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The Weyl Algebras THE 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 A1 29 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 th by A1 to be A(k). Define the n Weyl algebra for n > 1 by An = A(An−1) (note that this definition assumes that An−1 is a domain, this is proved later). For convenience assume A0 = k. Note that for n > 1 there are extra (implicit) relations: qipj − pjqi = 0 for i 6= j ie. the generators of different index commute. δf Definition 2.2. Define linear maps X, δ : k[x] → k[x] by X(f) = xf and δ(f) = 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 0 gives δ · X = X · δ + 1. Call this algebra A1. For n > 1 and 1 ≤ i ≤ n define linear maps Xi, δi : k[x1, . , xn] → k[x1, . , xn] δf by Xi(f) = xif and δi(f) = ie. formal partial differentiation with respect to xi. δxi 3 Once again, differenting the product yields the relations δiXj = Xjδi + 1 if i = j or 0 δiXj = Xjδi if i 6= j. Call this algebra An. Expressed as a quotient: 0 k < X1,...,Xn, δ1, . , δn > An = (δiXj − Xiδj − ∆ij) where 1 i = j ∆ij = 0 otherwise P i j Lemma 2.3. For any domain D, every x ∈ A(D) can be expressed as aijp q for some finite set {(i, j) ∈ N × N} and aij ∈ D. Proof. Since p, q generate An over D, every x ∈ An can be expressed as some finite sum X r(i,1) s(i,1) r(i,n ) s(i,n ) bip q . p i q i i + where bi ∈ D, ni ∈ Z and the leading or trailing coefficent (r(i,1) and s(i,ni) re- 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 #p(M) to be the number of p terms appearing in M. Define #q(M) similarly. Let I(M) be the number of ‘inversions’ 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 ( i Mi) to be maxi (I(Mi)). P Let R = i Mi be a represention of x in the form described above. If I(R) > 0, then for at least one monomial term Mi we must have I(Mi) > 0. Thus the 4 monomial Mi must contain at least one factor of the form qp ie. Mi = AqpB where A may be in k and B may be 1. Pick one such term and apply the identity qp = pq + 1 to give: 0 Mi = biApqB + biAB calculating gives: 0 I(Mi ) = max(I(ApqB),I(AB)) = max(I(mi) − 1,I(mi) − (#q(A) + #p(B)) − 1) 0 clearly therefore, I(Mi ) = I(Mi) − 1. 0 Inductively therefore, the sequence of manipulations Mi → Mi must terminate ∗ ∗ in some Mi with I(Mi ) = 0. Applying this to each term of R gives a representation in the required form. Corollary 2.3.1. Any x ∈ An can be expressed as X i1 in j1 jn ai1...inj1...jn p1 . pn q1 . qn Proof. Since k is a domain, the result is true for n = 1. An is defined recursively as A(An−1). Assuming that An−1 is a domain (again, this is proved shortly) and that the result holds for n − 1, the result follows by induction on n since the generators of different index commute. 0 P i1 in i1 in Lemma 2.4. Every x ∈ An can be expressed as aX1 ...Xn δ1 . δn for some finite set {(i, j) ∈ N × N}. 0 0 Proof. By writing An recursively as An−1 < Xn, δn > and using the relation δnXn = Xnδn + 1, the result follows as for (2.3.1). 0 Lemma 2.5. The k-linear map φ : An → An defined by φ(pi) = Xi and φ(qi) = δi is an algebra homomorphism. 5 Proof. By the universal property, it suffices to check the images of the defining relations qipi − piqi − 1 and qipj − pjqi for i 6= j: φ(qipi − piqi − 1) = δiXi − Xiδi − 1 = 0 φ(qipj − pjqi) = δiXj − Xjδi = 0 0 Lemma 2.6. For any element of An, the representation given in lemma 2.4 is unique. 0 Proof. Suppose that an element x ∈ An has two distinct representations X i1 in j1 jn ai1...inj1...jn X1 ...Xn δ1 . δn X i1 in j1 jn = bi1...inj1...jn X1 ...Xn δ1 . δn Cancel all equal terms in the above sums to give two differential operators (A, B) with coefficients ai1...inj1...jn and bi1...inj1...jn such that ai1...inj1...jn 6= bi1...inj1...jn for all ∗ i1 . inj1 . jn . For each 1 ≤ k ≤ n, pick jk to be minimal with respect to the ∗ ∗ ∗ ∗ property that ai1...inj1 ...jk jk+1...jn and bi1...inj1 ...jk jk+1...jn appear as coefficients of A ∗ ∗ j1 jn and B respectively, for some i1 . in, jk+1 . jn. Let p = x1 . xn ∈ k[x1, . , xn]. Apply the operators A and B to p. j∗ ∗ X i1 in j1 jn 1 jn Ap = ai1...inj1...jn X1 ...Xn δ1 . δn x1 . xn j∗ ∗ X i1 in j1 jn 1 jn Bp = bi1...inj1...jn X1 ...Xn δ1 . δn x1 . xn Consider a single term of the above sums: j∗ ∗ i1 in j1 jn 1 jn ta = ai1...inj1...jn X1 ...Xn δ1 . δn x1 . xn j∗ ∗ i1 in j1 jn 1 jn tb = bi1...inj1...jn X1 ...Xn δ1 . δn x1 . xn 6 ∗ If for all 0 ≤ k ≤ n, jk = jk , then ∗ ∗ i1 in ta = ai1...inj1...jn (j1 ! . jn !)(x1 . xn ) ∗ ∗ i1 in tb = bi1...inj1...jn (j1 ! . jn !)(x1 . xn ) ∗ Suppose that the jk and jk differ for some set of indices. Let l be the smallest such ∗ ∗ index. By the choice of the jk , we must have jl > jl . Since t contains a factor ∗ jl jl δl xl , t = 0. Therefore, X ∗ ∗ i i ∗ ∗ 1 n Ap = ai1...inj1 ...jn (j1 ! . jn !)(x1 . xn ) X ∗ ∗ i i ∗ ∗ 1 n Bp = bi1...inj1 ...jn (j1 ! . jn !)(x1 . xn ) Since the above are simply polynomials in x1, . , xn and Ap = Bp, we can equate coefficients which implies that ∗ ∗ ∗ ∗ ai1...inj1 ...jn = bi1...inj1 ...jn which is a contradiction.
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