A note on centralizers in q-deformed Heisenberg algebras

Volodymyr Mazorchuk

Max-Planck-Institut für Mathematik, Bonn, Germany^{∗} ^{†}

Abstract

We reprove and generalize several results (including the main one) from the recent monograph [3] using the technique of generalized Weyl algebras.

1 Introduction

Let 𝔨 be a field, J a set and q = (q_{i})_{i∈J} ∈ 𝔨^{J}. The authors of [3] define the q-deformed Heisenberg algebra as an associative unital 𝔨-algebra, H(q, J), generated over 𝔨 by {X_{i}, Y_{i}|i ∈ J} subject to the following relations: [A_{i}, A_{j}] = [B_{i}, B_{j}] = [A_{i}, B_{j}] = 0, i ≠ j; A_{i}B_{i}−q_{i}B_{i}A_{i} = 1, i ∈ J. Approx. 400 papers, where H(q, J), its properties, generalizations and several physical applications were studied, are cited in [3] and we refer the reader to [3] for these details.

One of the principal theorems in [3] states that, for |J| = 1 and q not a root of unity, any two commuting elements in H(q) = H(q, J) are algebraically dependent ([3, Theorem 7.4]).

The aim of this note is to show (in Section 2) how one can quickly obtain this result and even generalize it to a wider class of algebras, if one realizes that q-deformed Heisenberg algebras belong to the class of generalized Weyl algebras (GWAs), introduced by V.Bavula in the late 80’s. There are several advantages of this approach. First of all, this drastically simplifies the proof and avoids lengthy calculations. Then, next to a generalization of [3, Theorem 7.4] we get a generalization of another central result [3, Theorem 6.6], where the centralizer of an element in H(q) is described. We also get some additional information, e.g. the commutativity of the centralizer, which appears to be new. In Section 3 we consider the root of unity case, in which GWAs have large centers. In this case we obtain a generalization of [3, Theorem 7.5] and [3, Corollary 6.12]. Finally, in Section 4 we use highest weight modules over GWAs to construct their realizations by difference operators acting on a polynomial ring. This generalizes results from [3, Chapter 8].

Our arguments in the proof of Theorem 1 are very close to those of [2], but, formally, V.Bavula considers a slightly different class of algebras (e.g. 𝔨 is supposed to be of characteristic zero) and one has to insert a small preliminary step to be able to transfer his proof to the case we consider here.

Generalized Weyl algebras are associated with a ring, R, central elements 0 ≠ t_{i}, i ∈ J, and pairwise commuting automorphisms σ_{i}, i ∈ J, of R such that σ_{i}(t_{j}) = t_{j}, i ≠ j.

∗on leave from: Chalmers Tekniska Högskola och Göteborgs Universitet, Göteborg, Sweden, e-mail:

volody@math.chalmers.se

†Junior Associate of the ICTP

The corresponding generalized Weyl algebra A(R, {t_{i}}, {σ_{i}}) is defined as a ring, obtained by adjoining to R symbols {X_{i}, Y_{i}|i ∈ J} which satisfy the following relations: Y_{i}X_{i} = t_{i}, X_{i}Y_{i} = σ_{i}(t_{i}), i ∈ J; X_{i}a = σ_{i}(a)X_{i}, aY_{i} = Y_{i}σ_{i}(a), i ∈ J, a ∈ R; [X_{i}, Y_{j}] = [X_{i}, X_{j}] = [Y_{i}, Y_{j}] = 0, i ≠ j. This algebra possesses a natural ℤ^{J}-gradation. To get H(q, J) one should take R = 𝔨[h_{i}, i ∈ J], t_{i} = h_{i}, i ∈ J, and σ_{i} defined by σ_{i}(h_{j}) = h_{j}, i ≠ j and σ_{i}(h_{i}) = q_{i}h_{i} + 1 (this formally works only for q_{i} ≠ 0 as σ_{i} would not be an automorphism otherwise, however, the definition of GWA can be extended to the case when σ_{i} are endomorphisms, moreover, this does not affect our applications to results in [3], because q_{i} ≠ 0 is assumed there).

2 Main result

Consider the GWA A = A(R, t, σ), where R = 𝔨[H], t ∈ R \ 𝔨 and σ(H) = qH + 1 for q ≠ 0 not a root of unity from 𝔨 (in particular, in this case 𝔨 is infinite). As |J| = 1 we will write simply X and Y for X_{i}, Y_{i}. A is an integral domain and ℤ-graded with A_{0} = R, A_{i} = RX^{i} and A_{−i} = RY^{i}, i ∈ ℕ. We set A_{±} = ⊕_{i∈ℕ}A_{±i}.

