Weyl Algebra Modules

Viktor Bekkert ^{∗}, Georgia Benkart^{†}, and Vyacheslav Futorny^{‡}

Abstract

We investigate weight modules for finite and infinite Weyl algebras, classifying all such simple modules. We also study the representation type of the blocks of locally-finite weight module categories and describe indecomposable modules in tame blocks.

1 Introduction.

The nth Weyl algebra A_{n} is the unital associative algebra over a field 𝕂 with generators x_{i}, ∂_{i}, i = 1, 2, …, n, which satisfy the defining relations

[x_{i}, x_{j}] = 0 = [∂_{i}, ∂_{j}]  (1)
[∂_{i}, x_{j}] = δ_{i,j}1,  (2)

for 1 ≤ i, j ≤ n, expressed using the commutator [a, b] = ab − ba. Here n may be a positive integer or countably infinite, and δ_{i,j} is the Kronecker delta.

Critical to our investigations of modules for A_{n} is a realization of A_{n} as a generalized Weyl algebra D(\underline{σ}, \underline{a}) of degree n in the sense of [B1]. The ingredients for a generalized Weyl algebra D(\underline{σ}, \underline{a}) are a unital associative algebra D; an n-tuple of central elements \underline{a} = (a_{1}, …, a_{n}) of D; and an ntuple of commuting automorphisms \underline{σ} = (σ_{1}, …, σ_{n}) of D such that σ_{i}(a_{j}) = a_{j} if i ≠ j. The algebra A = D(\underline{σ}, \underline{a}) is generated over D by elements X_{i}, Y_{i}, i = 1, …, n, which satisfy the relations

∗Supported in part by CNPq #301183/00-7.

†Supported in part by National Science Foundation Grants #DMS–9810361 (at MSRI)

and #DMS–9970119.

‡Supported in part by CNPq #300679/1997-1, CRDF grant UM1-327, and National Science Foundation Grant #DMS–9810361 (at MSRI).

Mathematics Subject Classification: Primary 16D60, 16D90, 16D70; Secondary 17B65

X_{i}d = σ_{i}(d)X_{i}  Y_{i}d = σ_{i}^{−1}(d)Y_{i}  (3)
Y_{i}X_{i} = a_{i}  X_{i}Y_{i} = σ_{i}(a_{i})  (4)
[X_{i}, X_{j}] = 0 = [Y_{i}, Y_{j}]  1 ≤ i, j ≤ n  (5)
[Y_{i}, X_{j}] = 0  1 ≤ i ≠ j ≤ n,  (6)

for all d ∈ D. (Here also n may be a positive integer or countably infinite.)

The elements t_{i} = ∂_{i}x_{i} in the Weyl algebra A_{n} generate a polynomial algebra D = 𝕂[t_{1}, …, t_{n}]. By taking \underline{a} = (a_{1}, …, a_{n}) to be the tuple of elements a_{i} = t_{i} in D, setting X_{i} = x_{i}, and Y_{i} = ∂_{i}, and letting \underline{σ} = (σ_{1}, …, σ_{n}) be the tuple of commuting automorphisms of D given by σ_{i}(t_{j}) = t_{j} − δ_{i,j}1, where δ_{i,j} = 1 if i = j and is 0 otherwise, we obtain a realization of A_{n} as the generalized Weyl algebra D(\underline{σ}, \underline{a}).

In [Bl], Block classified the simple modules for the Weyl algebra A_{1} over an algebraically closed field 𝕂 of characteristic 0 along with the simple modules for the Lie algebra 𝔰𝔩_{2}(𝕂). An alternate approach using generalized Weyl algebras has been proposed in [B1], [B2], [BO1] when 𝕂 is arbitrary. A module V for a generalized Weyl algebra D(\underline{σ}, \underline{a}) is said to be a weight module if V = ⨁_{𝔪∈𝔪𝔞𝔵D} V_{𝔪}, where 𝔪𝔞𝔵D is the set of maximal ideals of D and V_{𝔪} = {v ∈ V | 𝔪v = 0}. Weight modules over A_{1} (and over some other generalized Weyl algebras of degree 1) for arbitrary fields have been studied in [DGO]. For algebraically closed fields of characteristic 0, simple holonomic modules over the Weyl algebra A_{2} have been classified in [BO2]. Weight and generalized weight modules for A_{n} for n < ∞ (and for some other generalized Weyl algebras) over an algebraically closed field have been investigated in [BBL] and [BB]. In particular, [BBL] has given an explicit description of the weight modules for the complex Weyl algebras A_{n} for n < ∞ and has used that to construct the weight modules having all weight spaces one-dimensional for the finite-dimensional simple complex Lie algebras. The paper [BB] has furnished a classification of the representation tame blocks in the category of locally-finite weight modules and described indecomposable modules in tame blocks. In this case all simple modules can be obtained as tensor products of simple modules over A_{1}.

In our paper we combine the techniques of [BB], [DGO], and [GP] to describe the simple weight A_{n}-modules for n ≤ ∞ over any field. We also classify the blocks of tame representation type in the category of all locallyfinite weight modules together with the indecomposable modules in each such block. From the classification of certain simple weight modules for A_{∞}, we obtain many examples of simple ℤ-graded modules for an infinitedimensional Heisenberg Lie algebra with infinite-dimensional homogeneous components and a nonzero central charge.

Here is a brief outline of the paper. Section 2 introduces the main object of study – weight modules for Weyl algebras. In Section 3 we define the category C_{O} and its skeleton. We determine all simple weight modules for Weyl algebras in Section 4. Various examples are described in Section 5. Using the classification of certain simple weight modules for A_{∞}, we discuss simple ℤ-graded modules with a nonzero central charge for the infinitedimensional Heisenberg algebra in Section 6. Finally, in the last section we describe the blocks of tame representation type and classify indecomposable modules in those blocks.

2 Weight modules over Weyl algebras.

Assume for 1 ≤ n ≤ ∞ that A = A_{n}, the nth Weyl algebra, and let D = 𝕂[t_{1}, …, t_{n}] for t_{i} = ∂_{i}x_{i} as above. Let G be the group generated by the automorphisms σ_{i} (1 ≤ i ≤ n) of D, where σ_{i}(t_{j}) = t_{j} − δ_{i,j}1. Then G acts on the set 𝔪𝔞𝔵D of maximal ideals of D.

An A-module V is a weight module if

V = ⨁_{𝔪∈𝔪𝔞𝔵D} V_{𝔪}, V_{𝔪} = {v ∈ V | 𝔪v = 0} ,

and if V_{𝔪} ≠ 0, we say that 𝔪 is a weight of V. The set 𝔰𝔲𝔭𝔭(V) = {𝔪 ∈ 𝔪𝔞𝔵D | V_{𝔪} ≠ 0} is the support of V. It is easy to see that x_{i} V_{𝔪} ⊆ V_{σ_{i}(𝔪)} and ∂_{i} V_{𝔪} ⊆ V_{σ_{i}^{−1}(𝔪)}.

Let W(A) be the category of weight A-modules. For each subset T ⊂ 𝔪𝔞𝔵 D we will denote by W_{T} (A) the full subcategory in W(A) consisting of all modules V with 𝔰𝔲𝔭𝔭(V) ⊂ T. Each weight module V can be decomposed into a direct sum of A-submodules:

V = ⨁_{O} V_{O}, V_{O} := ⨁_{𝔪∈O} V_{𝔪}

where O runs over the orbits of G on 𝔪𝔞𝔵D. Hence, the category W(A) decomposes into a sum of full subcategories corresponding to the orbits of G. In particular, if V is indecomposable, then its support belongs to a single orbit O.

An orbit is said to be cyclic (resp. linear) if τ(𝔪) = 𝔪 for some τ ∈ G, τ ≠ 1 (resp. τ(𝔪) ≠ 𝔪 for all τ ∈ G, τ ≠ 1). It is evident that if char𝕂 = 0, then each orbit is linear. On the contrary, if char𝕂 = p > 0, then each orbit O is cyclic and |O| < ∞ if n < ∞.

Remark 2.1. Let V ∈ W(A) be simple and assume 𝔪 ∈ 𝔰𝔲𝔭𝔭(V). Suppose that the corresponding orbit is linear. Following [BO1], we consider A(𝔪) = A/A𝔪 ∈ W(A) and the maximal submodule N(𝔪) ⊂ A(𝔪) which is the sum of all submodules that intersect D/𝔪 trivially. Then V ≅ A(𝔪)/N(𝔪). Hence, for linear orbits the problem is to describe the submodule N(𝔪).

Assume now that char𝕂 = p > 0. The algebra A = A_{n} is ℤ^{n}-graded: A = ∑_{\underline{j}∈ℤ^{n}} A\underline{j}. For \underline{j} = (j_{1}, …, j_{n}) ∈ ℤ^{n}, the homogeneous space A\underline{j} =

Dv\underline{j}, where v\underline{j} = v_{j_{1}}(1) … v_{j_{n}}(n), and v_{j}(i) = x_{i}^{j}, v_{−j}(i) = ∂_{i}^{j}, j > 0, v_{0}(i) = 1 for all i = 1, …, n. We define the Veronese subalgebra A_{[p]} = ∑_{\underline{j}∈ℤ^{n}} A_{p\underline{j}} (cf. [BO1]). If 𝔪 is a maximal ideal of D, then A_{[p]}𝔪 ⊆ A_{[p]} as D ⊆ A\underline{0}.

Remark 2.2. Let char𝕂 = p > 0 and V ∈ W(A) be a simple module for A = A_{n}. Assume V_{𝔪} ≠ 0 for some maximal ideal 𝔪 of D. Define an Amodule by A(𝔪) = A ⊗_{A_{[p]}} V_{𝔪}. Then V ≅ A(𝔪)/N(𝔪) where N(𝔪) is the maximal submodule in A(𝔪) which trivially intersects D/𝔪. The problem of describing such simple modules V reduces to determining the simple modules for A_{[p]}/A_{[p]}𝔪 and the maximal submodule N(𝔪).

Remarks 2.1 and 2.2 provide the construction of all the simple weight A_{n}- modules, reducing the problem of their classification to the description of the maximal submodule N(𝔪). Because determining the maximal submodule N(𝔪) can be highly nontrivial, in what follows we will adopt an alternate approach based on the ideas of [DGO] and [BB].

2.3. We say that the maximal ideal 𝔪 of D is a break with respect to i if t_{i} ∈ 𝔪 for some i ∈ {1, …, n}. An orbit O is degenerate with respect to i if it contains a break with respect to i (for some 𝔪 ∈ O). Often we simply say that O is degenerate without specifying i or 𝔪.

A maximal ideal 𝔪 of D is a maximal break with respect to \underline{I} = \underline{I}(𝔪) ⊂ \{1, …, n\} if t_{i} ∈ 𝔪 for each i ∈ \underline{I}, and t_{j} ∉ τ(𝔪) for each j ∉ \underline{I} and each τ ∈ G. We set \underline{I}^{c} := \{1, …, n\} \ \underline{I} and say that the maximal break has order |\underline{I}| (which may be infinite if n = ∞).

