Tilting Modules in Truncated Categories

We begin the study of a tilting theory in certain truncated categories of modules $\mathcal G(\Gamma)$ for the current Lie algebra associated to a finite-dimensional complex simple Lie algebra, where $\Gamma = P^+ \times J$, $J$ is an interval in $\mathbb Z$, and $P^+$ is the set of dominant integral weights of the simple Lie algebra. We use this to put a tilting theory on the category $\mathcal G(\Gamma')$ where $\Gamma' = P' \times J$, where $P'\subseteq P^+$ is saturated. Under certain natural conditions on $\Gamma'$, we note that $\mathcal G(\Gamma')$ admits full tilting modules.


Introduction
Associated to any finite-dimensional complex simple Lie algebra g is its current algebra g [t]. The current algebra is just the Lie algebra of polynomial maps from C → g and can be identified with the space g ⊗ C[t] with the obvious commutator. The study of the representation theory of current algebras was largely motivated by its relationship to the representation theory of affine and quantum affine algebras associated to g. However, it is also now of independent interest since the current algebra has connections with problems arising in mathematical physics, for instance the X = M conjectures, see [1,17,25]. Also, the current algebra, and many of its modules, admits a natural grading by the integers, and this grading gives rise to interesting combinatorics. For example, [22] relates certain graded characters to the Poincaré polynomials of quiver varieties.
Let P + be the set of dominant integral weights of g, Λ = P + × Z, and G the category of Z-graded modules for g [t] with the restriction that the graded pieces are finite-dimensional. Also, let G be the full subcategory of G consisting of modules whose grades are bounded above. Then Λ indexes the simple modules in G. In this paper we are interested in studying Serre subcategories G(Γ) where Γ ⊂ Λ is of the form P × J where J ⊂ Z is a (possibly infinite) interval and P ⊂ P + is closed with respect to a natural partial order. In particular, we study the tilting theories in these categories. This generalized the work of [3], where Γ was taken to be all of Λ.
The category G(Γ) contains the projective cover and injective envelope of its simple objects. Given a partial order on the set Γ, we can define the standard and costandard objects, as in [18]. The majority of the paper is concerned with a particular order, in which case the standard objects ∆(λ, r)(Γ) are quotients of the finite-dimensional local Weyl modules, and the costandard objects ∇(λ, r)(Γ) are submodules of (appropriately defined) duals of the infinitedimensional global Weyl modules. We recall (see, for example, [23]) that a module T is called This paper is a contribution to the Special Issue on New Directions in Lie Theory. The full collection is available at http://www.emis.de/journals/SIGMA/LieTheory2014.html arXiv:1307.3307v7 [math.RT] 2 May 2014 tilting if T admits a filtration by standard modules and a filtration by costandard modules. In our case, both sets of objects have been extensively studied (see [12,19,20,24] for the local Weyl modules, and [6,15], for the global Weyl modules). Both families of modules live in a subcategory G bdd (Γ) consisting of objects whose weights are in a finite union of cones (as in O) and whose grades are bounded above. The main goal of this paper is to construct another family of modules indexed by Γ and which are in G bdd (Γ). These modules are denoted by T (λ, r)(Γ), and admit an infinite filtration whose quotients are of the form ∆(µ, s)(Γ), for (µ, s) ∈ Γ. They also satisfy the homological property that Ext 1 G (∆(µ, s)(Γ), T (λ, r)(Γ)) = 0 for all (µ, s) ∈ Γ. We use the following theorem to prove that this homological property is equivalent to having a ∇(Γ)-filtration, proving that the T (λ, r)(Γ) are tilting. The theorem was proved in [4,2], and [10] for sl 2 [t], sl n+1 [t] and general g[t] respectively. Theorem 1.1. Let P (λ, r) denote the projective cover of the simple module V (λ, r). Then P (λ, r) admits a filtration by global Weyl modules, and we have an equality of filtration multiplicities [P (λ, r) : W (µ, s)] = [∆(µ, r) : V (λ, s)], where ∆(µ, r) is the local Weyl module.
The following is the main result of this paper. 2. Moreover, any indecomposable tilting module in G bdd (Γ) is isomorphic to T (λ, r)(Γ) for some (λ, r) ∈ Γ, and any tilting module in G bdd (Γ) is isomorphic to a direct sum of indecomposable tilting modules.
The majority of the paper is devoted to the case where Γ = P + × J. It is easy to see from the construction that the module T (λ, r)(Γ) has its weights bounded above by λ. It follows that if we let P ⊂ P + be saturated (downwardly closed with respect to the normal partial order on weights), and set Γ = P × J, then T (λ, r)(Γ ) = T (λ, r)(Γ).
Another purpose of this paper is the following. In [3], the tilting modules T (λ, r) are constructed for all (λ, r) ∈ Λ. It is normal to then consider the module T = (λ,r)∈Λ T (λ, r), the algebra A = End T , and use several functors to find equivalences of categories. However, it is not hard to see that if T is defined in this way, then T fails to have finite-dimensional graded components, and hence T / ∈ Ob G. One of the purposes of this paper is to find Serre subcategories with index sets Γ such that T (Γ) = (λ,r)∈Γ T (λ, r)(Γ) ∈ Ob G(Γ). It is not hard to see that (except for the degenerate case where Γ = {0} × J) a necessary and sufficient pair of conditions on Γ is that P be finite and J have an upper bound. It is natural to study the algebra End T (Γ) in the case that T (Γ) ∈ G(Γ), and this will be pursued elsewhere. We also note that in the case that Γ is finite then End T (Γ) is a finite-dimensional associative algebra.
We end the paper by considering other partial orders which can be used on Γ ⊂ Λ. In particular, we consider partial orders induced by the so-called covering relations. One tends to get trivial tilting theories in these cases (one of the standard-costandard modules is simple, and the other is projective or injective), but the partial orders are natural for other reasons, and we include their study for completeness. One of the reasons to study these other subcategories is that one can obtain directed categories as in [7] (in the sense of [16]).
The paper is organized as follows. In Section 2 we establish notation and recall some basic results on the finite-dimensional representations of a finite-dimensional simple Lie algebra. In Section 3 we introduce several important categories of modules for the current algebra. We also introduce some important objects, including the local and global Weyl modules. In Section 4 we state the main results of the paper and establish some homological results. Section 5 is devoted to constructing the modules T (λ, r)(Γ) and establishing their properties. Finally, in Section 6, we consider the tilting theories which arise when considering partial orders on Λ which are induced by covering relations.
We also provide for the reader's convenience a brief index of the notation which is used repeatedly in this paper.

