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    arXiv:1307

    .3307v2

    [math.RT]18Aug2013

    TILTING MODULES IN TRUNCATED CATEGORIES

    MATTHEW BENNETT AND ANGELO BIANCHI

    Abstract. We begin the study of a tilting theory in certain truncated category of modulesG() for the current Lie algebra associated to a finite dimensional complex simple Lie algebra,where = P+ J, J is an interval in Z, and P+ is the set of dominant weights of the simpleLie algebra. We use this to put a tilting theory on the category G() where = P J,where P P+ is saturated. Under certain natural conditions on , we note that G()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 identifiedwith the space gC[t] with the obvious commutator. The study of the representation theory ofcurrent algebras was largely motivated by its relationship to the representation theory of affineand quantum affine algebras associated to g. However, it is also now of independent interestsince the current algebra has connections with problems arising in mathematical physics, forinstance the X = M conjectures, see [1, 18, 25]. Also, the current algebra, and many of itsmodules, admits a natural grading by the integers, and this grading gives rise to interestingcombinatorics. For example, [22] relates certain graded characters to the Poincare polynomials

    of quiver varieties.Let P+ be the set of dominant integral weights of g, = P+ Z, and G the category ofZ-graded modules for g[t] with the restriction that the graded pieces are finite-dimensional.Then indexes the simple modules in G. In this paper we are interested in studying Serresubcategories G() where 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 studythe tilting theories in these categories. This generalized the work of [3], where was taken tobe all of .

    The category G() contains the projective cover and injective envelope of its simple objects.Given a partial order on the set , we can define the standard and costandard objects, as in [19].The majority of the paper is concerned with a particular order, in which case the standard ob-

    jects (, r)() are quotients of the finite-dimensional local Weyl modules, and the costandardobjects (, r)() are submodules of (appropriately defined) duals of the infinite-dimensionalglobal Weyl modules. Both sets of objects have been extensively studied (see [13, 20, 24] forthe local Weyl modules, and [7], for the global Weyl modules). Both families of modules livein a subcategory Gbdd() 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

    This work was partially supported by the FAPESP grants 2012/06923-0 (M.B) and 2011/22322-4 (A.B).

    1

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    2 MATTHEW BENNETT AND ANGELO BIANCHI

    another family of modules indexed by and which are in Gbdd(). These modules are denotedby T(, r)(), and admit an infinite filtration whose quotients are of the form (, s)(), for

    (, s) . They also satisfy the homological property that Ext1

    G((, s)(), T(, r)()) = 0for all (, s) . We use the following theorem to prove that this homological property isequivalent to having a () filtration, proving that the T(, r)() are tilting. The theoremwas proved in [4], [5], and [11] for sl2[t], sln+1[t] and general g[t] respectively.

    Theorem 1. Let P(, r) denote the projective cover of the simple V(, r). Then P(, r)admits a filtration by global Weyl modules, and we have an equality of filtration multiplicities[P(, r) : W(, s)] = [(, r) : V(, s)].

    The following is the main result of this paper.

    Theorem 2.

    (i) Given (, r) , there exists an indecomposable module T(, r)() Ob Gbdd() which

    admits a ()-filtration and a () filtration. FurtherT(, r)()[s] = 0 s > r, dim T(, r)()[r] = 1, wt T(, r)() conv W ,

    and T(, r)() = T(, s)() if and only if (, r) = (, s).(ii) Moreover any indecomposable tilting module in Gbdd() is isomorphic to T(, r)() for

    some (, r) . Finally any tilting module in Gbdd() is isomorphic to a direct sum ofindecomposable tilting modules.

    The majority of the paper is devoted to the case where = P+ J. It is easy to see fromthe construction that the module T(, r)() has its weights bounded above by . It followsthat if we let P P+ be saturated (downwardly closed with respect to the normal partialorder on weights), and set = P J, then T(, r)() = T(, r)().

