Thin Coverings of Modules

Yuly Billig^{1}^{∗} and Michael Lau^{2}^{∗}^{†}

1Carleton University, School of Mathematics and Statistics, 1125 Colonel By Drive, Ottawa, Ontario, Canada K1S 5B6 Email: billig@math.carleton.ca

2University of Ottawa, Department of Mathematics and Statistics, 585 King Edward Ave., Ottawa, Ontario, Canada K1N 6N5 Email: mlau@uottawa.ca

Abstract. Thin coverings are a method of constructing graded-simple modules from simple (ungraded) modules. After a general discussion, we classify the thin coverings of (quasifinite) simple modules over associative algebras graded by finite abelian groups. The classification uses the representation theory of cyclotomic quantum tori. We close with an application to representations of multiloop Lie algebras.

Keywords: graded module; associative algebra; multiloop Lie algebra; quantum torus MSC: 16W50, 17B70.

0 Introduction

In a recent series of papers [2, 3, 1], B. Allison, S. Berman, A. Pianzola, and J. Faulkner examined the structure of multiloop algebras. These algebras are formed by a generalization of the twisting process used in Kac’s construction [10] of the affine Lie algebras.

More specifically, for any algebra B over the complex field ℂ and finiteorder commuting automorphisms σ_{1}, …, σ_{N} of B of period m_{1}, …, m_{N}, respectively, the multiloop algebra L(B, σ_{1}, …, σ_{N}) is the following subalgebra of B ⊗ ℂ[t_{1}^{±}, …, t_{N}^{±}]:

∗Both authors gratefully acknowledge funding from the Natural Sciences and Engineering Research Council of Canada. Y.B. is partially supported by an NSERC Discovery Grant; M.L. is supported by an NSERC postdoctoral fellowship.

†Corresponding author.

L(B, σ_{1}, …, σ_{N}) = ⨁_{(i_{1},... ,i_{N})∈ℤ^{N}} B_{\overline{ι}_{1},... ,\overline{ι}_{N}} ⊗ ℂt_{1}^{i_{1}} ⋯ t_{N}^{i_{N}} ,

where B_{\overline{ι}_{1},... ,\overline{ι}_{N}} is the common eigenspace

{b ∈ B | σ_{j}b = ξ_{j}^{i_{j}}b for 1 ≤ j ≤ N}

for fixed primitive m_{j}th roots of unity ξ_{j}.

In particular, when B is a finite-dimensional simple Lie algebra, one can construct realizations for almost all extended affine Lie algebras by taking central extensions of the resulting multiloop algebras and adjoining appropriate derivations [1, 14].

While studying the representation theory of such extended affine Lie algebras [6], we considered methods of “twisting” simple B-modules into graded-simple modules for the multiloop algebras L(B, σ_{1}, …, σ_{N}). As part of our construction, we used structures called thin coverings of modules.

Let B = ⨁_{g∈G} B_{g} be an algebra graded by an abelian group G. A thin covering of a left B-module M is a family of subspaces {M_{g} | g ∈ G} with the following properties:

(i) M = ∑_{g∈G} M_{g},

(ii) B_{g}M_{h} ⊆ M_{g+h} for all g, h ∈ G,

(iii) If {N_{g} | g ∈ G} also satisfies (i) and (ii) and N_{g} ⊆ M_{g} for all g ∈ G, then N_{g} = M_{g} for all g ∈ G.

We consider finite-dimensional and infinite-dimensional quasifinite modules over a G-graded associative algebra A. Each thin covering of an (ungraded) left A-module M is associated with a graded left A-module M̃. Although thin coverings are generally not unique, the isomorphism class of M completely determines the graded-isomorphism class of M̃, under mild conditions. It is trivial to show that M̃ is graded-simple whenever M is simple.

An A-module may be twisted with an A-automorphism associated with its G-grading. Such twists do not change the isomorphism class of the graded modules coming from the thin covering construction, and they play a vital role in our classification of thin coverings.

After these applications to graded theory, we give an explicit characterization of thin coverings of quasifinite simple modules over arbitrary (unital) associative algebras graded by finite abelian groups. The classification uses the fact that any such module is also a module for a cyclotomic quantum torus coming from isomorphisms between twists of the module.

We conclude the paper with an application to the representation theory of multiloop Lie algebras. Our approach is an alternative to Clifford theory, where graded-simple A-modules are constructed by inducing from simple A_{0}-modules. (See [12], Thm. 2.7.2 and also [8, 9].)

This method gives an explicit procedure for constructing graded-simple A-modules from (ungraded) simple A-modules via the action of a cyclotomic quantum torus. For the applications we have in mind, it would be difficult to get a classification of simple A_{0}-modules, whereas ungraded simple Amodules are well-understood.

Note: Although we work over the field ℂ of complex numbers, all the results of this paper hold over an arbitrary algebraically closed field of characteristic zero. All modules are left modules unless otherwise indicated. Likewise, all hypotheses of simplicity and semisimplicity should be interpreted as left simplicity and left semisimplicity.

Acknowledgments. Y.B. would like to thank the University of Sydney, and M.L. the University of Alberta, for their warm hospitality during our visits. Y.B. also expresses grateful appreciation for helpful conversations with C. Dong.

1 Thin Coverings and Graded Modules

Let A be a (unital) associative algebra graded by an abelian group G. For any module M over A, a set of subspaces {M_{g} | g ∈ G} is a G-covering of M if ∑_{g∈G} M_{g} = M and A_{g}M_{h} ⊆ M_{g+h} for all g, h ∈ G. We will drop the prefix G from “G-covering” when there is no ambiguity. Two coverings {M_{g} | g ∈ G} and {M′_{g} | g ∈ G} are equivalent, and we write

{M_{g} | g ∈ G} ∼ {M′_{g} | g ∈ G}

if there is a fixed h ∈ G such that M_{g} = M′_{g+h} for all g ∈ G.

The set ℭ := ℭ(M, A, G) of coverings of the module M is partially ordered: for any {M_{g} | g ∈ G} and {N_{g} | g ∈ G} in ℭ, let

{M_{g} | g ∈ G} ⪯ {N_{g} | g ∈ G}

if the covering {M_{g}} is equivalent to a covering {M′_{g}} for which M′_{g} ⊆ N_{g} for all g ∈ G.

The minimal coverings in the poset ℭ are called thin coverings. If M is a finite-dimensional or infinite-dimensional quasifinite module (see Section 4), then it has a thin covering.

Remark 1.1.1 Let 𝔤 be a Lie algebra graded by an abelian group G, and let M be a 𝔤-module. The grading on 𝔤 extends naturally to a G-grading on the universal enveloping algebra U(𝔤). Note that the coverings of M as a 𝔤module coincide with its coverings as a U(𝔤)-module. Thus, by studying thin coverings of modules over associative algebras, we automatically obtain the corresponding results for Lie algebras. The same applies to Lie superalgebras as well, where we would need to consider the ℤ_{2}-graded version of a thin covering. (See the definition in Section 4.)

The following elementary, but important lemma says that the thin coverings of a simple A-module M are essentially determined by the simple A_{0}-submodules of M.

Lemma 1.1.2 Let {M_{g} | g ∈ G} be a covering for a simple module M over an associative algebra A graded by an abelian group G. This covering is thin if and only if

M_{g} = A_{g−h}u,

for all g, h ∈ G and any nonzero u ∈ M_{h}.

In particular, in a thin covering, every nonzero space M_{h} is a simple A_{0}-module.

Proof Since M is a simple A-module, ∑_{g∈G} A_{g−h}u = Au = M. This gives a covering of M with A_{g−h}u ⊆ M_{g} for all g ∈ G. Thus {M_{g}} is a thin covering if and only if A_{g−h}u = M_{g} for all g. □

Although the sum ∑_{g∈G} M_{g} of the spaces M_{g} in a thin covering need not be direct, the external direct sum of these subspaces always has a natural graded module structure. In fact, this construction “preserves simplicity”, as described in the following theorem.

Theorem 1.1.3 Let M be a simple (not necessarily graded) module over an associative algebra A graded by an abelian group G. Then for any thin covering {M_{g} | g ∈ G} of M, the space

M̃:= ⨁_{g∈G} M_{g}

is a graded-simple A-module.

Proof The space M̃ has an obvious graded module structure, namely M̃_{g} := M_{g} and A_{g} : M̃_{h} → M̃_{g+h} for all g, h ∈ G.

By Lemma 1.1.2, A_{h−g}u = M_{h} for all g ∈ G and nonzero elements u ∈ M_{g}. Likewise, A_{h−g}ũ = M̃_{h} for every h ∈ G, where ũ is the element of M̃ having the nonzero element u ∈ M_{g} in the M̃_{g}-component and 0 elsewhere.

Suppose Ñ = ⨁_{g∈G} Ñ_{g} is a nonzero G-graded submodule of M̃. Then for any nonzero component Ñ_{g} and h ∈ G, we have A_{h−g}Ñ_{g} = M̃_{h}. Hence Ñ = M̃. □

In the next theorem, we show that graded-simple A-modules come from thin coverings of simple (non-graded) A-modules. The assumption on the existence of a maximal submodule holds in the finite-dimensional case as well as for the infinite-dimensional quasifinite modules. (See Remark 4.4.2.)

