Generalized Weyl algebras and their global dimension V. V. Bavula 1 - - PDF document

Generalized Weyl algebras and their global dimension

• V. V. Bavula

1 Generalized Weyl algebras

Definition of the generalized Weyl algebras. Let D be a ring, σ = (σ1, ..., σn) a set

• f commuting automorphisms of D, (σiσj = σjσi), a = (a1, ..., an) a set of elements of the

centre Z(D) of D, such that σi(aj) = aj for all i ̸= j. The generalized Weyl algebra A = D(σ, a) (briefly GWA) of degree n with a base ring D is the ring generated by D and the 2n indeterminates X+

1 , ..., X+ n ,X− 1 , ..., X− n subject

to the defining relations: X−

i X+ i = ai,

X+

i X− i = σi(ai),

X±

i α = σ±1 i (α)X± i , ∀ α ∈ D,

[X−

i , X− j ] = [X+ i , X+ j ] = [X+ i , X− j ] = 0, ∀ i ̸= j,

where [x, y] = xy − yx. We say that a and σ are the sets of defining elements and automorphisms of A respectively. We use also the following notation: Xi := X+

i

and Yi := X−

i .

Z-grading on GWA. For an vector k = (k1, . . . , kn) ∈ Zn we put vk = vk1(1) · · · vkn(n) where for 1 ≤ i ≤ n and m ≥ 0: v±m(i) = (X±

i )m, v0(i) = 1.

In the case n = 1, we write vm for vm(1). It follows from the definition of the GWA that A = ⊕k∈ZnAk is a Zn-graded algebra (AkAe ⊆ Ak+e, for all k, e ∈ Zn), where Ak = Dvk = vkD. 1

The category of generalized Weyl algebras is closed under the tensor product (over a base field) of algebras: A ⊗ A′ = D ⊗ D′(σ ∪ σ′, a ∪ a′). Noetherian property. Proposition 1.1 Let A be a generalized Weyl algebra with base ring D. Then

• 1. if D is left (right) Noetherian, then A is left (right) Noetherian;
• 2. if D is a domain and ai ̸= 0, for all i = 1, ..., n, then A is a domain.

The Weyl algebra An. Define the n-th Weyl algebra, An = An(K), over a field (a ring) K to be the associative K-algebra with identity generated by the 2n indeterminates X1, ..., Xn, ∂1, ..., ∂n, subject to the relations: [Xi, Xj] = [∂i, ∂j] = [∂i, Xj] = 0 for i ̸= j, [∂i, Xi] = 1 for all i. The Weyl algebra An is the generalized Weyl algebra A = D(σ; a) of degree n where D = K[H1, ..., Hn] is a polynomial ring in n variables with coefficients in K. The sets of defining elements and automorphisms of A are {ai = Hi | 1 ≤ i ≤ n} and {σi | σi(Hj) = Hj − δij}, respectively, where δij is the Kronecker delta. Moreover, the map An → A, Xi → X+

i , ∂i → X− i , ∂iXi → Hi, i = 1, . . . , n,

is an algebra isomorphism. Examples of generalized Weyl algebras of degree 1. Let A = D(σ, a) be a generalized Weyl algebra of degree 1, a ∈ Z(D), σ ∈ Aut(D). The ring A is generated by D, X = X+

1 and Y = X− 1 subject to the defining relations:

Xα = σ(α)X and Y α = σ−1(α)Y, ∀α ∈ D, Y X = a and XY = σ(a). The algebra A = ⊕n∈Z An 2

is Z-graded, where An = Dvn, vn = Xn (n > 0), vn = Y −n (n < 0), v0 = 1. It follows from the above relations that vnvm = (n, m)vn+m = vn+m < n, m > for some (n, m) ∈ D. If n > 0 and m > 0 then n ≥ m : (n, −m) = σn(a) · · · σn−m+1(a), (−n, m) = σ−n+1(a) · · · σ−n+m(a), n ≤ m : (n, −m) = σn(a) · · · σ(a), (−n, m) = σ−n+1(a) · · · a, in other cases (n, m) = 1. Let A(i) = Di(σi, ai) (i = 1, ..., n) be a GWA of degree 1 over a field K and assume that each σi is a K-automorphism, then their tensor product ⊗n

