 = AŜ−1. �
Lemma 4.5. Let A be a noetherian domain, S1 ⊆ A \ {0} an Ore set, and for integers
j = 2, . . . , n let Sj be an Ore set in ((AS
1 ) · · · )S−1j−1 \ {0} with Sj−1 ⊆ Sj.
(1) Then there exists an Ore set T ⊆ A \ {0} such that AT−1 = (((AS−11 )S−12 ) · · · )S−1n .
(2) Suppose A is a k-algebra, S1 is generated by s1, and for j = 2, . . . , n, Sj is generated
by Sj−1 ∪ {sj}, where sisj = γijsjsi for some multiplicatively antisymmetric matrix
(γij) ∈ Mn(k×). Then there are a cyclic Ore set T̂ ⊆ A and an n-generator Ore set
Ŝ ⊆ A such that S1 ⊆ Ŝ, and ((AS−11 )S−12 ) · · ·S−1n = AT̂−1 = AŜ−1.
Proof. (1) The proof proceeds by induction on n. The case n = 1 is covered in the
lemma above. Suppose that for all j ≤ n− 1 there exists an Ore set Tj ⊆ A \ {0} such
that AT−1j = (((AS
2 ) · · · )S−1j . Then the equality
AT−1n−1 = (((AS
2 ) · · · )S−1n−1
identifies an Ore set Tn ⊆ AT−1n−1 \ {0} such that
(AT−1n−1)T
n = (((AS
2 ) · · ·S−1n−1)S−1n .
Furthermore, Lemma 4.4 implies the existence of an Ore set T ⊆ A \ {0} such that
AT−1 = (AT−1n−1)T
n = (((AS
2 ) · · ·S−1n−1)S−1n .
(2) Suppose, inductively, that there exist
(i) a cyclic Ore set T̂n−1 ⊆ A \ {0} generated by s1ā2 · · · ān−1
(ii) an (n− 1)-generator Ore set Ŝn−1 ⊆ A \ {0} with S1 ⊆ Ŝn−1 and generators
s1, ā2, ā3, . . . , ān−1
(iii) the āi scalar commute with s1 and with each other
(iv) ((AS−11 )S
2 ) · · ·S−1n−1 = AT̂−1n−1 = AŜ−1n−1 as subrings of D.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 19
Then sn = ān(s1ā2 · · · ān−1)−r for some ān ∈ A and r ∈ N. Using the relations
sisj = γijsjsi, routine calculations show that the āi scalar commute with the sj ,
and also with each other, for all i, j. Let T̂ be the multiplicative set generated by
s1ā2 · · · ān, and let Ŝ be the multiplicative set generated by s1, ā2, ā3, . . . , ān. Then
((AS−11 )S
2 ) · · ·S−1n = (AT̂−1n−1)S−1n = AT−1 from part (1). Using Lemma 4.4, we con-
clude that T̂ and Ŝ are Ore sets in A and that AT−1 = AT̂−1 = AŜ−1. �
In the proof of the main theorem, we will use without mention the facts gathered here.
For greater details on these statements, see [16, 10X, 10Y] and [10, 1.4].
(1) Given a noetherian ring A and a normal element x ∈ A, the multiplicative set
generated by x is an Ore set.
(2) The multiplicative set generated by a nonempty family of right Ore sets is right
(3) Let A = R[x; τ, δ], and S a right denominator set in R such that τ(S) = S.
Then S is a right denominator set in A and the identity map on AS−1 extends
to an isomorphism of AS−1 onto (RS−1)[x; τ, δ] sending x1−1 to x. Note that
if A is a k-algebra, τ , δ are k-linear, and τ(k×S) = k×S, then the result holds
because S is a denominator set if and only if k×S is a denominator set.
Theorem 4.6. Let R be a k-algebra and noetherian domain,
A = R[x1; τ1, δ1] · · · [xn; τn, δn],
where each τi is a k-linear automorphism of R〈xi, . . . , xi−1〉 such that τi(xj) = λijxj
for all i, j with 1 ≤ j < i ≤ n and some λij ∈ k×, and where each δi is a k-linear τi-
derivation. Assume that there exist elements qi ∈ k× with qi 6= 1 such that δiτi = qiτiδi,
and that δi extends to a locally nilpotent, iterative h.qi-s.τi-d. on R〈xi, . . . , xi−1〉 for
i = 1, . . . , n.
(1) Then there exists an Ore set T ⊆ A generated by n elements of A such that
AT−1 ∼= R[y±11 ; τ1][y±12 ; τ ′2] · · · [y±1n ; τ ′n]
where τ ′i |R = τi and τ ′i(yj) = λijyj for all i, j with 1 ≤ j < i ≤ n
(2) There is PI degree parity between A and R[y1; τ1][y2; τ
2] · · · [yn; τ ′n]. Moreover, these
algebras have isomorphic division rings of fractions.
Proof. (a) Suppose, inductively, that we have
R[x1; τ1, δ1][y
2 ; τ2] · · · [y±1n ; τ ′n] ∼= AS−12
where the restriction of τ ′i to R〈x1〉 coincides with τi, τ ′i(ym) = λimym for 2 ≤ i ≤ n and
1 < m < i, and S2 is an Ore set in A generated by n − 1 elements from A. Then by
20 HEIDI HAYNAL
Lemma 4.2
AS−12
∼= R[y±12 ; τ ′′2 ] · · · [y±1n ; τ ′′n ][x1; τ ′1, δ′1] (9)
where the restrictions of τ ′1 and δ
1 to R coincide with τ1 and δ1, τ
1(yj) = λ
j1 yj, δ
1(yj) =
0, and τ ′′i coincides with the restriction of τi to R〈y2, . . . , yi−1〉 for 2 ≤ i ≤ n. Observe
that by Lemmas 4.2 and 2.7 we also have δ′1τ
1 = q1τ
1, and that δ
1 extends to a
locally nilpotent iterative h.q1-s.τ -d. on R〈y±12 , . . . , y±1n 〉. Then applying the derivation
removing homomorphism to the right hand side of (9) gives an isomorphism
(AS−12 )T
∼= R[y±12 ; τ ′2] · · · [y±1n ; τ ′n][y±11 ; τ ′1]
where T1 ⊆ AS−12 is an Ore set generated by one element of AS−12 . Then Lemma 4.5
and a reordering of variables shows the existence of an Ore set T ⊆ A, generated by n
elements of A, such that AT−1 ∼= R[y±11 ; τ1][y±12 ; τ ′2] · · · [y±1n ; τ ′n].
(2) This follows from part (1). �
Corollary 4.7. Let A = k[x1; τ1, δ1] · · · [xn; τn, δn] with the hypotheses as in Theorem
4.6. Set λ = (λij). Then
(1) A and Oλ(kn) have isomorphic division rings of fractions.
(2) A is a PI-algebra if and only if all the λij are roots of unity, in which case A and
Oλ(kn) have the same PI degree.
In general, identification of the generators for the Ore set T in Theorem 4.6 is very
cumbersome. To illustrate the computations on a fairly short iterated skew polynomial
ring, we consider the multiparameter second quantized Weyl algebra A
2 (k). Here,
Q = (q1, q2) ∈ (k×)2, qi 6= 1 for all i, and Γ = (γij) ∈ M2(k×) with γii = 1 and
γ21 = γ
12 . The algebra A
2 (k) may be presented as an iterated skew polynomial ring
of the form k[y1][x1; τ2, δ2][y2; τ3][x2; τ4, δ4], where the τi are k-linear automorphisms and
the δ2i are k-linear τ2i-derivations such that
τ2(y1) = q1y1, δ2(y1) = 1
τ3(y1) = γ
12 y1
τ3(x1) = γ12x1
τ4(y1) = q1γ12y1, δ4(y1) = 0
τ4(x1) = q
1 γ21x1, δ4(x1) = 0
τ4(y2) = q2y2, δ4(y2) = (q1 − 1)y1x1 + 1.
For greater detail about this algebra, the reader is referred to [1], [23], [12], and [15].
Routine computations show that the pair (τ2, δ2) is a q1-skew derivation and that (τ4, δ4)
is a q2-skew derivation. To show that δ2 and δ4 are locally nilpotent, it suffices to check
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 21
for local nilpotence on a set of generators. Given their definitions, this is accomplished
by verifying their action on powers of y1 and y2:
δi2(y
1 ) =
(n)!q1
(n−i)!q1
yn−i1 i ≤ n
0 i > n
δi4(y
2 ) =
(n)!q2
(n−i)!q2
[δ4(y2)]
iyn−i2 i ≤ n
0 i > n
Using Theorem 2.8 we have a h.q1-s.τ2-d. {d2,i} extending δ2, and a h.q2-s.τ4-d. {d4,i}
extending δ4, both of which are iterative and locally nilpotent. Let S2 ⊆ AQ,Γ2 (k) be the
multiplicative set generated by x2. The derivation removing homomorphism induces an
isomorphism
Φ : k[y1][x1; τ2, δ2][y2; τ3][z
2 ; τ4] −→ A
2 (k)S
whose action on generators is given by
y1 7→ y1
x1 7→ x1
z2 7→ x2
y2 7→ y2 + (q2 − 1)−1
(q1 − 1)y1x1 + 1
x−12 .
For simplicity, label the domain of Φ as BZ−1. Let X1 ⊆ BZ−1 be the Ore set generated
by z2 and x1. Applying the derivation removing homomorphism to BZ
−1 induces an
isomorphism
Ψ : k[y1][z
1 ; τ2][y2; τ3][z
2 ; τ
4] −→ (BZ−1)X−11
whose action on generators is given by
z1 7→ z1
z2 7→ z2
y2 7→ y2
y1 7→ y1 + (q1 − 1)−1x−11 .
The derivation removing homomorphism need not be employed again to achieve the
result. Through iterated localization we find that there is an Ore set T ⊆ AQ,Γ2 (k) such
2 (k)T
−1 ∼= k[y±11 ][x±11 ; τ2][y±12 ; τ3][x±12 ; τ4]
and T is generated by the four elements x2, x1, y2x2(q2 − 1) + y1x1(q1 − 1) + 1, and
y1x1(q1 − 1) + 1. Note that we recover the result of [22, Theorem 5].
22 HEIDI HAYNAL
5. Examples
We will demonstrate how each of the following k-algebras satisfies all the conditions
of Theorem 2.8. Then Corollary 4.7 is applied to obtain an isomorphism of quotient
division rings (thereby confirming the quantum Gel’fand-Kirillov conjecture) and PI
degree parity with a multiparameter quantum affine space. When calculating the PI
degree of a quantum affine space, we encounter an antisymmetric, or skew-symmetric,
integral matrix. As proved in [30, Theorem IV.1], such a matrix is congruent to a matrix
in skew normal form.
Theorem 5.1. [Newman] Let A be a skew-symmetric matrix of rank r which belongs
to Mn(R), where the commutative principal ideal domain R is not of characteristic 2.
Then r = 2s and A is congruent to a matrix in block diagonal form