Theorem 1.1.4 Let M̃ be a graded-simple module over an associative algebra A graded by an abelian group G:

M̃ = ⨁_{g∈G} M̃_{g}.

Assume that as a non-graded A-module, M̃ has a maximal (non-graded) submodule and a simple (non-graded) quotient M:

φ : M̃ → M.

Let M_{g} = φ(M̃_{g}). Then {M_{g} | g ∈ G} is a thin covering of the module M and M̃ is graded-isomorphic to the graded A-module associated with this thin covering,

M̃ ≅ ⨁_{g∈G} M_{g},

as in Theorem 1.1.3.

Proof Since the map φ is surjective, we get that ∑_{g∈G} φ(M̃_{g}) = φ(M̃) = M,

and hence {φ(M̃_{g}) | g ∈ G} is a covering of M.

Moreover, the restrictions of φ to M̃_{g},

φ_{g} : M̃_{g} → M_{g},

are bijections. If these maps had non-trivial kernels, then ⨁_{g∈G} ker φ_{g} would

be a non-trivial graded submodule of M̃, which contradicts the fact that M̃ is graded-simple.

We now show that the covering {φ(M̃_{g}) | g ∈ G} is thin. Suppose {N_{g} | g ∈ G} is another covering of M with N_{g} ⊆ M_{g}. Let Ñ_{g} ⊆ M̃_{g} be the pre-image of N_{g} under the map φ_{g}. It is easy to see that

Ñ = ⨁_{g∈G} Ñ_{g}

is a graded A-submodule in M̃. To verify this, it is enough to check that A_{h}Ñ_{g} ⊆ Ñ_{g+h}. However, φ(A_{h}Ñ_{g}) = A_{h}N_{g} ⊆ N_{g+h}, and Ñ_{g+h} is the preimage of N_{g+h} under φ_{g+h}. Thus A_{h}Ñ_{g} ⊆ Ñ_{g+h}, and Ñ is a nonzero graded submodule of M̃. Since M̃ is graded-simple, we conclude that Ñ_{g} = M̃_{g} for all g ∈ G. This implies that N_{g} = M_{g} because the map φ_{g} is a bijection. Hence the covering {M_{g} | g ∈ G} is indeed thin.

Once again using bijectivity of φ_{g}, we see that the graded A-module associated with this covering is isomorphic to M̃:

⨁_{g∈G} M_{g} ≅ ⨁_{g∈G} M̃_{g}.

□

Gradings on A by finite abelian groups can be alternatively described via finite abelian subgroups of the group of automorphisms of A. Suppose A is graded by a finite abelian group G, and consider the dual group Ĝ = Hom(G, ℂ^{∗}). Then we can interpret an element σ ∈ Ĝ as the finite order A-automorphism defined by σ(a) = σ(g)a for a ∈ A_{g}. Although the groups G and Ĝ are isomorphic, there is no canonical isomorphism between them.

Conversely, a finite abelian group Ĝ of automorphisms of A defines a grading. The algebra A is graded by the group G = Hom(Ĝ, ℂ^{∗}):

A = ⨁_{g∈G} A_{g},

where A_{g} = {a ∈ A | σ(a) = g(σ)a, for all σ ∈ Ĝ}.

To be consistent with the additive notation for grading groups, we will slightly abuse notation and always treat G as an additive group. The sum of two elements g, h ∈ G is defined by

(g + h)(σ) = g(σ)h(σ) for σ ∈ Ĝ,  (1.1.5)

and we will denote the identity element of G by 0.

An A-module (M, ρ) can be twisted by an automorphism σ ∈ Aut(A), and we use the notation M^{σ} for the module (M, ρσ). That is, the module action in M^{σ} is given by ρσ : A → End_{ℂ}M.

The following proposition shows that a simple A-module and its twists with automorphisms in Ĝ yield isomorphic graded-simple modules.

Proposition 1.1.6 Let A be an associative algebra graded by a finite abelian group Ĝ. Let M be a simple A-module, and let σ ∈ Ĝ.

(i) Any thin covering {M_{g} | g ∈ G} of M is also a thin covering of M^{σ}. (ii) Let {M_{g} | g ∈ G} be a thin covering of M, and let {M_{g}^{σ} | g ∈ G} be

the same covering of M^{σ}. Then the following graded-simple A-modules are naturally isomorphic

⨁_{g∈G} M_{g}^{σ} ≅ (⨁_{g∈G} M_{g})^{σ} .

(iii) If M is a G-graded A-module, then M^{σ} ≅ M for all σ ∈ Ĝ.

Proof We first show that a covering of M is also a covering of M^{σ}. For all a ∈ A_{g} and m ∈ M_{h}, we have ρ(a)m ∈ M_{g+h}, so

ρσ(a)m = g(σ)ρ(a)m ∈ M_{g+h}.

This gives a bijection between the coverings of M and the coverings of M^{σ}. Clearly the thin coverings of M correspond to the thin coverings of M^{σ}.

Part (ii) is obvious.

To establish (iii), we define the isomorphism

θ : M = ⨁_{g∈G} M_{g} → M^{σ}

by θ(m) = g(σ)m for m ∈ M_{g}. Let us check that θ commutes with the action of A. For a ∈ A_{g}, m ∈ M_{h}, we have

ρσ(a)θ(m) = g(σ)h(σ)ρ(a)m = θ(ρ(a)m).
□

Remark 1.1.7 The following partial converse is true: if the modules M and M^{σ} are isomorphic for some finite order automorphism σ ∈ AutA, then M admits a grading by the cyclic group ⟨σ⟩. (See Example 3.3.25 below.) However, if we replace a cyclic group ⟨σ⟩ by a finite abelian group G, the analogous statement is, in general, false (cf. Example 3.3.26).

2 Thin coverings of modules over a semisimple algebra

Recall that an associative algebra A is semisimple if each module for A is completely reducible. In Theorem 2.2.2, we prove that if A is semisimple, 1 ∈ A_{g}A_{−g} for all g, and each A_{g} is an irreducible A_{0}-module, then the isomorphism class of the graded-simple module M̃ does not depend on the choice of the thin covering of M. The following lemma is well-known (cf. [12], Cor. 2.74, for instance):

Lemma 2.2.1 Let A be a semisimple associative algebra. If A is graded by an abelian group G, then its subalgebra A_{0} is also semisimple. □

Theorem 2.2.2 Let A be a semisimple associative algebra graded by an abelian group G. Assume that for each g ∈ G, the graded component A_{g} satisfies the following two properties:

(i) 1 ∈ A_{g}A_{−g},

(ii) A_{g} is irreducible as a A_{0}-module. Let {M_{g} | g ∈ G} and {N_{g} | g ∈ G} be thin coverings for an irreducible A-module M. Then for some h ∈ G,

⨁_{g∈G} M_{g} ≅ ⨁_{g∈G} N_{g−h}.

as graded A-modules.

Proof If M = 0, there is nothing to prove. Otherwise, choose h ∈ G so that N_{h} ≠ 0. Note that every N_{g} is nonzero, since

0 ≠ N_{h} ⊆ A_{h−g}A_{g−h}N_{h} ⊆ A_{h−g}N_{g}.

Likewise, M_{g} ≠ 0 for all g ∈ G. By Lemma 2.2.1, N_{0} has a direct complement H as an A_{0}-module. Since {M_{g} | g ∈ G} is a covering, there is an h ∈ G such that the projection

π_{h} : M_{h} → N_{0}

(given by the splitting M_{h} ⊆ M = N_{0} ⊕H) is nonzero. Since π_{h} is a nonzero homomorphism of simple A_{0}-modules (using Lemma 1.1.2), M_{h} ≅_{A_{0}} N_{0} by Schur’s Lemma.

Our next task is to show that

⨁_{g∈G} M_{g} ≅ A ⊗_{A_{0}} M_{h}

as graded A-modules. We begin by noting that for each g ∈ G, the A_{0}- module A_{g} ⊗_{A_{0}} M_{h} is irreducible.

Let 0 ≠ ∑_{i} a_{i} ⊗ m_{i} ∈ A_{g} ⊗_{A_{0}} M_{h}, and let m ∈ M_{h} be nonzero. Since M_{h} is irreducible as an A_{0}-module (Lemma 1.1.2), there exist b_{i} ∈ A_{0} such that b_{i}m = m_{i} for all i. Then

∑_{i} a_{i} ⊗ m_{i} = (∑_{i} a_{i}b_{i}) ⊗ m.

Since A_{g} is irreducible as a A_{0}-module, we have

A_{0} ((∑_{i} a_{i}b_{i}) ⊗ m) = A_{g} ⊗ m
⊆ A_{g}A_{0} ⊗ m
= A_{g} ⊗ A_{0}m
= A_{g} ⊗_{A_{0}} M_{h}.

Thus A_{g} ⊗_{A_{0}} M_{h} (and A_{g} ⊗_{A_{0}} N_{0}) is an irreducible A_{0}-module.

