Simple ambiskew polynomial rings

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by  David A. Jordan

SIMPLE AMBISKEW POLYNOMIAL RINGS
DAVID A. JORDAN AND IMOGEN E. WELLS
3
1
0 Abstract. Wedeterminesimplicitycriteriaincharacteristics0andpforaubiquitousclass
2
of iterated skew polynomial rings in two indeterminates over a base ring. One obstruction

n to simplicity is the possible existence of a canonical normal element z. In the case where
a
this element exists we give simplicity criteria for the rings obtained by inverting z and the
J
rings obtained by factoring out the ideal generated by z. The results are illustrated by
3
numerous examples including higher quantized Weyl algebras and generalizations of some
2
low-dimensional symplectic reflection algebras.

]
A
R
. 1. Introduction
h
t
a The first Weyl algebra A (F) over a field F of characteristic 0 is the best known example of
1
m
a simple skew polynomial ring of the form R[x;δ] where δ is a derivation of R. It illustrates
[ the well-known result that, for an algebra R over a field F of characteristic 0, R[x;δ] is simple

1 if and only if R contains no non-zero proper δ-invariant ideal and the derivation δ is outer.
v
See [9, Theorem 3.2], [29, Theorem 1.8.4] or [14, Proposition 2.1]. The algebra A (F) is
0 1
6 obtained on taking R = F[y] and δ = d .
dy
5
The situation for simple skew polynomial rings of the form R[x;α,δ] where α is an en-
5
. domorphism of R and δ is an α-derivation of R is less well understood and there are few
1
0 documented examples in which α is not an inner automorphism. One significant but difficult
3 example, due to Cozzens [8] and fully documented in [9], features a division ring in the role
1
of R and a non-surjective endomorphism in the role of α and has interesting asymmetric
:
v
properties. Detailed examples with an automorphism α that is not inner are surprisingly
i
X difficult to track down in the literature. Results are also mostly restricted to division rings;
r see the survey in the introduction to the recent paper [30]. Here we find simplicity criteria
a
for a class of iterated skew polynomial rings in two indeterminates x and y over a base ring A
which, for convenience, we assume to be an F-algebra for a field F. These have been studied,
ata variety of levels ofgeneralisation, ina sequence of papers including [17, 18, 19, 21, 22, 23]
and were given the name ambiskew polynomial ring in [23].
For our purposes here, the construction of an ambiskew polynomial ring R(A,α,v,ρ)
requires three commuting automorphisms α, β, γ of A, with β = α−1γ, a non-zero element
ρ of F and an element v of A such that va = γ(a)v for all a A and v = γ(v). The
∈
indeterminates satisfy the relations xy = ρyx + v and, for all a A, ya = α(a)y and
∈
xa = β(a)x. Thus y is adjoined in the formation of A[y;α] and the adjunction of x involves
2010 Mathematics Subject Classification. Primary 16D30; Secondary 16S36, 16W20, 16W25, 16U20.
Key words and phrases. Simple ring, skew polynomial ring.
1
2 DAVIDA. JORDANANDIMOGEN E. WELLS
extending β to an automorphism of A[y;α], with β(y) = ρy, and then forming A[y;α][x;β,δ]
where δ is a β-derivation of A[y;α] such that δ(A) = 0 and δ(y) = v. If v is central then
β = α−1. Taking A = F, α = β = id and ρ = v = 1 gives the first Weyl algebra A (F) and
F 1
iteration yields the higher Weyl algebras A (F), n 2, where A = A (F), α = β = id
n n−1 A
≥
and ρ = v = 1.
The development of the theory of ambiskew polynomial rings began in [17] where R is
commutative, ρ = 1 and v = u α(u) for some u A. Examples included the enveloping
− ∈
algebra U(sl ), its quantization U (sl ) and the coordinate ring (M (F)) of quantum
2 q 2 q 2
O
2 2 matrices. Each of these examples has a distinguished central element, namely the
×
quantum determinant in (M (F)) and the Casimir elements in U(sl ) and U (sl ). These
q 2 2 q 2
O
are examples of a general phenomenon. In the construction presented in [17], the element
xy u = yx α(u) is central and is called the Casimir element. The algebras covered in
− −
[17] include the algebras considered by Smith in [34] and that paper heavily influenced the
subsequent development of the theory of ambiskew polynomial rings.