Theorem 1. The centralizer C(f) of any non-scalar element f ∈ A is a commutative algebra and a free 𝔨[f]-module of finite rank r. Moreover, r divides both the maximal degree π_{+}(f) and the minimal degree π_{−}(f) in the graded decomposition of f.

This is a generalization of [2, Theorem 7] and Amitsur’s theorem on centralizers in Weyl algebra ([1]). Our proof follows closely [2, Chapter 7] (where the case q = 1 and char(𝔨) = 0 was considered) with some differences on the first stage caused by a different choice of σ. But before presenting it we give two immediate corollaries of Theorem 1:

Corollary 1. Two commuting elements of A, in particular, of H(q), are algebraically dependent.

Corollary 2. If f ∈ A is such that π_{+}(f) and π_{−}(f) are relatively prime then C(f) = 𝔨[f].

Proof of Theorem 1. Step 1. Let ℤ act on 𝔨 via 1(x) = qx + 1. We claim that the only finite orbit of this action is {(1 − q)^{−1}}.

Indeed, n(x) = q^{n}x+(q^{n} −1)/(q −1) and n(x) = x implies x = (q^{n} −1)/((q −1)(1−q^{n})) = (1 − q)^{−1}.

Step 2. σ extends to an automorphism of 𝔨(H). By abuse of notation we will denote this extension by σ as well. We claim that σ^{n}(p) = p for some p ∈ 𝔨(H) and some n ∈ ℕ implies p ∈ 𝔨.

Indeed, let p = q_{1}(H)/q_{2}(H). Adding to 𝔨 all roots of q_{1} and q_{2} if necessary, we may

assume that p = α\frac{(H − α_{1}) … (H − α_{i})}{(H − β_{1}) … (H − β_{j})}. As 𝔨 is infinite, σ(p) = p implies that the multisets

{α_{1}, …, α_{i}} and {β_{1}, …, β_{j}} are stable under the ℤ-action from Step 1. As they are finite we get that the only possibility is α_{s} = β_{s} = c = (1 − q)^{−1} and hence p = α(H − c)^{l}, l ∈ ℤ. Now, as q is not a root of unity, we compare the leading coefficients in p and σ(p) and conclude that l = 0.

The referee has pointed out that there is a shorter way to get the above result using the division algorithm and comparing the leading coefficients. However, we decided to keep the above proof as we will use the description of finite ℤ-orbits later in the proof of Theorem 3.

From now on, Bavula’s proof formally works without any change, but we repeat it for convenience of the reader.

Step 3. Let g ∈ A, π_{+}(g) = n > 0 and g_{1}, g_{2} ∈ C(g), m = π_{+}(g_{1}) = π_{+}(g_{2}) ≥ 0, such that m-th graded terms of g_{1} and g_{2} are equal b_{1}X^{m} and b_{2}X^{m}, b_{1}, b_{2} ∈ R, respectively. Then b_{1} and b_{2} are linearly dependent.

Let bX^{n} be the highest term of g, 0 ≠ b ∈ R. From [g, g_{i}] = 0 we get bσ^{n}(b_{i}) = σ^{m}(b)b_{i}, i = 1, 2. Hence σ^{n}(b_{1}/b_{2}) = b_{1}/b_{2} and the statement follows from Step 2.

Step 4. Denote m = π_{−}(f) and n = π_{+}(f). If m = n = 0 then f ∈ R and C(f) = R follows from the graded decomposition of A. Clearly, R is a free 𝔨[f]-module of rank deg_{R}(f) – the degree of the polynomial f.

Assume n > 0 (the case m < 0 is analogous). Then C(f)∩A_{−} = ∅. Otherwise there exists g ∈ C(f) ∩ A_{−} of largest possible degree π_{+}(g) = −k < 0. Then π_{+}(g^{ni}f^{2ki}) = π_{+}(f^{ki}) = nki ≥ 0 for all i ∈ ℕ but, as t ∉ 𝔨, the 𝔨[H]-coefficients of X^{nki} in the graded decomposition of g^{ni}f^{2ki} and f^{ki} have different degrees for sufficiently large i, which contradicts Step 3.