Simple Lie algebras and current algebras
We fix g, a complex simple finite-dimensional Lie algebra, and let h ⊂ g be a fixed Cartan subalgebra. Denote by {α i : i ∈ I} a set of simple roots of g with respect to the Cartan subalgebra h, where I = {1, . . . , dim h}. Let R ⊂ h * be the corresponding set of roots, R + the positive roots, P + the dominant integral weights, and Q + the positive root lattice. By θ we denote the highest root. Given λ, µ ∈ h * , we say that λ ≥ µ if and only if λ − µ ∈ Q + . The Weyl group of g is the subgroup W ⊂ Aut(h * ) generated by the simple reflections s i , and we let w • denote the unique longest element of W . For α ∈ R we write g α for the corresponding root space. Then the subspaces n ± = α∈R + g ±α , form Lie subalgebras of g. We fix a Chevalley basis {x ± α , h i | α ∈ R + , i ∈ I} of g, and for each α ∈ R + we set h α = [x α , x −α ]. Note that h α i = h i , i ∈ I, and we let ω i = h * i ∈ P + . For any Lie algebra a we can construct another Lie algebra a[t] = a ⊗ C[t], with bracket given by [x⊗t r , y ⊗t s ] = [x, y]⊗t r+s , which is the current algebra associated to a. It is well-known that the universal enveloping algebra U(a) is a Hopf algebra. In particular, it is equipped with a comultiplication defined by sending x → x⊗1+1⊗x for x ∈ a, and extending this assignment to be a homomorphism. In the case where a = b[t] or b[t] + , the comultiplication is a homomorphism of graded associative algebras. We note that if [a, a] = a (which holds for our Lie algebra g), then as a graded associative algebra, a and a ⊗ t generate U(a[t]).

