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Symmetry, Integrability and Geometry: Methods and Applications SIGMA 11 (2015), 016, 17 pages

Extension Fullness of the Categories

of Gelfand–Zeitlin and Whittaker Modules

† ‡

Kevin COULEMBIER and Volodymyr MAZORCHUK

† Department of Mathematical Analysis, Ghent University, Krijgslaan 281, 9000 Gent, Belgium

E-mail: coulembier@cage.ugent.be

URL: http://cage.ugent.be/~coulembier/

‡ Department of Mathematics, Uppsala University, Box 480, SE-751 06, Uppsala, Sweden E-mail: mazor@math.uu.se URL: http://www2.math.uu.se/~mazor/ Received September 25, 2014, in final form February 20, 2015; Published online February 24, 2015 http://dx.doi.org/10.3842/SIGMA.2015.016 Abstract. We prove that the categories of Gelfand–Zeitlin modules of g = gln and Whittaker modules associated with a semi-simple complex finite-dimensional algebra g are extension full in the category of all g-modules. This is used to estimate and in some cases determine the global dimension of blocks of the categories of Gelfand–Zeitlin and Whittaker modules.

Key words: extension fullness; Gelfand–Zeitlin modules; Whittaker modules; Yoneda extensions; homological dimension 2010 Mathematics Subject Classiﬁcation: 16E30; 17B10 1 Introduction Homological invariants are useful technical tools in modern representation theory. As classification of all modules of a given (Lie) algebra is a wild problem in almost all non-trivial and interesting cases (see e.g. [5, 11, 19]), the usual “reasonable” setup for the study of representations of a given (Lie) algebra assumes some fixed subcategory of the category of all modules.

Therefore, the problem to compare homological invariants for a given category and some of its subcategories is natural and important.

Given an Abelian category A and an Abelian subcategory B of A such that the natural inclusion B → A is exact, we say that B is extension full in A provided that the natural inclusion induces isomorphisms Extd (M, N ) ∼ Extd (M, N ) for all M, N ∈ B and all d ≥ 0, see = B A Subsection 2.2 for details (we write X ∈ C when M is an object of some category C). Extension fullness is a useful notion which allows one to freely transfer homological information between categories A and B. Recently this concept of extension fullness has also been studied by Herman in [25], where it appears under the name entirely extension closed.

Motivated by the so-called Alexandru conjecture from [17, 22] (a part of which asserts extension fullness of certain categories in Lie theory), in our previous paper [7] we proved that the category O associated with a semi-simple complex finite-dimensional Liealgebra g is extension full in the category of all weight g-modules, and that the thick version of O is extension full in the category of all g-modules. As a bonus, we determined the global dimension of the thick category O as well as projective dimensions of its simple objects.

Although category O is probably the most studied category of g-modules, there are several other natural and well-studied categories which have rather different flavor. One of them is the category G Z of so-called Gelfand–Zeitlin1 modules, introduced in [13] for the algebra sl3 (C), in [14] for the algebra gln (C) and in [34] for orthogonal Lie algebras. The category G Z can be seen as a generalization of O in the sense that it contains both O and the thick version of O. The study of Gelfand–Zeitlin modules attracted considerable attention, see e.g. [12, 18, 20, 21, 26, 28, 33, 35, 36, 37, 44, 45, 46] and references therein. As far as we know, simple generic Gelfand–Zeitlin modules give the richest known family of simple gln (C)-modules. This family depends on n(n+1) generic parameters. The first main result of this paper is the following

**statement proved in Section 3 (we refer to Sections 2 and 3 for more details):**

Theorem A. The category G Z is extension full in the category of all gln -modules.

The added difficulty of the category G Z in comparison with thick category O in [7] is that G Z is not a Serre subcategory generated by a well-known category which has enough projective objects (contrary to the relation between category O and thick category O). Therefore to prove Theorem A we have to modify and strengthen the

**Abstract**

results on extension fullness in [7]. Our arguments also heavily use some properties of Gelfand–Zeitlin modules established by Futorny and Ovsienko in [21].

