Table Of ContentTHE WEYL ALGEBRAS
David Cock
Supervisor: Dr. Daniel Chan
School of Mathematics,
The University of New South Wales.
November 2004
Submitted in partial fulfillment of the requirements of the degree of
Bachelor of Science with Honours
Contents
Chapter 1 Introduction 1
Chapter 2 Basic Results 3
Chapter 3 Gradings and Filtrations 16
Chapter 4 Gelfand-Kirillov Dimension 21
Chapter 5 Automorphisms of A 29
1
References 53
i
Chapter 1
Introduction
An important result in single-variable calculus is the so-called product rule. That
is, for two polynomials (or more generally, functions) f(x),g(x) : R → R:
δ δ δ
(fg) = ( f)g +f( g)
δx δx δx
It turns out that this formula, which is firmly rooted in calculus has very inter-
esting algebraic properties. If k[x] denotes the ring of polynomials in one variable
over a characteristic 0 field k, differentiation (in the variable x) can be considered as
a map δ : k[x] → k[x]. It is relatively straighforward to verify that the map δ is in
fact a k-linear vector space endomorphism of k[x]. Similarly, we can define another
k-linear endomorphism X by left multiplication by x ie. X(f) = xf. Consider the
expression (δ ·X)f(x). Expanding this gives:
(δ ·X)f(x) = δ(xf(x))
applying the product rule gives
(δ ·X)f(x) = δ(x)f(x)+xδf(x)
= f(x)+(X ·δ)f(x)
1
noting the common factor of f(x) gives us the relation (this time in the ring of
k-linear endomorphisms of k[x]):
δ ·X = X ·δ +1
where 1 is the identity map. This is the defining relation of the first Weyl algebra
which can be viewed as the ring of differential operators on k[x] with polynomial
coefficients. There also exist higher order Weyl algebras related to the polynomial
ring in n variables.
The Weyl algebras arise in a number of contexts, notably as a quotient of the
universal enveloping algebra of certain finite-dimensional Lie algebras (arising from
the Heisenberg group) which have links to quantum mechanics.
The second chapter of this paper covers some basic results on the Weyl alge-
bras, culminating in the proof that they are simple domains. The third chapter
covers gradings, filtrations and the concept of an associated graded algebra. The
fourth chapter introduces the concept of the Gelfand-Kirillov dimension which is a
useful invariant of finitely-generated associative algebras. The final chapter is an
exposition of a proof published in [1] that characterises the automorphisms of the
first Weyl algebra.
2
Chapter 2
Basic Results
In the following, k will always be a field of characteristic 0 and all ideals are two-
sided unless specifically stated otherwise.
Definition 2.1. Let D be a (not neccessarily commutative) domain. Define A(D)
as the non-commutative algebra over D on the two generators p,q with defining
relation
qp−pq = 1 (2.1)
ie.
D < p,q >
A(D) =
(qp−pq −1)
For a field k of characteristic 0, define the first Weyl algebra over k, denoted
by A to be A(k). Define the nth Weyl algebra for n > 1 by A = A(A )
1 n n−1
(note that this definition assumes that A is a domain, this is proved later). For
n−1
convenience assume A = k. Note that for n > 1 there are extra (implicit) relations:
0
q p −p q = 0 for i 6= j ie. the generators of different index commute.
i j j i
Definition 2.2. Define linear maps X,δ : k[x] → k[x] by X(f) = xf and δ(f) = δf
δx
ie. formal differentiation. X and δ generate a sub-algebra of the ring of k-linear
endomorphisms of k[x]. Applying Leibniz’ rule for the differentiation of a product
gives δ ·X = X ·δ +1. Call this algebra A0.