Arbitrary non-zero values for ρ were introduced in [19, 21], where v became u ρα(u) and
−
the Casimir element became xy u = ρ(yx α(u)). Although not central when ρ = 1, xy u
− − 6 −
is normal. Among the new examples included by this generalization were the quantized Weyl
algebra Aq(F), where, for q F 0,1 , xy qyx = 1, the dispin enveloping algebra, where
1 ∈ \{ } −
ρ = 1, and an algebra introduced by Woronowicz [37] in the context of quantum groups.
−
At this stage, the Weyl algebra A (F) was excluded and there were no simple examples,
1
due to the existence of the normal Casimir element which can never be a unit. In [24], A
remained commutative but no conditions were imposed on the element v. The condition
that v = u ρα(u) for some u A, giving rise to a normal Casimir element, was named
− ∈
conformal and its negation was named singular. Thus singularity is a necessary condition
for simplicity and A (F) is a singular ambiskew polynomial ring.
1
To allow for iteration, and application to the higher quantized Weyl algebras arising from
thequantum calculus ofMaltsiniotis [31], noncommutative coefficient ringswhere introduced
in[22]butonlyintheconformalcasewithv = u ρα(u)forsomeu Asuchthatua = γ(a)u
− ∈
Λ,q
for all a A. The higher quantized Weyl algebra A will be specified in detail in Section
n
∈
6 but a brief discussion of the second quantized Weyl algebra, which requires three non-zero
parameters q ,q and λ, will set the pattern and illustrate the role of Casimir elements in
1 2
iteration. Renaming the generators, the first quantized Weyl algebra Aq1(F) is generated by
1
x and y subject to the relation
1 1
x y q y x = 1. (1)
1 1 1 1 1
−
Let v = 1+(q 1)y x which, if q = 1, is a multiple of the first Casimir element and, for
1 1 1 1
all q , satisfies t−he equations vy = q6 y v and vx = q−1x v. The second quantized Weyl
1 1 1 1 1 1 1
algebra Aλ,q1,q2(F) is constructed using the F-automorphisms α and β such that α(x ) = λx ,
2 1 1
α(y ) = λ−1y , β(x ) = (q λ)−1x and β(y ) = q λy . Thus Aλ,q1,q2(F) is generated by
1 1 1 1 1 1 1 1 2
SIMPLE AMBISKEW POLYNOMIAL RINGS 3
x ,y ,x and y subject to (1) and the further relations
1 1 2 2
y x = λx y ,
2 1 1 2
y y = λ−1y y ,
2 1 1 2
x x = (q λ)−1x x ,
2 1 1 1 2
x y = q λy x ,
2 1 1 1 2
x y q y x = 1+(q 1)y x .
2 2 2 2 2 1 1 1
− −
When q = q = λ = 1 this is the Weyl algebra A (F). The element v +(q 1)y x , which
1 2 2 2 2 2
−
is either a multiple of a second Casimir element or equal to v, is normal and can be used for
further iteration.
Λ,q
The higher generalized Weyl algebra A , where q = (q ,...,q ) is an n-tuple of elements
n 1 n
of F, and Λ = (λ ) is an appropriate n n matrix over F, has 2n generators y ,x ,...,y ,x
i,j 1 1 n n
×
and is constructed by iteration. It has n distinguished normal elements v ,v ,...,v which
1 2 n
are distinct if each q = 1. It is shown in [22] that if q = 1 for all i, the algebra obtained by
i i
6 6
inverting these n normal elements is simple. We shall see that if each q = 1 and charF = 0
j
then the higher quantized Weyl algebra is itself simple. This follows from one of our main
results, Theorem 3.10, which says that if charF = 0 then the ambiskew polynomial ring
R(A,α,v,ρ) is simple if and only if
(i) A has no non-zero proper ideal invariant under α,
(ii) R is singular, and
(iii) for all m 1, the element m−1ρlαl(v) is a unit.