−h1 0 0
−h2 0
. . .
0 −hs 0

where hi | hi+1, 1 ≤ i ≤ s− 1.
The same result, in the language of alternating bilinear forms, can be found in [3, Section
5.1].
The matrix S in Theorem 5.1 is clearly equivalent to the more familiar Smith normal
form, diag(h1, h1, h2, h2, . . . , hs, hs, 0, 0, . . . , 0), where the diagonal entries are the in-
variant factors of the matrix A. In the examples that follow, we outline the operations
necessary to obtain the Smith normal form.
Definition 5.2. Let A = k[x1; τ1, δ1] · · · [xn; τn, δn] and A′ = k[x1; τ1] · · · [xn; τn] be
iterated skew polynomial rings. (1) If there exists Q = (q1, . . . , qn) ∈ (k×)n such that
δiτi = qiτiδi for i = 1, . . . , n, then A is called an iterated Q-skew polynomial ring. (2) If
there exist λji ∈ k× such that τj(xi) = λjixi for all i < j, then set λij = λ−1ji and λii = 1
for all i. We call Λ = (λij) ∈ Mn(k×) the matrix of relations for A′.
Lemma 5.3. Let C be a commutative k-algebra, A a C-algebra, B ⊆ A a C-subalgebra
generated by {b1, b2, . . . }. Let τ be a C-algebra automorphism of A, and δ a u-skew
τ -derivation on A for some unit u ∈ C. If τ(bj) ∈ B and δn(bj) ∈ (n)!uB for all j, n,
then δn(B) ⊆ (n)!uB for all n.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 23
Proof. Note that τ(bj) ∈ B for all j implies that τ(B) ⊆ B and hence we have
(j)!uB
⊆ (j)!uB for all j. Suppose that for integers m ≥ 1 and 1 ≤ ℓ ≤ m − 1, we
have δi(bj1 · · · bjℓ) ∈ (i)!uB for all i, and all choices of j1, . . . , jℓ. Then
δn(bj1 · · · bjm) =
τn−iδi(bj1 · · · bj(m−1))δ
n−i(bjm)
(i)!u(n− i)!uB ⊆ (n)!uB
for all n and all j1, . . . , jm by induction. �
For a first family of examples, we take odd-dimensional quantum Euclidean spaces. The
even-dimensional ones will be covered in Example 5.4.
5.1. The coordinate ring of odd-dimensional quantum Euclidean space; Oq(ok2n+1).
For q ∈ k×, assuming q has a (fixed) square root q1/2 ∈ k, the k-algebra Oq(ok2n+1)
may be presented as an iterated skew polynomial ring
k[w][y1; σ1][x1; τ1, δ1] · · · [yn; σn][xn; τn, δn]
with automorphisms σi, τi and derivations δi defined by
σi(w) = q
−1w all i
τi(w) = qw all i
σi(yj) = q
−1yj j < i
σi(xj) = q
−1xj j < i
τi(yj) = qyj i 6= j
τi(xj) = qxj j < i
τi(yi) = yi all i
δi(w) = δi(xj) = δi(yj) = 0 j < i
δi(yi) = (q
1/2 − q3/2)w2 + (1− q2)
yℓxℓ all i.
Quantum Euclidean spaces have been studied since 1990 when they were introduced by
Reshetikhin et al. in [36]. The three-dimensional case has applications to the structure
of space-time at small distances. Musson simplified the original set of relations in [29],
and Oh further simplified them, renaming the generators ω, xi, yi in [31]. Here, we have
made a change to Oh’s variables, yi 7→ qiyi, to obtain the relations in our presentation
of Oq(ok2n+1).
Routine computations show that τ−1i δiτi(yi) = q
−2δi(yi) for all i, and so we conclude
that each (τi, δi) is a q
−2-skew derivation. We may present the analogous k[t±1]-algebra
24 HEIDI HAYNAL
Ot(ok[t±1]2n+1) as an iterated skew polynomial ring with coefficient ring k[t±1] and
generators w, yi, xi for i = 1, . . . , n,
k[t±1][w][y1; σ̄1][x1; τ̄1, δ̄1] · · · [yn; σ̄n][xn; τ̄n, δ̄n]
where the automorphisms and derivations are defined analogously to those of the algebra
Oq(ok2n+1) with t ∈ k[t±1] replacing q ∈ k×. So each (τ̄i, δ̄i) is a t−2-skew derivation. It
is immediate that
Ot(ok[t±1]2n+1)/〈t− q〉 ∼= Oq(ok2n+1)
with each τ̄i and δ̄i reducing to τi and δi respectively.
Let Aj denote the k[t
±1]-subalgebra generated by w, ym, xm for m < j, and yj. To show
that δ̄ij(Aj) ⊆ (i)!t−2Aj, we apply Lemma 5.3 noting that δ̄ij(yj) has been given for i = 1
and is zero for i > 1. So, by Theorem 2.8, each δi in our presentation of Oq(ok2n+1)
extends to an iterative, locally nilpotent h.q−2-s.τi-d. on an appropriate subalgebra.
Then Corollary 4.7 gives
FractOq(ok2n+1) ∼= FractOB(k2n+1),
where the matrix of relations is