Finite-dimensional modules
The first category we consider is F(g) the category of finite-dimensional modules for g with morphisms g-module homomorphisms. It is well known that this is a semi-simple category, and that the simple objects are parametrized by λ ∈ P + . Letting V (λ) denote the simple module associated to λ, it is generated by a vector v λ ∈ V (λ) satisfying the defining relations for all h ∈ h, i ∈ I. This category admits a duality, which on simple modules is given by For any such V , we define the subset wt(V ) = {λ ∈ h * : V λ = 0} and define the character of V to be the sum ch V = dim V λ r λ . The following results are standard: Lemma 2.1. Let V ∈ F(g) and λ ∈ P + . Then,

The main category and its subcategories
In this section we introduce the main categories of study, and present several properties and functors between them. We will also introduce several families of modules which will play important roles. Most of these categories and objects have been studied elsewhere (see [2,3,7]).

The main category
We denote by G the category of Z-graded g[t] modules such that the graded components are finite-dimensional and where morphisms are degree zero maps of g[t]-modules.
we see that V [r] is a finite-dimensional g module, while z ⊗ t k .V [r] ⊂ V [r + k] for all z ∈ g, k ∈ Z ≥0 , and r ∈ Z. For M ∈ G, its graded character is the sum (formal, and possibly infinite) For V ∈ F(g) we make V an object in G, which we shall call ev V , in the following way. Set ev V [0] = V and ev V [r] = 0 for all r = 0. Then necessarily we have z ⊗ t k .v = δ k,0 z.v for z ∈ g, k ∈ Z + , v ∈ ev V . It is not hard to see that this defines a covariant functor ev : F(g) → G. Further, for s ∈ Z let τ s : G → G be the grade shift functor given by For (λ, r) ∈ P + × Z set V (λ, r) := τ r (ev(V (λ))) and v λ,r := τ r (v λ ).
Proposition 3.1. The isomorphism classes of simple objects in G are parametrized by pairs (λ, r) and we have Moreover, if V ∈ Ob G satisfies V = V [n] for some n ∈ Z, then V is semi-simple.
The category G admits a duality, where given M we define M * ∈ Ob G to be the module given by We note that M * * ∼ = M and that ch gr Denote by Λ = P + × Z and equip Λ with the lexicographic partial order ≤, i.e.

Some bounded subcategories of the main category
We let G ≤s be the full subcategory of G whose objects V satisfy V [r] = 0 for all r > s. Clearly G ≤s is a full subcategory of G ≤r for all s < r ∈ Z. Define G to be the full subcategory of G whose objects consist of those objects V satisfying V ∈ Ob G ≤s for some s ∈ Z. Finally, let G bdd be the full subcategory of G consisting of objects M satisfying the following condition: and a corresponding quotient V ≤s = V /V >s . Then it is clear that V ≤s ∈ Ob G ≤s , and indeed this is the maximal quotient of V in G ≤s . Any f ∈ Hom G (V, W ) naturally induces a morphism f ≤s ∈ Hom G ≤s (V ≤s , W ≤s ). The following is proved in [7].
V, W ∈ Ob G, define a full, exact and essentially surjective functor from G to G ≤r .

Projective and injective objects in the main category
and Clearly these are an infinite-dimensional Z-graded g[t]-module. Using the PBW theorem we have an isomorphism of graded vector spaces U(g[t]) ∼ = U(g[t] + ) ⊗ U(g), and hence we get This shows that P (λ, r) ∈ Ob G and also that P (λ, r) Proposition 3.3. Let (λ, r) ∈ Λ, and s ≥ r.
1. P (λ, r) is generated as a g[t]-module by p λ,r with defining relations for all h ∈ h, i ∈ I. Hence, P (λ, r) is the projective cover in the category G of its unique simple quotient V (λ, r).
2. The modules P (λ, r) ≤s are projective in G ≤s .

4.
Any injective object of G is also injective in G.

Local and global Weyl modules
The next two families of modules in G bdd we need are the local and global Weyl modules which were originally defined in [15]. For the purposes of this paper, we shall denote the local Weyl modules by ∆(λ, r), (λ, r) ∈ P + × Z. Thus, ∆(λ, r) is generated as a g[t]-module by an element w λ,r with relations: here i ∈ I, h ∈ h and s ∈ Z + . Next, let W (λ, r) be the global Weyl modules, which is g[t]-module generated as a g[t]-module by an element w λ,r with relations: where i ∈ I and h ∈ h. Clearly the module ∆(λ, r) is a quotient of W (λ, r) and moreover V (λ, r) is the unique irreducible quotient of W (λ, r). It is known (see [6] or [15]) that W (0, r) ∼ = C and that, if λ = 0, the modules W (λ, r) are infinite-dimensional and satisfy wt We note that ∆(λ, r) (resp. ∇(λ, r)) is the maximal quotient of P (λ, r) (resp. submodule of I(λ, r)) satisfying Hence these are the standard (resp. costandard) modules in G associated to (λ, r).