Another big class of g-modules, where now g is an arbitrary semi-simple Lie algebra with a fixed triangular decomposition g = n− ⊕ h ⊕ n+, is the class of so-called Whittaker modules introduced by Kostant in [29]. Simple Whittaker modules are simple g-modules on which the algebra U (n+ ) acts locally finitely, see also [3] for a general Whittaker setup. These modules were studied in [27, 38, 39, 40, 48] in the classical setup. Generalizations of these modules for (infinite-dimensional) Lie algebras and some related algebras attracted a lot of attention recently, see [2, 3, 4, 6, 24, 41, 42, 43] and references therein.

We define the category W of Whittaker modules of finite length and for this category we

**prove the next statement, which is our second main result (we refer to Section 4 for more details):**

Theorem B. The category W is extension full in the category of all g-modules.

Using adjunction, the study of W reduces to the study of locally finite modules over a certain Noetherian algebra. An added difficulty compared to the case of the category G Z is that a module in W does not decompose into a direct sum of finite-dimensional U (n+ )-modules despite the fact that the action of U (n+ ) is locally finite. Moreover, a module in W is not always finitely generated over U (n+ ). To be able to prove extension fullness, we crucially depend on a result of Donkin and Dahlberg from the 80’s (see [10] and [8]) asserting that essential extensions of locally finite modules over solvable finite-dimensional Lie algebras are locally finite.

An advantage of the result in Theorems A and B is that it allows to replace the calculation of extensions in a category without projective objects by the calculation of extensions in a category with enough projective objects. In particular, as a consequence we obtain that the global dimension of the categories G Z and W equals dim g. As both category O and its thick version are full subcategories in G Z or W (thick category O is even a Serre subcategory, as defined in [49, Subsection 10.3.2]), combining Theorems A and B with the results of [7] gives, in particular, a lower bound on the projective dimension of simple highest weight and Verma modules in the categories G Z and W.

The paper is organized as follows: in Section 2 we prove some preliminary homological algebra statements, Section 3 deals with the case of Gelfand–Zeitlin modules and Section 4 is devoted to the case of Whittaker modules.

The surname Zeitlin is spelled Ce˘ itlin in Russian. It appeared in different transliterations in Latin script, in particular, as Cetlin, Zetlin, Tzetlin and Tsetlin. However, it seems that the origin of this surname is the German word “Zeit” which justifies our present version.

Extension Fullness of the Categories of Gelfand–Zeitlin and Whittaker Modules 3

The set Extd (M, N ) has the natural structure of an Abelian group via the Baer sum. If A A is k-linear for some field k, then Extd (M, N ) has the structure of a k-vector space. We refer A to [49, Section 3.4] for further information and details.

For M ∈ A, the projective dimension proj.dim(M ) of M is defined as the maximal d such that

**there is N ∈ A with Extd (M, N ) = 0. If such maximal d does not exist, then proj.dim(M ) :=**

A ∞. Dually one defines the injective dimension inj.dim(N ) for N ∈ A. The global dimension gl.dim(A) ∈ Z≥0 ∪ {∞} is the supremum of projective dimensions taken over all objects in A.

The global dimension coincides with the supremum of injective dimensions taken over all objects in A (see [51, Lemma 5.11.11]).

2.2 Extension full subcategories Let A be an Abelian category and B a full Abelian subcategory of A in the sense that the Abelian structure of B is inherited from A. In particular, the natural inclusion functor ι : B → A is exact. Then, for every M, N ∈ B and every d ∈ Z≥0, the functor ι induces homomorphisms

** ιd d d M,N : ExtB (M, N ) → ExtA (M, N )**

of Abelian groups. In general, these homomorphisms ιd M,N are neither injective nor surjective.

We say that B is extension full in A provided that ιd M,N are bijective for all d ∈ Z≥0 and for all M, N ∈ B. We refer to [7, Section 2] for details.

Let 0 → K → M → N → 0 be a short exact sequence in B. Then, for Q ∈ B, application of HomB (Q, − ) and HomA (Q, − ) to this short exact sequence produces the usual long exact sequences in homology for the categories B and A, respectively. Moreover, the homomorphisms ιd − give rise to a homomorphism between these long exact sequences. A similar stateQ, ment is true for HomB (−, Q) and HomA (−, Q).