To see that maximal breaks exist for degenerate orbits when n < ∞, suppose O = O(𝔪) is degenerate. If 𝔫 ∈ O is a break with respect to i, then t_{i} ∈ 𝔫. Moreover, for any product τ = σ_{1}^{k_{1}} ⋯ σ_{n}^{k_{n}} ∈ G with the σ_{i} term omitted, we have t_{i} ∈ τ(𝔫) as σ_{j}(t_{i}) = t_{i} for j ≠ i. Suppose \underline{I} = {i_{1}, …, i_{s}} ⊆ {1, …, n} is a maximal set of indices such that O has breaks 𝔫_{i_{1}}, …, 𝔫_{i_{s}} relative to i_{1}, …, i_{s} respectively and no breaks with respect to any j ∈ \underline{I}^{c}. Assume 𝔫_{i} = σ_{1}^{r_{1,i}} ⋯ σ_{n}^{r_{n,i}}(𝔪) for each i ∈ \underline{I}. Then for i ∈ \underline{I}, t_{i} ∈ σ_{i}^{r_{i,i}}(𝔪), hence t_{i} ∈ 𝔫 := σ_{i_{1}}^{r_{i_{1},i_{1}}} ⋯ σ_{i_{s}}^{r_{is,is}} (𝔪), and 𝔫 is a maximal break in O with respect to \underline{I} = \{i_{1}, …, i_{s}\}.

2.4. Because it is not apparent that maximal breaks exist for A_{∞}, we will assume that the degenerate orbits O we consider for A_{∞} have a maximal break 𝔪. In this case, the break set \underline{I}(𝔪) may be finite or infinite.

2.5. In what follows, we let \underline{τ}(𝔪) = (τ_{1}, ⋯, τ_{n}), where τ_{i} = σ_{i} if σ_{i}(𝔪) = 𝔪 and τ_{i} = 1 otherwise. Each such τ_{i} induces an automorphism on D/𝔪.

3 Category C_{O} and its skeleton.

We begin this section with the abstract concepts needed to describe Weyl algebra modules, which can be found in ([GR], Chap. 2). Ultimately we specialize to very particular categories determined from these modules.

3.1. Categories

Let 𝔽 be a field. A category C is said to be an 𝔽-category if each morphism set C(α, β) is equipped with an 𝔽-bimodule structure, the composition is 𝔽-linear with respect to both the left and right 𝔽-module structures, and (αλ)β = α(λβ) for any possible morphisms α, β and λ ∈ 𝔽. If additionally αλ = λα for all morphisms α and λ ∈ 𝔽, then we say that C is an 𝔽-linear category. An 𝔽-algebra A gives rise to an 𝔽-category with just one object - say α such that C(α, α) = A, and the composition is multiplication in A.

With any category C, we can associate an 𝔽-linear category 𝔽C whose objects are the same 𝔬𝔟(C) = 𝔬𝔟(𝔽C), and the morphism space 𝔽C(α, β) has as its basis over 𝔽 the elements of C(α, β). The composition in 𝔽C is the 𝔽-linear extension of the composition in C.

An 𝔽-linear functor between two 𝔽-linear categories C and D is a functor F : C → D whose defining maps F (α, β) : C(α, β) → D(F α, F β) are 𝔽-linear for all α, β ∈ 𝔬𝔟C.

Let 𝔽 − 𝔪𝔬𝔡 denote the category of 𝔽-vector spaces and let 𝔄𝔟 be the category of abelian groups. Given an 𝔽-category C we denote by C-𝔪𝔬𝔡 the category of all additive functors M : C → 𝔄𝔟. The functors M are called C-modules, or more precisely left C-modules. For an α ∈ 𝔬𝔟C, the elements of M(α) are the elements of the module M (at α). To make the action more module-like, we shall write au instead of M(a)u for u ∈ M(α) and a ∈ C(α, β). For each α ∈ 𝔬𝔟C, the group M(α) becomes an 𝔽-vector space if we put λu = (λ1_{α})u for all u ∈ M(α) and λ ∈ 𝔽. By C-𝔣𝔡𝔪𝔬𝔡 we mean the full subcategory of all locally finite-dimensional objects M in C-𝔪𝔬𝔡, that is dim_{𝔽} M(α) < ∞ for all α ∈ 𝔬𝔟C.

If N is a C-submodule of M, then N(α) is a subspace of M(α) for all α ∈ 𝔬𝔟C and au ∈ N(β) for all a ∈ C(α, β) and u ∈ N(α). The module M is simple if it has no nontrivial C-submodules; while M = M_{1} ⊕ M_{2} if M_{1} and M_{2} are submodules such that M(α) = M_{1}(α) ⊕ M_{2}(α) for all α ∈ 𝔬𝔟C. The module M is indecomposable if it cannot be written as a direct sum M = M_{1} ⊕ M_{2} of proper submodules.

Right C-modules may be defined as C^{𝔬𝔭}-modules, where C^{𝔬𝔭} is the category opposite to C. In this case we write va for v ∈ M(β) and for a ∈ C(α, β) = C^{𝔬𝔭}(β, α). The category of right C-modules will be denoted by 𝔪𝔬𝔡-C and the full subcategory of locally finite-dimensional objects by 𝔣𝔡𝔪𝔬𝔡-C.

For 𝔽-categories C and D a C − D-bimodule is an additive functor B : C × D^{𝔬𝔭} → 𝔄𝔟. If u ∈ B(γ, δ), then u is an element of the module B with source γ and target δ, and we write aub rather than B(a, b)u for a ∈ C(γ, γ′) and b ∈ D(δ′, δ). Any 𝔽-category C can be viewed as a C − C-bimodule mapping a pair of objects (β, γ) to C(β, γ).

A category C is said to be basic if

• all its objects are pairwise nonisomorphic;

• for each object α there are no nontrivial idempotents in C(α, α). A full subcategory S is a skeleton of a category C if it is basic, and each object α ∈ 𝔬𝔟C is isomorphic to a direct summand of a (finite) direct sum of some objects of S. If the category C has a unique direct decomposition property, then it has a skeleton which is unique up to isomorphism. The natural inclusion functor I : S → C of a skeleton S into C, is an equivalence of categories. By this functor, C becomes a C − S-bimodule in an obvious way. Tensoring this bimodule over S furnishes equivalences S-𝔪𝔬𝔡 → C-𝔪𝔬𝔡 and S-𝔣𝔡𝔪𝔬𝔡 → C-𝔣𝔡𝔪𝔬𝔡. This is a reformulation of Morita equivalence in the categorical context.

3.2. Quivers

A quiver Q is a tuple (Q_{0}, Q_{1}, 𝔰, 𝔢) consisting of a set Q_{0} of vertices, a set Q_{1} of arrows, and maps 𝔰, 𝔢 : Q_{1} → Q_{0} which specify the starting and ending vertices. A path p in Q of length ℓ(p) = n ≥ 1 is a sequence of arrows a_{n}, …, a_{1} such that 𝔰(a_{i+1}) = 𝔢(a_{i}) for 1 ≤ i < n. Set 𝔰(p) = 𝔰(a_{1}) and 𝔢(p) = 𝔢(a_{n}). Then the concatenation p′p of two paths p, p′ is defined in the natural way whenever 𝔰(p′) = 𝔢(p). Every vertex a ∈ Q_{0} determines a path 1_{a} (of length 0) with 𝔰(1_{a}) = a and 𝔢(1_{a}) = a. A quiver Q determines a category 𝔭𝔞𝔱𝔥Q with objects the vertices of Q and morphisms from vertex a to vertex b being the paths from a to b. The composition in 𝔭𝔞𝔱𝔥Q of paths of positive length is just concatenation, and the path 1_{a} acts as the identity on all paths for which composition makes sense. The corresponding path algebra, which we will abbreviate simply 𝔽Q, has an 𝔽-basis consisting of the paths of Q with multiplication given by concatenation of paths. If R is the ideal of 𝔽Q generated by a family {ρ_{i}} of morphisms, we say that 𝔽Q/R is the 𝔽-linear category defined by the quiver Q and the relations {ρ_{i}}.

3.3. Category C_{O}

In this section we prove that the category W_{O}(A) of weight modules for the Weyl algebra A = A_{n} (1 ≤ n ≤ ∞) having support in the orbit O, (which is assumed to have a maximal break if O is degenerate and n = ∞), is equivalent to C_{O}-𝔪𝔬𝔡, the category of modules over a certain category C_{O} first introduced in [DGO].

Assume 𝔪 is a fixed maximal break of the orbit O if O is degenerate and is any fixed element of O otherwise. For a given 𝔫 ∈ O we denote by σ_{𝔫} the element of G/𝔰𝔱𝔞𝔟(𝔪) (where 𝔰𝔱𝔞𝔟(𝔪) is the stabilizer subgroup of 𝔪 in G) such that 𝔫 = σ_{𝔫}(𝔪). Thus, σ_{𝔫} induces an isomorphism D/𝔪 → D/𝔫, which we again denote σ_{𝔫}, and its inverse induces an isomorphism σ_{𝔫}^{−1} : D/𝔫 → D/𝔪.

We define C_{O} as the D/𝔪-category with 𝔬𝔟C_{O} = O, generated over D/𝔪 by the set of morphisms {X_{𝔫,i}, Y_{𝔫,i} | 𝔫 ∈ O, 1 ≤ i ≤ n}, where X_{𝔫,i} : 𝔫 → σ_{i}(𝔫) and Y_{𝔫,i} : σ_{i}(𝔫) → 𝔫, subject to the relations:

• X_{𝔫,i}λ = λX_{𝔫,i}, Y_{𝔫,i}λ = λY_{𝔫,i}, Y_{𝔫,i}X_{𝔫,i} = σ_{𝔫}^{−1}(t̄_{i})1_{𝔫} and X_{𝔫,i}Y_{𝔫,i} = σ_{𝔫}^{−1}(t̄_{i})1_{σ_{i}(𝔫)}, t̄_{i} = t_{i} +𝔫, for each λ ∈ D/𝔪, 𝔫 ∈ O, and each 1 ≤ i ≤ n such that σ_{i}(𝔪) ≠ 𝔪;

• X_{𝔫,i}λ = σ_{i}(λ)X_{𝔫,i}, Y_{𝔫,i}σ_{i}(λ) = λY_{𝔫,i}, Y_{𝔫,i}X_{𝔫,i} = σ_{𝔫}^{−1}(t̄_{i})1_{𝔫} and X_{𝔫,i}Y_{𝔫,i} = σ_{i}σ_{𝔫}^{−1}(t̄_{i})1_{σ_{i}(𝔫)}, t̄_{i} = t_{i} + 𝔫, for each λ ∈ D/𝔪, 𝔫 ∈ O, and each 1 ≤ i ≤ n such that σ_{i}(𝔪) = 𝔪;

• U_{𝔫,i}V_{𝔭,j} − V_{𝔮,j}U_{𝔯,i} = 0 for all j ≠ i and all possible U, V ∈ {X, Y }, 𝔫, 𝔭, 𝔮, 𝔯 ∈ O, for which the last equality makes sense.