≥ l=0
A more complex criterion for non-zero characteristic appears in Theorem 3.15.
P
When charF = 0, the simple quantized Weyl algebra Aλ,1,1(F) has a claim to being the
2
most accessible example of a simple skew polynomial ring of the form B[x;α,δ] with α not
inner. Two competitors, whose simplicity follows from Theorem 3.10, are skew versions of
the first Weyl algebra that have not, to our knowledge, previously appeared in the literature.
For these, view C as an R-algebra with conjugation as an R-automorphism α. We shall see
that the R-algebra extension of C generated by x and y subject to the relations
xi = ix, yi = iy, xy yx = 1
− − −
isa simple ambiskew polynomial ring, as isthe R-algebrain which the thirdof these relations
is replaced by xy +yx = i.
New examples of ambiskew polynomial rings have continued to emerge, including the
down-up algebras of Benkart and Roby [6], the generalized down-up algebras of Cassidy
and Shelton [7] and, most recently, the augmented down-up algebras of Terwilliger and
Worawannatoi [35]. To accommodate the non-Noetherian down-up algebras, the definition
of ambiskew polynomial ring was amended to allow α to be non-bijective and ρ to be zero
but here we shall assume that α is bijective and ρ = 0.
4 DAVIDA. JORDANANDIMOGEN E. WELLS
In a higher quantized Weyl algebra with each q = 1, the only barrier to simplicity is
i
6
the existence of the normal Casimir elements. The localization obtained by inverting these
elements is simple. In Section 4, we find a simplicity criterion for the localization of a
conformal ambiskew ring at the powers of the Casimir element z to be simple. The barrier
canalso beremoved by factoringouttheideal generatedby z which yields a generalized Weyl
algebra in the sense of Bavula [3]. A mild generalization is needed to cover the possibility
that v is not central. In Section 5, we give a simplicity criterion generalizing simplicity
criteria from [20, 6.1 Theorem] and its subsequent generalization [5, Theorem 4.2]. Section
6 presents the application of results from Sections 3, 4 and 5 to examples.
All the examples mentioned so far are domains but simple ambiskew polynomial rings
may have zero-divisors. There has been much interest in the symplectic reflection algebras
of Etingof and Ginzburg [12]. In common with enveloping algebras, quantized enveloping
algebras and quantum matrices, it turns out that low-dimensional examples of these algebras
are ambiskew polynomial rings or iterated ambiskew polynomial rings over the group algebra
of a finite cyclic group. Two such examples are documented as Examples 6.17 and 6.20.
Although simplicity criteria for these algebras are known, they provide nice illustrations of
Theorem 3.10 and it is interesting to see how they fit into this picture. In both cases the
parameter ρ is 1 but we shall also discuss some similar examples in which ρ = 1.
6
Some of the results presented here appeared in the PhD thesis [36] of the second author.
2. Preliminaries
Definitions 2.1. Let γ be an automorphism of a ring A and let v A. If vA = Av then v
∈
is normal in A and if va = γ(a)v for all a A and γ(v) = v, we shall say that v is γ-normal.
∈
If v is a regular normal element then v is γ-normal for a unique automorphism γ, while 0 is
γ-normal for every automorphism γ of A.
Let Γ be a set of automorphisms of A. An ideal I of A is a Γ-ideal of A if γ(I) I for
⊆
all γ Γ. The ring A is Γ-simple if 0 and A are the only Γ-ideals of A. If Γ = α is a
∈ { }
singleton, we write α-ideal and α-simple rather than α -ideal and α -simple.