1 q q−1 q q−1 · · · q q−1
q−1 1 1 q q−1 · · · q q−1
q 1 1 q q−1 · · · q q−1
q−1 q−1 q−1 1 1 · · · q q−1
q q q 1 1 · · · q q−1
. . .
q−1 q−1 q−1 q−1 q−1 · · · 1 1
q q q q q · · · 1 1

If q ∈ k× is a root of unity, we may assume without loss of generality that it is a
primitive rth root of unity. Then the powers of q from the matrix B become the entries
of a (2n+ 1)× (2n+ 1) integer matrix

0 1 −1 1 −1 · · · 1 −1
−1 0 0 1 −1 · · · 1 −1
1 0 0 1 −1 · · · 1 −1
−1 −1 −1 0 0 · · · 1 −1
1 1 1 0 0 · · · 1 −1
. . .
−1 −1 −1 −1 −1 · · · 0 0
1 1 1 1 1 · · · 0 0

Now, PIdegOq(ok2n+1) can be computed from Theorem 1.2(2) using the matrix B′. The
cardinality of the image will not be changed if we first perform some row reductions on
B′. Letting N = 2n+ 1, n > 2, we manipulate the rows as follows.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 25
• For i = 2, 4, 6, . . . , N − 1, replace row i with row i + row (i+ 1).
• For i = N,N − 2, N − 4, . . . , 5, replace row i with row i − row (i− 2).
• Replace row 5 with row 5 − row 1.
• For i = 2, 4, 6, . . . , N − 5, replace row i with row i − 2row (i+ 5).
• Multiply the even numbered rows, except row 2n− 2, by −1.
The resulting matrix has 2n pivots and one zero row. We put the rows in this order
3, 1, 5, 7, 2, 9, 4, 11, 6, 13, . . . , 2i, 2i+ 7, . . . , N,N − 5, N − 3, N − 1
to place the pivots on the main diagonal and the zero row in the last position. Then we
have a matrix of this form

1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 1 −1 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 0 2 ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 0 0 1 1 ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 0 0 0 4 ∗ ∗ ∗ ∗ ∗ ∗ ∗
0 0 0 0 0 1 1 ∗ ∗ ∗ ∗ ∗
0 0 0 0 0 0 4 ∗ ∗ ∗ ∗ ∗
0 0 0 0 0 0 0
. . . ∗ ∗ ∗ ∗
0 0 0 0 0 0 0 0 1 1 ∗ ∗
0 0 0 0 0 0 0 0 0 4 ∗ ∗
0 0 0 0 0 0 0 0 0 0 2 −2
0 0 0 0 0 0 0 0 0 0 0 0

The diagonal entries of this echelon matrix do not yet reveal the size of its image because
the pivot in row three does not divide all of the (suppressed) entries in its row when
n ≥ 3. So more row reduction is needed.
First replace row 3 with row 3 +
row(4i+ 2).
For n even and j = 5, 7, 9, . . . , 2n− 3, replace row j as follows:
for j = 4p+ 1, p ≥ 1, use row j +
i=p+1
2 · row(4i) + row(2n);
for j = 4p+ 3, p ≥ 1, use row j +
i=p+1
2 · row(4i+ 2).
26 HEIDI HAYNAL
For n odd and j = 5, 7, 9, . . . , 2n− 5, replace row j as follows:
for j = 4p+ 1, p ≥ 1, use row j +
i=p+1
2 · row(4i) + 2 · row(2n);
for j = 4p+ 3, p ≥ 1, use row j +
i=p+1
2 · row(4i+ 2) + row(2n).
Then add row(2n) to row(2n − 3), and add 2·row(2n) to row(2n − 1). For integers
4 ≤ j ≤ 2n − 1, with j 6≡ 2(mod 4), add (−1)jcol 3 to col j. Subtract col(2n + 1)
from col 3; add row 3 to row(2n − 2); and subtract 2·row 3 from row(2n). The result
is an upper echelon matrix in which each pivot divides all the nonzero entries in its
row. So it is trivial to diagonalize by column operations. The Smith normal form for n
odd is diag(1, 1, . . . , 1, 4, 4, . . . , 4, 0) with n+1 ones and n− 1 fours. The Smith normal
form for n even is diag(1, 1, . . . , 1, 2, 2, 4, 4, . . . , 4, 0) with n ones, two twos, and n − 2
fours.
For the cases n = 1, 2, the row-reduced matrices are, respectively,
1 0 0
0 1 −1
0 0 0