1A(i) = (⊗n 1Di)((σi), (ai))

is a GWA of degree n over K. This construction allows us to build a great deal of examples of generalized Weyl alge- bras of degree n. For example, the n-th Weyl algebra An(K) can be written in this way as An(K) = A1 ⊗ · · · ⊗ A1, n times. Example 1. A = K[H](σ, a), where σ is an arbitrary automorphism of K[H], i.e. σ(H) = λH + µ, λ ̸= 0, µ ∈ K, a ∈ K[H]. When σ(H) = H − 1 and a = H we get the 1st Weyl algebra: A1 ≃ K[H](σ, a = H). Let Fm = AG

1 be the fixed ring where G is the cyclic group of order m, acting on the

Weyl algebra A1 as follows: ω : A1 → A1, ∂ → ω∂, X → ω−1X, where ω is a primitive m’th root of unity. Then Fm = K < ∂m, ∂X, Xm >≃ K[H](σ, a = mmH(H − 1/m) . . . (H − (m − 1)/m), 3

∂m ↔ Y, Xm ↔ X, ∂X/m ↔ H, is a GWA of degree 1, where σ(H) = H − 1, char K = 0. Case, µ = 0. Let Λ = K < X, Y | XY = λY X >, the quantum plane, then Λ ≃ K[H](σ, a = H), σ(H) = λH. Let A(S2

λ)

= K < X, Y, H | XH = λHX, Y H = λ−1HY, λ1/2Y X = −(c − H)(d + H), λ−1/2XY = −(c − λH)(d + λH) > be the algebra of functions on the quantum 2-dimensional sphere, then A(S2

λ) ≃ K[H](σ, a = −λ−1/2(c − H)(d + H)), σ(H) = λH.

The quantum Weyl algebra A1(q) =< x, ∂ | ∂x − qx∂ = 1 >

• f degree 1 over K (q ̸= 0 ∈ K) is the GWA

A1(q) = K[H](σ, a = H)

• f degree 1 where σ(H) = q−1(H − 1).

Example 2. A = D(σ, a), where D = K[H, (H − µ/(1 − λ))−1], σ(H) = λH + µ, λ ̸= 0, 1, µ ∈ K, a ̸= 0 ∈ D. In particular, when µ = 0 we have A = K[H, H−1](σ, a), σ(H) = λH. Example 3. Consider the K-algebra Λ(b), deformation of Usl(2) which is generated by X, Y, Z subject to the relations: [H, X] = X, [H, Y ] = −Y, [X, Y ] = b ̸= 0 ∈ K[H]. Then Λ(b) ∼ = K[H, C](σ, a = C − α) 4

where σ : K[H, C] → K[H, C], H → H − 1, C → C, and α ∈ K[H] is a solution of the equation α − σ(α) = b. For b = 2H, Λ(b) = Usl(2). If K is a field of characteristic zero, then the center of Λ(b) is K[C] . For any λ ∈ K the factor algebra Λ(b, λ) := Λ(b)/Λ(b)(C − λ) is isomorphic to the GWA from Example 1 with the defining element λ − α. Let U(λ) := Usl(2)/Usl(2)(C − λ) ≃ K[H](σ, λ − H(H + 1)) be the infinite dimensional primitive factor of Usl(2). Example 4. The quantum Heisenberg algebra: Hq = K < X, Y, H | XH = q2HX, Y H = q−2HY, XY − q−2Y X = q−1H >, q ∈ K, q4 ̸= 1 is isomorphic to the GWA of degree 1: Hq ≃ k[H, C](σ; a = ρ−1C − µH = q2(C − H/q(1 − q4)), σ(H) = q2H, σ(C) = q−2C. The element Ω = HC belongs to the centre of Hq. For each λ ̸= 0 ∈ K the factor algebra Hq(λ) := Hq/(Ω − λ) is the GWA of degree 1: Hq(λ) ≃ K[H, H−1](σ; a = q2(λH−1 − H/q(1 − q4)), σ(H) = q2H. The ambiskew polynomial rings E are GWAs. Let D be an ring, σ ∈Aut(D). Suppose that elements b and ρ belong to the centre of D, moreover, ρ is invertible and σ-stable, i.e. σ(ρ) = ρ. Then the ambiskew polynomial ring E = D < σ; b, ρ > is obtained by adjoining to D two symbols X and Y subject to the relations: Xα = σ(α)X, Y α = σ−1(α)Y, ∀α ∈ D; XY − ρY X = b. 5