2.3 Checking extension fullness In this section we formulate and prove three propositions which will be useful for our study of extension fullness later in the paper. The following statement is a modification of [7, Lemma 4] which also allows for a somewhat stronger formulation.

Proposition 1. Let A be an Abelian category and B a Serre subcategory of A. Assume that B

**contains a full subcategory B0 with the following properties:**

(a) Every object in B is a quotient of an object in B0.

4 K. Coulembier and V. Mazorchuk

0→K→M →Q→0 with M ∈ B0 and K ∈ B, which exists by condition (a). Applying HomB (−, N ) and HomA (−, N )

**gives for each d the following commutative diagram with exact rows:**

The proof of the following proposition is dual to that of [7, Corollary 5] concerning acyclicness (see [49, Subsection 2.4.3]) of projective objects.

Proposition 2. Let A be an Abelian category and B a full Abelian subcategory of A and assume that they both have enough injective objects. If every injective object in B is acyclic for the functor HomA (K, − ) for any K ∈ B, then B is extension full in A.

of functors from the category B to the category Sets.

By assumption, the exact functor ι maps injective objects in B to injective objects in A.

Injective objects in A are acyclic for the functor HomA (K, − ), that is for all such objects I we have Extd (K, I) = 0 for all d 0. The classical Grothendieck spectral sequence, see [49, A Section 5.8], therefore implies the statement.

Extension Fullness of the Categories of Gelfand–Zeitlin and Whittaker Modules 5 Let us fix the following notation: for an associative algebra A over a field k denote by A-Mod the category of all A-modules. We also denote by A-mod the full subcategory of A-Mod consisting of all finitely generated modules. We denote by A-lfmod the full subcategory of A-Mod consisting of all modules on which the action of A is locally finite. Finally, we denote by A-fmod the full subcategory of A-mod consisting of all finite-dimensional modules.

Proposition 3. Consider an associative algebra A, a full Abelian subcategory A in A-Mod, and

**a full Abelian subcategory B of A. Assume that these data satisfy the following conditions:**

For i = 1, 2,..., d, let Qi denote the submodule of Zi generated by the images of Yi and Xi.

Since B is a Serre subcategory of A-Mod, we have that Qi belongs to B. Then diagram (1) restricts to the commutative diagram

with exact rows and such that the second row is in B. This means that the natural map (2) is surjective and hence bijective, completing the proof.

**As an immediate corollary from Proposition 3 we obtain:**

Corollary 1. For an associative algebra A we have that (i) A-fmod is extension full in A-lfmod, (ii) A-mod is extension full in A-Mod provided that A is Noetherian.

2.4 Adjunction lemma The following statement is standard when dealing with categories with enough projective or injective objects. We failed to find it in the literature in the generality we need, so we provide a proof without the use of projective or injective objects.

Proposition 4 (adjunction lemma). Let A and B be two Abelian categories and (F, G) an adjoint pair of exact functors F : A → B and G : B → A. Then for every d ∈ Z≥0, N ∈ A and M ∈ B there are isomorphisms

with exact rows. Here the homomorphism from the first to the second row is given by the natural embedding K → FG and the third row is just the corresponding cokernel with the morphism from the second to the third row being the canonical projection. In particular, M := FG(M )/K(M ).

The homomorphism from M to M is the natural inclusion (coming from the definition of K) and, finally, X is defined as the push-out and the map to F(X2 ) is given by the universal property of push-outs. Functoriality of the construction yields a group homomorphism

Y is defined as the pullback and the map from G(Yd−1 ) is given by the universal property of pullbacks. Functoriality of the construction yields a group homomorphism Ψ : Extd (F(N ), M ) → Extd (N, G(M )).

B A

Here the second row is obtained from the first one by applying GF and the homomorphism from the first to the second row is given by adjunction IdA → GF. The second and the third rows and the homomorphism between them are given by applying the exact functor G to the two middle rows of (3). Note that, by construction, application of G identifies the two last rows of (3), that is G(M ) ∼ G(M ) and G(X ) ∼ G(X ) (this follows from surjectivity of the = = natural transformation GFG → G given by adjunction). Consequently, from the adjunction identities we have that the composition of the maps in the first column is the identity on G(M ).

Finally, the last row and the homomorphism to the third row are given by the definition of Ψ.