Note that A ⊗_{A_{0}} M_{h} = ⨁_{g∈G} (A_{g} ⊗_{A_{0}} M_{h}), and there is a well-defined graded map

ϕ := ⨁_{g∈G} ϕ_{g} : ⨁_{g∈G} (A_{g} ⊗_{A_{0}} M_{h}) → ⨁_{g∈G} M_{g}

given by ϕ_{g}(a⊗m) = am ∈ M_{g} for all a ∈ A_{g−h} and m ∈ M_{h}. The map ϕ_{g} is a nonzero homomorphism between simple A_{0}-modules, so is an A_{0}-module isomorphism. Thus ϕ is a bijection. Also

ϕ(ab ⊗ m) = abm
= aϕ(b ⊗ m)

for any homogeneous a, b ∈ A and m ∈ M_{h}, so ϕ is an isomorphism of graded A-modules. Likewise,

A ⊗_{A_{0}} N_{0} = ⨁_{g∈G} A_{g} ⊗_{A_{0}} N_{0} ≅ ⨁_{g∈G} N_{g}

as graded A-modules.

Since the A_{0}-modules M_{h} and N_{0} are isomorphic, we get that A ⊗_{A_{0}} M_{h} ≅ A ⊗_{A_{0}} N_{0}, and thus

⨁_{g∈G} M_{g} ≅ ⨁_{g∈G} N_{g−h}.

as graded A-modules. □

While this method of associating graded modules M̃ with modules M is not functorial (it is not even well-defined if M is not simple), it is surjective in the sense that not only does every graded module come from a thin covering, but (in the context of the previous theorem) every graded-simple module comes from a thin covering of a simple module.

The first assertion (“graded modules come from thin coverings”) is trivial. Let M = ⨁_{g∈G} M_{g} be a graded module of a graded associative algebra A = ⨁_{g∈G} A_{g}. Forgetting the graded structure on M gives an (ungraded) A-module M with thin covering {M_{g} | g ∈ G}. The second claim (“gradedsimples come from simples”) is the following theorem. (See also Theorem 1.1.4.)

Theorem 2.2.3 Let M = ⨁_{g∈G} M_{g} be a graded-simple module for a semisimple associative algebra A = ⨁_{g∈G} A_{g} graded by an abelian group G. Assume that A satisfies conditions (i) and (ii) of Theorem 2.2.2. Then there is a simple module N for A so that N has a thin covering {N_{g} | g ∈ G} with

⨁_{g∈G} N_{g} ≅ ⨁_{g∈G} M_{g}

as graded A-modules.

Proof Let N be a simple A-submodule of M, and let N′ be a simple A_{0}-submodule of N. Since N is simple,

N = AN′ = ∑_{g∈G} A_{g}N′.

Clearly A_{g}A_{h}N′ ⊆ A_{g+h}N′, so {A_{g}N′ | g ∈ G} is a covering of N. By the argument in the proof of Theorem 2.2.2, the A_{0}-modules A_{g}N′ are simple, as are the components N_{g} in any subcovering {N_{g} | g ∈ G}. The N_{g} are all nonzero by the proof in Theorem 2.2.2. Thus the covering {A_{g}N′ | g ∈ G} is thin.

Since N is nonzero, the projection

π_{h} : N → M_{h}

is nonzero for some h ∈ G. Choose h′ ∈ G so that the restriction of π_{h} to A_{h′}N′ is also nonzero. For each g ∈ G, let

C_{g} = A_{g−h+h′}N′.

Then {C_{g} | g ∈ G} is a thin covering of N with the property that the projection map

π_{h} : C_{h} → M_{h}

is nonzero. Since C_{g} = A_{g−h}C_{h} and M_{g} = A_{g−h}M_{h}, we see that the projection π_{g} : C_{g} → M_{g} is also nonzero.

Let

π = ⨁_{g∈G} π_{g} : ⨁_{g∈G} C_{g} → ⨁_{g∈G} M_{g},

where the restriction of π to C_{g} is the projection π_{g}. Then since each π_{g} is a nonzero homomorphism between simple A_{0}-modules C_{g} and M_{g}, the maps π_{g} are A_{0}-module isomorphisms. Thus π is a bijection.

Finally, π is also a graded homomorphism since

π(an) = π_{g+g′}(an) = aπ_{g′}(n) = aπ(n)

for all a ∈ A_{g} and n ∈ C_{g′}. Thus

⨁_{g∈G} C_{g} ≅ ⨁_{g∈G} M_{g}

as graded A-modules. □

3 Classification of Thin Coverings

In this section, we will assume that A is a (unital) associative algebra over ℂ, and M is a finite-dimensional irreducible A-module whose action is given by the homomorphism ρ : A → End_{ℂ}M.

In order to find the thin coverings of a module M, it is important to know for which σ ∈ Ĝ, the modules M and M^{σ} are isomorphic.

Lemma 3.3.1 Let Ĝ be a subgroup of Aut A and let

Ĥ = {σ ∈ Ĝ | M^{σ} ≅ M as A-modules} .

Then (i) Ĥ is a subgroup of Ĝ. (ii) For σ_{1}, σ_{2} ∈ Ĝ we have M^{σ_{1}} ≅ M^{σ_{2}} if and only if σ_{1}σ_{2}^{−1} ∈ Ĥ.

Proof An isomorphism between A-modules M and M^{σ} is a map T_{σ} ∈ GL(M) satisfying

T_{σ}ρ(a)u = ρ(σ(a))T_{σ}u,

for all a ∈ A and u ∈ M. Equivalently,

ρ(σ(a)) = T_{σ}ρ(a)T_{σ}^{−1}.

The first part of the lemma follows from this relation.

The fact that the modules M^{σ_{1}} and M^{σ_{2}} are isomorphic is equivalent to the existence of a T ∈ GL(M), such that ρ(σ_{1}(a)) = T ρ(σ_{2}(a))T^{−1}. Letting b = σ_{2}(a), we get that ρ(σ_{1}σ_{2}^{−1}(b)) = T ρ(b)T^{−1}, which means that σ_{1}σ_{2}^{−1} ∈ Ĥ. This establishes the second claim of the lemma. □

Consider the operator T_{σ} introduced in the above proof. If σ ∈ Ĥ has order m, then for all a ∈ A,

ρ(a) = ρ(σ^{m}(a)) = T_{σ}^{m}ρ(a)T_{σ}^{−m}.

By Schur’s Lemma, T_{σ}^{m} is a scalar operator, and we may normalize T_{σ} to get T_{σ}^{m} = id. It is often convenient to use this normalization of T_{σ}.

Recall that there is an equivalent description of gradings and coverings in terms of the dual groups H = Hom(Ĥ, ℂ^{∗}) and G = Hom(Ĝ, ℂ^{∗}) (cf. Section 1). Our strategy will be to describe the thin H-coverings of M, and then use them to determine the thin G-coverings. In order to understand the structure of thin H-coverings, we need to better understand the H-grading on A and on its image ρ(A) = End_{ℂ}(M). This can be done in two ways. One possibility is to apply the theorem of Bahturin et al. ([4], Thm. 6) on gradings of a matrix algebra. A second, equivalent approach, is to use the classification of cyclotomic quantum tori [13, 1]. We will follow the second approach here, since it can also be adapted for the infinite-dimensional (quasifinite) set-up.

����

Let us recall the definition of a quantum torus. A quantum torus is the unital associative algebra R_{q} = R_{q} ⟨t_{1}^{±}, …, t_{r}^{±}⟩, whose generators t_{i} are subject to the defining relations

t_{i}t_{j} = q_{ij}t_{j}t_{i}, i, j = 1, …, r,

where q = (q_{ij}) is a complex r × r matrix satisfying q_{ii} = 1, q_{ij} = q_{ji}^{−1}.

We say that the quantum torus R_{q} is cyclotomic if all q_{ij} are complex roots of 1.

Lemma 3.3.2 Let η_{1}, …, η_{r} be a set of generators of the group Ĥ of orders s_{1}, …, s_{r}, respectively, and let T_{1}, …, T_{r} be invertible operators on M satisfying

ρ(η_{j}(a)) = T_{j}ρ(a)T_{j}^{−1}, for all a ∈ A, j = 1, …, r.  (3.3.3)

Then the map t_{i} ↦ T_{i} defines a representation of a cyclotomic quantum torus R_{q} ⟨t_{1}^{±}, …, t_{r}^{±}⟩, with

q_{ij}^{gcd(s_{i},s_{j})} = 1.

Proof The automorphisms η_{i} and η_{j} commute. Thus for all a ∈ A,

T_{i}T_{j}ρ(a)T_{j}^{−1}T_{i}^{−1} = T_{j}T_{i}ρ(a)T_{i}^{−1}T_{j}^{−1},

or equivalently,

T_{i}^{−1}T_{j}^{−1}T_{i}T_{j}ρ(a) = ρ(a)T_{i}^{−1}T_{j}^{−1}T_{i}T_{j}.