If D = K[H] is the polynomial ring, ρ = 1, b = 2H, and σ(H) = H − 1, we get the universal enveloping algebra Usl(2)). The ring E is the iterated skew polynomial ring E = D[Y ; σ−1][X; σ, ∂] where ∂ is the σ−derivation of D[Y ; σ−1] such that ∂D = 0 and ∂Y = b (here σ is extended from D to D[Y ; σ−1] by the rule: σ(Y ) = ρY ). Lemma 1.2 shows that the rings E = D < σ; b, ρ > are generalized Weyl algebras of degree 1. Lemma 1.2 Each iterated skew polynomial ring E is the generalized Weyl algebra of degree 1 with base polynomial ring D[H] and defining automorphism σ : σ(H) = ρ(H) + b (σ acts

• n D as before):

D < σ; b, ρ >≃ D[H](σ; a = H), X ↔ X, Y ↔ Y, d ↔ d (∀d ∈ D), Y X ↔ H. An element d of a ring D is normal if dD = Dd. Lemma 1.3 The following are equivalent.

• 1. C = ρ(Y X + α) = XY + σ(α) is normal in D < σ; b, ρ >;
• 2. ρα − σ(α) = b for some α ∈ D;
• 3. D[H] = D[C1] for some C1 ∈ D[H] such that σ(C1) = γC1 where γ ∈ D is invertible

and σ(γ) = γ. Corollary 1.4 Let E be as in Lemma 1.3. Then D < σ; b, ρ >≃ D[C](σ, a = ρ−1C − α), σ(C) = ρC. Putting ρ = 1 we obtain the following Lemma and Corollary. Lemma 1.5 The following are equivalent.

• 1. C = Y X + α = XY + σ(α) is central in D < σ; b, ρ = 1 >;
• 2. α − σ(α) = b for some α ∈ D;
• 3. D[H] = D[C1] for some C1 ∈ D[H] such that σ(C1) = C1 (then C1 = γC for some

central invertible σ-stable element γ). Corollary 1.6 Suppose that Lemma 1.5 holds. Then D < σ; b, ρ = 1 >≃ D[C](σ, a = C − α), σ(C) = C. 6

Further Examples. For q, h = q − q−1 ∈ K = C, the algebra Uq = Uqsl(2) is generated by X, Y, H−, H+ subject to the relations: H+H− = H−H+ = 1, XH± = q±1H±X, Y H± = q∓1H±Y, [X, Y ] = (H2

+ − H2 −)/h.

It follows from the relations that: Uq ≃ K[C, H, H−1](σ, a = C + {H2/(q2 − 1) − H−2/(q−2 − 1)}/2h) where σ(H) = qH, σ(C) = C. Woronowicz’s deformation V is generated by V0, V−, V+ subject to the relations: [V0, V+]s2 ≡ s2V0V+ − s−2V+V0 = V+, [V−, V0]s2 = V−, [V+, V−]1/s ≡ s−1V+V− − sV−V+ = V0. The algebra V is the GWA : V ≃ K[u, v](σ, a = v), V± ↔ X±, V0 ↔ u, V−V+ ↔ v, where σ : u → s2(s2u − 1), v → s2v + su, is the automorphism of the polynomial ring K[u, v] in two variables u and v. If we put H = u + s2/(1 − s4), and Z = v + u/s(1 − s2) + s3/(1 − s2)(1 − s4), then σ(H) = s4H, σ(Z) = s2Z and K[u, v] = K[H, Z]. So, V ≃ K[H, Z](σ, a = Z + αH + β), V± ↔ X±, V0 ↔ H − s2/(1 − s4), where σ : H → s4H, Z → s2Z; α = −1/s(1 − s2) and β = s/(1 − s4). Witten’s first deformation E is the algebra generated by E0, E−, E+ : [E0, E+]p ≡ pE0E+ − p−1E+E0 = E+, [E−, E0]p = E−, [E+, E−] = E0 − (p − 1/p)E2