Truncated subcategories
In this section, we recall the definition of certain Serre subcategories of G.
Given Γ ⊂ Λ, let G(Γ) be the full subcategory of G consisting of all M such that The subcategories G(Γ) and G bdd (Γ) are defined in the obvious way. Observe that if (λ, r) ∈ Γ, then V (λ, r) ∈ G(Γ), and we have the following trivial result.

A specif ic truncation
We now focus on Γ of the form Γ = P + × J, where J is an interval in Z with one of the forms (−∞, n], [m, n], [m, ∞) or Z, where n, m ∈ Z. We set a = inf J and b = sup J. Throughout this section, we assume that (λ, r) ∈ Γ. Let P (λ, r)(Γ) be the maximal quotient of P (λ, r) which is an object of G(Γ) and let I(λ, r)(Γ) be the maximal submodule of I(λ, r) which is an object of G(Γ). These are the indecomposable projective and injective modules associated to the simple module V (λ, r) ∈ G(Γ).
For an object M ∈ G, let M Γ be the subquotient Remark 3.6.
Clearly M Γ ∈ G(Γ), and because morphisms are graded, this assignment defines a functor from G to G(Γ). It follows from Lemma 3.2 that Γ is exact.
If we define another subset Γ = P + × {−J}, then it follows from the definition of the graded duality that if M ∈ Ob G(Γ) then M * ∈ Ob G(Γ ).
The following proposition summarizes the properties of ∆(λ, r)(Γ) which are necessary for this paper. They can easily be derived from the properties of the functor Γ. 1. The module ∆(λ, r)(Γ) is generated as a g[t]-module by an element w λ,r with relations: for all i ∈ I, h ∈ h and s ∈ Z + , where if b = ∞, then the final relation is empty relation.

The truncated global Weyl modules
Here we collect the results on W (λ, r)(Γ) which we will need for this paper.
1. The module W (λ, r)(Γ) is generated as a g[t]-module by an element w λ,r with relations: where, if b = ∞, then the final relation is empty relation. Here i ∈ I and h ∈ h.

The costandard modules
The following proposition summarizes the main results on ∇(λ, r)(Γ) that are needed for this paper. All but the final result can be found by considering the properties of the functor Γ and the paper [3]. Proof . We prove the final item. As a vector space we have Since W (−ω 0 λ, −r)(Γ ) is a quotient of W (−ω 0 λ, −r), its dual must be a submodule of ∇(λ, r). By the definition of the graded dual, we see that, as a vector space, Hence, as vector spaces, we see that ∇(λ, r)(Γ) ∼ = W (−ω 0 λ, −r)(Γ ) * . Now, the fact that W (−ω 0 λ, −r)(Γ ) * is a submodule completes the proof. 4 The main theorem and some homological results Definition 4.1. We say that M ∈ Ob G(Γ) admits a ∆(Γ) (resp. ∇(Γ))-filtration if there exists an increasing family of submodules 0 ⊂ M 0 ⊂ M 1 ⊂ · · · with M = k M k , such that for some choice of m k (λ, r) ∈ Z + . We do not require k ≥ 0, then we say that M admits a finite ∆(Γ) (resp. ∇(Γ))-filtration. Because our modules have finite-dimensional graded components, we can conclude that the multiplicity of a fixed ∆(λ, r)(Γ) (resp. ∇(λ, r)(Γ)) in a ∆(Γ)-filtration (resp. ∇(Γ)-filtration) must be finite, and The main goal of this paper is to understand tilting modules in G bdd (Γ). (The case where J = Z was studied in [3].) In the case of algebraic groups (see [18,23]) a crucial necessary result is to give a cohomological characterization of modules admitting a ∇(Γ)-filtration. The analogous result in our situation is to prove the following statement: It is not hard to see that the forward implication is true. The converse statement however requires one to prove that any object of G bdd (Γ) can be embedded in a module which admits a ∇(Γ)-filtration. This in turn requires Theorem 5.8. Summarizing, the first main result that we shall prove in this paper is:  1. Given (λ, r) ∈ Γ, there exists an indecomposable module T (λ, r)(Γ) ∈ Ob G bdd (Γ) which admits a ∆(Γ)-filtration and a ∇(Γ)-filtration. Further, and T (λ, r)(Γ) ∼ = T (µ, s)(Γ) if and only if (λ, r) = (µ, s).