    We use the convention that (, r)() is simply written (, r), and similarly for otherobjects. Keeping = P+ J, there is a natural functor taking M Ob G to M Ob G().For (, r) this functor preserves many objects, and in particular we have ( , r) =(, r)() and (, r) = (, r)(). So it is natural to ask if T(, r) = T(, r)(). Theanswer is no, and is a result of the following phenomena: for ( , s) / , the module (, s) isnot in general zero, and does not correspond to any simple module. Hence (, s) can not beconsidered costandard. In particular, it is shown in [2] that the costandard filtration ofT(4, 0)includes a subquotient of the form (2, 3). If we let J = (, 0] Z, then (2, 3) = 0, andthe induced filtration on T(4, 0) is not a costandard filtration.

    Another purpose of this paper is the following. In [3], the tilting modules T(, r) areconstructed for all (, r) . It is normal to then consider the module T = (,r)T(, 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-dimensionalgraded components, and hence T / Ob G. One of the purposes of this paper is to find Serresubcategories with index sets such that T() = (,r)T(, r)() Ob G(). It is not hardto see that (except for the degenerate case where = {0} J) a necessary and sufficient pairof conditions on is that P be finite and J have an upper bound . It is natural to study thealgebra End T() in the case that T() G(), and this will b e pursued elsewhere. We alsonote that in the case that is finite then End T() is a finite-dimensional associative algebra.

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    We end the paper by considering other partial orders which can be used on . Inparticular, 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, andthe other is projective or injective), but the partial orders are natural for other reasons, andwe include their study for completeness. One of the reasons to study these other subcategoriesis that one can obtain directed categories as in [8] (in the sense of [17]).

    The paper is organized as follows. In the first section we establish notation and recallsome basic results on the finite-dimensional representations of a finite-dimensional simple Liealgebra. In the second section, we introduce several important categories of modules for thecurrent algebra. We also introduce some important objects, including the local and global Weylmodules. In section 3 we state the main results of the paper and establish some homologicalresults. Section 4 is devoted to constructing the modules T(, r)() and establishing theirproperties. Finally, in section 5, we consider the tilting theories which arise when consideringpartial orders on which are induced by covering relations.

    Acknowledgments: The authors are grateful for the stimulating discussions with ProfessorAdriano Moura as well as the hospitality of the Institute of Mathematics of the State Universityof Campinas where most of this work was completed.

    1. Preliminaries

    1.1. Simple Lie algebras and current algebras. Throughout the paper g denotes afinite-dimensional complex simple Lie algebra and h a fixed Cartan subalgebra of g. SetI = {1, . . . , dim h} and let {i : i I} h

    be a set of simple roots of g with respect toh. Let R h (respectively, R+, P+, Q+) be the corresponding set of roots (respectively,positive roots, dominant integral weights, the Z+-span of R

    +) and let R+ be the highest

    root. Given , h, we say that iff Q+. Let W Aut(h) be the Weylgroup ofg and w be the longest element of W. For R denote by g the correspondingroot space. The subspaces n =

    R+ g, are Lie subalgebras ofg. Fix a Chevalley basis

    {x , hi | R+, i I} ofg and for R+ set h = [x, x]. Note that hi = hi, i I.

    For i I, let i P+ be defined by i(hj ) = ij for all j I.

    Given any Lie algebra a let a[t] = a C[t] be the current algebra of a. Let a[t]+ be theLie ideal a tC[t]. Both a[t] and a[t]+ are Z+-graded Lie algebras with the grading given bypowers oft. Let U(a) denote the universal enveloping algebra ofa. Then U(a[t]) has a naturalZ+-grading as an associative algebra and we let U(a)[k] be the k

    th-graded piece. It followsfrom the PBW basis that dim U(a)[k] < for all k. The algebra U(a) is a Hopf algebra, thecomultiplication being given by extending the assignment x x 1 + 1 x for x a to analgebra homomorphism ofU(a). In the case ofU(a[t]) the comultiplication is a map of gradedalgebras. We shall repeatedly use the fact that U(a[t]) is generated as a graded algebra by aand a t without further comment.