1 0 0 1 −1
0 1 −1 1 −1
0 0 2 −2 2
0 0 0 2 −2
0 0 0 0 0

Hence we have, for all n > 0,
PIdegOq(ok2n+1) =
rn, r odd
rn/2⌊
⌋, r even, r /∈ 4Z
rn/2n−1, r ∈ 4Z
5.2. The multiparameter quantized Weyl algebras; AQ,Γn (k). For a fixed n-tuple
Q = (q1, . . . , qn) ∈ (k×)n and Γ = (γij) a multiplicatively antisymmetric n × n matrix
over k, the algebra AQ,Γn (k), studied in [23] and [26], may be presented as an iterated
skew polynomial ring
k[y1][x1; τ1, δ1][y2; σ2][x2; τ2, δ2] · · · [yn; σn][xn; τn, δn]
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 27
where the automorphisms and derivations are defined by
σi(yj) = γjiyj j < i
σi(xj) = γijxj j < i
τi(yj) = qjγjiyj j < i
τi(xj) = q
j γijxj j < i
τi(yi) = qiyi all i
δi(xj) = δi(yj) = 0 j < i
δi(yi) = 1 +
(qℓ − 1)yℓxℓ all i.
Routine computations show that τ−1i δiτi(yi) = qiδi(yi) for all i, and so we conclude
that each (τi, δi) is a qi-skew derivation. We may present the k[t
1 , . . . , t
n ]-algebra
AT,Γn (k[t
1 , . . . , t
n ]) as an iterated skew polynomial ring
k[t±11 , . . . , t
n ][y1][x1; τ̄1, δ̄1][y2; σ̄2][x2; τ̄2, δ̄2] · · · [yn; σ̄n][xn; τ̄n, δ̄n]
where the automorphisms and derivations are defined analogously to those of AQ,Γn (k)
with ti ∈ k[t±11 , . . . , t±1n ] replacing qi ∈ k. So each (τ̄i, δ̄i) is a ti-skew derivation. It is
immediate that
AT,Γn (k[t
1 , . . . , t
n ])/〈t1 − q1, . . . , tn − qn〉 ∼= AQ,Γn (k)
with each τ̄i and δ̄i reducing to τi and δi respectively.
Let Aj denote the k[t
1 , . . . , t
n ]-subalgebra generated by ym, xm for m < j, and yj.
To show that δ̄ij(Aj) ⊆ (i)!tjAj, it suffices to check δ̄ij(yj) by Lemma 5.3. But this is
given by definition for i = 1 and is zero for i > 1. So, by Theorem 2.8, each δi in
our presentation of AQ,Γn (k) extends to an iterative, locally nilpotent h,qi-s.τi-d. on the
appropriate subalgebra. Then Corollary 4.7 gives FractAQ,Γn (k)
∼= FractOΛ(k2n), where
the 2n× 2n matrix of relations Λ is comprised of 2× 2 blocks
Bii =
1 q−1i
, for all i;
Bij =
γji q
i γji
γij qiγij
, for i < j;
Bij =
γji γij
qjγji q
j γij
, for i > j.
If γij and qi are roots of unity for all i, j, then OΛ(k2n) is a PI algebra. Assuming that
γij is an r
ij root of unity and that qi is an r
i root of unity, we let
r = lcm{rij, ri | i, j = 1, . . . , n}.
28 HEIDI HAYNAL
Then there exists a primitive rth root of unity q ∈ k and integers bi, bij such that qi = qbi
and γij = q
bij for i, j = 1, . . . , n. The powers of this q from the matrix Λ give a 2n× 2n
integer matrix Λ′ comprised of 2× 2 blocks
B′ii =
0 −bi
, for all i;
B′ij =
bji bji − bi
bij bij + bi
, for i < j;
B′ij =
bji bij
bj + bji bij − bj
, for i > j.
Then PIdegAQ.Γn (k) can be computed using the matrix Λ
′ in Theorem 1.2 (2).
Consider the single parameter case, denoted Aqn(k), where qi = q for all i, and γij = 1
for i < j, relegating the σi to identity maps. Assuming that q is a primitive r
th root of
unity, then δi(y
i ) = 0 and τi(y
i ) = y
i for all i, implying that y
i is central. The definition
of the τi, along with the q-Liebnitz rule, implies that x
i is central for all i. So the algebra
Aqn(k) is a finitely generated module over the central subring k[y
i , x
1, . . . , y
n]. To
find the PI degree in this case, the integer matrix becomes

0 −1 0 −1 . . . 0 −1
1 0 0 1 0 1
0 0 0 −1 0 −1
1 −1 1 0 0 1
. . .
0 0 0 0 . . . 0 −1
1 −1 1 −1 . . . 1 0

which is seen to have a trivial kernel after these row reductions:
• Replace row 2n with row 2n− row (2n− 2)− row (2n− 3)
• For j = n−1, n−2, . . . , 2, replace row 2j with row 2j−row (2j−2)−row (2j−3)
• Rearrange the rows to order 2, 1, 4, 3, 6, 5 . . . , 2n, 2n− 1.
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 29
The resulting matrix has the form

0 −1 ∗
. . .
0 1 0

thus verifying that PIdegAqn(k) = r
5.3. The multiparameter coordinate ring of quantum n×nmatrices; Oλ,p
Mn(k)
The multiparameter coordinate ring of quantum n×n matrices was introduced by Artin,
Schelter, and Tate in [2]. The k-algebra Oλ,p
Mn(k)
is defined by generators xij for
i, j = 1, . . . , n and relations
xℓmxij =
pℓipjmxijxℓm + (λ− 1)pℓiximxlj (ℓ > i, m > j)
λpℓipjmxijxℓm (ℓ > i, m ≤ j)
pjmxijxℓm (ℓ = i, m > j),
where λ ∈ k× and p = (pij) ∈ Mn2(k×) is multiplicatively antisymmetric. It can also
be presented as an iterated skew polynomial ring
k[x11][x12; τ12] · · · [xij ; τij, δij ] · · · [xnn; τnn, δnn]
where each τℓm and δℓm is k-linear and satisfies
τℓm(xij) =
pℓipjmxij when ℓ > i and m 6= j
λpℓipjmxij when ℓ > i and m = j
pjmxij when ℓ = i and m > j,
δℓm(xij) =
(λ− 1)pℓiximxℓj when ℓ > i and m > j
0 otherwise.
Routine computations show τ−1ℓm δℓmτℓm(xij) = λ
−1δℓm(xij) as in [9, Section 5], and so
we conclude that each (τℓm, δℓm) is a λ
−1-skew derivation. We may present the k[t±1]-
algebra Ot,p
Mn(k[t
as an iterated skew polynomial ring with generators xij for
i, j = 1, . . . , n
k[t±1][x11][x12, τ̄12] · · · [xij ; τ̄ij, δ̄ij ] · · · [xnn; τ̄nn, δ̄nn]
30 HEIDI HAYNAL
where the automorphisms and derivations are defined analogously to those of the algebra
Mn(k)
with t ∈ k[t±1] replacing λ ∈ k. So each (τ̄ℓm, δ̄ℓm) is a t−1-skew derivation.
It is immediate that
Mn(k[t
/〈t− λ〉 ∼= Oλ,p
Mn(k)
with each τ̄ℓm and δ̄ℓm reducing to τℓm and δℓm respectively.
Let A−ℓm denote the k[t
±1]-subalgebra generated by the xij with (i, j) < (ℓ,m) in the
lexicographic order. Lemma 5.3 allows us to to verify that δ̄sℓm(A
ℓm) ⊆ (s)!t−1(A−ℓm) by
checking only that δ̄sℓm(xij) is contained in A
ℓm. This is immediate from the formula for
δ̄ℓm given above. Thus, by Theorem 2.8, each δℓm in our presentation of Oλ,p
Mn(k)
extends to an iterative, locally nilpotent h.λ−1-s.τℓm-d. on the appropriate k-subalgebra.
Then Corollary 4.7 gives
FractOλ,p
Mn(k)
) ∼= FractOΛ(kn
where the matrix of relations Λ = (bij) ∈ Mn2(k) is comprised of n× n blocks
Bii =