{ } { }
Definition 2.2. The level of generality of the construction of ambiskew polynomial ring
varies through the papers [17, 18, 19, 21, 22, 23]. Here we shall work with a common
generalization of the constructions given in [22] and [24]. Let F be a field, and let A be an F-
algebra. Let ρ F 0 and let v be a γ-normal element of A for some F-automorphism γ of
∈ \{ }
A. Let α Aut A be such that αγ = γα and let β := α−1γ = γα−1, so that αβ = γ = βα.
F
∈
Extend β toanF-automorphismofA[y;α]bysetting β(y) = ρy. By [14, Exercise 2ZC], there
is a β-derivation δ of A[y;α] such that δ(A) = 0 and δ(y) = v. The ambiskew polynomial
ring R(A,α,v,ρ) is the iterated skew polynomial ring A[y;α][x;β,δ]. Thus
ya = α(a)y for all a A,
∈
xa = β(a)x for all a A and
∈
xy = ρyx+v.
SIMPLE AMBISKEW POLYNOMIAL RINGS 5
Strictly speaking, we should write R(A,α,γ,v,ρ). However if v is regular then v determines
γ and the only zero-divisor appearing as v in any example that we shall consider is 0, in
which case we take γ = id . There is some symmetry in the roles of x and y inasmuch that
A
R(A,α,v,ρ) can also be presented as A[x;β][y;α,δ′], where α(x) = ρ−1x, δ′(A) = 0 and
δ′(x) = ρ−1(v). If v is central then the actions of x and y on A involve twisting using α
−
from opposite sides: ya = α(a)y and ax = xα(a). This is the reason for the name.
Let v(0) := 0 and v(m) := m−1ρlαl(v) for m N. In particular v(1) = v. Each v(m) is
l=0 ∈
γ-normal and it is easily checked, by induction, that, for m 0,
P ≥
xym ρmymx = v(m)ym−1 and (2)
−
xmy ρmyxm = xm−1v(m) = βm−1(v(m))xm−1. (3)
−
Notation 2.3. Let w := xy = ρyx+v. For all a A,
∈
wa = (xy)a = βα(a)xy = γ(a)w
and the subring of R generated by A and w may be identified with A[w;γ]. Note that,
as γ(v) = v, w commutes with v, so that v is normal in A[w;γ]. Also γ extends to an
F-automorphism of A[w;γ], with γ(w) = w, and v is γ-normal in A[w;γ] as well as in A.
The F-automorphisms α and β can be extended to A[w;γ] by setting α(w) = ρ−1(w v) and
−
β(w) = α−1(w) = ρw+α−1(v). Then yw = α(w)y, xw = β(w)x and the following identities,
in which m 1, are easily established by induction on m:
≥
αm(w) = ρ−m(w v(m)); (4)
−
α−m(w) = ρmw +βm(v(m)); (5)
m−1
xmym = α−l(w); (6)
l=0
Y
m
ymxm = αl(w). (7)
l=1
Y
The factors in the products on the right hand sides of (6) and (7) commute so the products
are well-defined.
Definitions 2.4. Suppose that there is a γ-normal element u A such that v = u ρα(u).
∈ −
Then the element z := xy u = ρ(yx α(u)) is such that zy = ρyz, zx = ρ−1xz, za = γ(a)z
− −
for all a A and zu = uz. Hence z is γ-normal in R, where γ is extended to an F-
∈
automorphism of R by setting γ(y) = ρy and γ(x) = ρ−1x. If such an element u exists
then it will be called a splitting element and we shall say that the 4-tuple (A,α,v,ρ) is
conformal and that R is a conformal ambiskew polynomial ring; otherwise (A,α,v,ρ) and R
are singular. In the conformal case there need not be a unique splitting element. In general,
if u is a splitting element and u′ A then u′ is a splitting element if and only if u u′ is
∈ −
γ-normal and α(u u′) = ρ(u u′). For example if ρ = 1, γ = id and u is a splitting
A
− −
element then u+λ is a splitting element for all λ F. If u is not unique we shall adopt a
∈
6 DAVIDA. JORDANANDIMOGEN E. WELLS
convention of choosing one splitting element u and fixing it subsequently. We then refer to
the element z := xy u = ρ(yx α(u)) as the Casimir element of R. In the conformal case,
− −
note that A[z;γ] = A[w;γ], that α(z) = ρ−1z and that v(m) = u ρmαm(u) for all m 0.