    1.2. Finite dimensional modules. Let F(g) be the category of finite-dimensional g-modules with the morphisms being maps of g-modules. In particular, we may write Homgfor HomF(g). The set P

    + parametrizes the isomorphism classes of simple objects in F(g).

    For P+, let V() be the simple module in the corresponding isomorphism class which is

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    generated by an element v V() satisfying the defining relations:

    n+v = 0, hv = (h)v, (xi)

    (hi)+1v = 0,

    for all h h, i I. The module V(w) is the g-dual of V(). IfV F(g), write

    V =h

    V,

    where V = {v V : hv = (h)v, h h}. Set wt(V) = { h : V = 0}. Finally, recall

    also that the category F(g) is semi-simple, i.e. any object in F(g) is isomorphic to a directsum of the modules V(), P+. We shall use the following standard results without furthermentioning (cf. [26] for (iv)).

    Lemma.

    (i) Let P+. Then wt(V()) Q+.(ii) Let V F(g). Then w wt(V) wt(V) and dim V = Vw for all w W.

    (iii) Let V F(g). Then

    dim Homg(V(), V) = dim{v V : n+v = 0}.

    (iv) Let , P+. Then the module V(w) V() is generated as a U(g)-module by theelement v = v v with defining relations:

    (x+i)(hi)+1v = (xi)

    (hi)+1v = 0 and hv = ( )(h)v,

    for all i I and h h.

    2. The main category G and its subcategoriesIn this section we define the categories which are our objects of study and present several

    properties and functors between them. We also introduce several families of modules whichwill be relevant. Most of these have been studied elsewhere.

    2.1. The main category G. Let G be the category whose objects are Z-graded g[t]-modules with finite-dimensional graded pieces and where the morphisms are 0-graded maps of

    g[t]-modules. More precisely, if V ObG thenV =

    rZ

    V[r],

    where V[r] is a finite-dimensional subspace of V such that (xtk)V[r] V[r + k] for all x g

    and r, k Z. In particular, V[r] Ob F(g) and, if V, W Ob

    G, then

    Hom G(V, W) = {f Homg[t](V, W) : f(V[r]) W[r], r Z}.

    For M G, we define its graded character to bechgrM =

    rZ

    chM[r]ur.

    Define a covariant functor ev : F(g) G by the requirements:ev(V)[0] = V and ev(V)[r] = 0, for all l0 = r Z,

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    and with g[t]-action given by

    (xtk)v = k,0xv, for all lx g, k Z+, v V

    andHom G(ev(V), ev(W)) = Homg(V, W). (2.1)

    For s Z let s be the grading shift given by

    (sV)[k] = V[k s], for all k Z, V ObG.Clearly s(V) ObG.

    For P+ and r Z, setV(, r) := r(ev(V()). (2.2)

    Proposition. The isomorphism classes of simple objects in

    G are parametrized by pairs (, r)

    and we have

    Hom G(V(, r), V(, s)) = 0, if (, r) = (, s),C, if (, r) = (, s).Moreover, if V ObG is such that V = V[n] for some n Z, then V is semi-simple.

    For M ObG we define its (graded) dual to be the module M ObG, whereM = rZM

    [r] and M[r] = M[r]

    and equipped with the usual action where

    (xtr)m(v) = m(xtr.v).

    We note that M = M and that chgrM =

    rZ ch(M[r])u

    r.

    From now on we set = P+ Z.

    and we equip with the dictionary partial order , i.e.

    (, r) (, s) either or = and r s.

    2.2. Some bounded subcategories of G. Let Gs be the full subcategory of G whoseobjects V satisfy

    V[r] = 0, for all r > s,

    and let G be the full subcategory of G consisting of V ObG such that V Ob Gs for somes Z. It follows from the definition that Gs is a full subcategory of Gr for all s < r Z+.Further, let Gbdd be the full subcategory of G consisting of objects M satisfying the followingcondition: there exists 1, . . . , s P

    + (depending on M) such that

    wt M s

    =1

    conv W ,

    Given s Z and V ObG, defineV>s =

    r>s

    V[r] and Vs = V /V>s.