1 p21 p31 · · · pn1
p12 1 p32 · · · pn2
p13 p23 1 · · · pn3
. . .
p1n p2n p3n · · · 1

for all i,
Bij =

λ−1pij pijp21 pijp31 · · · pijpn1
λ−1pijp12 λ
−1pij pijp32 · · · pijpn2
λ−1pijp13 λ
−1pijp23 λ
−1pij · · · pijpn3
. . .
λ−1pijp1n λ
−1pijp2n λ
−1pijp3n · · · λ−1pij

, for i < j,
Bij =

λpij λpijp21 λpijp31 · · · λpijpn1
pijp12 λpij λpijp32 · · · λpijpn2
pijp13 pijp23 λpij · · · λpijpn3
. . .
pijp1n pijp2n pijp3n · · · λpij

, for i > j.
If λ and pij are roots of unity for all i, j, then OΛ(kn
) is a PI algebra. In this case we
may assume that λ is an sth root of unity and that pij is an r
ij root of unity, and let
r = lcm{s, rij | i, j = 1, . . . , n}. Then there exists a primitive rth root of unity q ∈ k
and integers b, bij such that λ = q
b and pij = q
bij . The powers of this q from the matrix
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 31
Λ provide entries for an n2 × n2 integer matrix Λ′ made up of n× n blocks
B′ii =

0 b21 b31 · · · bn1
b12 0 b32 · · · bn2
b13 b23 0 · · · bn3
. . .
b1n b2n b3n · · · 0

for all i,
B′ij =

bij − b bij + b21 bij + b31 · · · bij + bn1
bij + b12 − b bij − b bij + b32 · · · bij + bn2
bij + b13 − b bij + b23 − b bij − b · · · bij + bn3
. . .
bij + b1n − b bij + b2n − b bij + b3n − b · · · bij − b

, for i < j,
B′ij =

bij + b bij + b21 + b bij + b31 + b · · · bij + bn1 + b
bij + b12 bij + b bij + b32 + b · · · bij + bn2 + b
bij + b13 bij + b23 bij + b · · · bij + bn3 + b
. . .
bij + b1n bij + b2n bij + b3n · · · bij + b

, for i > j.
Then PIdegOλ,p
Mn(k)
can be calculated using Λ′ in Theorem 1.2 (2).
The single parameter quantized coordinate ring of n × n matrices, Oq(Mn(k)), is de-
fined over k analogously to Oλ,p(Mn(k)), but with relations that are recovered by setting
λ = q−2 and pij = q for all i > j. When k has characteristic zero and q is a primitive m
root of unity for m odd, Jakobsen and Zhang found in [20] that
PIdegOq(Mn(k)) = m
n(n−1)
2 by using De Concini’s and Procesi’s tool given in Theo-
rem 1.2. This result is reproved in [19] using results of De Concini and Procesi and also
Jøndrup’s work from [21]. Now we can recover PIdegOq(Mn(k) without the assumption
that k has characteristic zero.
In the single parameter case of n × n quantum matrices, the matrix that we use to
calculate the PI degree is

An In In In · · · In
−In An In In · · · In
−In −In An In · · · In
−In −In −In −In · · · An

32 HEIDI HAYNAL
where

0 1 1 1 · · · 1
−1 0 1 1 · · · 1
−1 −1 0 1 · · · 1
−1 −1 −1 · · · −1 0

is n× n and In is the n× n identity matrix.
For any n, the characteristic polynomial of An is the sum of the terms of degree ≡ n
(mod 2) in the binomial expansion of (x+1)n, so in fact χn(x) =
(x+1)n+ 1
(x− 1)n.
But there is also a recursion formula for the characteristic polynomial for n ≥ 3 given
χn(x) = χn−1(x)(x+ 1)− (x− 1)n−1,
which will be useful in the linear algebra that follows.
We will perform the following row reductions on the rows of blocks of Λ′. For ease of
notation, we’ll denote the jth row of blocks as BRj , the interchange of BRi and BRj as
BRi ↔ BRj , and the addition of a multiple of BRi to BRj as MBRi + BRj 7→ BRj ,
where M ∈ Mn(Z).
• BR1 ↔ BRn.
• −InBR1 7→ BR1.
• For i = 2, . . . , n− 1, BR1 +BRi 7→ BRi.
• BRn − AnBR1 7→ BRn.
This yields the matrix

In In In In · · · −An
0 An + In 2In 2In · · · In −An
0 0 An + In 2In · · · In −An
. . .
0 0 0 · · · An + In In −An
0 In −An In −An In −An · · · In + A2n

which can be reduced further by n − 2 block row operations, each of which produces
one zero block in the nth row. We list the first three here along with the resulting (n, n)
block.
• (An + In)BRn − (In − An)BR2 7→ BRn : A3n + 3An
• (An + In)BRn + (In −An)2BR3 7→ BRn : A4n + 6A2n + In
• (An + In)BRn − (In − An)3BR4 7→ BRn : A5n + 10A3n + 5An
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 33
In general, the block row operations that we need to perform in order to obtain a block
upper triangular matrix are:
• For i = 2, . . . , n− 1, (An + In)BRn + (−1)i−1(In − An)i−1BRi 7→ BRn.
These row operations are justified when m is odd because An + In is invertible in
Mn(Z/mZ) in that case, as will be shown below. After applying this step to the i
th row,
the (n, n) block is χi+1(An). So the resulting block upper triangular matrix is

In In In In · · · −An
0 An + In 2In 2In · · · In −An
0 0 An + In 2In · · · In −An
. . .
0 0 0 · · · An + In In −An
0 0 0 0 · · · χn(An)

where χn(An) is the n× n zero matrix. Each block on the diagonal is
An + In =

1 1 1 1 · · · 1
−1 1 1 1 · · · 1
−1 −1 1 1 · · · 1
−1 −1 −1 −1 · · · 1

which can be row reduced just by adding row 1 to rows 2 through n to yield the
matrix 