Note that the category C_{O} is 𝔽-linear when σ_{i}(𝔪) ≠ 𝔪 for all i; in particular, C_{O} is always 𝔽-linear when 𝔽 has characteristic 0.

Proposition 3.4. Let A = A_{n}, the nth Weyl algebra over the field 𝕂, and let O be an orbit of 𝔪𝔞𝔵D (which is assumed to have a maximal break if O is degenerate and n = ∞). Then W_{O}(A) ≅ C_{O}-𝔪𝔬𝔡.

Proof. The proof is similar to that of Proposition 2.2 in [DGO]. We assume 𝔪 is the designated maximal ideal of O. Let V = ⨁_{𝔫∈O} V_{𝔫} belong to W_{O}(A). For each 𝔫 ∈ O set M_{V} (𝔫) = V_{𝔫}. Using the isomorphism σ_{𝔫} : D/𝔪 → D/𝔫, we can view M_{V} (𝔫) as a D/𝔪-vector space via d̄v := σ_{𝔫}(d̄)v, (d̄ = d + 𝔪). For v ∈ M_{V} (𝔫) and w ∈ M_{V} (σ_{i}(𝔫)), we define X_{𝔫,i}v := x_{i}v ∈ M_{V} (σ_{i}(𝔫)) and Y_{𝔫,i}w := ∂_{i}w ∈ M_{V} (𝔫). Then if d̄∈ D/𝔪, we have

X_{𝔫,i}d̄v = x_{i}σ_{𝔫}(d̄)v = σ_{i}(σ_{𝔫}(d̄))x_{i}v
= \{ \begin{matrix}d̄X_{𝔫,i}v & if σ_{i}(𝔪) ≠ 𝔪 \\ σ_{i}(d̄)X_{𝔫,i}v & if σ_{i}(𝔪) = 𝔪, \end{matrix}

as σ_{i}(𝔫) = 𝔫 for all 𝔫 ∈ O, and σ_{i}(σ_{𝔫}(d̄)) = σ_{𝔫}(σ_{i}(d̄)), whenever σ_{i}(𝔪) = 𝔪. Likewise Y_{𝔫,i}d̄w = d̄Y_{𝔫,i}w. We also have Y_{𝔫,i}X_{𝔫,i}v = ∂_{i}x_{i}v = t_{i}v = (t_{i} + 𝔫)v = σ_{𝔫}^{−1}(t̄_{i})v for t̄_{i} = t_{i} + 𝔫, and X_{𝔫,i}Y_{𝔫,i}w = x_{i}∂_{i}w = σ_{i}(t_{i})w = (σ_{i}(t_{i}) + σ_{i}(𝔫))w = σ_{𝔫}^{−1}(t̄_{i})w. The reason for the last equality is that there is an action of G on G/𝔰𝔱𝔞𝔟(𝔪), and under this action σ_{i}σ_{𝔫} = σ_{σ_{i}(𝔫)}. Hence, M_{V} (M_{V} : C_{O} → D/𝔪-𝔪𝔬𝔡) is a C_{O}-module, and we have a functor

F : W_{O}(A) → C_{O}-𝔪𝔬𝔡,  V ↦ M_{V} .  (7)

Conversely, for each M ∈ C_{O}-𝔪𝔬𝔡, we define the A-module V_{M} := ⊕_{𝔫∈O}M(𝔫) where dv := σ_{𝔫}^{−1}(d̄)v, d̄ = d + 𝔫 ∈ D/𝔫, x_{i}v := X_{𝔫,i}v, and ∂_{i}v := Y_{σ_{i}^{−1}(𝔫),i}v for v ∈ M(𝔫). This gives a functor

F′ : C_{O}-𝔪𝔬𝔡 → W_{O}(A),  M ↦ V_{M}  (8)

which is inverse to F.

3.5. Skeleton of C_{O}

For a given orbit O we define the set 𝔅_{O} according to the following rule: If O is nondegenerate, then 𝔅_{O} = {𝔪} for the fixed 𝔪 ∈ O used to define C_{O}, and \underline{I}(𝔪) = ∅. If O is a linear degenerate orbit, then 𝔪 ∈ O is the fixed maximal break used in the construction of C_{O}. We assume that the set of breaks is \underline{I} = \underline{I}(𝔪) = \{i_{1}, …, i_{s}\}. We set 𝔅_{O} = \{σ_{i_{1}}^{δ_{1}} ⋯ σ_{i_{s}}^{δ_{s}}(𝔪) | δ_{j} ∈ \{0, 1\}\} (where only finitely many δ_{j} are nonzero when n = ∞). In this case, for each maximal ideal 𝔫 = σ_{i_{1}}^{δ_{1}} ⋯ σ_{i_{s}}^{δ_{s}}(𝔪) ∈ 𝔅_{O} we set

O_{𝔫} := \{σ_{1}^{γ_{1}} ⋯ σ_{n}^{γ_{n}}(𝔫) | γ_{i} = (−1)^{δ_{i}+1}k, k ∈ ℕ if i ∈ \underline{I}, and γ_{i} ∈ ℤ if i ∈ \underline{I}^{c}\},

(9)

(where only finitely many γ_{i} are nonzero when n = ∞). If O is a cyclic degenerate orbit, and 𝔪 ∈ O is the maximal break used to define C_{O}, then 𝔅_{O} := {𝔪} and O_{𝔪} := O.

3.6. We introduce an equivalence relation ∼ on the set of maximal ideals 𝔫 ∈ 𝔪𝔞𝔵D. This relation is the transitive extension of the relation specified by 𝔫 ∼ σ_{i}(𝔫) if and only if t_{i} ∉ 𝔫.

Lemma 3.7. Assume 𝔫 and 𝔭 belong to O. Then 𝔫 ∼ 𝔭 if and only if 𝔫 and 𝔭 are isomorphic in C_{O}.

Proof. Let 𝔪 be the fixed maximal break of O if O is degenerate and be the unique element of 𝔅_{O} if O is nondegenerate. We assume for each 𝔫 ∈ O that σ_{𝔫} ∈ G/𝔰𝔱𝔞𝔟(𝔪) satisfies σ_{𝔫}(𝔪) = 𝔫 as in the proof of Proposition 3.4.

It suffices to argue for any 𝔫 ∈ O that 𝔫 ∼ σ_{i}(𝔫) if and only if 𝔫 and σ_{i}(𝔫) are isomorphic in C_{O}.

(⟹) Assume 𝔫 ∼ σ_{i}(𝔫). Then t_{i} ∉ 𝔫 so t_{i} + 𝔫 ≠ 0. As σ_{𝔫}^{−1} : D/𝔫 → D/𝔪 is an isomorphism, σ_{𝔫}^{−1}(t_{i} + 𝔫) ≠ 0 and by (3.3), X_{𝔫,i} and Y_{𝔫,i} are invertible in C_{O}. Hence 𝔫 and σ_{i}(𝔫) are isomorphic in C_{O}.

(⟸) Assume 𝔫 and σ_{i}(𝔫) are isomorphic in C_{O}. It follows from the definition of the morphisms in C_{O} that C_{O}(𝔫, σ_{i}(𝔫)) = (D/𝔪)X_{𝔫,i} and C_{O}(σ_{i}(𝔫), 𝔫) = (D/𝔪)Y_{𝔫,i} for 1 ≤ i ≤ n. Hence X_{𝔫,i} and Y_{𝔫,i} are isomorphisms. If t_{i} ∈ 𝔫, then σ_{𝔫}^{−1}(t_{i} + 𝔫) = 0 ∈ D/𝔪 and Y_{𝔫,i}X_{𝔫,i} = σ_{𝔫}^{−1}(t_{i} + 𝔫)1_{𝔫} = 0, a contradiction.

Remark 3.8. For 𝔫 ∈ 𝔅_{O}, the equivalence class of 𝔫 is exactly O_{𝔫}, and O is the disjoint union of the sets O_{𝔫}, 𝔫 ∈ 𝔅_{O}.

Corollary 3.9. The full subcategory S_{O} with 𝔬𝔟S_{O} = 𝔅_{O} is a skeleton of the category C_{O}.

Proof. Assume 𝔪 is the designated maximal ideal in O, and the corresponding set of breaks is \underline{I} = \underline{I}(𝔪) = \{i_{1}, …, i_{s}\} (which is empty if O is nondegenerate). In the degenerate case, the maximal ideal σ_{i_{1}}^{δ_{1}} ⋯ σ_{i_{s}}^{δ_{s}}(𝔪) has breaks at {i_{k} | δ_{k} = 0}. Thus, these ideals are pairwise nonisomorphic. Moreover, if 𝔫 = σ_{1}^{r_{1}} ⋯ σ_{n}^{r_{n}}(𝔪) ∈ O (with only finitely many r_{i} nonzero when n = ∞), then by Lemma 3.7, 𝔫 is isomorphic to σ_{i_{1}}^{δ_{1}} ⋯ σ_{i_{s}}^{δ_{s}}(𝔪) ∈ 𝔅_{O} such that δ_{i} = 1 whenever i ∈ \underline{I} and r_{i} ≥ 1, and δ_{i} = 0 otherwise. In the nondegenerate case, 𝔅_{O} = {𝔪} and every element of O is isomorphic to 𝔪 by the lemma. Consequently, in both cases 𝔅_{O} is a skeleton of C_{O}.

3.10. Algebraic description of the skeleton

For a given field 𝔽 and an arbitrary subset I of the set ℕ of positive integers, we define the category A = A(𝔽, I) as the 𝔽-linear category with the set of objects 𝔬𝔟A := {0, 1}^{|I|} (which we assume have only finitely many nonzero components when |I| = ∞) generated (over 𝔽) by the set of morphisms, A_{1} := {a_{α,i}, b_{α,i} | α ∈ 𝔬𝔟A, i ∈ I, and α_{i} = 0}, where a_{α,i} : α → β and b_{α,i} : β → α, such that β_{j} = α_{j} for all j ≠ i and β_{i} = 1, subject to the relations:

• a_{α,i}b_{α,i} = b_{α,i}a_{α,i} = 0 for each a_{α,i}, b_{α,i} ∈ A_{1};

• u_{α,i}v_{β,j} − v_{γ,j}u_{δ,i} = 0 for all j ≠ i and all possible u, v ∈ {a, b}, α, β, γ, δ ∈ 𝔬𝔟A, for which the last equality makes sense.

When I is empty, let A (𝔽, ∅) be the category with a unique object, say ω, and with morphism set 𝔽1_{ω}.

