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«Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 016, 17 pages Extension Fullness of the Categories of ...»

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In particular, from the construction we have that the image of N in the second row (coming from the first row) and in the third row (coming from the last row) coincide. Hence diagram (4) gives rise to a diagram / G(M ) / X1 / X2 / ··· / Xd−1 / Xd /N /0

–  –  –

with exact rows. The last diagram shows that the extensions given by the first and the last rows coincide, which proves that ΨΦ is the identity map.

The claim that ΦΨ is the identity map is proved similarly. This completes the proof.

Extension Fullness of the Categories of Gelfand–Zeitlin and Whittaker Modules 9 3 Gelfand–Zeitlin modules Notation. For a Lie algebra a we denote by U (a) its universal enveloping algebra and by Z(a) the center of U (a).

3.1 Gelfand–Zeitlin subalgebra of gln For k ∈ Z0 denote by gk the Lie algebra glk (C). Set Uk = U (gk ) and Zk = Z(gk ). We consider the usual chain

–  –  –

of embeddings of associative algebras. The subalgebra Γ of U := Un generated by all centers Zk, where k = 1, 2,..., n, is called the Gelfand–Zeitlin subalgebra. We set g := gn.

The algebra Γ is obviously commutative, moreover, Γ is a polynomial algebra in n(n+1) vari-2 ables (these can be taken to be generators of Zk for k = 1, 2,..., n), see [23] or [50, Chapter X].

Considered as a subalgebra of U, Γ is a Harish-Chandra subalgebra in the sense of [12, Section 1], which means that every finitely generated Γ-subbimodule of U is already finitely generated both as a left and as a right Γ-module, see [12, Theorem 24]. Furthermore, U is free both as a left and as a right Γ-module, see [44]. Consequently, the usual induction and coinduction functors

–  –  –

are exact.

3.2 Gelfand–Zeitlin modules The category G Z of Gelfand–Zeitlin-modules for U is defined as the full subcategory of U -mod (the category of finitely generated U -modules), which consists of those M ∈ U -mod on which the action of Γ is locally finite, that is dim(Γv) ∞ for all v ∈ M. The category G Z is a Serre subcategory of U -mod as U is Noetherian.

Write Specm(Γ) for the set of maximal ideals in Γ. As Γ is a polynomial algebra, we have the decomposition

–  –  –

where Γ-fmodm denotes the full subcategory of Γ-fmod consisting of all objects annihilated by some power of m. From this decomposition, consider the functor

–  –  –

defined as the restriction of IndU to Γ-fmodm.

Γ Note that modules in G Z are usually infinite-dimensional and hence the usual restriction ResU ends up in Γ-lfmod and not in Γ-fmod. However, from [21, Corollary 5.3(a)] it follows that Γ for any M ∈ G Z and m ∈ Specm(Γ) the space

–  –  –

of the coinduction functor to Γ-fmodm.

3.3 The main result

Our main result in this section is the following:

Theorem 1. The category G Z is extension full in U -Mod.

–  –  –

Proof. By additivity, we may assume N ∈ Γ-fmodm for some m ∈ Specm(Γ). The image of the functor IndU : Γ-fmodm → A belongs to B. We show that for every d ∈ Z≥0, any Γ,m N ∈ Γ-fmodm and any Q ∈ G Z we have the isomorphisms

–  –  –

Since, by construction, these three isomorphisms together with the morphism ιd M,Q will yield d a commutative square, we get that ιM,Q is an isomorphism.

Isomorphism (6) follows from [7, Lemma 17]. Isomorphisms (5) and (7) follow from adjunction lemma (Proposition 4). Note that adjunction lemma applies here since U is free over Γ.

Theorem 1 now follows.

Extension Fullness of the Categories of Gelfand–Zeitlin and Whittaker Modules 11 The global dimension of G Z 3.4 Corollary 2. We have gl.dim(G Z ) = gl.dim(U -Mod) = dim g.

Proof. It is well-known, see e.g. [49, Corollary 7.7.3], that the trivial g-module C has maximal possible projective dimension in U -Mod, namely dim g. Since we obviously have C ∈ G Z, the claim follows from Theorem 1.

Remark 1. The category G Z decomposes into a direct sum of indecomposable blocks, see [12, Theorem 24].

The proof of Corollary 2 implies that the block containing the trivial g-module has global dimension dim g = n2. However, most of the blocks in G Z, namely all the socalled strongly generic blocks in the sense of [36, Section 3], are equivalent to the category of finite dimension modules over Γm, the completion of Γ with respect to a maximal ideal m (this equivalence is induced by IndU ), and hence have global dimension n(n+1), that is the Krull Γ,m 2 dimension of Γ.