2.
Moreover, any indecomposable tilting module in G bdd (Γ) is isomorphic to T (λ, r)(Γ) for some (λ, r) ∈ Γ, and any tilting module in G bdd (Γ) is isomorphic to a direct sum of indecomposable tilting modules.
5 Proof of Proposition 4.2

Initial homological results
We begin by proving the implication (1) =⇒ (2) from Proposition 4.2. In order to do this, we first establish some homological properties of the standard and costandard modules which will be used throughout the paper.
The proofs for parts (3) and (4) are similar to that for part (1) and are omitted.
The proof of the following lemma is standard (see, for example, [3]). where λ i > λ j implies i > j. In particular if µ is maximal such that M µ = 0, then there exists s ∈ Z and a surjective map M → ∇(µ, s)(Γ) such that the kernel has a ∇(Γ)-filtration.
We can now prove the implication (1) =⇒ (2) from Proposition 4.2.  By assumption we have Ext 1 G(Γ) (M k+1 /M k , N ) = 0. Let U k ⊂ U be the pre-image of M k , which contains N because 0 ∈ M k . Note that U k+1 /U k ∼ = M k+1 /M k . Now, consider the short exact sequence 0 → N → U k → M k → 0. This sequence defines an element of Ext 1 G(Γ) (M k , N ). Since M k has a finite ∆(Γ)-filtration it follows that Ext 1 G(Γ) (M k , N ) = 0. Hence the sequence splits and we have a retraction ϕ k : U k → N . We want to prove that ϕ k+1 : U k+1 → N can be chosen to extend ϕ k . For this, applying N ) → 0, which shows that we can choose ϕ k+1 to lift ϕ k . Now defining ϕ : U → N by ϕ(u) = ϕ k (u), for all u ∈ U k , we have the desired splitting of the original short exact sequence.

Towards understanding extensions between the standard and costandard modules
Together with Proposition 5.1 and taking N = ∇(λ, r)(Γ) in the proposition above, we now have:

A natural embedding
In this section we show that every M ∈ Ob G bdd (Γ) embeds into an injective module I(M ) ∈ Ob G(Γ). Let soc M ⊂ M be the maximal semi-simple submodule of M .

o-canonical f iltration
In this section we shall establish a finite filtration on modules M ∈ Ob G bdd (Γ) where the successive quotients embed into direct sums of ∇(Γ). We then use the filtration to establish lower and upper bounds on the graded character of M . We use the character bounds to prove Proposition 4.2. Now fix an ordering of P + = {λ 0 , λ 1 , . . . } such that λ r > λ s implies that r > s. For M ∈ Ob G(Γ) we set M s ⊂ M as the maximal submodule whose weights lie in {conv W λ r | r ≤ s}. Evidently M s−1 ⊂ M s . We call this the o-canonical filtration, because it depends on the order. This is a finite filtration for M ∈ Ob G bdd (Γ) and we set k(M ) to be minimal such that

A homological characterization of costandard modules
Note that even if N / ∈ Ob G bdd , we can still define the submodules N s , and by definition N s ∈ Ob G bdd . The following result, or more precisely a dual statement about projective modules and global Weyl filtrations, was proved for g = sl 2 in [4], for g = sl n+1 in [2] and for general g in [10]. In particular, we note that the argument in [2, Section 5.5] works in general. We combine this with equation (5.2) and the linear independence of the graded characters of the ∇(λ, r)(Γ) to see that [I(λ, r) p : ∇(µ, s)] = dim Hom G(Γ) (∆(λ s , r), I(λ, r) p ).
As a consequence of the theorem and the exactness of the functor Γ, we conclude that I(λ, r) p (Γ) has a ∇(Γ)-filtration. It is easy to see that I(λ, r) p (Γ) ∈ G bdd (Γ).
For M ∈ G bdd (Γ), let p be minimal such that M p = M . Then it is clear that we can refine the embedding from Lemma 5.6 to M → I(λ, r) p (Γ). We can conclude the following: We now complete the proof of Proposition 4.2 following the argument in [3]. Let M ∈ G bdd (Γ) and assume that Ext where the final equality is from the exactness of (5.3). We now get that the character bound in (5.2) is an equality for M , and, hence, that M has a ∇(Γ)-filtration. Finally, we can prove the following.
Proposition 5.10. The following are equivalent for a module M ∈ Ob G bdd (Γ)