The 𝔽-algebra corresponding to the category A(𝔽, I) above consists of finite 𝔽-linear combinations of morphisms in the category, and the product is simply composition of morphisms whenever it is defined and is 0 otherwise. It has a unit element if I is finite. We adopt the same notation A(𝔽, I) for the algebra, as it will be evident from the context which one is meant. The algebra A(𝔽, I) is finite-dimensional when |I| = s < ∞, since for I = {i_{1}, ⋯, i_{s}}, A(𝔽, I) ≅ A(𝔽, {i_{1}}) ⊗ ⋯ ⊗ A(𝔽, {i_{s}}). It is easy to see that algebra A(𝔽, \{i\}) is isomorphic to the algebra \overline{Q}_{1} := 𝔽Q_{1}/R corresponding to the following quiver and relations:

[figure: Q_{1} :; a; b; 1; 2; ab = ba = 0.; ∘; ∘]

As \overline{Q}_{1} is generated over 𝔽 by 1_{1}, 1_{2}, a, b, modulo the relations ab = 0 = ba, it has dimension 4. (Here and throughout the paper we do not list obvious relations such as a^{2} = 0, 1_{1}^{2} = 1_{1}, a1_{1} = a, etc.)

3.11. For subsets I and J of ℕ and an I∪J-tuple \underline{τ} = (τ_{i})_{i∈I∪J} of commuting automorphisms of 𝔽, let B(𝔽, I, J, \underline{τ}) be the 𝔽-category with a unique object, say ω, whose endomorphism algebra is a unital associative algebra over 𝔽 generated by a_{i}, b_{i}, c_{j}, c_{j}^{−1}, i ∈ I, j ∈ J, subject to the relations: a_{i}b_{i} = b_{i}a_{i} = 0, a_{i}λ = τ_{i}(λ)a_{i}, λb_{i} = b_{i}τ_{i}(λ) for i ∈ I and λ ∈ 𝔽, c_{j}c_{j}^{−1} = 1 = 1_{ω}, c_{j}λ = τ_{j}(λ)c_{j} for j ∈ J and λ ∈ 𝔽, and u_{s}v_{t} = v_{t}u_{s} for u, v ∈ {a, b, c} and all s, t ∈ I ∪ J.

Proposition 3.12. Let A_{n} be the nth Weyl algebra and assume O is an orbit in D = 𝕂[t_{1}, …, t_{n}] under the automorphism group G (which is assumed to have a maximal break if O is degenerate and n = ∞). Let 𝔪 be the unique element of 𝔅_{O} in the nondegenerate case, and the designated maximal break of O in the degenerate case.

1. If char 𝕂 = 0, then S_{O} ≅ A(D/𝔪, \underline{I}(𝔪));

2. If char 𝕂 = p > 0, then S_{O} ≅ B(D/𝔪, \underline{I}(𝔪), \underline{I}(𝔪)^{c}, \underline{τ}(𝔪)).

Proof. Corollary 3.9 shows that the category S_{O} with objects 𝔅_{O} is a skeleton of C_{O}.

Assume that char 𝕂 = 0. When \underline{I}(𝔪) ≠ ∅, define the functor G : A(D/𝔪, \underline{I}(𝔪)) → S_{O} as follows:

G(α) = σ^{α}(𝔪),  G(a_{α,i}) = X_{σ^{α}(𝔪),i},  G(b_{α,i}) = Y_{σ^{α}(𝔪),i}  (10)

where σ^{α} = ∏_{i} σ_{i}^{α_{i}}. From (3.3) it is easy to see that this is an isomorphism. Now suppose that \underline{I}(𝔪) = ∅. The orbit is nondegenerate, and in this case, the functor G : A(D/𝔪, ∅) → S_{O} defined by

G(ω) = 𝔪,  G(1_{ω}) = 1_{𝔪}  (11)

is an isomorphism.

We suppose that char 𝕂 = p, and for 1 ≤ i ≤ n, let r_{i} denote the minimal positive integer such that σ_{i}^{r_{i}}(𝔪) = 𝔪. Thus, r_{i} = p or 1. Then we define

G : B(D/𝔪, \underline{I}(𝔪), \underline{I}(𝔪)^{c}, \underline{τ}(𝔪)) → S_{O}
G(ω) = 𝔪
G(a_{i}) = X_{σ_{i}^{r_{i}−1}(𝔪),i} X_{σ_{i}^{r_{i}−2}(𝔪),i} ⋯ X_{σ_{i}(𝔪),i} X_{𝔪,i}  (12)
G(b_{i}) = Y_{𝔪,i} Y_{σ_{i}(𝔪),i} ⋯ Y_{σ_{i}^{r_{i}−1}(𝔪),i}
G(c_{j}) = X_{σ_{j}^{r_{j} −1}(𝔪),j} X_{σ_{j}^{r_{j} −2}(𝔪),j} ⋯ X_{σ_{j}(𝔪),j} X_{𝔪,j}.

It can be easily checked that this determines an isomorphism.

Remark 3.13. The algebras A(D/𝔪, \underline{I}(𝔪)) are finite-dimensional when |\underline{I}(𝔪)| < ∞; whereas the algebras B(D/𝔪, \underline{I}(𝔪), \underline{I}(𝔪)^{c}, \underline{τ}(𝔪)) are always infinite-dimensional.

4 Simple weight modules for Weyl algebras.

The purpose of this section is to describe the simple weight modules for the Weyl algebra A_{n} with 1 ≤ n ≤ ∞. These results generalize those obtained in [BB], where the underlying field 𝕂 was assumed to be algebraically closed. The case of the first Weyl algebra over an arbitrary field has been treated previously in for example ([DGO], [B1], [B2] and [BO1]).

4.1. Proposition 3.12 allows us to focus on the algebras of the form A(𝔽, I) in the characteristic 0 case. Thus, we assume 𝔽 is a field of characteristic 0. For each α ∈ 𝔬𝔟A(𝔽, I), define a simple A(𝔽, I)-module S_{α} such that S_{α}(β) = δ_{α,β}𝔽 for all objects β ∈ 𝔬𝔟A(𝔽, I), and let all morphisms be trivial.

In order to deal with the case n = ∞ we will need the following wellknown result, which can be found in [DOF].

Lemma 4.2. Let C be a category, and assume M is a simple C-module and M(α) ≠ 0 for some α ∈ 𝔬𝔟C. Then M(α) is simple as a C(α, α)-module. Conversely for any simple C(α, α)-module N, there exists a unique (up to isomorphism) simple C-module M such that M(α) ≅ N as C(α, α)-modules.

Proposition 4.3. Any simple module over the algebra A = A(𝔽, I) is isomorphic to S_{α} for some object α ∈ 𝔬𝔟A(𝔽, I).

Proof. If n < ∞ and I ⊆ {1, …, n}, then the statement is obvious since A is finite-dimensional. Suppose n = ∞, and let M be a simple A-module. For an object α ∈ 𝔬𝔟A denote by M(α), the module M at α. Assume that M(α) ≠ 0. By Lemma 4.2, M(α) is a simple A(α, α)-module. Suppose u ∈ A(α, α), u ≠ 0, 1_{α}, and u = (u_{1})_{β^{1},i_{1}} ⋯ (u_{m})_{β^{m},i_{m}} where u_{j} ∈ {a, b}, β^{j} ∈ 𝔬𝔟A, and i_{j} ∈ I for each j = 1, …, m. Since u ∈ A(α, α), then for each j, there exists k such that {u_{j}, u_{k}} = {a, b} and i_{j} = i_{k}. Using the second set of relations (the commuting relations) in A(𝔽, I), we can rewrite u as product u = ∏_{j} a_{β^{j},s_{j}}b_{β^{j},s_{j}}. But then the first set of relations in A(𝔽, I) implies that u = 0, which contradicts our assumption. Hence, A(α, α) = 𝔽1_{α}, and M(α) is one-dimensional because it is simple as an A(α, α)-module.

Consider the set J of all β ∈ 𝔬𝔟A such that α_{j} = β_{j} for all j ∈ I except one and M(β) ≠ 0. When β ∈ J and β_{i} ≠ α_{i}, we say that β is an α-sink if a_{α,i} ≠ 0 (hence b_{α,i} = 0) and is an α-source if b_{α,i} ≠ 0 (hence a_{α,i} = 0). Let J_{1} (resp. J_{2}) be the set of all α-sinks in J (resp. the set of all αsources in J) and J_{3} = J \ (J_{1} ∪ J_{2}). Consider the following submodule of M: M′ = A( ∑_{β∈J_{1}∪J_{3}} M(β)). We claim that M′ ∩ M(α) = 0. Indeed, if u ∈ A and uM(β) = M(α) for some β ∈ J_{1} ∪ J_{3}, then it follows from the commuting relations that u contains b_{α,i}, where β_{i} = 1 and β_{j} = α_{j} for j ≠ i. But because β ∈ J_{1} ∪ J_{3}, it must be that b_{α,i} = 0, and hence u = 0. We conclude that M′ ∩ M(α) = 0, so that M′ = 0 by simplicity. This argument shows that whenever M(α) ≠ 0, then there are no α-sinks. But then by simplicity, we must have M = M(α), and hence M = S_{α} in this case.

4.4. In the characteristic p > 0 case, it suffices to consider the algebras B(𝔽, I, J, \underline{τ}) where I, J ⊆ ℕ by Proposition 3.12. Assume Γ ⊆ I, and ξ : Γ → {0, 1}. Associated to this data is the subalgebra

ℜ_{Γ,ξ} = 𝔽[d_{i}, c_{j}^{±1}, \underline{τ} | i ∈ Γ, j ∈ J]  (13)

of B(𝔽, I, J, \underline{τ}) where d_{i} = a_{i} if ξ(i) = 0 and d_{i} = b_{i} if ξ(i) = 1. (In (13), only the automorphisms τ_{i} with i ∈ Γ ∪ J are used, and ℜ_{Γ,ξ} is a skew polynomial algebra.) Then for a maximal ideal 𝔑 in ℜ_{Γ,ξ}, define the simple module S_{Γ,ξ,𝔑} = ℜ_{Γ,ξ}/𝔑. Note that when I = ∅, then Γ = ∅; and whenever Γ = ∅, we assume there is just one ξ and ℜ_{Γ,ξ} := 𝔽[c_{j}^{±1}, \underline{τ} | j ∈ J]. Again in this case, for every maximal ideal 𝔑 of ℜ_{Γ,ξ}, we define S_{Γ,ξ,𝔑} = ℜ_{Γ,ξ}/𝔑.

Proposition 4.5. The modules S_{Γ,ξ,𝔑} constitute an exhaustive list of pairwise nonisomorphic simple B (𝔽, I, J, \underline{τ})-modules.