Let κ be the composition of π_{+} with ℤ → ℤ/nℤ. Then G = κ(C(f) \ {0}) is a cyclic group of order r, which divides n. Let G = {m_{1} = 0, …, m_{r}}. For each m_{i} choose g_{i} ∈ C(f) such that κ(g_{i}) = m_{i} and the number π_{+}(g_{i}) be the smallest possible. From C(f) ∩ A_{−} = ∅ and Step 3 it follows that such g_{i} do exist and their highest terms are unique up to non-zero scalars. In particular, we can set g_{1} = 1. Assume ∑_{i} g_{i}φ_{i} = 0 for some φ_{i} ∈ 𝔨[H] and not all φ_{i} are zero. Then there should exist i, j such that π_{+}(g_{i}φ_{i}) = π_{+}(g_{j}φ_{j}). But then κ(g_{i}) = κ(g_{j}) and we obtain a contradiction. Thus the right K[f]-module M, generated by {g_{i}}, is free.

Step 5. Now we claim that C(f) = M, in fact, we need C(f) ⊂ M. If g ∈ C(f) and π_{+}(g) = 0 then Step 3 and C(f) ∩ A_{−} = ∅ imply g ∈ 𝔨 = g_{1}𝔨. If π_{+}(g) = k > 0, then there exists i such that κ(g) = κ(g_{i}) and π_{+}(g_{i}) ≤ k. Hence k = π_{+}(g_{i}f^{s}) for some s ∈ ℤ_{+}. Applying Step 3 one more time we get λ ∈ 𝔨 such that π_{+}(g − λg_{i}f^{s}) < k and the proof is completed by induction on k.

Step 6. Finally, we claim that C(f) is commutative. Choose g ∈ C(f) such that κ(g) is a generator of G. Denote by E ⊂ C(f) the commutative subalgebra, generated by f and g. By Step 3, Step 4 and the induction on the degree of elements, the 𝔨[f]-module C(f)/E is finitedimensional, hence for any u ∈ C(f) there is 0 ≠ P ∈ 𝔨[f] such that P u ∈ E. Let v ∈ C(f) be arbitrary and Qv ∈ E for some 0 ≠ Q ∈ 𝔨[f]. Then P Quv = (P u)(Qv) = (Qv)(P u) = P Qvu. Since A is an integral domain, uv = vu and the proof is complete.

In the same way as in [2], Theorem 1 immediately implies the following.

Corollary 3. 1. Any maximal commutative subalgebra of A has the form C(f) for some non-scalar element f ∈ A.

2. If f, g ∈ A commute then C(f) = C(g).

3. If C is a maximal commutative subalgebra of A and f ∈ A such that p(f) ∈ C for some p(f) ∈ 𝔨[f] then f ∈ C.

4. The intersection of two distinct maximal commutative subalgebras of A is 𝔨.

5. The center of A equals 𝔨.

3 Root of unity case

Assume now that q^{l} = 1, l ∈ ℕ \ {1}, and q^{i} ≠ 1, i = 1, …, l − 1 or q = 1 and char(𝔨) = l > 0. Then the center of the algebra A is quite big and can be completely described. Set R ∋ F = ∏_{i=0}^{l−1} σ(H) and W = ⟨σ⟩. The next Theorem is a generalization of [3, Corollary 6.12].

Theorem 2. The center Z(A) of A equals B = ⟨F, X^{l}, Y^{l}⟩ = ⟨X^{l}, Y^{l}⟩ and A is a finitelygenerated Z(A)-module.

Proof. Step 1. B = ⟨F, X^{l}, Y^{l}⟩ = ⟨X^{l}, Y^{l}⟩ ⊂ Z(A).

If we put n = l into the formula for n(x) in Step 1 of Theorem 1, we get l(x) = x and hence σ^{l}(f) = f for any f ∈ R. Hence σ^{l} = 1. From the definition of A we get fX^{l} = X^{l}f and fY^{l} = Y^{l}f for all f ∈ R. Moreover, X^{l}Y = X^{l−1}(XY) = X^{l−1}σ(t) = tX^{l−1} = (Y X)X^{l−1} = Y X^{l}. Analogously, Y^{l}X = XY^{l}. As σ^{l} = 1, we have σ(F) = F and hence F X = XF and F Y = Y F. Thus the subalgebra of A, generated by F, X^{l} and Y^{l} is contained in Z(A). The equality ⟨F, X^{l}, Y^{l}⟩ = ⟨X^{l}, Y^{l}⟩ follows from F = X^{l}Y^{l} = Y^{l}X^{l}.

Step 2. R ∩ Z(A) = {f ∈ R|σ(f) = f} = 𝔨[F].