Extensions between simple modules
Our final result before constructing the tilting modules T (λ, r)(Γ) shows that the space of extensions between standard modules is always finite-dimensional. The proof is analogous to the proof in [3].
6 Construction of tilting modules 6.1 Def ining a subset which can be appropriately enumerated In this section we construct a family of indecomposable modules in the category G bdd (Γ), denoted by {T (λ, r)(Γ) : (λ, r) ∈ Γ}, each of which admits a ∆(Γ)-filtration and satisfies It follows that the modules T (λ, r)(Γ) are tilting and we prove that any tilting module in G bdd (Γ) is a direct sum of copies of T (λ, r)(Γ), (λ, r) ∈ Γ. The construction is a generalization of the one from [3], and the ideas are similar to the ones given in [23]. One of the first difficulties we encounter when trying to construct T (λ, r)(Γ), using the algorithm given in [23], is to find a suitable subset (depending on (λ, r)) of Γ which can be appropriately enumerated. Hence we assume the following result, whose proof we postpone to Section 6.6.
Furthermore, there exists an injection η : Without loss of generality we may assume that η(λ, r) = 0 and the image of η is an interval. We need the following elementary result.

Initial properties of tilting modules
The next result is an analog of Fitting's lemma for the infinite-dimensional modules T (λ, r)(Γ). Proof . Since ψ preserves both weight spaces and graded components it follows that ψ(M s ) ⊂ M s for all s ≥ 0. Moreover, since M s is indecomposable and finite-dimensional it follows from Fitting's lemma that the restriction of ψ to M s , s ≥ 0 is either nilpotent or an isomorphism. If all the restrictions are isomorphisms then since T (λ, r)(Γ) is the union of M s , s ≥ 0, it follows that ψ is an isomorphism. On the other hand, if the restriction of ψ to some M s is nilpotent, then the restriction of ψ to all M , ≥ 0 is nilpotent which proves that ψ is locally nilpotent.
In the rest of the section we shall complete the proof of the main theorem by showing that any indecomposable tilting module is isomorphic to some T (λ, r)(Γ) and that any tilting module in G bdd (Γ) is isomorphic to a direct sum of indecomposable tilting modules. Let T ∈ G bdd (Γ) be a fixed tilting module. Then we have Ext 1 G (T, ∇(λ, r)(Γ)) = 0 = Ext 1 G (∆(λ, r)(Γ), T ), (λ, r) ∈ Γ, (6.3) where the first equality is due to Corollary 5.5.