Proof. As the case I = ∅ is apparent, we may assume S is a simple B(𝔽, I, J, \underline{τ})-module where I ≠ ∅. It is clear that both a_{i}S and b_{i}S are submodules for any i ∈ I. Because S is simple, each of them coincides either with S or with 0. But a_{i}S = S implies b_{i}S = 0 and vice versa. Therefore, either a_{i}S = b_{i}S = 0, or a_{i}S = S, b_{i}S = 0, or a_{i}S = 0, b_{i}S = S for any i ∈ I. Let Γ = {i ∈ I | either a_{i}S ≠ 0 or b_{i}S ≠ 0}. Define ξ : Γ → {0, 1} by ξ(i) = 0 if a_{i}S ≠ 0, ξ(i) = 1 if b_{i}S ≠ 0. Then B(𝔽, I, J, \underline{τ}) modulo the annihilator of S in 𝔽[a_{i}, b_{i}, \underline{τ} | i ∈ I] is isomorphic to ℜ_{Γ,ξ}. The result then follows.

4.6. Next we will describe the simple modules for the Weyl algebra A = A_{n} corresponding to Propositions 4.3 and 4.5. As before, we assume D = 𝕂[t_{1}, …, t_{n}] where t_{i} = ∂_{i}x_{i}.

Assume first that char 𝕂 = 0 and that O is a nondegenerate orbit of G on 𝔪𝔞𝔵 D. Recall in this case that 𝔅_{O} = {𝔪}. Set

S(O) = ⨁_{𝔫∈O}D/𝔫  (14)

and define a left A-module structure on S(O) by specifying for i = 1, …, n

x_{i}(d + 𝔫) := σ_{i}(d) + σ_{i}(𝔫),  ∂_{i}(d + 𝔫) := t_{i}σ_{i}^{−1}(d) + σ_{i}^{−1}(𝔫). (15)

As S(O) is generated by 1 + 𝔪, we have that S(O) ≅ A/A𝔪 where 1 + 𝔪 ↦ 1 + A𝔪.

Now suppose that char 𝕂 = 0, O is degenerate, and 𝔪 is the fixed maximal break. For 𝔭 ∈ 𝔅_{O} set

S(O, 𝔭) := ⨁_{𝔫∈O_{𝔭}} D/𝔫,  (16)

where O_{𝔭} is as in (9). One can define a structure of a left A-module on S(O, 𝔭) by the same formulae as in (15), but when the image is not in S(O, 𝔭), the result is 0. Assuming 𝔭 = σ_{i_{1}}^{δ_{1}} ⋯ σ_{i_{s}}^{δ_{s}}(𝔪), where \underline{I}(𝔪) = {i_{1}, …, i_{s}}, we have in this case S(O, 𝔭) ≅ A/A(𝔭, Z_{i_{1}}, …, Z_{i_{s}}) where Z_{k} = x_{k} if 𝔭 is a break with respect to k, and Z_{k} = ∂_{k} otherwise. The isomorphism is given by 1 + 𝔭 ↦ 1 + A(𝔭, Z_{i_{1}}, …, Z_{i_{s}}). It follows from the construction that S(O) and S(O, 𝔭) are simple A-modules.

Suppose now that char 𝕂 = p > 0. Let 𝔪 be the designated maximal ideal when the orbit O is nondegenerate, and let 𝔪 be the fixed maximal break when O is degenerate. Assume \underline{I} = \underline{I}(𝔪) is the set of breaks of 𝔪, and let Γ be a subset of \underline{I}. Choose ξ : Γ → \{0, 1\}. Then for 𝔫 ∈ O, define

ℜ_{Γ,ξ}(𝔫) = (D/𝔫)[d_{i}, c_{j}^{±1}, \underline{τ}(𝔫) | i ∈ Γ, j ∈ \underline{I}^{c}],  (17)

where d_{i} := a_{i} if ξ(i) = 0 and d_{i} := b_{i} if ξ(i) = 1 as in (4.4). Note that if Γ = ∅, then ℜ_{Γ,ξ}(𝔫) = (D/𝔫)[c_{j}^{±1}, \underline{τ}(𝔫) | j = 1, …, n] (and there is only one possible ξ). Let 𝔑 be a maximal ideal in the ring ℜ_{Γ,ξ}(𝔪).

Set

S(O, Γ, ξ, 𝔑) := ⨁_{𝔫∈O}ℜ_{Γ,ξ}(𝔫)/σ_{𝔫}(𝔑),  (18)

where σ_{𝔫}(𝔑) means apply σ_{𝔫} to the coefficients of elements of 𝔑. Define a structure of a left A-module on S(O, Γ, ξ, 𝔑) by specifying for each i ∈ {1, 2, …, n},

x_{i}(f+σ_{𝔫}(𝔑)) = \{ \begin{matrix}σ_{i}(f) + σ_{i}σ_{𝔫}(𝔑) & if 𝔫 ≠ σ𝔪 for any σ ∈ Ĝ_{i}; \\ b_{i}σ_{i}σ_{𝔫}(f) + σ_{i}σ_{𝔫}(𝔑) & if 𝔫 = σ𝔪 for some σ ∈ Ĝ_{i}, \\ & i ∈ Γ, and ξ(i) = 1; \\ c_{i}σ_{i}σ_{𝔫}(f) + σ_{i}σ_{𝔫}(𝔑) & if 𝔫 = σ𝔪 for some σ ∈ Ĝ_{i} \\ & and i ∈ \underline{I}(𝔪)^{c}; \\ 0 & in all other cases; \end{matrix} (19)

∂_{i}(f+σ_{𝔫}(𝔑)) = \{ \begin{matrix}t_{i}σ_{i}^{−1}σ_{𝔫}(f) + σ_{i}^{−1}σ_{𝔫}(𝔑) & if 𝔫 ≠ σ_{i}σ𝔪 for any σ ∈ Ĝ_{i}; \\ a_{i}t_{i}σ_{i}^{−1}σ_{𝔫}(f) + σ_{i}^{−1}σ_{𝔫}(𝔑) & if 𝔫 = σ_{i}σ𝔪 for some \\ & σ ∈ Ĝ_{i}, i ∈ Γ,& ξ(i) = 0; \\ c_{i}^{−1}t_{i}σ_{i}^{−1}σ_{𝔫}(f) + σ_{i}^{−1}σ_{𝔫}(𝔑) & if 𝔫 = σ_{i}σ𝔪 for some \\ & σ ∈ Ĝ_{i} and i ∈ \underline{I}(𝔪)^{c}, \\ 0 & in all other cases; \end{matrix}

(20)

where Ĝ_{i} is the subgroup of G generated by the σ_{j} for all j ≠ i. It follows from the construction that S(O, Γ, ξ, 𝔑) is simple A-module.

The next theorem provides a classification of simple weight A_{n}-modules, 1 ≤ n ≤ ∞.

Theorem 4.7. Let A_{n} be the nth Weyl algebra for 1 ≤ n ≤ ∞ over a field 𝕂,

(i) If char 𝕂 = 0, then the modules S(O) and S(O, 𝔭), where 𝔭 ∈ 𝔅_{O}, constitute an exhaustive list of pairwise non-isomorphic simple weight A_{n}-modules with support in O.

(ii) If char 𝕂 = p > 0 then the modules S(O, Γ, ξ, 𝔑) constitute an exhaustive list of pairwise non-isomorphic simple weight A_{n}-modules with support O.

(In (i) and (ii), the orbit O is assumed to have a maximal break when O is degenerate and n = ∞.)

Proof. We fix an orbit O of 𝕂[t_{1}, …, t_{n}] under the group G and let C_{O} and S_{O} be the associated category and skeleton as (3.3) and (3.5). We can identify S_{O} with an algebra A(D/𝔪, \underline{I}(𝔪)) or B(D/𝔪, \underline{I}(𝔪), \underline{I}(𝔪)^{c}, \underline{τ}(𝔪)) using the functor G defined by (10), (11), or (12) in the proof of Proposition 3.12. We will use the determination of modules for those algebras in Propositions 4.3 and 4.5. We denote by E the equivalence functor C_{O} ⊗_{S_{O}} − and let F′ be given by (8).

(i) char 𝕂 = 0: When the orbit O is nondegenerate, it is easy to see that F′EG(S_{ω}) ≅ S(O). When the orbit is degenerate, and 𝔪 is the designated maximal break with break set \underline{I}(𝔪), we have for each α ∈ A(D/𝔪, \underline{I}(𝔪)) that F′EG(S_{α}) ≅ S(O, σ^{α}(𝔪)) where σ^{α} = ∏_{i∈\underline{I}(𝔪)} σ_{i}^{α_{i}}.

(ii) char 𝕂 = p: For the simple modules S_{Γ,ξ,𝔑} for B(D/𝔪, \underline{I}(𝔪), \underline{I}(𝔪)^{c}, \underline{τ}(𝔪)) we have that F′EG(S_{Γ,ξ,𝔑}) ≅ S(O, Γ, ξ, 𝔑). The statement in this case is a consequence of Proposition 4.5.

5 Examples.

5.1. Case A_{1}

Let A_{1} = 𝕂[t](σ, t) be the first Weyl algebra, where t = ∂x and σ(t) = t − 1. Simple weight A_{1}-modules have been described in [Bl], [B1], [B2] and [BO1], and they are the following:

1. Assume char 𝕂 = 0, O is a nondegenerate orbit of 𝔪𝔞𝔵(𝕂[t]) under σ, and 𝔪 ∈ O. Then 𝔪 is a principal ideal generated by an irreducible polynomial different from t over 𝕂 and S(O) = A/A𝔪 is the corresponding simple A-module.

2. Assume char 𝕂 = 0, O is degenerate, and 𝔪 ∈ O is the fixed maximal break (which is unique here). Then 𝔪 = (t), 𝔅_{O} = {𝔪, σ(𝔪)}, S(O, 𝔪) ≅ A/Ax and S(O, σ(𝔪)) ≅ A/A∂.

3. Assume char 𝕂 = p > 0, O is nondegenerate, and 𝔪 is the fixed maximal ideal in O. If σ(𝔪) ≠ 𝔪, then ℜ_{Γ,ξ}(𝔪) = (𝕂[t]/𝔪)[c^{±1}] (here Γ = ∅ and there is just one ξ). Then the simple A-modules S(O, ∅, ξ, 𝔑) are parametrized by the maximal ideals 𝔑 = (f) of (𝕂[t]/𝔪)[c^{±1}] generated by irreducible polynomials f of (𝕂[t]/𝔪)[c] different from c. When σ(𝔪) = 𝔪, (for example, when 𝔪 is generated by a polynomial of the form f(t) = t^{p} − t − ν for some nonzero ν ∈ 𝕂 and f is irreducible), then ℜ := ℜ_{Γ,ξ}(𝔪) = (𝕂[t]/𝔪)[c^{±1}, σ] is a skew polynomial algebra. In this case, the simple A-modules S(O, ∅, ξ, 𝔑) are parametrized by the maximal ideals 𝔑 of ℜ. Any such maximal ideal is principal, 𝔑 = (f), where f is irreducible in ℜ, and ℜ/(f) ≅ ℜ/(g) if and only if f and g are similar. (See for example, [DGO].)