− ≥
Lemma 2.5. Suppose that v is regular in A and let u A be such that, for some n 1,
∈ ≥
ua = γn(a)u for all a A. Then γ(u) = u. In particular, if ua = γ(a)u for all a A and
∈ ∈
v = u ρα(u) then u is a splitting element.
−
Proof. In this situation, uv = γn(v)u = vu = γ(u)v so, by the regularity of v, γ(u) = u.
In the particular case, γ(u) = u is the remaining condition required for u to be a splitting
(cid:3)
element
Remark 2.6. The use of the word conformal here is a possible source of confusion, due
to the conformal SL algebras of [26], which take their name from physical considerations
2
and happen to be ambiskew polynomial rings. In [32], the term J-conformal is used for this
reason.
Notation 2.7. For a ring A, the group of units of A will be denoted U(A).
3. Simplicity of R(A,α,v,ρ)
Throughout this section, R will denote an ambiskew polynomial ring R(A,α,v,ρ) where
A is an algebra over a field F. The set = yi is a β-invariant right and left Ore set
i≥1
Y { }
in A[y;α] and, by [13, Lemma 1.4], it is a right and left Ore set in R. The localization
R is generated, as a ring extension of A, by y, y−1 and w or, in the conformal case, by y,
Y
y−1 and z. It can be identified with A[w;γ][y±1;α], where α(w) = ρ−1(w v), or, in the
−
conformal case, with A[z;γ][y±1;α], where α(z) = ρ−1z. Similarly = xi is a right
i≥1
X { }
and left Ore set in R and the localization R can be identified with A[w;γ][x±1;β], with
X
β(w) = ρw +α−1(v) or, in the conformal case, with A[z;γ][x±1;β], with β(z) = ρz.
The following lemma, which in the Noetherian case is an immediate consequence of [29,
2.1.16(vi)], will beuseful in handling the passage between R andR and insimilar situations
X
later in the paper.
Lemma 3.1. Let B be a ring with a regular element y such that := yi is a right and
i≥1
Y { }
left Ore set and let C = B = B −1 = −1B be the localization of B at . If C is simple
Y
Y Y Y
and I is a non-zero ideal of B then ys I for some s 0.
∈ ≥
Proof. Let J be the set of elements of C that are finite sums of elements of the form y−aiy−b
where i I and a,b 0. As y−aiy−b = y−(a+1)(yi)y−b = y−a(iy)y−(b+1), any element of J
∈ ≥
can be written as a single term y−miy−n. Using the right (resp. left) Ore condition it is
readily checked that J is a right (resp. left) ideal of C. By the simplicity of C, 1 = y−miy−n
for some i I and some m,n 0, whence ym+n I. (cid:3)
∈ ≥ ∈
Lemma 3.2. The ring R is simple if and only if R is simple and v(m) is a unit of A for
Y
all m 1.
≥
SIMPLE AMBISKEW POLYNOMIAL RINGS 7
Proof. Suppose that R is simple. Then R is simple by [29, 2.1.16(iii)] and its left analogue.
Y
Suppose that, for some m 1, v(m) is not a unit, and let I = v(m)A which, as v(m) is normal,
≥
is an ideal of A. Every element of R has a unique expression as a sum of elements of the
form xia yj, where a A,i 0 and j 0. Let J be the subspace of R spanned by the
i,j i,j
∈ ≥ ≥
elements of the form xiayj where i > 0 or j m or a I. Clearly Jy J, JA J and,
≥ ∈ ⊆ ⊆
as v(m) is not a unit, 1 / J. Also xiayjx J if i > 0 or, by (2), if j > m. By (2) and the
∈ ∈
γ-normality of v(j),
ayjx = ρ−j(xβ−1(a)yj v(j)γ(a)yj−1)
−
for all a A. It follows that ayjx J if either j = m, in which case v(j)γ(a) I, or a I.
∈ ∈ ∈ ∈
Consequently J is a proper right ideal of R. Moreover Rym J so ann (R/J) is a non-zero
R
⊂
proper ideal of R, contradicting the simplicity of R. Hence v(m) is a unit for all m 1.