However, by Schur’s Lemma, the centralizer of a finite-dimensional simple module is a multiple of the identity map, so

T_{i}T_{j} = q_{ij}T_{j}T_{i}

for some nonzero q_{ij} ∈ ℂ.

Moreover, the relation η_{i}^{s_{i}} = 1 implies that T_{i}^{s_{i}} is a multiple of the identity. Since T_{i}^{s_{i}}T_{j} = q_{ij}^{s_{i}}T_{j}T_{i}^{s_{i}}, we conclude that

q_{ij}^{s_{i}} = 1.

Likewise, q_{ij}^{s_{j}} = 1, so q_{ij}^{gcd(s_{i},s_{j})} = 1. □

13

We will need elements of the representation theory of cyclotomic quantum tori. We call an R_{q}-module U diagonalizable if every generator t_{i} is diagonalizable on U. Whenever R_{q} is noncommutative, these operators clearly cannot be diagonal with respect to the same basis. Let C be the category of finite-dimensional diagonalizable R_{q}-modules. Since T_{i}^{s_{i}} is a multiple of the identity on M, we see that M belongs to Category C.

Lemma 3.3.4 Let R_{q} be a cyclotomic quantum torus, and let U be a module in Category C. Then for any m = (m_{1}, …, m_{r}) ∈ ℤ^{r}, the monomial t^{m} = t_{1}^{m_{1}} … t_{r}^{m_{r}} is diagonalizable on U.

Proof Choose a positive integer b such that q_{ij}^{b} = 1 for all i, j = 1, …, r. Then the elements t_{i}^{±b} belong to the centre of R_{q}. Since each of them is diagonalizable on U, they are simultaneously diagonalizable. Moreover, (t^{m})^{b} belongs to the subalgebra generated by t_{1}^{±b}, …, t_{r}^{±b}, and hence diagonalizable. This implies that t^{m} is diagonalizable as well. □

Proposition 3.3.5 Every module in Category C for the cyclotomic quantum torus R_{q} = R_{q} ⟨t_{1}^{±}, …, t_{r}^{±}⟩ is completely reducible, and all simple R_{q}- modules in Category C have the same dimension. More explicitly, fix an arbitrary simple R_{q}-module N in C. Then all other simple R_{q}-modules in C can be obtained from N as twists by the rescaling automorphisms of R_{q}:

t_{j} ↦ α_{j}t_{j}, j = 1, …, r,

where α_{1}, …, α_{r} ∈ ℂ^{∗}.

Proof To prove this proposition, we will put R_{q} in a normal form, using the classification of cyclotomic quantum tori [13, 1]. If we make a change of variables in a quantum torus using a matrix (a_{ij}) ∈ GL_{r}(ℤ):

t̄_{i} = t_{1}^{a_{i1}}t_{2}^{a_{i2}} … t_{r}^{a_{ir}},

we will get another quantum torus with a new matrix q̄_{ij}. The classification theorem states that by means of such a change of variables, a cyclotomic quantum torus can be brought to the normal form

R_{q} ⟨t_{1}^{±}, …, t_{r}^{±}⟩ ≅ R_{ζ_{1}} ⟨x_{1}^{±}, y_{1}^{±}⟩ ⊗ … ⊗ R_{ζ_{ℓ}} ⟨x_{ℓ}^{±}, y_{ℓ}^{±}⟩ ⊗ ℂ[z_{2ℓ+1}^{±}, …, z_{r}^{±}], (3.3.6)

where the first ℓ tensor factors are rank two cyclotomic quantum tori, and the last tensor factor is a commutative algebra of Laurent polynomials. By Lemma 3.3.4, an R_{q}-module’s membership in Category C is independent of the choice of the generators t_{i}^{±} of R_{q}.

The rank two quantum tori R_{ζ} = R_{ζ} ⟨x^{±}, y^{±}⟩ that appear in (3.3.6) have the following defining relation

xy = ζyx,

where ζ is a primitive complex dth root of unity. The centre of R_{ζ} is generated by x^{±d} and y^{±d}. Hence the centre of R_{q} is the Laurent polynomial algebra:

Z(R_{q}) = ℂ[x_{1}^{±d_{1}}, y_{1}^{±d_{1}}, …, x_{ℓ}^{±d_{ℓ}}, y_{ℓ}^{±d_{ℓ}}, z_{2ℓ+1}^{±}, …, z_{r}^{±}].

Let U be a finite-dimensional diagonalizable R_{q}-module. Since the action of Z(R_{q}) on U is diagonalizable, we may decompose U into the direct sum of R_{q}-submodules corresponding to various central characters χ : Z(R_{q}) → ℂ:

U = ⨁_{χ} U^{χ}.

Note that a central character χ is determined by its (nonzero) values χ(x_{i}^{d_{i}}), χ(y_{i}^{d_{i}}), and χ(z_{i}). Hence, by means of the rescaling automorphisms of R_{q}, we may twist a module with central character χ into a module with any other central character.

We will show that, up to isomorphism, there exists a unique finitedimensional simple R_{q}-module for each central character.

In order to prove that the module U is is completely reducible, it is enough to consider its single component U^{χ}. This component is a module for the quotient R_{q}^{χ} of R_{q} by the central ideal:

R_{q}^{χ} = R_{q}/⟨z − χ(z)1 | z ∈ Z(R_{q})⟩.

The algebra R_{q}^{χ} decomposes into the tensor product:

R_{q}^{χ} ≅ R_{ζ_{1}}^{χ} ⟨x_{1}^{±}, y_{1}^{±}⟩ ⊗ … ⊗ R_{ζ_{ℓ}}^{χ} ⟨x_{ℓ}^{±}, y_{ℓ}^{±}⟩ .

The rank two factors here have the following structure:

R_{ζ}^{χ} = R_{ζ} ⟨x^{±}, y^{±}⟩ /⟨x^{d} − χ(x^{d})1, y^{d} − χ(y^{d})1⟩ .

In fact, R_{ζ}^{χ} is isomorphic to the matrix algebra M_{d}(ℂ) under the following isomorphism:

x ⟼ (χ(x^{d}))^{1/d} ( \begin{matrix}1 & 0 & · & · & 0 \\ 0 & ζ & · & · & 0 \\ ⋮ & ⋮ & & ... & ⋮ \\ 0 & 0 & · & · & ζ^{d−1} \end{matrix} )

and y ⟼ ( \begin{matrix}0 & 0 & · & 0 & χ(y^{d}) \\ 1 & 0 & · & 0 & 0 \\ 0 & 1 & · & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & · & 1 & 0 \end{matrix} ) .  (3.3.7)

This implies that the algebra R_{q}^{χ} is isomorphic to the matrix algebra of rank d_{1} ⋯ d_{ℓ}. It is well-known that the matrix algebra is semisimple and has a unique, up to isomorphism, finite-dimensional simple module of dimension equal to the rank of the matrix algebra. □

The following corollary is an immediate consequence of Proposition 3.3.5.

Corollary 3.3.8 Let N be a simple module in Category C for the cyclotomic quantum torus R_{q} = R_{q} ⟨t_{1}^{±}, …, t_{r}^{±}⟩. Then every module M in Category C may be written as a finite sum

M ≅ N ⊗ ( ⨁_{α∈(ℂ^{∗})^{r}} V^{α}) ,  (3.3.9)

where the action of R_{q} on N ⊗ V^{α} is given by the formula

t_{i}(u ⊗ v^{α}) = t_{i}(u) ⊗ α_{i}v^{α}, for u ∈ N, v^{α} ∈ V^{α}.  (3.3.10)

Proof If we take a one-dimensional space V^{α}, then N ⊗ V^{α} is a simple R_{q}- module, which is a twist of N by the rescaling automorphism of R_{q} given by α ∈ (ℂ^{∗})^{r}. Thus the decomposition (3.3.9) is just an isotypic decomposition of the module M. □

We now look at the representation of R_{q} described in Lemma 3.3.2. We show that in this module, the elements α appearing in the isotypic decomposition of M are not arbitrary, but belong to the group H.

Lemma 3.3.11 Let M be a finite-dimensional simple module for an Hgraded algebra A. Suppose that T_{1}, …, T_{r} are operators on M that satisfy (3.3.3) and define the action of a cyclotomic quantum torus R_{q} on M. Let N be a simple R_{q}-submodule of M. Then the isotypic decomposition (3.3.9) for M can be written as

M ≅ N ⊗ (⨁_{h∈H} V^{h}) ,  (3.3.12)

where the action of R_{q} on N ⊗ V^{h} is given by the formula

t_{i}(u ⊗ v^{h}) = t_{i}(u) ⊗ h(η_{i})v^{h}, for u ∈ N, v^{h} ∈ V^{h}.  (3.3.13)

Proof The H-grading of the algebra A yields an H-grading of the matrix algebra

End_{ℂ}(M) = ρ(A) = ⨁_{h∈H} ρ(A_{h}),  (3.3.14)

where the endomorphisms in ρ(A_{h}) satisfy

T_{j}ρ(a)T_{j}^{−1} = ρ(η_{j}(a)) = h(η_{j})ρ(a).  (3.3.15)

This can be viewed as an eigenspace decomposition of End_{ℂ}(M) with respect to conjugations by the T_{j}’s.