4. Assume char 𝕂 = p > 0, O is degenerate, and 𝔪 ∈ O is the fixed maximal break. Then 𝔪 = (t), and ℜ_{Γ,ξ}(𝔪) = 𝕂[t]/𝔪 ≅ 𝕂 if Γ = ∅ and ℜ_{Γ,ξ}(𝔪) ≅ 𝕂[d] if Γ ≠ ∅. Hence in the first case, there is a unique simple module; while in the second, the simple modules S(O, Γ, ξ, 𝔑) are parametrized by maximal ideals 𝔑 ⊂ 𝕂[d].

It follows from (3) and (4) that if 𝕂 is an algebraically closed field of characteristic p, then each irreducible module S(O, Γ, ξ, 𝔑) has dimension p, as O has p elements and each ℜ_{Γ,ξ}(𝔫)/σ_{𝔫}(𝔑) is one-dimensional (see [C] for another approach to these modules). This need not be true if 𝕂 is not algebraically closed. For example, if 𝕂 = ℤ_{2}, and 𝔪 is the maximal ideal of 𝕂[t] generated by the irreducible polynomial t^{2} + t + 1, then σ(𝔪) = 𝔪 so O = {𝔪}. The maximal ideal 𝔑 of (𝕂[t]/𝔪)[c^{±1}, σ] generated by the irreducible polynomial c^{3}+c+1 gives an irreducible A_{1}-module S(O, ∅, ξ, 𝔑) of dimension 6 over ℤ_{2}.

5.2. Case A_{2}

Here we describe all simple weight A-modules for the second Weyl algebra A = A_{2}. If 𝕂 is algebraically closed and char 𝕂 = 0, then any simple weight A-module is a tensor product of two simple A_{1}-modules (cf. [BO2]). But in general this is not the case. We view A as a generalized Weyl algebra A = 𝕂[t_{1}, t_{2}]((σ_{1}, σ_{2}), (t_{1}, t_{2})) as above, where t_{i} = ∂_{i}x_{i}, σ_{i}(t_{j}) = δ_{i,j}(t_{i} − 1) for i, j = 1, 2.

(i) Suppose that char 𝕂 = 0, O is degenerate, and 𝔪 ∈ O is the fixed maximal break. Then we have the following three possibilities.

1. 𝔪 = (t_{1}, t_{2}) and 𝔅_{O} = {𝔪, σ_{1}(𝔪), σ_{2}(𝔪), σ_{1}σ_{2}(𝔪)},

S(O, 𝔪) ≅ A/A(x_{1}, x_{2}),

S(O, σ_{1}(𝔪)) ≅ A/A(∂_{1}, x_{2}),

S(O, σ_{2}(𝔪)) ≅ A/A(x_{1}, ∂_{2}),

S(O, σ_{1}σ_{2}(𝔪)) ≅ A/A(∂_{1}, ∂_{2}).

2. 𝔪 = (t_{1}, f), where f ∈ 𝕂[t_{2}] is an irreducible polynomial such that t_{2} ∉ (f), and 𝔅_{O} = {𝔪, σ_{1}(𝔪)},

S(O, 𝔪) ≅ A/A(f, x_{1}),

S(O, σ_{1}(𝔪)) ≅ A/A(f, ∂_{1}).

3. 𝔪 = (f, t_{2}), where f ∈ 𝕂[t_{1}] is an irreducible polynomial such that t_{1} ∉ (f), and 𝔅_{O} = {𝔪, σ_{2}(𝔪)},

S(O, 𝔪) ≅ A/A(f, x_{2}),

S(O, σ_{2}(𝔪)) ≅ A/A(f, ∂_{2}).

Note that in all cases the simple modules are tensor products of simple modules for A_{1}.

(ii) Assume now that O is a nondegenerate orbit in the characteristic 0 case, and 𝔪 ⊂ 𝕂[t_{1}, t_{2}] is the fixed maximal ideal in O. Then t_{1} ∉ 𝔪 and t_{2} ∉ 𝔪, and S(O) = A/A𝔪 is the corresponding simple A-module.

(iii) Assume char 𝕂 = p > 0, and 𝔪 is the designated maximal ideal in O when O is nondegenerate, and 𝔪 is the fixed maximal break with break set \underline{I} = \underline{I}(𝔪) when O is degenerate. Then the simple A-modules S(O, Γ, ξ, 𝔑) are parametrized by subsets Γ ⊆ \underline{I}, maps ξ : Γ → \{0, 1\}, and maximal ideals 𝔑 of ℜ_{Γ,ξ}(𝔪) = (𝕂[t_{1}, t_{2}]/𝔪)[d_{i}, c_{j}^{±1}, \underline{τ}(𝔪) | i ∈ Γ, j ∈ \underline{I}^{c}] where d_{i} = a_{i} if ξ(i) = 0 and d_{i} = b_{i} if ξ(i) = 1. This leads to the following possibilities.

1. If |\underline{I}| = 0 and Γ = ∅, then the simple A-modules S(O, ∅, ξ, 𝔑) are parametrized by maximal ideals 𝔑 ⊂ (𝕂[t_{1}, t_{2}]/𝔪)[c_{j}^{±1}, \underline{τ}(𝔪) | j = 1, 2].

2. If |\underline{I}| = 1 and Γ = ∅, then ℜ_{∅,ξ}(𝔪) ≅ (𝕂[t_{1}, t_{2}]/𝔪)[c_{j}^{±1}, \underline{τ}(𝔪)], and the modules S(O, ∅, ξ, 𝔑) are parametrized by maximal ideals 𝔑 ⊂ (𝕂[t_{1}, t_{2}]/𝔪)[c_{j}^{±1}, \underline{τ}(𝔪)], where 𝔑 ≠ (c_{j}).

3. If |\underline{I}| = 1 and Γ = \underline{I} = \{i\}, then 𝔪 = (t_{i}, f) for some polynomial f = f(t_{j}) ∉ 𝕂t_{j} for j ≠ i. The simple A-modules S(O, Γ, ξ, 𝔑) are parametrized by maximal ideals 𝔑 ⊂ (𝕂[t_{j}]/(f))[d_{i}][c_{j}^{±1}, \underline{τ}(𝔪)] and maps ξ : Γ → {0, 1} (which determine the A-action in (19) and (20)).

4. If |\underline{I}| = 2 and Γ = ∅, then 𝔪 = (t_{1}, t_{2}), ℜ_{Γ,ξ} ≅ 𝕂[t_{1}, t_{2}]/𝔪 ≅ 𝕂, and the simple A-modules S(O, ∅, ξ, 𝔑) are parametrized by maximal ideals 𝔑 ⊂ 𝕂, that is, by 𝔑 = 0.

5. If |\underline{I}| = 2 and Γ = \{i\}, then ℜ_{Γ,ξ} ≅ 𝕂[d_{i}], and the corresponding simple A-modules S(O, Γ, ξ, 𝔑) are parametrized by maximal ideals 𝔑 ⊂ 𝕂[d_{i}] and maps ξ.

6. If |\underline{I}| = 2 and Γ = \underline{I}, then ℜ_{Γ,ξ} ≅ 𝕂[d_{1}, d_{2}], and the simple A-modules S(O, Γ, ξ, 𝔑) are parametrized by maximal ideals 𝔑 ⊂ 𝕂[d_{1}, d_{2}] and maps ξ.

5.3. Case of separable ideals

If 𝔪 = (𝔪_{1}, …, 𝔪_{n}), where 𝔪_{i} is a maximal ideal of 𝕂[t_{i}] for each i, then 𝕂[t_{1}, …, t_{n}]/𝔪 ≅ 𝕂[t_{1}]/𝔪_{1} ⊗ ⋯ ⊗ 𝕂[t_{n}]/𝔪_{n}. In particular, this is always the case when 𝕂 is algebraically closed (as in [BB]). It implies that a skeleton S_{O} for A_{n} is a tensor product of corresponding skeletons for A_{1}. Hence, in particular, any simple weight module is a tensor product of simple A_{1}-modules if 𝕂 is algebraically closed or if 𝕂 has characteristic 0.

6 Simple modules over the Heisenberg algebra.

Let 𝕂 = ℂ, the complex numbers. Here we study modules for the Heisenberg algebra which are ℤ-graded with infinite-dimensional homogeneous components and have a nonzero central charge. Let H = ℂc ⊕ ⨁_{i∈ℤ\{0}} ℂe_{i} be an infinite-dimensional Heisenberg Lie algebra, where [e_{i}, e_{j}] = δ_{i,−j}c, [e_{j}, c] = 0 for all i ≥ 1 and all j. Let H_{ℤ} denote the category of all ℤ-graded H-modules V such that V = ⨁_{i∈ℤ} V_{i} and e_{i}V_{j} ⊂ V_{i+j}. If V ∈ H_{ℤ} is simple, then c acts as a scalar, called the (central) charge of V. All simple modules with a zero charge have been classified in [Ch]. Simple modules for which the charge is nonzero but 0 < dim_{ℂ} V_{i} < ∞ for at least one i have been described in [F]. There are relatively few known examples of simple modules V with a nonzero charge and dim_{ℂ} V_{i} = ∞ for all i. Here we will construct a series of such simple modules using the classification of simple weight A_{∞}-modules. Without loss of generality we can assume that a simple module V ∈ H_{ℤ} has central charge 1. Because the universal enveloping algebra of H modulo the ideal generated by c − 1 is isomorphic to A_{∞}, any simple module V ∈ H_{ℤ} with central charge 1 becomes a module for A = A_{∞} by identifying ∂_{i} = e_{i} and x_{i} = e_{−i} for all i > 0. Any orbit O of A is linear. Suppose O is nondegenerate and 𝔪 is the designated maximal ideal, and consider the simple A-module S(O) = A/A𝔪 = ⊕_{𝔭∈O}D/𝔭, where D = ℂ[t_{1}, t_{2}, …] and D/𝔭 ≅ ℂ for any 𝔭. Then S(O) is a ℤ-graded simple H-module,

S(O) = ∑_{i∈ℤ}S^{(−i)}(O),

where

S^{(−i)}(O) = ∑_{ℓ=0}^{∞} ∑ ; i_{1}, …, i_{ℓ} ∈ ℤ ; ∑_{k=1}^{ℓ} ki_{k} = i, i_{ℓ} ≠ 0 D/σ_{1}^{i_{1}} … σ_{ℓ}^{i_{ℓ}}(𝔪).

The ℓ = 0 summand is D/𝔪 if i = 0 and is 0 otherwise. Clearly, S^{(−i)}(O) is infinite-dimensional for each i. Note that S(O) remains simple as a ℤ-graded module.

7 Representation type and indecomposable modules for A_{n}, 1 ≤ n ≤ ∞.

In this section we classify the tame blocks in the category of locally-finite weight modules for the finite and infinite Weyl algebras A_{n} over a field 𝕂. We also determine the indecomposable modules in the tame blocks. These results generalize those in [BB] and [DGO].

We will use the following concepts which can be found in [D].