≥
For the converse, suppose that R is simple and that each v(m) is a unit of A. Let I be
Y
a non-zero ideal of R. By Lemma 3.1, yd I for some d 0. Choose the least such d. If
∈ ≥
d > 0, then, by (2),
yd−1 = (v(d))−1(xyd ρdydx) I,
− ∈
contradicting the minimality of d. So d = 0, 1 I, and I = R, whence R is simple. (cid:3)
∈
The following result is well-known; see [29, 1.8.5] or [16, Theorem 1].
Theorem 3.3. If σ is an automorphism of a ring T, the skew Laurent polynomial ring
T[y±1;σ] is simple if and only if T is σ-simple and there does not exist m 1 such that σm
≥
is inner on T.
For the following variation, see [16, Theorem 0].
Theorem 3.4. If σ is an automorphism of a ring T, the skew Laurent polynomial ring
T[y±1;σ] is simple if and only if T is σ-simple and there does not exist m 1 for which
≥
there is a unit d of T such that σ(d) = d and σm(a) = dad−1 for all a T.
∈
Corollary 3.5. The localization R is simple if and only if A[w;γ] is α-simple and αm is
Y
outer on A[w;γ] for all m 1.
≥
Proof. Given that R = A[w;γ][y±1;α], this is immediate from Theorem 3.3. (cid:3)
Y
Lemma 3.6. Let m 1. If αm is inner on A[w;γ] then v(m) = 0.
≥
Proof. Suppose that αm is inner on A[w;γ]. Thus there is a unit c = c wn+...+c w+c of
n 1 0
A[w;γ] such that αm(w)c = cw. By (4), ρ−m(w v(m))c = cw whence γ(c)w v(m)c = ρmcw,
− −
where γ is extended to A[w;γ] with γ(w) = w. Comparing constants in A[w;γ] shows that
v(m)c = 0. But as c is a unit in A[w;γ], c is a unit in A so v(m) = 0. (cid:3)
0 0
Remark 3.7. As each v(m) is γ-normal, if some v(m) is a unit then γ(a) = v(m)a(v(m))−1 for
all a A and every ideal of A is a γ-ideal.
∈
There are different simplicity criteria for R in characteristic 0 and characteristic p. The
next lemma is common to both.
8 DAVIDA. JORDANANDIMOGEN E. WELLS
Lemma 3.8. Suppose that A is α-simple and let J be a non-zero α-ideal of A[w;γ]. There
exists a non-negative integer m and an element f = a wm+a wm−1+...+a w+a J
m m−1 1 0
∈
such that a = 1 and, for 0 j m,
m
≤ ≤
m−j
m i
aj = j− ρiα(am−i)(−v)m−i−j, (8)
i=0 (cid:18) (cid:19)
X
and, for all a A,
∈
aa = a γj−m(a). (9)
j j
Proof. For m 0, let
≥
τ (J) = 0 a A : awm +a wm−1 +...+a w +a J .
m m−1 i 0
{ }∪{ ∈ ∈ }
Then τ (J) is an ideal of A and if a ,...,a A then
m 0 m
∈
m m
α a wi = ρ−iα(a )(w v)i.
i i
−
!
i=0 i=0
X X
Thus τ (J) is an α-ideal of A, and, by α-simplicity, τ (J) = A or τ (J) = 0. Let m 0 be
m m m
≥
minimal such that τ (J) = 0. Since τ (J) = A, there exist a ,a ...,a A, with a = 1,
m m 0 1 m m
6 ∈
such that f := a wm+a wm−1+...+a w+a J. Then f ρmα(f) J. As w and v
m m−1 1 0
∈ − ∈
commute,
m
f ρmα(f) = (a wl ρm−lα(a)(w v)l)
l l
− − −
l=0
X
m l
l
= (a wl ρm−lα(a)) ( v)l−jwj
l l
− j −
!
l=0 j=0 (cid:18) (cid:19)
X X
m m−j
m i
= a − ρiα(a )( v)m−i−j wj.
j m−i
− j −
!
j=0 i=0 (cid:18) (cid:19)
X X
The coefficient of wm is 0 so, by the minimality of m, f ρmα(f) = 0 and (8) follows.