0,

where p ̸= 0, ±1, ±i ∈ K. The Casimir operator which commutes with all generators is : C = E−E+ + E0(E0 + p)/p(p2 + 1). 7

Witten’s first deformation is the GWA : E ≃ K[C, H](σ, a = C − H(H + 1)/(p + p−1)), E± ↔ X±, E0 ↔ pH, where σ : C → C, H → p2(H − 1). The quantum group Oq2(so(k, 3)) = K[H] < σ; b = (q − q−1)H, ρ = 1 >, σ(H) = q2H, q ∈ K, by Corollary 1.6 is isomorphic to the GWA of degree 1: Oq2(so(k, 3)) = K[H, C](σ, a = C + H2/q(1 + q2)), σ(H) = q2H, σ(C) = C.

2 Simplicity Criteria for GWAs

Theorem 2.1 Let A = D(σ, a) be a GWA of degree 1. The algebra A is a simple algebra iff

• 1. the ring D has no proper σ-stable ideal,
• 2. no power of σ is an inner automorphism of D,
• 3. For all n ≥ 1, D = Da + Dσn(a), and
• 4. a is not a zero divisor in D.

Theorem 2.2 Let A = D(σ, a) be a GWA of degree n and D is a domain. The algebra A is a simple algebra iff

• 1. the ring D has no proper σ-stable ideal,
• 2. the subgroup of the group of outer automorphisms Aut(R)/Inn(R) of R generated by

the automorphisms σ1, . . . , σn is a free abelian group of rank n, and

• 3. For all m ≥ 1 and i = 1, . . . , n, D = Dai + Dσm

i (ai).

3 The global dimension of GWAs

Let A = D(σ, a) be a GWA. Denote by n = lgd D the left global dimension of the ring D.

• (Theorem) Then lgd A = n, n + 1 or ∞.
• (Theorem) If D is Noetherian and lgd A < ∞. Then gld A = max{n, µ′} where

µ′ = sup {pdM| AM is simple and finitely generated over D}. 8

• (Corollary) If D is semiprime Noetherian, n < ∞, lgd A < ∞, then gld A = n + 1

if and only if there is a simple A-module M which is finitely generated over D and with pdDM = n.

• (Theorem) If D is a commutative Noetherian domain of finite global dimension n,

a ̸= 0. Then gld A < ∞ if and only if pdA/A(X, p) < ∞ for all prime ideals p of D which contain a.

• (Theorem) If D is a commutative Noetherian ring, n < ∞, a is regular, gld A < ∞.

Then gld A = n + 1 if and only if either there is a semistable (i.e. σi(m) = m for some i ≥ 1) maximal ideal m of D of height n or there are maximal ideals p, q of D

• f height n such that σi(p) = q for some i ̸= 0 ∈ Z and a ∈ p, q.

In many cases the considered algebras are the generalized Weyl algebras A = D(σ, a)

• f degree 1 where D is a Dedekind ring D. By Theorem 3.1 we can compute their global

homological dimension. Theorem 3.1 Let A = D(σ, a) be the generalized Weyl algebra of degree 1 where D is a Dedekind ring, Da = pn1

1 . . . pns n is the product of the distinct maximal ideals of D. Then

the global dimension of A is gl.dim A =            ∞ , if a = 0 or ni ≥ 2 for some i; 2 , if a ̸= 0, n1 = . . . = ns = 1, s ≥ 1 or a is invertible and there exists an integer k ≥ 1 such that either σk(pi) = pj for some i, j

• r σk(q) = q for some maximal ideal q of D;