On the other hand, we can construct such an eigenspace decomposition using the isotypic decomposition (3.3.9). In particular, let N ⊗ V^{0} be the isotypic component of N in M, and let N ⊗ V^{α} be another nontrivial component in (3.3.9). We claim that id_{N} ⊗ Hom_{ℂ}(V^{0}, V^{α}) can be viewed as the eigenspace (with eigenvalue α_{j}) for the conjugation action of T_{j} on End_{ℂ}(M). Indeed, extend S ∈ Hom_{ℂ}(V^{0}, V^{α}) to End_{ℂ}(⨁_{β} V^{β}) by setting S (V^{β}) = 0 when β ≠ 0. Then for n ∈ N and v ∈ V^{0}, we have

T_{j}(id_{N} ⊗ S)T_{j}^{−1}(n ⊗ v) = T_{j}(id_{N} ⊗ S)(T_{j}^{−1}(n) ⊗ v)
= T_{j}(T_{j}^{−1}(n) ⊗ Sv)
= α_{j}n ⊗ Sv.

Thus T_{j}(id_{N} ⊗S)T_{j}^{−1} = α_{j}(id_{N} ⊗S). By (3.3.14) and (3.3.15), all eigenvalues of the conjugation action of T_{j} on End_{ℂ}(M) correspond to the elements of H, so there exists h ∈ H such that α_{j} = h(η_{j}) for all j = 1, …, r. □

Next we explicitly describe the eigenspace decomposition of End_{ℂ}(M). In order to do this, we introduce the group homomorphism

γ : ℤ^{r} → H,

defined by the formula

γ(a)(η^{b}) = ∏_{i,j=1}^{r} q_{ij}^{a_{i}b_{j}}, for a, b ∈ ℤ^{r}.

Here we use the multi-index notation η^{b} = η_{1}^{b_{1}} … η_{r}^{b_{r}} and T^{b} = T_{1}^{b_{1}} … T_{r}^{b_{r}}. The fact that γ(a′ + a′′) = γ(a′) + γ(a′′) is obvious (see (1.1.5) for the definition of the additive notation), so all we need to verify is that this map is well-defined. That is, we should check that γ(a)(η^{b′}) = γ(a)(η^{b′′}) whenever η^{b′} = η^{b′′} in H.

Taking b = b′ − b′′, it is enough to show that γ(a)(η^{b}) = 1 whenever η^{b} = 1 in H. However,

T^{a}T^{b} = γ(a)(η^{b})T^{b}T^{a}.

If η^{b} = 1, then T^{b} is a multiple of the identity and thus commutes with T^{a}. This implies that γ(a)(η^{b}) = 1.

Note that the kernel of γ corresponds to the centre of the quantum torus R_{q}:

Z(R_{q}) = Span {t^{a} | a ∈ ker γ} .

By Schur’s Lemma, the centre Z(R_{q}) acts on a finite-dimensional simple R_{q}-module N by scalar operators.

The following proposition is essentially equivalent to the classification of gradings on the matrix algebra ([4], Thm. 6).

Proposition 3.3.16 Let M = N ⊗( ⨁_{p∈H} V^{p}) be the isotypic decomposition

of M as an R_{q}-module. Then the space End_{ℂ}(M) decomposes into a direct sum of eigenspaces with respect to conjugation by the operators T_{j}, j = 1, …, r:

End_{ℂ}(M) = ⨁_{h∈H} ρ(A_{h}),

where

ρ(A_{h}) = ∑_{a∈ℤ^{r}} T^{a} ⊗ (⨁_{p∈H} Hom_{ℂ} (V^{p}, V^{p+h+γ(a)})) ,

is the eigenspace with eigenvalue h(η_{j}) with respect to conjugation by T_{j}.

Proof It is easy to see that the subspaces

T^{a} ⊗ (⨁_{p∈H} Hom_{ℂ} (V^{p}, V^{p+h+γ(a)}))

span End_{ℂ}(M) = End_{ℂ}(N) ⊗ End_{ℂ} (⨁_{h∈H} V^{h}). It remains only to verify that these subspaces are the eigenspaces for the conjugation by T_{j}’s.

Let S ∈ Hom_{ℂ} (V^{p}, V^{p+h+γ(a)}), n ∈ N, v ∈ V^{p}. Then

T_{j}(T^{a} ⊗ S)T_{j}^{−1}(n ⊗ v) = T_{j}(T^{a} ⊗ S)(T_{j}^{−1}(n) ⊗ p^{−1}(η_{j})v)
= T_{j}(T^{a}T_{j}^{−1}(n) ⊗ p^{−1}(η_{j})Sv)
= p(η_{j})h(η_{j})γ(a)(η_{j})p^{−1}(η_{j})T_{j}T^{a}T_{j}^{−1}(n) ⊗ Sv
= h(η_{j})(T^{a} ⊗ S)(n ⊗ v).
□

Using this result, we can now describe the thin H-coverings of the module M.

Theorem 3.3.17 Let M be a finite-dimensional simple A-module, and let Ĥ be a finite abelian subgroup of Aut A, such that the twisted modules M^{η} are isomorphic to M for all η ∈ Ĥ. Let

M = ⨁_{h∈H} (N ⊗ V^{h})

be the isotypic decomposition of M with respect to the action of the cyclotomic quantum torus R_{q} given by (3.3.13). Then up to equivalence, thin H-coverings of M are parametrized by one-dimensional subspaces in N. Namely, let n ∈ N, n ≠ 0. Define

M_{h} = ∑_{a∈ℤ^{r}} T^{a}(n) ⊗ V^{h+γ(a)}.  (3.3.18)

Then {M_{h} | h ∈ H} is a thin H-covering of M, and every thin H-covering of M is equivalent to a covering of this type.

Remark 3.3.19 The sum in (3.3.18) may be replaced with a finite sum taken over a ∈ ℤ^{r}/ ker γ.

Proof By Proposition 3.3.16,

ρ(A_{m}) = ∑_{b∈ℤ^{r}} T^{b} ⊗ (⨁_{p∈H} Hom_{ℂ}(V^{p}, V^{p+m+γ(b)}))

for any m ∈ H. Thus

ρ(A_{m})M_{h} = ∑_{a,b∈ℤ^{r}} T^{b}T^{a}(n) ⊗ V^{h+m+γ(a)+γ(b)}  (3.3.20)
= ∑_{c∈ℤ^{r}} T^{c}(n) ⊗ V^{h+m+γ(c)}  (3.3.21)
= M_{h+m}.  (3.3.22)

The subspaces M_{h} clearly span M:

∑_{h∈H} M_{h} = ∑_{h∈H} ∑_{a∈ℤ^{r}} T^{a}(n) ⊗ V^{h+γ(a)}

= ( ∑_{a∈ℤ^{r}} T^{a}(n)) ⊗ (⨁_{h∈H} V^{h})

= N ⊗ (⨁_{h∈H} V^{h})
= M.

Thus {M_{h} | h ∈ H} is an H-covering of M.

We will use Lemma 1.1.2 to show that this covering is thin. All that remains is to determine the simple ρ(A_{0})-submodules of M. Let L ⊆ M be a simple ρ(A_{0})-submodule, and let u be a nonzero vector in L. Fix a basis {v_{hj} | 1 ≤ j ≤ dim V^{h}} for each of the spaces V^{h}. Expand u according to this basis:

u = ∑_{h,j} n_{hj} ⊗ v_{hj} with n_{hj} ∈ N.

Since the algebra ρ(A_{0}) contains a subalgebra

id_{N} ⊗ (⨁_{h∈H} End_{ℂ} (V^{h})) ,

we see that L contains n_{hj} ⊗ v_{hj} for all h, j. Thus without loss of generality, we may assume that u = n ⊗ v, where n ∈ N, v ∈ V^{h}. Since

ρ(A_{0}) = ∑_{a∈ℤ^{r}} T^{a} ⊗ (⨁_{p∈H} Hom_{ℂ} (V^{p}, V^{p+γ(a)})) ,

L contains the subspace

U^{h}(n) = ∑_{a∈ℤ^{r}} T^{a}(n) ⊗ V^{h+γ(a)}.  (3.3.23)

The subspace U^{h}(n) is ρ(A_{0})-invariant, and hence every simple ρ(A_{0})-sub-module in M coincides with U^{h}(n) for some n ∈ N, h ∈ H, n ≠ 0, V^{h} ≠ 0.

Let us show that U^{h}(n) is, in fact, a simple ρ(A_{0})-module for all n ∈ N, h ∈ H such that n ≠ 0 and V^{h} ≠ 0. Fix b ∈ ℤ^{r}. The intersection of U^{h}(n) with N ⊗ V^{h+γ(b)} is

∑_{a∈ker γ} T^{b}T^{a}(n) ⊗ V^{h+γ(b)}.