The first equality is obvious and hence it is enough to prove that for f ∈ R the equality σ(f) = f implies f ∈ 𝔨[F]. Let f(H) ∈ 𝔨[H] be non-constant and 𝔨̂ be the decomposition field of f. Then f = β ∏_{j=1}^{k}(H −α_{j}) and it is enough to consider the case f = ∏_{w∈W} w(H −α). If q = 1, then σ^{i}(H −s) = H +i−s = H −s if and only if l|s and hence the orbit of (H −s) under the W action contains precisely l elements. If q^{l} = 1 then σ^{i}(H −s) = q^{i}H +(q^{i}−1)/(q−1)−s and again we get that each orbit contains precisely l elements. In particular, deg(f) ≥ l. So, it is enough to prove the statement for f = ∏_{i=0}^{l−1} σ^{i}(H − s). But F = ∏_{i=0}^{l−1} σ^{i}(H) and deg(f − F) < l, hence f − F is constant.

Step 3. Z(A) is a graded subalgebra of A, Z(A)_{i} = Z(A) ∩ A_{i} ≠ 0 if and only if l|i and Z(A)_{i} = X^{i}𝔨[F]. In particular, Z(A) = B.

If z ∈ Z(A) and z = ∑_{i∈ℤ} z_{i} is a graded decomposition of z, from zX = Xz, zY = Y z, zH = Hz we get z_{i}X = Xz_{i}, z_{i}Y = Y z_{i} and z_{i}H = Hz_{i} and hence all z_{i} ∈ Z(A). Therefore Z(A) is also graded. If l does not divide i, then σ^{i}(H) ≠ H and we have X^{i}H ≠ HX^{i} and Y^{i}H ≠ HY^{i}. Hence Z(A)_{i} = 0. If l|i then for f ∈ R from (X^{i}f)X = X(X^{i}f) it follows that σ(f) = f and hence f ∈ 𝔨[F] by Step 2.

Step 4. A is a finitely generated B-module.

As a system of 2l^{2} generators of A over B one cane take, e.g. Y^{i}H^{j}, X^{i}H^{j}, 0 ≤ i, j ≤ l−1. Theorem is proved.

From Theorem 2 we get the following generalization of [3, Theorem 7.5].

Corollary 4. If f, g ∈ A such that fg = gf then there is P (x, y) ∈ Z(A)(x, y) such that P (f, g) = 0.

I would like to finish this section with a counterexample to the conjecture on [3, page 126], where the authors ask if two commuting elements α, β ∈ H(q), whose degrees are relatively prime with l, will be algebraically dependent over 𝔨. Take X and XY^{l}, the elements of degrees 1 and l − 1 respectively, both relatively prime with l. For 0 ≠ p(x, y) ∈ 𝔨[x, y] each summand of p(X, XY^{l}) is homogeneous in A and p(X, XY^{l}) = 0 should be checked on all homogeneous components. As A is an integral domain and X is itself homogeneous, we can assume that deg_{A}(p(X, XY^{l})) = 0. Then p(X, XY^{l}) = ∑_{i} a_{i}X^{il}Y^{il} with a_{i} ∈ 𝔨. Set f_{i} = X^{il}Y^{il}. As t ∉ 𝔨, deg f_{i} = deg(t)^{il} > 0 and hence deg(f_{i}) ≠ deg(f_{j}), i ≠ j. From this we get that p(X, XY^{l}) is non-zero as a sum of polynomials with increasing degrees.

4 Realization by q-difference operators

Here we assume R to be commutative. Let A = A(R, t_{i}, σ_{i}) be a GWA and 𝔫 be a maximal ideal of R containing t_{i} for all i ∈ J. Set X = (X_{i}), Y = (Y_{i}), and for l = (l_{i}) ∈ ℤ_{+}^{J} put X^{l} = ∏_{i} X_{i}^{l_{i}}. Let R_{𝔫} = R/𝔫 and φ : R → R/𝔫 be the canonical projection. Let I_{𝔫} denote the left ideal in A, generated by 𝔫 and all X_{i}. From t_{i} ∈ 𝔫 it follows that I_{𝔫} ∩ A_{0} = 𝔫 and thus for l ∈ ℤ_{+}^{J} there holds I_{𝔫} ∩ A_{−l} = Y^{l}𝔫. Denote by M(𝔫) the left module A/I_{𝔫}, which is non-zero since (A/I_{𝔫})_{0} ≠ 0. As I_{𝔫} is a ℤ^{J}-graded ideal, the module M(𝔫) is also ℤ^{J}-graded and is naturally identified with the polynomial ring R_{𝔫}[Y_{i}] (with right R_{𝔫}-coefficients). If Y^{l} ∈ R_{𝔫}[Y_{i}] is a monomial, the action of A on Y^{l} is defined by Y_{j}(Y^{l}) = Y_{j}Y^{l}, r(Y^{l}) = Y^{l}φ((∏_{i} σ_{i}^{l_{i}})(r)) and X_{j}(Y^{l}) = (1 − δ_{l_{j},0}) ∏_{i} Y^{l_{i}−δ_{i,j}}φ(σ_{j}^{l_{j}}(t_{j})).