Definition 7.1. Let 𝔽 be an algebraically closed field, and let 𝔽 ⟨x, y⟩ be the free associative algebra with 1 over 𝔽 on two generators x, y.

• An 𝔽-category C is called wild if there exists a C − 𝔽 ⟨x, y⟩- bimodule M, free of finite rank as a right 𝔽 ⟨x, y⟩-module, such that the functor M ⊗_{𝔽⟨x,y⟩} − preserves indecomposability and isomorphism classes.

• An 𝔽-category C is called tame if, for each dimension vector \underline{d}, there exist a localization R = 𝔽[x]_{f} with respect to some f ∈ 𝔽[x] and a finite number of C − R-bimodules B_{1}, ⋯, B_{n} such that each B_{j} is free of finite rank as a right R-module, and such that every indecomposable X ∈ C-𝔣𝔡𝔪𝔬𝔡 with dim_{𝔽} X(α) = \underline{d}_{α} is isomorphic to B_{j} ⊗_{R} S for some j and some simple R-module S.

In the case of an arbitrary field 𝔽 we will say that an 𝔽-category C is wild (resp., tame) if it is wild (resp., tame) over the algebraically closure \overline{𝔽} of 𝔽.

We will consider weight A_{n}-modules V which are locally-finite, i.e. all V_{𝔪} are finite-dimensional vector spaces over D/𝔪 where D = 𝕂[t_{1}, …, t_{n}] and t_{i} = ∂_{i}x_{i} as before. We let W^{lf}(A_{n}) be the category of locally-finite weight A_{n}-modules. Also for each subset T ⊂ 𝔪𝔞𝔵D we denote by W_{T}^{lf}(A_{n}) the full subcategory in W^{lf}(A_{n}) consisting of all modules V with 𝔰𝔲𝔭𝔭(V) ⊂ T. The indecomposable locally-finite weight modules over the Weyl algebra A_{1} have been described in [DGO] (see also [BB] for the case of A_{n} (n < ∞) over an algebraically closed field and [C] for induced module constructions for Weyl algebras A_{n} (n < ∞) and quantum Weyl algebras). Because of the category isomorphisms of Sections 3 and 4, we can focus on the categories A(𝔽, I) and B(𝔽, I, J, \underline{τ}) for 𝔽 a field (which eventually we specialize to D/𝔪) and consider locally-finite indecomposable modules for them.

7.2. It follows from ([BB], Sec. 2.6) that B(𝔽, I, J, \underline{τ}) is wild if |I| + |J| > 1, and A(𝔽, I) is wild for |I| > 2. Therefore, it is enough to describe the indecomposables for the categories A(𝔽, I) with |I| ≤ 2 and for B(𝔽, I, J, \underline{τ}) with |I| + |J| = 1, which corresponds to the n = 1 case.

In (4.1) we have constructed the simple modules for A(𝔽, I). The category A(𝔽, ∅) has a unique object ω and morphism set 𝔽1_{ω}. Thus it is a simple algebra, and any finite-dimensional indecomposable module for A(𝔽, ∅) is simple.

7.3. Indecomposables for A(𝔽, I), |I| = 1

Recall that the category A(𝔽, I), |I| = 1, is isomorphic to the category \overline{Q}_{1} = 𝔽Q_{1}/R corresponding to the following quiver and relations:

[figure: Q_{1} :; a; b; 1; 2; ab = ba = 0.; ∘; ∘]

We denote by S_{i}, i = 1, 2, the simple modules for \overline{Q}_{1} (compare (4.1)). Now let M_{a} (resp., M_{b}) be the \overline{Q}_{1}-module such that M_{a}(1) = M_{b}(1) = 𝔽e_{1}, M_{a}(2) = M_{b}(2) = 𝔽e_{2}, where the action is given by ae_{1} := e_{2} and be_{2} := 0 (resp., ae_{1} := 0 and be_{2} := e_{1}). The following proposition is evident.

Proposition 7.4. The modules S_{1}, S_{2}, M_{a}, M_{b} constitute an exhaustive list of pairwise nonisomorphic finite-dimensional indecomposable \overline{Q}_{1}-modules.

7.5. Indecomposables for A(𝔽, I), |I| = 2

It is easy to see that category A(𝔽, I), |I| = 2, is isomorphic to the category \overline{Q}_{2} := 𝔽Q_{2}/R corresponding to the following quiver and relations:

[figure: Q_{2} :; a_{2}; b_{2}; b_{0}; a_{0}; a_{1}; b_{1}; b_{3}; a_{3}; 1; 2; 0; 3; a_{i}b_{i} = b_{i}a_{i} = 0, for i ∈ {0, 1, 2, 3}; a_{i}a_{j} = b_{ℓ}b_{m} for i, j, ℓ, m ∈ {0, 1, 2, 3},; whenever this is possible; ∘; ∘; ∘; ∘]

We regard the objects 0, 1, 2, 3 as the elements of ℤ_{4} = ℤ/4ℤ and denote the corresponding simple modules by S_{i}, i ∈ ℤ_{4}.

7.6. Now suppose M_{i}, i ∈ ℤ_{4}, is the \overline{Q}_{2}-module such that M_{i}(j) = 𝔽e_{j} for j ∈ ℤ_{4}, where for M_{i} the action is given by a_{i}e_{i} = e_{i+1}, a_{i+1}e_{i+1} = e_{i+2}, b_{i−1}e_{i} = e_{i−1}, b_{i−2}e_{i−1} = e_{i−2}, and u_{j}e_{k} = 0 for all other instances of u ∈ {a, b} and j, k ∈ ℤ_{4}.

7.7. For each n ∈ ℕ, n > 1, and j ∈ ℤ_{4} define the \overline{Q}_{2}-module M_{n,j,0} (resp., M_{n,j,1}) as follows. Consider n elements e_{1}, …, e_{n}. For ℓ ∈ ℤ_{4}, an 𝔽-basis of the vector space M_{n,j,0}(ℓ) (resp. M_{n,j,1}(ℓ)) is the set of e_{k} such that j + k − 1 ≡ ℓ (mod 4). The action of a_{ℓ} and b_{ℓ−1} on M_{n,j,0}(ℓ) (resp. M_{n,j,1}(ℓ)) is given by the rules:

a_{ℓ}e_{k} = \{ \begin{matrix}e_{k+1} & if ℓ is even (resp., odd), k < n, & j + k − 1 ≡ ℓ ( mod 4); \\ 0 & otherwise. \end{matrix}

b_{ℓ−1}e_{k} = \{ \begin{matrix}e_{k−1} & if ℓ is even (resp., odd), k > 1 & j + k − 1 ≡ ℓ ( mod 4); \\ 0, & otherwise. \end{matrix}

In all other cases, we have u_{j}e_{k} = 0 for u ∈ {a, b} and j, k ∈ ℤ_{4}.

7.8. We denote by Irr_{0}𝔽[x] the set of monic irreducible polynomials f ≠ x in 𝔽[x], and let Ind_{0}𝔽[x] = {f^{n} | f ∈ Irr_{0}𝔽[x] and n ∈ ℕ}. For each f(x) = x^{e} + ζ_{e}x^{e−1} + ⋯ + ζ_{1} ∈ Ind_{0}𝔽[x], define the \overline{Q}_{2}-module M_{f,1} (resp., M_{f,2}) as follows: Set M_{f,1}(i) := 𝔽^{e} (resp. M_{f,2}(i) = 𝔽^{e}) for i ∈ ℤ_{4}, and define

M_{f,1}(a_{0}) = M_{f,1}(a_{2}) = M_{f,1}(b_{1}) = I_{e}
M_{f,1}(b_{0}) = M_{f,1}(b_{2}) = M_{f,1}(a_{1}) = M_{f,1}(a_{3}) = 0,
M_{f,1}(b_{3}) =  𝔉_{f},
M_{f,2}(b_{0}) = M_{f,2}(b_{2}) = M_{f,2}(a_{1}) = I_{e},
M_{f,2}(a_{0}) = M_{f,2}(a_{2}) = M_{f,2}(b_{1}) = M_{f,2}(b_{3}) = 0,
M_{f,2}(a_{3}) =  𝔉_{f},

where 𝔉_{f} is the Frobenius (companion) matrix corresponding to the polynomial f:

𝔉_{f} = (\begin{matrix}0 & 0 & ⋯ & 0 & −ζ_{1} \\ 1 & 0 & ⋯ & 0 & −ζ_{2} \\ 0 & 1 & ⋯ & 0 & ⋮ \\ ⋮ & ⋮ & ... & ⋮ & −ζ_{e−1} \\ 0 & 0 & ⋯ & 1 & −ζ_{e} \end{matrix})

As a consequence of Proposition 3.3.1 in [BB] we have

Proposition 7.9. The modules S_{i}, M_{i}, M_{n,i,0}, M_{n,i,1}, M_{f,1}, M_{f,2} where f ∈ Ind_{0} 𝔽[x], i ∈ ℤ_{4}, and n ∈ ℕ, n > 1, constitute an exhaustive list of pairwise nonisomorphic finite-dimensional indecomposable \overline{Q}_{2}-modules.

Theorem 7.10. Let O be an orbit of G on 𝔪𝔞𝔵 D for the Weyl algebra A = A_{n} (1 ≤ n ≤ ∞) over 𝕂 (which is assumed to have a maximal break when O is degenerate and n = ∞).

(i) If char 𝕂 = 0, then the category W_{O}^{lf}(A) is tame if and only if either O is nondegenerate or the order of the maximal break is less than or equal to 2;

(ii) If char 𝕂 = p > 0, then the category W_{O}^{lf}(A) is tame if and only if n = 1.

Proof.

(i) (char 𝕂 = 0)

“⇒” This statement follows from (7.2).

“⇐” If O is nondegenerate with fixed maximal ideal 𝔪, then W_{O}^{lf}(A) ≅ A (D/𝔪, ∅)-𝔣𝔡𝔪𝔬𝔡 ≅ (D/𝔪)-𝔣𝔡𝔪𝔬𝔡 by Propositions 3.4 and 3.12, hence W_{O}^{lf}(A) is tame. If O is degenerate with designated maximal break 𝔪 having order |\underline{I}| = 1, then W_{O}^{lf}(A) ≅ A (D/𝔪, \underline{I})-𝔣𝔡𝔪𝔬𝔡 by Propositions 3.4 and 3.12, hence W_{O}^{lf}(A) is tame by Proposition 7.4. If O is degenerate with maximal break 𝔪 of order 2, then W_{O}^{lf}(A) ≅ A (D/𝔪, \underline{I})- 𝔣𝔡𝔪𝔬𝔡, |\underline{I}| = 2, by Propositions 3.4 and 3.12, hence W_{O}^{lf}(A) is tame by Proposition 7.9.

(ii) (char 𝕂 = p > 0)

“⇒” If B(𝔽, I, J, \underline{τ}) is tame, then |I| + |J| = 1 by (7.2). This corresponds to the case n = 1 which has been treated in [DGO] and Sec. 4.1 of [BB]. “⇐” This is a consequence of Theorem 5.7 in [DGO].