−
For all a A, af fγ−m(a) J and
∈ − ∈
m m
af fγ−m(a) = (aa wj a wjγ−m(a)) = (aa a γj−m(a))wj.
j j j j
− − −
j=0 j=0
X X
The coefficient of wm here is 0, so by the minimality of m, af fγ−m(a) = 0 for all a A
− ∈ (cid:3)
and (9) follows.
Lemma 3.9. Suppose that charF = 0, that v is regular in A and that every α-ideal of A is
a γ-ideal. Then A[w;γ] is α-simple if and only if A is α-simple and the 4-tuple (A,α,v,ρ)
is singular.
SIMPLE AMBISKEW POLYNOMIAL RINGS 9
Proof. Suppose that A[w;γ] is α-simple. If I is a non-zero proper α-ideal of A then, as
I is also a γ-ideal of A, IA[w;γ] is a non-zero proper ideal of A[w;γ]. It is clearly an α-
ideal so IA[w;γ] = A[w;γ], whence I = A and A is α-simple. If (A,α,v,ρ) is conformal,
with v = u ρα(u), then the Casimir element z = w u is a non-zero normal non-unit of
− −
A[w;γ], and α(z) = ρ−1z, so zA[w;γ] is a non-zero proper α-ideal of A[w;γ], contradicting
the α-simplicity of A[w;γ]. Hence (A,α,v,ρ) is singular
Conversely, suppose that A is α-simple and that (A,α,v,ρ) is singular. Let J be a non-
zero α-ideal of A[w;γ] and let m and f be as in Lemma 3.8. Suppose that m = 0. By (8)
6
with j = m 1,
−
a ρα(a ) = mv. (10)
m−1 m−1
− −
By (9) with j = m 1, a a = γ(a)a for all a A. Let u = a /m. Then
m−1 m−1 m−1
− ∈ −
ua = γ(a)u for all a A and, by (10), v = u ρα(u). By Lemma 2.5, u is a splitting
∈ −
element, contradicting the singularity of (A,α,v,ρ). Therefore m = 0, 1 = f J, and
J = A[w;γ], which is therefore α-simple. ∈ (cid:3)
We are now in a position to give a simplicity criterion for R when F has characteristic 0.
Theorem 3.10. Suppose that charF = 0. The ring R is simple if and only if
(i) A is α-simple;
(ii) the 4-tuple (A,α,v,ρ) is singular;
(iii) for all m 1, v(m) is a unit of A.
≥
Proof. Suppose that (i), (ii), (iii) hold. By (iii) v = v(1) is regular and Remark 3.7, every α-
ideal is a γ-ideal so, by (i), (ii) and Lemma 3.9, A[w;γ] is α-simple. By (iii) and Lemma 3.6,
each αm is outer on A[w;γ]. By Lemma 3.2 and Corollary 3.5, R is simple.
Conversely, suppose that R is simple. By Lemma 3.2, R is simple and v(m) is a unit
Y
for m 1. So (iii) holds and, as above, v is regular and every α-ideal is a γ-ideal. By
Corolla≥ry 3.5, A[w;γ] is α-simple and, by Lemma 3.9, (i) and (ii) hold. (cid:3)
Remark 3.11. In the conformal case an ambiskew polynomial ring has a normal element,
namely the Casimir element, and this allows for iteration. The next result shows that
iteration is also often available in the simple case. Examples of both types of iteration will
appear in Section 6.
Theorem 3.12. Suppose that charF = 0 and let S = R(A,α,v,ρ) be a simple ambiskew
polynomial ring such that v is an eigenvector for α with eigenvalue µ. Let λ F∗ and extend
∈
the automorphism α to an automorphism of S by setting α(y) = λy and α(x) = µλ−1x. The
iterated ambiskew polynomial ring R(S,α,v,ρ) is simple.