4 Whittaker modules

4.1 The category of Whittaker modules Let g be a semi-simple finite-dimensional complex Lie algebra with a fixed triangular decomposition g = n− ⊕ h ⊕ n+. Consider the subalgebra R = Z(g)U (n+ ) in U (g). Note that R is not commutative unless g is a direct sum of copies of sl2. Set U = U (g).

Denote by W the full subcategory of U -mod (the category of finitely generated U -modules) consisting of all g-modules which are locally finite with respect to the action of R (cf. [38, Definition 1.5]). Objects in W will be called Whittaker modules, which is a slight modification of the original notion from [29].

4.2 Simple f inite-dimensional R-modules In order to better understand the category W, we start with a classification of simple finitedimensional R-modules. For this we would need the following fact.

Proposition 5. The algebra R is isomorphic to Z(g) ⊗C U (n+ ).

Proof. From the PBW Theorem it follows that the multiplication map U (h) ⊗C U (n+ ) U (h)U (n+ ) (8) is bijective. Injectivity of the Harish-Chandra homomorphism Z(g) → U (h) yields that the U (h)components of different elements in Z(g) are different. Hence (8) implies that the surjective homomorphism Z(g) ⊗C U (n+ ) R given by multiplication is, in fact, injective, and hence an isomorphism (see also [29, Section 3.3]).

–  –  –

Proof. We have that Vm,χ is a Z(g) ⊗C U (n+ )-module by construction. Hence claim (i) follows from Proposition 5. Claim (iii) is clear by construction.

To prove claim (ii), we note that R is a finitely generated complex algebra and hence every simple R-module admits a central character by Dixmier’s version of Schur’s lemma, see [9, Proposition 2.6.8]. Therefore, from Proposition 5 it follows that simple finite-dimensional Rmodules are exactly simple finite-dimensional U (n+ )-modules with the action of Z(g) given via the projection Z(g)/m ∼ C for some m. Since n+ is nilpotent, all simple finite-dimensional n+ = have dimension one by Lie’s theorem, see [9, Corollary 1.3.13]. The claim follows.

4.3 The categories R-fmod and R-lfmod For a maximal ideal m in Z(g) and a linear map χ : n+ /[n+, n+ ] → C denote by R-fmodm,χ the full subcategory of R-fmod consisting of all modules for which all simple composition subquotients are isomorphic to Vm,χ. Define similarly the subcategory R-lfmodm,χ of R-lfmod.

Proposition 7. We have decompositions

–  –  –

As χ = χ, there is a ∈ n+ whose action on M has two different eigenvalues χ(a) = χ (a). Let v and w be the corresponding non-zero eigenvectors in M (note that each of them is unique up to a non-zero scalar). Then Cw coincides with the image of Vm,χ in M. We claim that Cv is an R-submodule (which would mean that the short exact sequence (10) splits thus completing the proof).

Indeed, we have Z(g)Cv ⊂ Cv as Cv = {x ∈ M : ax = χ(a)x} and Z(g) commutes with a.

Consider some filtration

–  –  –

such that [a, Xi ] ⊂ Xi−1 for all i. Such filtration exists as n+ is nilpotent. The elements of X1 commute with a and hence X1 Cv ⊂ Cv. Let b ∈ X2 and assume that bv = αv + βw. Then, on the one hand, abv = χ(a) αv + βw + (χ (a) − χ(a))βw = χ(a)bv + (χ (a) − χ(a))βw.

On the other hand, abv = χ(a)bv + [a, b]v. As [a, b] ∈ X1, X1 v ⊂ Cv and v and w are linearly independent, we get [a, b]v = (χ (a) − χ(a))βw = 0, that is β = 0. Therefore bv ∈ Cv and thus X2 v ⊂ Cv. Proceeding inductively, we get n+ v ⊂ Cv and the proof is complete.

Note that the algebra R is isomorphic to the enveloping algebra of the nilpotent Lie algebra n+ ⊕ a, where a is an Abelian Lie algebra of dimension dim(h), see Proposition 5. In particular R ∼ U (n) for some solvable Lie algebra n. Therefore we will be able to make use of the following = result taken from [8, Proposition 1], see also [10] and [16].

Extension Fullness of the Categories of Gelfand–Zeitlin and Whittaker Modules 13 Lemma 2. Let n be a solvable Lie algebra. Any module V ∈ U (n)-lfmod has an injective hull IV in U (n)-Mod, moreover, IV ∈ U (n)-lfmod.

We note that existence of injective hulls in U (n)-Mod is due, in much bigger generality, to Baer [1], see also [15]. Lemma 2 implies the following.

Proposition 8. Let n be a finite-dimensional solvable Lie algebra. Then the category U (n)-lfmod has enough injective objects and, moreover, U (n)-lfmod is extension full in U (n)-Mod.