The centre Z(R_{q}) acts on the simple R_{q}-module N by scalars. Thus for all a ∈ ker γ, T^{a}(n) is a multiple of n, so the intersection of U^{h}(n) with N ⊗ V^{h+γ(b)} is T^{b}(n) ⊗ V^{h+γ(b)}. By the argument given above, every nonzero vector in U^{h}(n) generates U^{h}(n). Therefore the spaces U^{h}(n) with V^{h} ≠ 0, n ≠ 0, exhaust all the simple ρ(A_{0})-submodules in M.

This proves that the components M_{h} of the H-covering (3.3.18) are simple ρ(A_{0})-modules whenever M_{h} ≠ 0. Thus for any nonzero m ∈ M_{h}, we have

ρ(A_{g−h})m ⊇ ρ(A_{g−h})ρ(A_{0})m = ρ(A_{g−h})M_{h} = M_{g},

using (3.3.20). Hence the covering {M_{h} | h ∈ H} is thin by Lemma 1.1.2.

Since the described components of the thin coverings exhaust all the simple ρ(A_{0})-submodules in M, we obtain that up to equivalence, these are all the thin H-coverings of M. □

Corollary 3.3.24 Let M be a finite-dimensional simple A-module, and let Ĥ be a finite abelian subgroup of Aut A, such that the twisted modules M^{η} are isomorphic to M for all η ∈ Ĥ. Let {M_{h} | h ∈ H} and {M′_{h} | h ∈ H} be two thin coverings of M. Then for some g ∈ H, the graded-simple modules

M̃ = ⨁_{h∈H} M_{h} and M̃′ = ⨁_{h∈H} M′_{h+g}

are isomorphic as graded A-modules.

Proof Clearly, we are allowed to replace the thin coverings {M_{h}} and {M′_{h}} with any equivalent thin coverings. By Theorem 3.3.17, we may assume that for all h ∈ H,

M̃_{h} = ∑_{a∈ℤ^{r}} T^{a}(n) ⊗ V^{h+γ(a)} and M̃′_{h} = ∑_{a∈ℤ^{r}} T^{a}(n′) ⊗ V^{h+γ(a)}

for some nonzero n, n′ ∈ N.

We construct a grading-preserving isomorphism ψ : M̃ → M̃′ defined on M̃_{h} by

ψ (T^{a}(n) ⊗ v) = T^{a}(n′) ⊗ v

for all h ∈ H, a ∈ ℤ^{r}, v ∈ V^{h+γ(a)}. Using Proposition 3.3.16, it is easy to see that ψ commutes with the action of A. □

We now consider some basic examples of thin H-coverings.

Example 3.3.25 Suppose A is graded by a finite cyclic group Ĥ = ⟨η⟩ ⊆ AutA, and suppose that there is a (normalized) isomorphism T : M → M^{η}. Then, up to equivalence, there is a unique thin H = Hom(Ĥ, ℂ^{∗})-covering {M_{h}} of M, where M_{h} is the eigenspace

M_{h} = {u ∈ M | T u = h(η)u}.

In this case, the module M admits an H-grading: M = ⨁_{h∈H} M_{h}.

Proof Since the sum M = ∑_{h∈H} M_{h} is direct, the thinness of the covering {M_{h}} in Example 3.3.25 is obvious. However, even in this simplest of examples, the uniqueness of the thin covering requires Theorem 3.3.17. Note that, in this case, the quantum torus R_{q} is just the algebra of Laurent polynomials in a single variable, so dim N = 1, and M may be identified with ⨁_{h∈H} V_{h}. □

From the statement of Theorem 3.3.17, it is clear that uniqueness of thin coverings is the exception, rather than the rule, as is illustrated by the next pair of examples.

Example 3.3.26 Let A = M_{n}(ℂ) be the associative algebra of n×n matrices. If A = ⨁_{g∈G} A_{g} is any grading satisfying conditions (i) and (ii) of Theorem 2.2.2, then by Theorem 2.2.3, every graded-simple module for A comes from a thin covering of a simple module for A. By the Artin-Wedderburn Theorem, there is only one simple module for A; and by Theorem 2.2.2, there is only one graded-simple module (up to graded-isomorphism) that can come from the thin coverings of this module. Therefore (up to graded isomorphism), A has only one graded-simple module.

In spite of the uniqueness of the graded-simple module for A, the thin coverings for A are far from unique. For example, A has a ℤ_{n} × ℤ_{n}-grading with

A_{(\overline{a},\overline{b})} := Span\{E^{a}F^{b}\}

where

E := ( \begin{matrix}1 & 0 & · & · & 0 \\ 0 & ζ & · & · & 0 \\ ⋮ & ⋮ & & ... & ⋮ \\ 0 & 0 & · & · & ζ^{d−1} \end{matrix} ) , F := ( \begin{matrix}0 & 0 & · & 0 & 1 \\ 1 & 0 & · & 0 & 0 \\ 0 & 1 & · & 0 & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ \\ 0 & 0 & · & 1 & 0 \end{matrix} ) ,

ζ is a primitive nth root of unity, and \overline{a} (resp., \overline{b}) is the image of a ∈ ℤ (resp., b ∈ ℤ) under the canonical homomorphism ℤ → ℤ_{n} = ℤ/nℤ.

Then the graded ring A satisfies conditions (i) and (ii) of Theorem 2.2.2, so any thin covering of the natural module M = ℂ^{n} induces the same (unique) graded-simple module of A (relative to the grading group G = ℤ_{n} ×ℤ_{n}). Let v be a nonzero vector in M. Then since M is irreducible,

M_{g} := A_{g}v

defines a covering of M. Let {N_{g} | g ∈ G} be another covering of M with N_{g} ⊆ M_{g} for all g ∈ G. Then N_{h} ≠ 0 for some h ∈ G, so N_{g} = A_{g−h}.N_{h} ≠ 0 for all g ∈ G, since the one-dimensional space A_{g−h} is spanned by the invertible matrix E^{a}F^{b} (where g − h = (\overline{a}, \overline{b})). Moreover, N_{g} is a nonzero subspace of the one-dimensional space M_{g}, so N_{g} = M_{g} for all g ∈ G. Thus the covering {M_{g}} is thin.

Any choice of v results in a thin covering consisting of n^{2} one-dimensional subspaces M_{g} of M. The space M_{0} is the span of the vector v. Since there are infinitely many distinct one-dimensional subspaces we could choose for M_{0}, there are infinitely many nonequivalent thin coverings of M.

More generally, using Theorem 3.3.17, we have the following example.

Example 3.3.27 Suppose Ĥ is generated by two automorphisms η_{1}, η_{2} of order d. Let T_{1}, T_{2} be isomorphisms from M to M^{η_{1}}, M^{η_{2}}, respectively. Assume that these operators satisfy the relation

T_{1}T_{2} = ζT_{2}T_{1},

where ζ is a primitive root of unity of order d. Consider the isotypic decomposition

M = N ⊗ (⨁_{h∈H} V^{h})

of M as an R_{ζ}-module. We obtain from Theorem 3.3.17 that the thin Hcoverings are parametrized by one-dimensional subspaces in a d-dimensional space N. For any nonzero vector n ∈ N we get a thin H-covering with

M_{h} = ∑_{a∈ℤ^{2}} T^{a}(n) ⊗ V^{h+γ(a)}.

Up to equivalence, these are all thin H-coverings of M.

We are finally ready to describe the thin G-coverings of M in the general case–that is, when the twists of M by automorphisms in Ĝ are not necessarily isomorphic to M. The following theorem shows that G-coverings of M can be described via H-coverings, where Ĥ is the subgroup of Ĝ defined in Lemma 3.3.1. In order to distinguish between G-coverings and H-coverings (resp. gradings), we will add an appropriate superscript to our notation.

The inclusion Ĥ ⊆ Ĝ yields a natural epimorphism

ψ : Hom(Ĝ, ℂ^{∗}) → Hom(Ĥ, ℂ^{∗})

by restriction of the maps to Ĥ.

Theorem 3.3.28 Let M be a finite-dimensional simple A-module, and let Ĝ be a finite abelian subgroup of Aut A. Let Ĥ be its subgroup defined in Lemma 3.3.1. Then

(i) ρ(A_{g}^{G}) = ρ(A_{ψ(g)}^{H}) for all g ∈ G.

(ii) There is a bijective correspondence between thin G-coverings of M and thin H-coverings of M. Given a thin H-covering {M_{h}^{H}} we get a thin G-covering {M_{g}^{G}} with M_{g}^{G} = M_{ψ(g)}^{H}. Every thin G-covering is of this form.

(iii) All graded-simple A-modules associated with various thin coverings of M (as in Theorem 1.1.3) are isomorphic (up to shifts in the gradings on the modules).

Proof Let τ_{1}, …, τ_{ℓ} be representatives of the cosets of Ĥ in Ĝ, where ℓ = |Ĝ|/|Ĥ|. By Lemma 3.3.1, the modules M^{τ_{1}}, …, M^{τ_{ℓ}} are pairwise non-isomorphic. Consider the representation ρ̂ of A on

M̂ = M^{τ_{1}} ⊕ ⋯ ⊕ M^{τ_{ℓ}}.