7.11. Indecomposables for the tame blocks

By Theorem 7.10, A_{1} is the only Weyl algebra in characteristic p having tame blocks, and indecomposable modules in these blocks have been determined in ([BB], Sec. 4.1). Thus, in what follows we concentrate on the characteristic 0 case and construct weight indecomposable A_{n}-modules for the tame blocks, using Propositions 3.4 and 3.12, and the description of A(𝔽, I)-modules in case |I| ≤ 2.

Definition 7.12. Assume char 𝕂 = 0, and the orbit O has a maximal break 𝔪 of order 1 with respect to i.

• M(O, 𝔭) := S(O, 𝔭) for 𝔭 ∈ 𝔅_{O}. (see (16) for the definition);

• M(O, x_{i}) := ⨁_{𝔫∈O} D/𝔫, where for each j:

x_{j}(d + 𝔫) := σ_{j}(d) + σ_{j}(𝔫) and

∂_{j}(d + 𝔫) := \{ \begin{matrix}0 & if j = i, σ_{i}^{−1}(𝔫) ∈ O_{𝔪}, \\ & and 𝔫 ∈ O_{σ_{i}(𝔪)}; \\ t_{j}σ_{j}^{−1}(d) + σ_{j}^{−1}(𝔫) & otherwise. \end{matrix}

• M(O, ∂_{i}) := ⨁_{𝔫∈O} D/𝔫, where for each j = 1, …, n:

x_{j}(d + 𝔫) := \{ \begin{matrix}0 & if j = i, 𝔫 ∈ O_{𝔪}, \\ & and σ_{i}(𝔫) ∈ O_{σ_{i}(𝔪)}; \\ σ_{j}(d) + σ_{j}(𝔫) & otherwise, \end{matrix}

and

∂_{j}(d + 𝔫) := t_{j}σ_{i}^{−1}(d) + σ_{j}^{−1}(𝔫).

Remark 7.13. Let O, 𝔪, i be as in Definition 7.12. It is easy to see that

• M(O, 𝔪) ≅ A/A(𝔪, x_{i});

• M(O, σ_{i}(𝔪)) ≅ A/A(σ_{i}(𝔪), ∂_{i});

• M(O, x_{i}) ≅ A/A𝔪;

• M(O, ∂_{i}) ≅ A/Aσ_{i}(𝔪).

7.14. Now suppose O is an orbit of D = 𝕂[t_{1}, …, t_{n}] under the automorphism group G of A_{n} having maximal break 𝔪 with respect to i, j ∈ {1, …, n} (i ≠ j). Assuming 𝔽 = D/𝔪, define a map γ : 𝔅_{O} → ℤ_{4} by the following rule: γ(𝔪) = 0, γ(σ_{i}(𝔪)) = 3, γ(σ_{j}(𝔪)) = 1 and γ(σ_{i}σ_{j}(𝔪)) = 2.

To M a \overline{Q}_{2}-module, we associate a corresponding weight module 𝔐 for A_{n} according to the following procedure: For 𝔭 ∈ 𝔅_{O} suppose 𝔐(𝔭) ≅ M(γ(𝔭)) (as D/𝔪-vector spaces); via the map that takes v ∈ M(γ(𝔭)) to v_{𝔭} ∈ 𝔐(𝔭). For each 𝔫 ∈ O_{𝔭} (compare (9)), suppose 𝔐(𝔫) ≅ 𝔐(𝔭) via v_{𝔭} ↦ v_{𝔫} for all v ∈ M(γ(𝔭)).

As σ_{𝔫}^{−1} induces an isomorphism from D/𝔫 to D/𝔪, we can consider 𝔐(𝔫) as a D/𝔫-module; hence as a D-module that 𝔫 annihilates. Therefore 𝔐 = ⨁_{𝔫∈O} 𝔐(𝔫) where 𝔐(𝔫) = {u ∈ 𝔐 | 𝔫u = 0}. For the A_{n}-action define:

• if 𝔫 ∈ O_{𝔪} and σ_{i}(𝔫) ∈ O_{σ_{i}(𝔪)}:

x_{i}v_{𝔫} = (M(b_{3})v)_{σ_{i}(𝔫)},  ∂_{i}v_{σ_{i}(𝔫)} = (M(a_{3})v)_{𝔫};
x_{i}v_{σ_{j}(𝔫)} = (M(a_{1})v)_{σ_{i}σ_{j}(𝔫)},  ∂_{i}v_{σ_{i}σ_{j}(𝔫)} = (M(b_{1})v)_{σ_{j}(𝔫)};

• if 𝔫 ∈ O_{𝔪} and σ_{j}(𝔫) ∈ O_{σ_{j}(𝔪)}:

x_{j}v_{𝔫} = (M(a_{0})v)_{σ_{j}(𝔫)},  ∂_{j}v_{σ_{j}(𝔫)} = (M(b_{0})v)_{𝔫};
x_{j}v_{σ_{i}(𝔫)} = (M(b_{2})v)_{σ_{i}σ_{j}(𝔫)},  ∂_{j}v_{σ_{i}σ_{j}(𝔫)} = (M(a_{2})v)_{σ_{i}(𝔫)};

• in all other cases:

x_{ℓ}v_{𝔫} = v_{σ_{ℓ}(𝔫)}, ∂_{ℓ}v_{𝔫} = t_{ℓ}v_{σ_{ℓ}^{−1}(𝔫)}.

For 𝔭 ∈ 𝔅_{O}, we denote by S(O, 𝔭), M(O, 𝔭), M(O, n, 𝔭, 0), M(O, n, 𝔭, 1), M(O, f, 1), and M(O, f, 2) the weight A_{n}-module which corresponds via this process to the \overline{Q}_{2}-module S_{γ(𝔭)}, M_{γ(𝔭)}, M_{n,γ(𝔭),0}, M_{n,γ(𝔭),1}, M_{f,1} and M_{f,2} respectively as in Proposition 7.9.

Remark 7.15. Let O, 𝔪, i, j, and 𝔭 ∈ 𝔅_{O} be as in (7.14). It is easy to see that

• S(O, 𝔪) ≅ A/A(𝔪, x_{i}, x_{j});

• S(O, σ_{i}(𝔪)) ≅ A/A(σ_{i}(𝔪), ∂_{i}, x_{j});

• S(O, σ_{j}(𝔪)) ≅ A/A(σ_{j}(𝔪), ∂_{j}, x_{i});

• S(O, σ_{i}σ_{j}(𝔪)) ≅ A/A(σ_{i}σ_{j}(𝔪), ∂_{i}, ∂_{j});

• M(O, 𝔭) ≅ A/A𝔭.

Theorem 7.16. Let O be an orbit of G on 𝔪𝔞𝔵 D for the Weyl algebra A_{n} (1 ≤ n ≤ ∞) over a field 𝕂 of characteristic 0 (which is assumed to have a maximal break if O is degenerate and n = ∞).

• If O is nondegenerate, then the simple module S(O) (see Section 3 for definition), is the unique (up to isomorphism) indecomposable module in W_{O}^{lf}(A_{n});

• If O has a maximal break 𝔪 of order 1 with respect to i, then the modules S(O, 𝔪), S(O, σ_{i}(𝔪)), M(O, x_{i}) and M(O, ∂_{i}), constitute an exhaustive list of pairwise nonisomorphic indecomposable modules in W_{O}^{lf}(A_{n});

• If O has a maximal break 𝔪 of order 2 with respect to i and j, then the modules S(O, 𝔭), M(O, 𝔭), M(O, n, 𝔭, 0), M(O, n, 𝔭, 1), M(O, f, 1), and M(O, f, 2), where f ∈ Ind_{0}(D/𝔪)[x], 𝔭 ∈ 𝔅_{O} and n ∈ ℕ, n > 1, constitute an exhaustive list of pairwise nonisomorphic indecomposable modules in W_{O}^{lf}(A_{n}).

Proof. As in the proof of Theorem 4.7, we use the functor F′EG, where E is the equivalence functor C_{O} ⊗_{S_{O}} −, F′ is given by (8), and G is defined by (10), (11), or (12).

(i) If the orbit O is nondegenerate, then it follows from Propositions 3.4 and 3.12, that W_{O}^{lf}(A_{n}) ≅ A(D/𝔪, ∅)-𝔣𝔡𝔪𝔬𝔡 ≅ D/𝔪-𝔣𝔡𝔪𝔬𝔡. Therefore every indecomposable module is simple in this case, and the statement follows from Theorem 7.10.

(ii) Assume O has a maximal break 𝔪 of order 1 with respect to i. Then it is easy to see that

F′EG(S_{1}) ≅ S(O, 𝔪), F′EG(S_{2}) ≅ S(O, σ_{i}(𝔪)),
F′EG(M_{a}) ≅ M(O, x_{i}), F′EG(M_{b}) ≅ M(O, ∂_{i}).

Consequently, the result in this case follows from Propositions 3.4, 3.12, and 7.4.

(iii) Suppose O has a maximal break 𝔪 of order 2 with respect to i and j. Let γ : \overline{Q}_{2} → 𝔅_{O} be as in (7.14). From

F′EG(S_{k}) ≅ S(O, γ^{−1}(k)), F′EG(M_{k}) ≅ M(O, γ^{−1}(k)),

F′EG(M_{n,k,ℓ}) ≅ M(O, n, γ^{−1}(k), ℓ), F′EG(M_{f,s}) ≅ M_{O,f,s},

where k ∈ ℤ_{4}, ℓ ∈ {0, 1}, and s ∈ {1, 2}, we obtain the desired result from Propositions 3.4, 3.12, and 7.9.

Remark 7.17. Because the algebras A(D/𝔪, ∅) and \overline{Q}_{1} are finite-dimensional and have only finitely many nonisomorphic finite-dimensional indecomposable modules, it follows from [A1], [A2] that any indecomposable module for those algebras is finite-dimensional. Therefore, in (i) and (ii) of Theorem 7.16 we obtain a classification of all indecomposable weight modules not just the locally-finite ones.

Acknowledgment

We gratefully acknowledge the support and hospitality of the following institutions:

• Universidade Federal do Rio Grande do Norte (the first author)

• the Mathematical Sciences Research Institute, Berkeley and the Fields Institute, Toronto (the second and third authors)

• the University of Wisconsin at Madison (the third author).

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Departamento de Matemática, CCET Universidade Federal do Rio Grande do Norte Campus Universitário, Lagoa Nova CEP 59072-970, Natal-RN, Brasil e-mail address: bekkert@ccet.ufrn.br

Department of Mathematics University of Wisconsin Madison, WI 53706 USA e-mail address: benkart@math.wisc.edu

Institute of Mathemática e Estatiśtica Universidade de São Paulo Caixa Postal 66281 São Paulo, CEP 05311-970 Brasil e-mail address: futorny@ime.usp.br