Proof. Using the relations ya = α(a)y, xa = β(a)x and xy ρyx = v it is routine to check
−
that the specified extension of α to S is indeed an automorphism, that γ and β also extend
to S with γ(y) = µ−1y, γ(x) = µx, β(y) = (λµ)−1y and β(x) = λx, and that v is γ-normal
in S. As R is simple, it is α-simple and, as ρ and v are unchanged, v(m) is a unit for all
m 1. Finally if R has a splitting element u = u yixj then u is a splitting element in
i,j 0,0
A a≥nd no such element exists. By Theorem 3.10, R(S,α,v,ρ) is simple. (cid:3)
P
10 DAVIDA. JORDANANDIMOGEN E. WELLS
Remark 3.13. Thesituationinfinitecharacteristic pismorecomplexthaninTheorem3.10.
We shall need to handle binomial coefficients modulo p and this will be done using Lucas’
Congruence [11, p 271] which has the consequence that if n r N have p-ary representa-
≥ ∈
tions
n = n ps +n ps−1 +...+n p+n and r = r ps +r ps−1 +...+r p+r
s s−1 1 0 s s−1 1 0
where 0 n ,r p 1 for s i 0, then
i i
≤ ≤ − ≥ ≥
n
0 mod p n r for all i.
i i
r 6≡ ⇔ ≥
(cid:18) (cid:19)
Singularity does not appear explicitly in the following analogue of Lemma 3.9 but is
covered by condition (ii), with n = 0, in which case (a) is vacuous and (b) states that u is
γ-normal and v = ρα(u) u.
−
Lemma 3.14. Suppose that charF = p = 0, that v is regular in A and that every α-ideal of
6
A is a γ-ideal. Then A[w;γ] is α-simple if and only if
(i) A is α-simple;
(ii) there do not exist a non-negative integer n and elements u and b , 0 i n 1, of
i
≤ ≤ −
A such that
(a) for 0 i n 1, γ(b ) = b , b is γpn−pi-normal and α(b ) = ρpi−pnb ;
i i i i i
≤ ≤ −
(b) γ(u) = u, u is γpn-normal and ρpnα(u) u = vpn + n−1b vpi.
− 0 i
Proof. Suppose that A[w;γ] is simple. As in the proof of Lemma 3P.9, A is α-simple. Suppose
that (ii) does not hold, and let h := wpn + ( n−1b wpi) + u where the elements u and b ,
i=0 i i
0 i n 1, have the properties specified in (a) and (b). Routine calculations show that
ha≤= ≤γpn(a−)h for all a A, that hw = wh aPnd that ρpnα(h) = h. Therefore hA[w;γ] is a
∈
non-zero proper α-ideal of A[w;γ], contradicting the α-simplicity of A[w;γ]. Thus (ii) holds.
For the converse, suppose that (i) and (ii) hold. Let J be a non-zero α-ideal of A[w;γ]
and let m and f be as in Lemma 3.8. Suppose that m = 0. Let J be a non-zero α-
6
ideal of A[w;γ] and proceed as in the proof of Lemma 3.9 to obtain a non-zero element
f = wm + a wm−1 + ... + a of J with minimal degree, to establish (8) and (9) and to
m−1 0
show that f ρmα(f) = 0 and that af fγ−ma = 0 for all a A. By (9) and Lemma 2.5,
− − ∈
γ(a ) = a for all i. As in the proof of Lemma 3.9, applying (8) and (9) with j = m 1 shows
i i
−
that a ρα(a ) = mv and a a = γ(a)a for all a A. If p does not divide m
m−1 m−1 m−1 m−1
− − ∈
then a /m is a splitting element, contradicting (ii) with n = 0. Hence p divides m.
m−1
−
Express m in the form pnr, where p does not divide r and let q = m pn. By (8) with
−
j = q,
pn
m i
a = − ρiα(a )( v)pn−i. (11)
q m−i
q −
i=0 (cid:18) (cid:19)
X
We claim that, for 0 i pn, if a = 0 then i = pn or i = pn pt for some t with
m−i
≤ ≤ 6 −
1 t n and that α(a ) = ρ−pta for 1 t n. To see that the result will
m−pn+pt m−pn+pt
≤ ≤ ≤ ≤