1 , otherwise. Tensor homological minimal algebras. Denote by Alg (respectively, LN; LFN; SC) the class of all (respectively, left Noetherian; left Noetherian and finitely generated; somewhat commutative) algebras over a fixed field K. The tensor product ⊗ means ⊗K and a module means a left module. Denote by d one of the following dimensions of rings: wd, the weak dimension; lgd, the left homological dimension; kd, the Krull dimension (in the sense of Gabriel-Rentschler). We give for a large class of (infinite dimensional non-commutative) algebras the answer to the question:

• when the dimension of the tensor product of algebras is the sum of dimensions of the

multiples, d(Λ1 ⊗ · · · ⊗ Λn) = dΛ1 + · · · + dΛn. (1.1) Usually, (1.1) is not true. For example, if Ln is the division ring of the polynomial ring Pn = K[X1, . . . , Xn] in n variables, then lgd Ln ⊗ Ln = n ̸= 0 = lgd Ln + lgd Ln. 9

Therefore, we should put on the algebras Λi some restricted conditions for (1.1) to be

• satisfied. It is well known, that for all above dimensions d=wd, kd (respectively, lgd)

d(A ⊗ B) ≥ dA + dB, for all (respectively, left Noetherian) algebras A, B. (1.2) So it is natural to give the following Definition. An algebra A is a tensor d-minimal with respect to some class of algebras Ω, if d(A ⊗ B) = dA + dB, for each B ∈ Ω. (1.3) In the case wd (lgd) we say that the algebra A is tensor weak minimal (tensor homological minimal), briefly, TWM (THM). The polynomial ring Pn is tensor homological minimal with respect to Alg, since for each algebra B lgd(Pn ⊗ B) = n + lgd B = lgd Pn + lgd B (Hilbert′s syzygy theorem). Lemma 3.2 Let Λi (i = 1, . . . , n) be tensor d-minimal algebras with respect to Ω such that Λi ⊗ Ω ⊆ Ω for all i. Then the tensor product ⊗n

i=1Λi is a tensor d-minimal algebra with

respect to Ω and d(⊗n

i=1Λi ⊗ B) = n

∑

i=1

dΛi + dB for each B ∈ Ω. Theorem 3.3 Let K be an algebraically closed uncountable field. Consider the following generalized Weyl algebras of degree 1 with non-zero defining element a.

• 1. char K = 0,

K[H](σ, a), σ(H) = H − µ, µ ̸= 0 ∈ K (if σ(H) = H − 1 we have the Weyl algebra A1 ≃ K[H](σ, a = H) and all prime quotients of Usl(2) : U(λ) = Usl(2)/(C − λ) ≃ K[H](σ, a = λ − H(H + 1)), where C is the Casimir element); 2. K[H, (H − µ/(1 − λ))−1](σ, a), σ(H) = λH + µ, λ ̸= 0, 1 ∈ K is not a root of 1, µ ∈ K (if σ(H) = λH, λ ̸= 0, 1, then K[H, H−1](σ, a = H) is a localization of the quantum plane Λ = K < X, Y |XY = λY X > at the mul- tiplicatively closed set S = {(Y X)n, n = 0, 1, . . .}). 10

Each tensor product Λ = ⊗n

i=1Λi of these algebras is a tensor homological minimal algebra

with respect to LFN , therefore lgd(⊗n

i=1Λi ⊗B) =

∑

n i=1lgdΛi+lgd B, for any left Noetherian finitely generated algebra B.

If K in not necessarily uncountable, then Λ, where all algebras Λi from 1, is a tensor homological minimal algebra with respect to SC. In particular, the left global dimension of the Weyl algebra is lgd (An) = lgd (A1 ⊗ · · · ⊗ A1) = n · lgd A1 = n.

References

[1] V. V. Bavula, Finite-dimensionality of Extn and Torn of simple modules over a class of

• algebras. Funct. Anal. Appl. 25 (1991), no. 3, 229–230.

[2] V. V. Bavula, Generalized Weyl algebras and their representations. (Russian) Algebra i Analiz 4 (1992), no. 1, 75–97; translation in St. Petersburg Math. J. 4 (1993), no. 1, 71–92. Department of Pure Mathematics University of Sheffield Hicks Building Sheffield S3 7RH UK email: v.bavula@sheffield.ac.uk

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