1
For n > 1 and 1 ≤ i ≤ n define linear maps X ,δ : k[x ,...,x ] → k[x ,...,x ]
i i 1 n 1 n
by X (f) = x f and δ (f) = δf ie. formal partial differentiation with respect to x .
i i i δxi i
3
Once again, differenting the product yields the relations δ X = X δ +1 if i = j or
i j j i
δ X = X δ if i 6= j. Call this algebra A0 . Expressed as a quotient:
i j j i n
k < X ,...,X ,δ ,...,δ >
A0 = 1 n 1 n
n (δ X −X δ −∆ )
i j i j ij
where
1 i = j
∆ =
ij
0 otherwise
Lemma 2.3. For any domain D, every x ∈ A(D) can be expressed as Pa piqj
ij
for some finite set {(i,j) ∈ N×N} and a ∈ D.
ij
Proof. Since p,q generate A over D, every x ∈ A can be expressed as some finite
n n
sum
X
bipr(i,1)qs(i,1)...pr(i,ni)qs(i,ni)
i
where b ∈ D,n ∈ Z+ and the leading or trailing coefficent (r and s re-
i i (i,1) (i,ni)
spectively) may be 0. Note that p and q both commute with elements of the base
domain D.
For a monomial product term M, define # (M) to be the number of p terms
p
appearing in M. Define # (M) similarly. Let I(M) be the number of ‘inversions’
q
in the term M. That is, the sum over every q term in M of the number of p terms
which occur to the right. For example:
I(pmqn) = I(λ ∈ k) = 0
I(qp) = 1
I(q2p) = I(qp2) = 2
I(q2p2) = I(qp4) = 4
P
Define I( M ) to be max (I(M )).
i i i i
P
Let R = M be a represention of x in the form described above. If I(R) > 0,
i i
then for at least one monomial term M we must have I(M ) > 0. Thus the
i i
4
monomial M must contain at least one factor of the form qp ie. M = AqpB
i i
where A may be in k and B may be 1. Pick one such term and apply the identity
qp = pq +1 to give:
M0 = b ApqB +b AB
i i i
calculating gives:
I(M0) = max(I(ApqB),I(AB))
i
= max(I(m )−1,I(m )−(# (A)+# (B))−1)
i i q p
clearly therefore, I(M0) = I(M )−1.
i i
Inductively therefore, the sequence of manipulations M → M0 must terminate
i i
in some M∗ with I(M∗) = 0. Applying this to each term of R gives a representation
i i
in the required form.
Corollary 2.3.1. Any x ∈ A can be expressed as
n
X
a pi1...pinqj1...qjn
i1...inj1...jn 1 n 1 n
Proof. Since k is a domain, the result is true for n = 1. A is defined recursively as
n
A(A ). Assuming that A is a domain (again, this is proved shortly) and that
n−1 n−1
the result holds for n−1, the result follows by induction on n since the generators
of different index commute.
Lemma 2.4. Every x ∈ A0 can be expressed as PaXi1...Xinδi1...δin for some
n 1 n 1 n
finite set {(i,j) ∈ N×N}.
Proof. By writing A0 recursively as A0 < X ,δ > and using the relation δ X =
n n−1 n n n n
X δ +1, the result follows as for (2.3.1).
n n
Lemma 2.5. The k-linear map φ : A → A0 defined by φ(p ) = X and φ(q ) = δ
n n i i i i
is an algebra homomorphism.
5
Proof. By the universal property, it suffices to check the images of the defining
relations q p −p q −1 and q p −p q for i 6= j:
i i i i i j j i
φ(q p −p q −1) = δ X −X δ −1 = 0
i i i i i i i i
φ(q p −p q ) = δ X −X δ = 0
i j j i i j j i
Lemma 2.6. For any element of A0 , the representation given in lemma 2.4 is
n
unique.