Theorem 3. Let A = A(R, t_{i}, σ_{i}), where R = 𝔨[H_{i}]_{i∈J} and σ_{i} are defined as follows: σ_{i}(H_{j}) = H_{j}, j ≠ i, σ_{i}(H_{i}) = q_{i}H_{i} + 1. Assume that h_{i} − (1 − q_{i})^{−1} ∉ 𝔫, q_{i} ≠ 1, and for all i the parameter q_{i} is either not a root of unity or possibly equals 1 if char(𝔨) = 0. Then the annihilator Ann_{A}(M(𝔫)) is zero. In all other cases it is non-zero.

Proof. Clearly, Ann_{A}(M(𝔫)) is a ℤ^{J}-graded ideal of A and we need Ann_{A}(M(𝔫)) ∩ A_{l} = 0, l ∈ ℤ^{j}, only. Then Ann_{A}(1) ∩ R = 𝔫 and hence Ann_{A}(Y^{l}) ∩ R = (∏_{i} σ_{i}^{−l_{i}})(𝔫). By Step 1 and Step 2 of Theorem 1, the condition h_{i}−(1−q_{i})^{−1} ∉ 𝔫 guarantees that the orbit of 𝔫 under W = ⟨σ_{i}⟩ is infinite and hence ∩_{w∈W} w(𝔫) = 0. This implies, in particular, R ∩ Ann_{A}(M(𝔫)) = 0. As Ann_{A}(M(𝔫)) is a ℤ^{J}-graded ideal and A is an integral domain, this automatically implies Ann_{A}(M(𝔫)) = 0.

If h_{i}−(1−q_{i})^{−1} ∈ 𝔫, we have (h_{i}−(1−q_{i})^{−1}) ∈ 𝔫 and w((h_{i}−(1−q_{i})^{−1})) = (h_{i}−(1−q_{i})^{−1}) for any w ∈ W. Hence (h_{i} − (1 − q_{i})^{−1}) ⊂ Ann_{A}(M(𝔫)). If q_{i}^{l} = 1 or q_{i} = 1 and char(𝔨) = l, then X_{i}^{l} ∈ Ann_{A}(M(𝔫)) by Theorem 2. This completes the proof.

Now, if we write H(q, J) as the GWA from Section 1 and set 𝔫 = (h_{i}), the formula X_{j}(Y^{l}) = (1 − δ_{l_{j},0}) ∏_{i} Y^{l_{i}−δ_{i,j}}φ(σ_{j}^{l_{j}}(t_{j})) will read X_{j}(Y^{l}) = (1 − δ_{l_{j},0})(∑_{s=0}^{l_{j}−1} q_{j}^{s}) ∏_{i} Y^{l_{i}−δ_{i,j}}, which is precisely the q_{j}-difference operator on 𝔨[Y] and we get the following refinement of [3, Theorem 8.1, Theorem 8.3]:

Corollary 5. Let 𝔫 = (h_{i}). Then Ann_{A}(M(𝔫)) = 0 if and only if all q_{i} are either non-roots of unity or some q_{i} = 1 and char(𝔨) = 0. In particular, in these cases H(q, J) can be realized via q-difference operators acting on 𝔨[Y].

Acknowledgment. This paper was written during the authors visit to Max-Planck-Institute für Mathematik in Bonn, whose financial support and hospitality are gratefully acknowledged. I also thank L.Turowska and V.Bavula for helpful conversations and the referee for useful remarks that led to the improvements in the paper.

References

[1] S.A.Amitsur, Commutative linear differential operators, Pacific J. Math. 8 (1958), 1-10.

[2] V.Bavula, Generalized Weyl algebras and their representations, St. Petersburg Math. J. 4 (1993), 71- 92.

[3] Lars Hellström and Sergej D. Silvestrov, Commuting elements in q-deformed Heisenberg algebras, World Scientific Publishing Co. Pte. Ltd., Singapore, 2000.

[AMA - Algebra Montpellier Announcements - 01 - 2001] [July 2001]
Received February 2001, revised June 2001.