An element a ∈ A_{g}^{G} acts on M̂ by

ρ̂(a) = (g(τ_{1})ρ(a), …, g(τ_{ℓ})ρ(a)) .

Thus dim ρ̂(A_{g}^{G}) = dim ρ(A_{g}^{G}). Moreover, by the definition of the map ψ, A_{g}^{G} ⊆ A_{ψ(g)}^{H}, so ρ(A_{g}^{G}) ⊆ ρ(A_{ψ(g)}^{H}). Hence

dim ρ̂(A) ≤ ∑_{g∈G} dim ρ̂(A_{g}^{G})  (3.3.29)
= ∑_{g∈G} dim ρ(A_{g}^{G})  (3.3.30)
≤ ∑_{g∈G} dim ρ(A_{ψ(g)}^{H})  (3.3.31)
= (|G|/|H|) ∑_{h∈H} dim ρ(A_{h}^{H})  (3.3.32)
≤ ℓ dim End_{ℂ}(M).  (3.3.33)

However, by the density theorem,

ρ̂(A) = End_{ℂ}(M^{τ_{1}}) ⊕ ⋯ ⊕ End_{ℂ}(M^{τ_{ℓ}}).

This implies that all inequalities in (3.3.29)–(3.3.33) are in fact equalities, and in particular,

dim ρ(A_{g}^{G}) = dim ρ(A_{ψ(g)}^{H}).

This completes the proof of Part (i) of the theorem.

To prove the second part, we note that ρ(A_{0}^{G}) = ρ(A_{0}^{H}). This means that the simple ρ(A_{0}^{G})-submodules in M are the simple ρ(A_{0}^{H})-submodules. Let {M_{g}^{G}} be a thin G-covering of M. Without loss of generality, we may assume that M_{0}^{G} ≠ 0. Then by Lemma 1.1.2, M_{0}^{G} is a simple ρ(A_{0}^{H})- submodule. We have seen in the proof of Theorem 3.3.17 that every simple ρ(A_{0}^{H})-submodule in M occurs as a component of a thin H-covering. Since we can always replace such a covering by an equivalent thin covering, we see that every simple ρ(A_{0}^{H})-submodule in M occurs as the 0-component of a thin H-covering. Thus there exists a thin H-covering {M_{h}^{H}}, such that M_{0}^{H} = M_{0}^{G}. Moreover, M_{h}^{H} = ρ (A_{h}^{H}) M_{0}^{H} for all h ∈ H. By Lemma 1.1.2 and Part (i), we get that

M_{g}^{G} = ρ (A_{g}^{G}) M_{0}^{G} = ρ (A_{ψ(g)}^{H}) M_{0}^{H} = M_{ψ(g)}^{H}.

Thus every thin G-covering comes from a thin H-covering. Conversely, if we construct a G-covering from a thin H-covering by taking M_{g}^{G} = M_{ψ(g)}^{H}, then Lemma 1.1.2 implies that this G-covering is thin.

Part (iii) follows from (i), (ii), and Corollary 3.3.24. □

When Ĝ is finite cyclic, Theorem 3.3.28 reduces to the following eigenspace decomposition.

Example 3.3.34 Suppose Ĝ is a finite cyclic group generated by η ∈ AutA, Ĥ ⊆ Ĝ is generated by η^{s}, and T : M → M^{η^{s}} is a (normalized) module isomorphism. Then (up to equivalence) M has a unique thin G-covering given by

M_{g} = {u ∈ M | T u = g(η)^{s}u}.

4 Thin coverings of quasifinite modules

In this section we generalize our results to an infinite-dimensional setting. We will assume that the algebra A has an additional grading by an abelian group Z,

A = ⨁_{k∈Z} A_{k},

and that M is a Z-graded A-module,

M = ⨁_{n∈Z} M_{n},

with finite-dimensional graded components M_{n}. Such modules are called quasifinite.

Note that M may be infinite-dimensional, since we allow Z to be infinite. We are primarily interested in the case when Z = ℤ, but our results hold for arbitrary abelian grading groups.

We will work in the categories of Z-graded algebras and modules and consider only those automorphisms σ which preserve the Z-grading. That is,

σ(A_{k}) = A_{k}.

Let M be a quasifinite graded-simple module (with no proper Z-graded submodules). In this section, we show that Theorems 3.3.17 and 3.3.28 still hold in this more general setting. We start with some obvious adjustments to the definition of a thin G-covering.

Let Ĝ be a finite abelian group of (Z-grading-preserving) automorphisms of A. Then A is graded by the group G = Hom(Ĝ, ℂ^{∗}), and the two gradings are compatible:

A = ⨁_{(k,g)∈Z×G} A_{k,g}.

A covering of M is a set of Z-graded subspaces {M_{g} | g ∈ G} of M that span M and satisfy

ρ(A_{g})M_{h} ⊆ M_{g+h}, for all g, h ∈ G.  (4.4.1)

As before, the set of coverings is partially ordered, and the minimal elements of the poset of coverings are called thin coverings.

For all n, k ∈ Z, let ρ_{nk} be the restriction of the action of A_{k−n} to M_{n}:

ρ_{nk} : A_{k−n} → Hom_{ℂ}(M_{n}, M_{k}).

It is easy to see that G-coverings can be defined in terms of ρ_{nk}. Indeed, condition (4.4.1) can be replaced by

ρ_{nk}(A_{k−n,h})M_{n,g} ⊆ M_{k,g+h},

and “local” information about ρ_{nk}(A_{k−n}) ⊆ Hom_{ℂ}(M_{n}, M_{k}) can substitute for “global” information about ρ(A) ⊆ End_{ℂ}(M). The proofs of the previous section are essentially based on Schur’s Lemma and the Jacobson Density Theorem. We will use the quasifinite version of the density theorem presented in the appendix.

Remark 4.4.2 The assumption on the existence of a maximal submodule made in Theorem 1.1.4 holds for the quasifinite modules described above.

Proof Let M be a Z-graded quasifinite module, which also has a compatible G-grading and is graded-simple (has no proper Z × G-homogeneous submodules). Fix n ∈ Z such that M_{n} ≠ 0. Note that by Zorn’s Lemma, M has a maximal Z-graded submodule. Indeed, if U is a proper Z-graded submodule in M then U_{n} ≠ M_{n}. Otherwise, if U_{n} = M_{n} then the space U_{n} is homogeneous both with respect to Z and G and will generate the whole module M since it is graded-simple.

Now if we consider an increasing chain of Z-graded submodules in M,

⋯ ⊆ U^{(i)} ⊆ U^{(i+1)} ⊆ ⋯,

the corresponding chain of components

⋯ ⊆ U_{n}^{(i)} ⊆ U_{n}^{(i+1)} ⊆ ⋯ ⊆ M_{n}

will stabilize, since dim M_{n} < ∞. Thus the union ⋃_{i} U^{(i)} is a Z-graded

submodule in M, with nth component properly contained in M_{n}. Applying Zorn’s Lemma, we conclude that the partially ordered set of Z-graded proper submodules of M contains a maximal element. □

Our main results about quasifinite modules, Theorems 4.4.3 and 4.4.4, can be proven using the same arguments as for the finite case. Only the proof of Theorem 4.4.4(i) requires nontrivial modification, since the corresponding finite result, Theorem 3.3.28(i), used a dimension argument. We write down the details of the modified proof below, leaving the other proofs as straightforward exercises to the reader.

Theorem 4.4.3 Let M be a Z-graded-simple quasifinite module for a Zgraded algebra A. Let Ĥ be a finite abelian group of automorphisms of A preserving the Z-grading, such that there exist Z-grading-preserving isomorphisms T_{η} between M and M^{η} for all η ∈ Ĥ. Let

M = ⨁_{h∈H} N ⊗ V^{h}

be the isotypic decomposition of M with respect to the action of the cyclotomic quantum torus R_{q} given by the operators T_{η}. In this decomposition, each subspace V^{h} is Z-graded. Then up to equivalence, thin H-coverings of M are parametrized by one-dimensional subspaces in N. Namely, for n ∈ N, n ≠ 0, define

M_{h} = ∑_{a∈ℤ^{r}} T^{a}(n) ⊗ V^{h+γ(a)}.

Then {M_{h} | h ∈ H} is a thin H-covering of M, and every thin H-covering of M is equivalent to a covering of this type. □

Theorem 4.4.4 Let M be a Z-graded-simple quasifinite module for a Zgraded algebra A. Let Ĝ be a finite abelian group of automorphisms of A preserving the Z-grading. Let Ĥ be its subgroup defined in Lemma 3.3.1. Then

(i) ρ_{nk}(A_{k−n,g}^{G}) = ρ_{nk}(A_{k−n,ψ(g)}^{H}) for all g ∈ G.

(ii) There is a bijective correspondence between thin G-coverings of M and thin H-coverings of M. Given a thin H-covering {M_{h}^{H}} we get a thin G-covering {M_{g}^{G}} with M_{g}^{G} := M_{ψ(g)}^{H}. Every thin G-covering of M is of this form.

(iii) All graded-simple A-modules associated with various thin coverings of M (as in Theorem 1.1.3) are isomorphic (up to shifts in the G-gradings on the modules).