1 1 1 1 · · · 1
0 2 2 2 · · · 2
0 0 2 2 · · · 2
. . .
0 0 0 0 · · · 2

In particular this shows that An + In is invertible in Mn(Z/mZ) for m odd. Hence
Λ′ can be reduced through row operations to an upper triangular n2 × n2 matrix with
2n− 2 ones, (n− 1)(n− 2) twos, and n zeroes on the diagonal. Assuming that q ∈ k is
a primitive mth root of unity, and recalling Theorem 1.2, the cardinality of the image
in (Z/mZ)n
is mn
2−n if m is odd. Thus we conclude that PIdegOqMn(k) = m
n(n−1)
recovering the result of Jakobsen and Zhang [20] in characteristic zero. By similar
methods, one can show that PIdegOqMn(k) = m
n(n−1)
(n−1)(n−2)
2 when m is even. For
details on this result see [20] or [17].
5.4. The algebra K
n,Γ (k), which generalizes the coordinate rings of even-
dimensional quantum Euclidean space and quantum symplectic space. For
P = (p1, . . . , pn) and Q = (q1, . . . , qn) in (k
×)n with pi 6= qi for all i = 1, . . . , n, and
34 HEIDI HAYNAL
Γ = (γij) ∈ Mn(k×) multiplicatively antisymmetric, the k-algebra KP,Qn,Γ (k) introduced
in [18] is defined by generators xi, yi for i = 1, . . . , n and relations
yiyj = γijyjyi all i, j
xixj = qip
j γijxjxi i < j
xiyj = pjγjiyjxi i < j
xiyj = qjγjiyjxi i > j
xiyi = qiyixi +
(qℓ − pℓ)yℓxℓ all i.
This algebra may be presented in the form of an iterated skew polynomial ring
k[y1][x1; τ1][y2; σ2][x2; τ2, δ2] · · · [yn; σn][xn; τn, δn]
where the automorphisms τi, σi and derivations δi are defined by
σi(yj) = γijyj j < i
σi(xj) = p
i γjixj j < i
τi(yj) = qjγjiyj j < i
τi(xj) = q
j piγijxj j < i
τi(yi) = qiyi all i
δi(xj) = δi(yj) = 0 j < i
δi(yi) =
(qℓ − pℓ)yℓxℓ all i.
Routine computations show that τ−1i δiτi(yi) = qip
i δi(yi) for all i, and so we conclude
that each (τi, δi) is a qip
i -skew derivation. For ease of notation we now shall let
k = k[t±11 , . . . , t
n , u
1 , . . . , u
n ] with T = (t1, . . . , tn) ∈ k and U = (u1, . . . , un) ∈ k.
We may present the k-algebra K
n,Γ (k) as an iterated skew polynomial ring
k[y1][x1; τ̄1][y2; σ̄2][x2; τ̄2, δ̄2] · · · [yn; σ̄n][xn; τ̄n, δ̄n]
where the automorphisms and derivations are defined analogously to those of K
n.Γ (k)
with ti replacing pi and ui replacing qi. Let I ⊆ KT,Un,Γ (k) be the ideal generated by the
2n monomials ti − pi, ui − qi for i = 1, . . . , n. It is immediate that
n,Γ (k)/I
∼= KP,Qn,Γ (k),
with each τ̄i, δ̄i, σ̄i reducing to τi, δi, σi respectively.
Let Aj denote the subalgebra of K
n,Γ (k) generated by ym, xm for m < j and yj. To
show that δ̄ij(Aj) ⊆ (i)!ujt−1j Aj , it suffices to check that δ̄
j(yj) is an element of (i)!ujt−1j
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 35
by Lemma 5.3. This is given for i = 1 by the formula for δ̄j and is zero for i > 1. So,
by Theorem 2.8, each δi in our presentation of K
n,Γ (k) extends to an iterative, locally
nilpotent h.qip
i -s.τi-d. on the appropriate subalgebra. Then Corollary 4.7 gives
FractK
n,Γ (k)
∼= FractOΛ(k2n)
where the 2n× 2n matrix of relations Λ = (Bij) is comprised of 2× 2 blocks
Bii =
1 q−1i
, for all i;
Bij =
γij q
i γji
pjγji qip
j γij
, for i < j;
Bij =
γij p
i γij
qjγji q
j piγij
, for i > j.
If the qi, pi and γi are all roots of unity, then OΛ(k2n) is a PI algebra. Suppose qi is an
rthi root of unity, pi is an s
i root of unity, and γij is an r
ij root of unity for all i, j. Let
r = lcm{ri, si, γij | i, j = 1, . . . , n}. Then there extsis a primitive rth root of unity q ∈ k
and integers bi, ci, bij such that qi = q
bi , pi = q
ci, and γij = q
bij for all i, j. The powers
of q from the matrix Λ provide the entries for an integer matrix Λ′ comprised of 2 × 2
blocks
B′ii =
0 −bi
, for all i;
B′ij =
bij bji − bi
bji + cj bi + bij − cj
, for i < j;
B′ij =
bij bij − ci
bji + bj bij + ci − bj
, for i > j.
Then PIdegK
n,Γ (k) can be calculated using Λ
′ in Theorem 1.2 (2).
The coordinate ring of quantum Euclidean 2n-space over k, Oq(ok2n), is formed by
setting qi = 1, pi = q
−2 for all i, and γij = q
−1 for i < j in the parameters Q, P , and Γ
36 HEIDI HAYNAL
(see [18], Example 2.6). Then the integer matrix, Λ′, is

0 0 −1 1 −1 1 . . . −1 1
0 0 −1 1 −1 1 . . . −1 1
1 1 0 0 −1 1 −1 1
−1 −1 0 0 −1 1 ...
1 1 1 1 0 0
−1 −1 −1 −1 0 0
. . .
1 1 1 1 1 . . . 0 0
−1 −1 −1 −1 −1 . . . 0 0

We perform the following row reductions that preserve the size of the image of the
homomorphism Z2n −→ Z2n given by Λ′:
• For j = 2n, 2n− 1, 2n− 2, . . . , 4, replace row j with row j + row (j − 1)
• Replace row 2 with row 2− row 1
• Replace the (new) row 5 with row 5 + row 1
• For j = 4, 6, 8, . . . , 2n− 4, replace row j with row j + 2row (j + 3)
• For n ≥ 4, rearrange the rows to order 3, 1, 5, 7, 4, 9, 6, 11, . . . , 2i, 2i+ 5, . . . ,
2n− 4, 2n− 2, 2, 2n.
The resulting matrix has the form