Proof. Suppose that an element x ∈ A0 has two distinct representations
n
X
a Xi1...Xinδj1...δjn
i1...inj1...jn 1 n 1 n
X
= b Xi1...Xinδj1...δjn
i1...inj1...jn 1 n 1 n
Cancel all equal terms in the above sums to give two differential operators (A, B)
with coefficients a and b such that a 6= b for all
i1...inj1...jn i1...inj1...jn i1...inj1...jn i1...inj1...jn
i ...i j ...j . For each 1 ≤ k ≤ n, pick j∗ to be minimal with respect to the
1 n 1 n k
property that a and b appear as coefficients of A
i1...inj1∗...jk∗jk+1...jn i1...inj1∗...jk∗jk+1...jn
and B respectively, for some i ...i , j ...j . Let p = xj1∗...xjn∗ ∈ k[x ,...,x ].
1 n k+1 n 1 n 1 n
Apply the operators A and B to p.
Ap = Xa Xi1...Xinδj1...δjnxj1∗...xjn∗
i1...inj1...jn 1 n 1 n 1 n
Bp = Xb Xi1...Xinδj1...δjnxj1∗...xjn∗
i1...inj1...jn 1 n 1 n 1 n
Consider a single term of the above sums:
t = a Xi1...Xinδj1...δjnxj1∗...xjn∗
a i1...inj1...jn 1 n 1 n 1 n
t = b Xi1...Xinδj1...δjnxj1∗...xjn∗
b i1...inj1...jn 1 n 1 n 1 n
6
If for all 0 ≤ k ≤ n, j = j ∗, then
k k
t = a (j ∗!...j ∗!)(x i1...x in)
a i1...inj1...jn 1 n 1 n
t = b (j ∗!...j ∗!)(x i1...x in)
b i1...inj1...jn 1 n 1 n
Suppose that the j and j ∗ differ for some set of indices. Let l be the smallest such
k k
index. By the choice of the j ∗, we must have j > j ∗. Since t contains a factor
k l l
δ jlx jl∗, t = 0. Therefore,
l l
X
Ap = ai1...inj1∗...jn∗(j1∗!...jn∗!)(x1i1...xnin)
X
Bp = bi1...inj1∗...jn∗(j1∗!...jn∗!)(x1i1...xnin)
Since the above are simply polynomials in x ,...,x and Ap = Bp, we can equate
1 n
coefficients which implies that
a = b
i1...inj1∗...jn∗ i1...inj1∗...jn∗
which is a contradiction.
Corollary 2.6.1. For any element of A , the representation given in lemma 2.3 is
n
unique.
Proof. This follows from considering the homomorphism φ defined above. If x ∈ A
n
has distinct representations
X
a p i1...p inq i1...q in
i1...inj1...jn 1 n 1 n
and
X
b p i1...p inq i1...q in
i1...inj1...jn 1 n 1 n
7
then
(cid:16)X (cid:17)
φ a p i1...p inq i1...q in
i1...inj1...jn 1 n 1 n
X
= a X i1...X inδ i1...δ in
i1...inj1...jn 1 n 1 n
and
(cid:16)X (cid:17)
φ b p i1...p inq i1...q in
i1...inj1...jn 1 n 1 n
X
= b X i1...X inδ i1...δ in
i1...inj1...jn 1 n 1 n
are distinct representations of φ(x) ∈ A0 , a contradiction.
n
Lemma 2.7. A ’ A0
n n
Proof. Take the homomorphism φ as above. By Lemma 2.4, any x0 ∈ A0 can be
n
expressed in the form
X
a X i1...X inδ j1...δ jn
i1...inj1...jn 1 n 1 n
Let
X
x = a p i1...p inq j1...q jn
i1...inj1...jn 1 n 1 n
Clearly, φ(x) = x0. Therefore φ is surjective.
Take y ∈ A and suppose φ(y) = 0. By lemma 2.3, write y as
n
X
a p i1...p inq j1...q jn
i1...inj1...jn 1 n 1 n
The image, φ(y), is therefore
X
a X i1...X inδ j1...δ jn
i1...inj1...jn 1 n 1 n
Since the representation of φ(y) is unique, we can equate coefficients which implies
that all of the a are zero. Thus φ is injective.
i1...inj1...jn
8