Proof. As U (n)-lfmod is a Serre subcategory of U (n)-Mod, the injective hulls in U (n)-Mod of Lemma 2 are automatically injective hulls in U (n)-lfmod. This proves that U (n)-lfmod has enough injective objects.

As U (n)-lfmod is a locally Noetherian Grothendieck category (see [30, Appendix A] or [47]), it follows that each injective object in this category is a coproduct of indecomposable injective objects and this decomposition is unique up to isomorphism. Since n is finite-dimensional, U (n) is Noetherian and hence a coproduct of injective objects in U (n)-Mod is injective, see [32, Proposition 1.2]. This implies that any injective object in U (n)-lfmod is also injective when regarded as a module in U (n)-Mod. The extension fullness thus follows from Proposition 2.

As a consequence, we obtain the following statement.

Corollary 3. The category R-lfmod is extension full in R-Mod.

4.4 R versus U (g) Proposition 9. The algebra U (g) is free both as a left and as a right R-module.

Proof. We prove the statement for the right module structure. The claim for the left module structure then follows by applying the canonical antiautomorphism of U (g) generated by x → −x for x ∈ g.

Choose a basis Y1, Y2,..., Yk in n−, a basis H1, H2,..., Hl in h and a basis X1, X2,..., Xk in n+. Then

–  –  –

plus a linear combination of elements v from the basis (11) that have strictly higher degree than the degree of the homogeneous element (12). Consider the multiplication map

–  –  –

Using the induction on the total degree of the Cartan part, it is easy to check that this map is surjective. At the same time, we have the induced map

–  –  –

For x ∈ B and y ∈ B, by taking the unique element from (11) of minimal degree which appears in the expression of ϕ(x, y) with a non-zero coefficient, induces a bijection from B × B to B.

This implies that all elements in BB are linearly independent and hence B is a basis of U (g) as a free right R-module.

Proposition 10. For every finite-dimensional R-module V the induced g-module M (V ) :=

U (g) IndR V has finite length.

Proof. It is enough to prove the claim for V = Vm,χ, where m is a maximal ideal in Z(g) and χ : n+ /[n+, n+ ] → C. In this case the statement follows from [38, Theorem 2.8].

Corollary 4. Every object in W has finite length.

Proof. Each M ∈ W is generated, as a g-module, by some finite-dimensional R-submodule V. By adjunction, M is thus a quotient of M (V ) and hence the claim follows from Proposition 10.

4.5 The main result

Our main result in this section is the following:

Theorem 2. The category W is extension full in U -Mod.

Proof. We start by proving extension fullness of the category W which is defined as the full subcategory of U -Mod consisting of all modules which are locally R-finite. The difference between W and W is that we drop the condition of being finitely generated.

We apply Proposition 1 for A = U -Mod, B = W and B0 being the full subcategory of B consisting of all U -modules isomorphic to M (V ) for some V ∈ R-lfmod.

Lemma 3. Let Q ∈ W and V ∈ R-lfmod.

Then ιd (V ),Q is an isomorphism for every d ∈ Z≥0.

M Proof. The image of the functor

–  –  –

belongs to R-lfmod. Therefore, to prove our lemma, it is enough to show that for every d ∈ Z≥0, any N ∈ R-lfmod and any Q ∈ W we have the isomorphisms

–  –  –

Isomorphisms (13) and (15) follow from adjunction lemma (Proposition 4) which applies thanks to Proposition 9, while isomorphism (14) is Proposition 3. The claim follows.

Lemma 3 and Proposition 1 thus imply that W is extension full in U -Mod. Hence, to complete the proof of Theorem 2, it remains to note that, by Proposition 3, the category W is extension full in W.

The global dimension of W 4.6 Corollary 5. We have gl.dim(W ) = gl.dim(U -Mod) = dim g.

Proof. Mutatis mutandis Corollary 2.

Remark 2. The category W has a decomposition

–  –  –

where Wm,χ is the full subcategory consisting of all modules which restrict to R-lfmodm,χ, see [3, Theorem 9]. Corollary 5 says that one of these blocks, namely the one corresponding to the trivial central character and trivial χ has global dimension dim g. This particular block contains many simple objects. Most of the blocks contain only one simple object and are expected to have smaller global dimension. Note also that thick category O is a Serre subcategory of W.

Acknowledgements KC is a Postdoctoral Fellow of the Research Foundation – Flanders (FWO). VM is partially supported by the Swedish Research Council. A substantial part of this work was done during the visit of the authors to the CRM Thematic Semester “New Directions in Lie Theory” in Montreal. We thank CRM for hospitality and partial support. We thank the referees for helpful comments.

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