0 −1 1
0 2 ∗
0 1 1
. . .
0 1 1
0 0 4
0 −2 2

When n is even, the pivot in the third row does not divide all the entries in its row,
so more elementary row and column operations are needed before it becomes clear that
the matrix can be diagonalized. By a method similar to that used in Example 5.1,
suppressed here in the interest of saving space but listed explicitly in [17], we obtain the
Smith normal form diag(1, 1, . . . , 1, 4, 4, . . . , 4, 0, 0) with n ones and n− 2 fours when n
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 37
is even; and diag(1, 1, . . . , 1, 2, 2, 4, 4, . . . , 4, 0, 0), with n− 1 ones and n− 3 fours when
n is odd. Thus we have
PIdegOq(ok2n) =
rn−1, r odd
rn−1/2⌊
⌋, r even /∈ 4Z
rn−1/2n−2, r ∈ 4Z
. (10)
The low-dimensional cases do not fit the same pattern, but the matrices for the cases
n = 2 and n = 3 are readily transformed to
1 1 0 0
0 0 −1 1
0 0 0 0
0 0 0 0
 and

1 1 0 0 −1 1
0 0 −1 1 −1 1
0 0 0 2 0 0
0 0 0 0 −2 2
0 0 0 0 0 0
0 0 0 0 0 0

respectively. Therefore, formula (10) holds for all n ≥ 2.
As a specific case ofK
n,Γ (k), quantum symplectic spaceOq(sp(k2n)) is formed by setting
qi = q
−2 and pi = 1 for all i, and γij = q for i < j (see [18], Example 2.4). With these
parameters, the 2n× 2n integer matrix Λ′ is

0 2 1 1 1 1 . . . 1 1
−2 0 −1 −1 −1 −1 . . . −1 −1
−1 1 0 2 1 1 1 1
−1 1 −2 0 −1 −1 −1 −1
−1 1 −1 1 0 2 ...
−1 1 −1 1 −2 0
. . .
−1 1 −1 1 −1 1 . . . 0 2
−1 1 −1 1 −1 1 . . . −2 0

We perform the following row reductions that preserve the size of the image of the
homomorphism Z2n −→ Z2n given by Λ′:
• For j = 2n, 2n− 1, . . . , 4, replace row j with row j − row (j − 1)
• Replace row 2 with −(row 2− 2row 3 + row 1)
• For j = 4, 6, 8, . . . , 2n− 2, replace row j with row j + 2row (j + 1)
• For n ≥ 3, order the rows 3, 1, 5, 2, 7, 4, 9, . . . , 2j, 2j + 5, . . . , 2n− 4, 2n, 2n− 2.
38 HEIDI HAYNAL
This yields a matrix whose image is more easily measured:

0 1 1 ∗
. . .
0 0 4
−2 −2

But the pivot in row 2 is problematic because it does not always divide the other
entries in its row. With further elementary row and column operations, full details
of which can be found in [17], we can bring this matrix into Smith normal form
diag(1, 1, . . . , 1, 4, 4, . . . , 4) with n ones and n fours when n is even; or the form
diag(1, 1, . . . , 1, 2, 2, 4, 4, . . . , 4) with n − 1 ones, two twos, and n − 1 fours when n
is odd.
For n = 1, 2, the row reduced matrices are, respectively,
−1 1 0 2
0 1 1 1
0 0 −4 −4
0 0 0 −4
 .
Hence we have, for all n,
PIdegOq(sp(k2n)) =
rn, r odd
rn/2⌊
⌋, r even, r /∈ 4Z
rn/2n, r ∈ 4Z
6. Prime Factor Localizations
In this section we present a structure theorem for completely prime factors of iterated
skew polynomial rings analogous to the main theorem of section four. Applying this
result to the algebras studied in section five, we’d like to strengthen it to the form
of the quantum Gel’fand-Kirillov conjecture. Recall that the assumptions about skew
polynomial rings from section one are still in effect.
Theorem 6.1. Let A = R[x; τ, δ], where R is noetherian and δτ = qτδ for some
q ∈ k×. Assume that δ extends to a locally nilpotent, iterative h.q-s.τ -d., {di}, on R.
Let P ∈ specA be completely prime. Then
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 39
(1) there exists a cyclic Ore set S in A/P such that (A/P )S−1 ∼=
R[y; τ ]/Q
Y −1 for
some completely prime Q ∈ specR[y; τ ] and cyclic Ore set Y ,
(2) FractA/P ∼= FractR[y; τ ]/Q.
Proof. The completely prime ideal P naturally satisfies one of two cases: x ∈ P or
x /∈ P . If x ∈ P , then xA ⊆ P and Ax ⊆ P . So the relation xr = τ(r)x + δ(r)
implies that δ(r) ∈ P for all r ∈ R. Hence, there is a completely prime ideal I ∈ R
such that A/P ∼= R/I ∼= R[y; τ ]/(I + 〈y〉). In this case, we can take S = Y = {1} and
localize. If x /∈ P , then xi /∈ P for all i ∈ N ∪ {0} because A/P is a domain. Letting
S = {1, x, x2, . . . }, which is a known denominator set in A, we have P ∩ S = ∅. Since
extension and contraction provide inverse bijections between the sets specAS−1 and
{I ∈ specA | I ∩ S = ∅}, we know that P e ∈ specAS−1. From Theorem 3.7, we
have AS−1 ∼= R[y±1; τ ], a localization of R[y; τ ]. So there is a completely prime ideal
Q̄⊳R[y±1; τ ] such that AS−1/P e ∼= R[y±1; τ ]/Q̄. Setting Y = {1, y, y2, . . . , }, contraction
to R[y; τ ] gives a completely prime ideal Q, where Q∩ Y = ∅, such that R[y±1; τ ]/Q̄ is
isomorphic to (R[y; τ ]/Q)Y −1. The canonical projection π : AS−1 −→ (A/P )S−1 gives
AS−1/P e ∼= (A/P )S−1. Thus (A/P )S−1 ∼=
R[y; τ ]/Q
Y −1. �
Theorem 6.2. Let R be a noetherian k-algebra, and let
A = R[x1, τ1, δ1] · · · [xn; τn, δn]
be an iterated skew polynomial ring where, for j < i and λij ∈ k×, τi(xj) = λijxj, and δi
is a qi-skew τi-derivation, qi 6= 1, which extends to a locally nilpotent, iterative h.qi-s.τi-d.
{di,p}∞p=0 on R[x1; τ1, δ1] · · · [xi−1; τi−1, δi−1] for all i. Let A′ = R[y1; τ ′1][y2; τ ′2] · · · [yn; τ ′n]
where τ ′i(yj) = λijyj for all i with j < i and the same units λij as above. Let P be a
completely prime ideal in A. Then
(1) there exists a finitely generated Ore set Sn in A/P such that (A/P )S
n is isomorphic
Y −1n for some completely prime ideal Q ⊆ A′ and finitely generated Ore set
(2) FractA/P ∼= FractA′/Q.
Proof. The case n = 1 has been established in Theorem 6.1. Suppose the result holds
for the case n− 1, and let An−1 = R[x1, τ1, δ1] · · · [xn−1; τn−1, δn−1] ⊆ A. Then we have
A = An−1[xn; τn, δn]. If xn ∈ P , then as in Theorem 6.1 there is a completely prime
ideal I ⊆ An−1 such that A/P ∼= An−1/I ∼= An−1[yn; τ ′n]/(I + 〈yn〉). The induction
hypothesis and Lemma 4.2 imply that
An−1[yn; τ
n]/(I + 〈yn〉)
S−1 ∼=
Y −1 for
some finitely generated Ore sets S and Y . Hence there is a finitely generated Ore set
Sn in A such that (A/P )S
∼= (A′/Q)Y −1. If xn /∈ P , let Sn = {1, xn, x2n, . . . } ⊆ A
and Yn = {1, yn, y2n, . . . } ⊆ An−1[yn; τn]. Then from the single-variable result, it follows
that (
An−1[yn; τ
n]/Q̄
Y −1n , (11)
40 HEIDI HAYNAL
for a completely prime ideal Q̄ ⊆ An−1[yn; τ ′n]. From Lemma 4.2, we have
An−1[yn; τ
n] = R[yn; τ
n][x1; τ
1] · · · [xn−1; τ ′n−1, δ′n−1],
which is an iterated skew polynomial ring in n − 1 variables over the coefficient ring
R[yn; τ
n] that satisfies the current assumptions. So, we apply the induction hypothesis
and rearrange variables to obtain
An−1[yn; τn]/Q̄
Y −1n
R[yn; τ
n][y1; τ
1] · · · [yn−1; τ ′n−1]/Q
R[y1; τ
1][y2; τ
2] · · · [yn; τ ′n]/Q
for a completely prime ideal Q ⊆ R[y1; τ ′1][y1; τ ′1] · · · [yn; τ ′n] and a denominator set
Z ⊆ R[y1; τ ′1][y1; τ ′1] · · · [yn; τ ′n]/Q. This, along with isomorphism (11) gives the re-
sult. �
When R is replaced by k, we have the following result.
Corollary 6.3. Let A = k[x1, τ1, δ1] · · · [xn; τn, δn], where τi(xj) = λijxj and δiτi = qiτiδi,
qi 6= 1, for λij , qi ∈ k× and all i with j < i. Assume that each δi extends to a locally
nilpotent, iterative h.qi-s.τi-d. {di,m}∞m=0 on the subalgebra k[x1; τ1, δ1] · · · [xi−1; τi−1, δi−1].
Let P be a completely prime ideal in A and set λii = 1 and λji = λ
ij . Then for
λ = (λij) ∈ Mn(k), and an appropriate completely prime ideal Q ⊆ Oλ(kn), we have
FractA/P ∼= FractOλ(kn)/Q.
We summarize how this applies to the k-algebras of quantized coordinate type.
Corollary 6.4. Let A be any of the examples discussed in sections 5.1 - 5.4, and let P be
a completely prime ideal of A. Then there exist a positive integer N , a multiplicatively
antisymmetric N ×N matrix λ over k, and a completely prime ideal Q ∈ Oλ(kN) such
that FractA/P ∼= FractOλ(kN)/Q.
To complete the question posed by the corollary, one might ask how far the quantum
Gel’fand-Kirillov conjecture extends to prime factor algebras. For instance:
Question 6.5. Find conditions under which we can conclude that for any positive in-
teger n, multiplicatively antisymmetric matrix λ ∈ Mn(k×), and completely prime ideal
Q ∈ specOλ(kn), we have
FractOλ(kn)/Q ∼= FractOp(Km)
for some field extension K ⊇ k, integer m ≤ n, and m×m matrix p over K.
The case n = 1 is trivial. When n = 2 and Q contains x1 or x2, then FractOλ(k2)/Q is
isomorphic either to FractOp(k(y)) where p = (1), or to k itself. In fact, for any n, if Q
is generated by a subset S of {x1, . . . , xn}, then the result holds, with p the submatrix
of λ formed by deleting the ith row and column for xi ∈ S, and K = k. When xi /∈ Q
PI DEGREE PARITY IN q-SKEW POLYNOMIAL RINGS 41
for all i, answering the question fully will likely require different methods depending on
the presence of roots of unity among the λij. A positive answer in the generic case has
been provided in the proof of [13, Theorem 2.1]:
Theorem 6.6. [Goodearl - Letzter] Let k be a field, λ = (λij) a multiplicatively anti-
symmetric n× n matrix over k×, and Λ the subgroup of k× generated by the λij. If Λ
is torsionfree, then all of the prime ideals Q of Oλ(kn) are completely prime.
In their proof, they showed that FractOλ(kn)/Q ∼= FractOp(Km), and identified K as
the quotient field of a commutative domain embedded in the center of Oλ((k×)n)/Q′,
where Q′ is the prime ideal in Oλ((k×)n) induced by localization.
Quantum affine space is included in a class called quantum solvable algebras by A. N. Panov.
The main theorem of [34, Section 3], states that when the group generated by the λij is
torsionfree, then FractOλ(kn)/Q is isomorphic to the quotient division ring of a quan-
tum torus. The main theorem of [35, Section 3], allows roots of unity and states that
when Q satisfies the extra condition of being stable under a certain set of derivations,
then FractOλ(kn)/Q is isomorphic to the quotient division ring of a quantum torus.
Cauchon’s work may also be specialized to apply to quantum affine space when the
group generated by the λij is torsionfree. The result of [5, Theorem 6.1.1], indicates
that FractOλ(kn)/Q is isomorphic to FractOp(Km) which specializes to this result.
But the division ring of real quaternions provides an example showing that Question
6.5 needs to have some conditions imposed. Note that
H ∼= Oλ(R3)/Q, where λ =
1 −1 −1
−1 1 −1
−1 −1 1
 , andQ = 〈x21 + 1, x22 + 1, x23 + 1〉.
Therefore, we cannot obtain the desired isomorphism of quotient division rings in this
case, illustrating the necessity of an extra condition such as the one imposed by Panov
in [35].
acknowledgments
The author thanks her dissertation advisor, Ken Goodearl, for his direction that was so
freely given in many inspiring discussions.
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Department of Mathematics, University of California, Santa Barbara, California
93106
E-mail address : heidi@softerhardware.com
	1. Introduction
	2. Higher q-Skew -Derivations
	3. The -Derivation Removing Homomorphism
	4. Main Theorem
	5. Examples
	5.1. The coordinate ring of odd-dimensional quantum Euclidean space; Oq (o k2n+1)
	5.2. The multiparameter quantized Weyl algebras; AnQ, (k)
	5.3. The multiparameter coordinate ring of quantum n n matrices; O, bold0mu mumu pppppp(to.Mn(k))to.
	5.4. The algebra Kn, P, Q (k), which generalizes the coordinate rings of even-dimensional quantum Euclidean space and quantum symplectic space
	6. Prime Factor Localizations
	acknowledgments
	References