Proof We will give a proof for Part (i) of the theorem. Parts (ii) and (iii) may be proven analogously to Theorem 3.3.28.

Let τ_{1}, …, τ_{ℓ} be representatives of the cosets of Ĥ in Ĝ, where ℓ = |Ĝ|/|Ĥ|. The modules M^{τ_{1}}, …, M^{τ_{ℓ}} do not admit (Z-grading-preserving) isomorphisms between any distinct pair of them.

Consider the representation ρ̂ of A on

M̂ = M^{τ_{1}} ⊕ ⋯ ⊕ M^{τ_{ℓ}}.

For n, k ∈ Z, let ρ̂_{nk} be the restriction of ρ̂(A_{k−n}) to M̂_{n} = M_{n}^{τ_{1}} ⊕⋯⊕M_{n}^{τ_{ℓ}}:

ρ̂_{nk} : A_{k−n} → Hom_{ℂ}(M̂_{n}, M̂_{k}).

An element a ∈ A_{k−n,g}^{G} acts on M̂_{n} by

ρ̂_{nk}(a) = (g(τ_{1})ρ_{nk}(a), …, g(τ_{k})ρ_{nk}(a)).

Thus dim ρ̂_{nk}(A_{k−n,g}^{G}) = dim ρ_{nk}(A_{k−n,g}^{G}). Hence

dim ρ̂_{nk}(A_{k−n}) ≤ ∑_{g∈G} dim ρ̂_{nk}(A_{k−n,g}^{G}) = ∑_{g∈G} dim ρ_{nk}(A_{k−n,g}^{G})

≤ ∑_{g∈G} dim ρ_{nk}(A_{k−n,ψ(g)}^{H}) = (|G|/|H|) ∑_{h∈H} dim ρ_{nk}(A_{k−n,h}^{H}),  (4.4.5)

using the fact that A_{k−n,g}^{G} ⊆ A_{k−n,ψ(g)}^{H} by the definition of ψ. The space ρ_{nk}(A_{k−n}) decomposes into a direct sum

ρ_{nk}(A_{k−n}) = ⨁_{h∈H} ρ_{nk}(A_{k−n,h}^{H})

(cf. Proposition 3.3.16), so

(|G|/|H|) ∑_{h∈H} dim ρ_{nk}(A_{n−k,h}^{H}) = ℓ dim ρ_{nk}(A_{k−n}).

But by the Quasifinite Density Theorem (A.A.2),

ρ_{nk}(A_{k−n}) = Hom_{ℂ}(M_{n}, M_{k})

and

ρ̂_{nk}(A_{k−n}) = ⨁_{j=1}^{ℓ} Hom_{ℂ}(M_{n}^{τ_{j}}, M_{k}^{τ_{j}}),

so dim ρ̂_{nk}(A_{n−k}) = ℓ dim ρ_{nk}(A_{n−k}).

Therefore all the inequalities in (4.4.5) are in fact equalities, and in particular,

dim ρ_{nk}(A_{k−n,g}^{G}) = dim ρ_{nk}(A_{k−n,ψ(g)}^{H}).

This completes the proof of Part (i). □

5 Application to multiloop Lie algebras

Let Ĝ be a finite abelian group of grading-preserving automorphisms of a Z-graded complex Lie algebra 𝔤. Consider the dual group G = Hom(Ĝ, ℂ^{∗}). The Lie algebra 𝔤 is then graded by the group Z × G. By choosing a set of N generators of G, we get an epimorphism ℤ^{N} → G. We will denote by \overline{a} the image of an element a ∈ ℤ^{N} under this map.

Consider the multiloop Lie algebra L(𝔤, Ĝ):

L(𝔤, Ĝ) = ⨁_{a∈ℤ^{N}} 𝔤\overline{a} ⊗ t^{a},

which is a Lie subalgebra of 𝔤 ⊗ ℂ[t_{1}^{±}, …, t_{N}^{±}].

Let M be a quasifinite Z-graded-simple module for 𝔤. Using Theorem 4.4.4, we can construct a G-covering {M_{g}} of M. Then the space

⨁_{a∈ℤ^{N}} M\overline{a} ⊗ t^{a}

becomes a (Z×ℤ^{N})-graded-simple module for the multiloop algebra L(𝔤, Ĝ).

A Appendix

In this appendix, we prove the quasifinite density theorem used in the proof of Theorem 4.4.4.

Let M be a Z-graded module over a Z-graded associative ring R. We call such a module graded-semisimple if it is a direct sum of a finite number of graded-simple modules. Consider the set of endomorphisms of M that shift the grading by an element p ∈ Z:

End (M)_{p} = {X ∈ End (M) | XM_{n} ⊆ M_{n+p} for all n ∈ Z} .

In this notation, End (M)_{0} is the ring of grading-preserving endomorphisms of M.

The following is a graded version of the usual density theorem of Jacobson and Chevalley (cf. [11], Thm. 4.11.16, for instance):

Theorem A.A.1 (Graded Density Theorem) Let M be a Z-graded-semi-simple module over a Z-graded associative ring R. Let K = End_{R}(M)_{0} and let E = ⨁_{p∈Z} End_{K}(M)_{p}. Suppose that {x_{1}, …, x_{m}} ⊆ M_{s} for some

s ∈ Z. Then for every f ∈ E there exists an element r ∈ R such that rx_{j} = f(x_{j}) for j = 1, 2, …, m.

Proof The argument given in [11] is valid in the graded set-up as well. □

Theorem A.A.2 (Quasifinite Density Theorem) Let A be an associative algebra graded by an abelian group Z. Let M^{(1)}, ..., M^{(n)} be Z-graded-simple quasifinite A-modules, such that there is no grading-preserving isomorphism between any distinct pair of them. Fix s, p ∈ Z, and for each i = 1, …, n, let {x_{j}^{i}}_{j=1,... ,m_{i}} be a ℂ-linearly independent subset of M_{s}^{(i)}, and let {y_{j}^{i}}_{j=1,... ,m_{i}} be a set of vectors in M_{s+p}^{(i)}. Then there exists an element a ∈ A_{p}, such that ax_{j}^{i} = y_{j}^{i} for all i, j.

Proof By Schur’s Lemma, every nonzero grading-preserving homomorphism of graded-simple modules is an isomorphism. Thus

Hom_{A} (M^{(i)}, M^{(ℓ)})_{0} = 0 for i ≠ ℓ.

Applying Schur’s Lemma again, we see that every grading-preserving endomorphism of a quasifinite graded-simple module is a multiple of the identity. Take M = M^{(1)} ⊕ ⋯ ⊕ M^{(n)}. Then

K = End_{A}(M)_{0} = ℂπ_{1} ⊕ ⋯ ⊕ ℂπ_{n},

where π_{j} is the projection π_{j} : M → M^{(j)}. It follows that

End_{K}(M)_{p} = ⨁_{j=1}^{n} End_{ℂ}(M^{(j)})_{p},

and now Theorem A.A.2 follows immediately from Theorem A.A.1. □

References

[1] B. Allison, S. Berman, J. Faulkner, A. Pianzola, Realization of gradedsimple algebras as loop algebras, preprint arXiv:math.RA/0511723.

[2] B. Allison, S. Berman, A. Pianzola, Covering algebras I: Extended affine Lie algebras, J. Algebra 250 (2002), 485–516.

[3] B. Allison, S. Berman, A. Pianzola, Covering algebras II: Isomorphism of loop algebras, J. Reine Angew. Math. 571 (2004), 39–71.

[4] Y.A. Bahturin, S.K. Sehgal, M.V. Zaicev, Group gradings on associative algebras, J. Algebra 241 (2001), 677–698.

[5] Y.A. Bahturin, M.V. Zaicev, Group gradings on simple Lie algebras of type “A”, preprint arXiv:math.RA/0506055.

[6] Y. Billig and M. Lau, Irreducible modules for extended affine Lie algebras, in preparation.

[7] N. Bourbaki, Éléments de Mathématique. Fasc. XXIII, Livre II: Algèbre. Chapitre 8: Modules et anneaux semi-simples, Hermann, Paris, 1958.

[8] A.H. Clifford, Representations induced in an invariant subgroup, Ann. of Math. 38 (1937), 535–550.

[9] E.C. Dade, Group-graded rings and modules, Math. Z. 174 (1980), 241– 262.

[10] V.G. Kac, Automorphisms of finite order of semi-simple Lie algebras, Funct. Anal. Appl. 3 (1969), 252–254.

[11] T.Y. Lam, A First Course in Noncommutative Rings, Grad. Texts in Math., vol. 131, Springer, New York, 1991.

[12] C. Năstăsescu and F. Van Oystaeyen, Methods of Graded Rings, Lecture Notes in Math., vol. 1836, Springer, Berlin, 2004.

[13] K.-H. Neeb, On a normal form of rational quantum tori and their automorphism groups, preprint arXiv:math.RA/0511263.

[14] E. Neher, Extended affine Lie algebras, C. R. Math. Acad. Sci. Soc. R. Can. 26 (2004), 90–96.