Completing the proof of the main theorem
The preceding lemma illustrates one of the difficulties we face in our situation. Namely, we cannot directly conclude that T 1 has a ∆(Γ)-filtration from the vanishing Ext-condition by using the dual of Proposition 4.2. However, we have the following, whose proof is given in [3]. Proposition 6.7. Suppose that N ∈ G bdd (Γ) has a ∇(Γ)-filtration and satisfies The following is immediate. Note that this also shows that our construction of the indecomposable tilting modules is independent of the choice of enumeration of P + , the set S(λ, r) and η. Corollary 6.8. Any indecomposable tilting module is isomorphic to T (λ,r)(Γ) for some (λ,r) ∈ Γ. Further if T ∈ Ob G bdd (Γ) is tilting there exists (λ, r) ∈ Γ such that T (λ, r)(Γ) is isomorphic to a direct summand of T .
Proof . Since T and T (λ, r)(Γ) are tilting they satisfy (6.3) and the corollary follows.
We can now prove the following theorem. Theorem 6.9. Let T ∈ Ob G bdd (Γ). The following are equivalent.
6.6 Proof of Proposition 6.1 We construct here the set S(λ, r) and the enumeration η.
The set S(λ, r). Recall that Γ = P + × J, and that a = inf J and b = sup J. Using the enumeration of P + , let λ = λ k and define integers r k ≤ r k−1 ≤ · · · ≤ r 0 recursively by setting r k = r and Note that because ∆(λ i , r i )(Γ)[p] = 0 for any p > r i−1 , and r i ≤ r j for all j < i, we have ∆(λ i , r i )(Γ)[p] = 0 if p > r j for any j < i. Then, it follows that ∆(λ i , s)(Γ)[p] = 0 for any s ≤ r i < p. (6.4) We set S(λ, r) = {(λ i , s)|i ≤ k, s ≤ r i }, and note that it satisfies conditions (1) and (2) of Proposition 6.1 by construction.
The enumeration η. It remains to define the enumeration η. The case where J = Z is done in [3], and the case where J is a finite or of the form [a, ∞) (in which case S(λ, r) is in fact finite), we use the enumeration defined by the following rules Suppose that i < j and let (µ i , p i ) = η −1 (i) and (µ j , p j ) = η −1 (j). If µ i = µ j , then rule (2) implies that p j < p i , and Proposition 5.1 says that Ext 1 G (∆(µ i , p i )(Γ), ∆µ j , p j )(Γ)) = 0. Otherwise, we have µ j ≤ µ i , and the result again follows by Proposition 5.1.
We are left with the case where J = (−∞, b]. In this case η will in fact be a bijection. Note that it is enough to define a bijective, set theoretic inverse η −1 . We recursively define another set of integers {r i } by setting r k = r k and letting r i = max{r|∆(λ i+1 , r i+1 )[r] = 0}. It is easy to see that r i ≤ r i . If r i < r i , then we must have r i > b, which implies that r i = b. This implies that r j = b for all j < i. We note that ∆(λ j , b)(Γ) = V (λ j , b). Set a s := r s − r s+1 . Proof . We first prove that under these conditions Ext 1 G (∆(λ i , c), ∆(λ s , d)) = 0. Note that we can shift by −d + r s , and so we examine Ext 1 G (∆(λ i , c − d + r s ), ∆(λ s , r s )). According to our hypothesis, we have c−d+r s ≥ r s−1 +1. It follows by the definition of r s−1 that ∆(λ s , r s ) be a pre-image of w λ i , it is clear that m satisfies the defining relations of w λ i . Therefore, the sequence splits. Finally, observe that ∆(λ s , d)(Γ)[c] = 0, again by the definition of the r i and noting that ∆(λ s , d)(Γ) is a quotient of ∆(λ s , d). The same argument as above also shows that Ext 1 G (∆(λ i , c)(Γ), ∆(λ s , d)(Γ)) = 0.
To prove that Ext 1 G (∆(µ i , p i )(Γ), ∆(µ j , p j )(Γ)) = 0 if i < j, we assume that µ i ≤ µ j . If µ i = µ j then this follows from Proposition 5.1. So we assume that µ i < µ j , and lets say that µ j = λ . In this case we must have p i − p j > a −1 and so the result follows by Lemma 6.10.
7 Some dif ferent considerations on truncated categories Throughout this section we discuss some "trivial" tilting theories for the category G(Γ) by considering different type of orders on the set Λ. These categories, equipped with the orders described below, have already appeared in the literature (see [5,7,8,9,11] and references therein).
Notice that for any (µ, s) ∈ Λ the set of (λ, r) ∈ Λ such that (µ, s) covers (λ, r) is finite. Let be the unique partial order on Λ generated by this covering relation.
One of the main inspirations to consider this relation comes from the following proposition: In other words, Ext 1 G (V (λ, r), V (µ, s)) = 0 except when (µ, s) covers (λ, r).

Truncated categories related to restricted Kirillov-Reshetikhin modules
One of the goals of [5,9] was to study the modules P (λ, r) Γ (and their multigraded version) under certain very specific conditions on Γ. In these papers it was shown that the modules P (λ, r) Γ are giving in terms of generators and relations which allows us to regard these modules as specializations of the famous Kirillov-Reshetikhin modules (in the sense of [13,14]). These papers develop a general theory over a Z + -graded Lie algebra a = i∈Z g[i] where g 0 is a finitedimensional complex simple Lie algebra and its non-zero graded components g[i], i > 0, are finite-dimensional g 0 -modules. By focusing in these algebras with g[i] = 0 for i > 1, we have a ∼ = g V , where V is a g-module, and in this context a very particular tilting theory can be described as follows. Remark 7.5. Such subsets are precisely those contained in a proper face of the convex polytope determined by wt(V ) conform [21].