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    6 MATTHEW BENNETT AND ANGELO BIANCHI

    Then Vs Ob Gs. Furthermore, if f Hom G(V, W), then V>s is contained in the kernel

    of the canonical morphism f : V Ws and hence we have a natural morphism fs

    Hom Gs(Vs, Ws).

    Lemma. The assignments V Vr for all V ObG and f fr for all f Hom G(V, W),V, W ObG define a full, exact and essentially surjective functor from G to Gr.

    Given V Ob G we denote by [V : V(, r)] the multiplicity of V(, r) in a composition

    series for V. Furthermore, given W ObG, we set [W : V(, r)] := [Wr : V(, r)]. Observethat [V : V(, r)] equals the g-module multiplicity of V() in V[r]. For any V ObG, define

    (V) = {(, r) : [V : V(, r)] = 0}.

    2.3. Projective and injective objects in

    G and G. Given (, r) , set

    P(, r) = U(g[t]) U(g) V(, r) and I(, r)= P(w, r)

    .Clearly, P(, r) is an infinite-dimensional Z-graded g[t]-module. Using the PBW theorem wehave an isomorphism of graded vector spaces

    U(g[t]) = U(g[t]+) U(g),

    and hence we getP(, r)[k] = U(g[t]+)[k r] V(, r), (2.3)

    where we understand that U(g[t]+)[k r] = 0 if k < r. This shows that P(, r) ObG andalso that

    P(, r)[r] = 1 V(, r).

    Set p,r = 1 v,r.

    Proposition. Let(, r) , and s r.

    (i) P(, r) is generated as ag[t]-module by p,r with defining relations

    (n+)p,r = 0, hp,r = (h)p,r, (xi)

    (hi)+1p,r = 0,

    for all h h, i I. Hence, P(, r) is the projective cover in the category G of its uniquesimple quotient V(, r).

    (ii) The modules P(, r)s are projective in Gs.

    (iii) LetV ObG. Then dim Hom G(P(, r), V) = [V : V(, r)].(iv) Any injective object of G is also injective in G(v) Let(, r) . The object I(, r) is the injective envelope of V(, r) in G or

    G.

    2.4. Local and Global Weyl modules. The next two families of modules in Gbdd weneed are the local and global Weyl modules which were originally defined in [16].

    For the purposes of this paper, we shall denote the local Weyl modules by ( , r), (, r) P+ Z. Thus, (, r) is generated as a g[t]-module by an element w,r with relations:

    n+[t]w,r = 0, (xi )

    (hi)+1w,r = 0,

    (h ts)w,r = s,0(h)w,r,

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    here i I, h h and s Z+.

    Next, let W(, r) be the global Weyl modules W(, r), which is g[t]-module generated as a

    g[t]-module by an element w,r with relations:n+[t]w,r = 0, (x

    i )

    (hi)+1w,r = 0,

    hw,r = (h)w,r,

    where i I and h h. Clearly the module (, r) is a quotient of W(, r) and moreoverV(, r) is the unique irreducible quotient of W(, r). It is known (see [7] or [16] ) thatW(0, r) = C and that if = 0, the modules W(, r) are infinite-dimensional and satisfy

    wt W(, r) conv W and W(, r)[s] = 0 iff s r,

    from which we see that W(, r) / Ob G. It follows that, if we set

    (, r) = W(w0, r),

    then (, r) Ob Gbdd and soc (, r) = V(, r).We note that (, r) (resp. (, r)) is the maximal quotient of P(, r) (resp. submodule

    of I(, r)) satisfying[(, r) : V(, s)] = 0 = (, s) (, r).

    ( resp., [(, r) : V(, s)] = 0 = (, s) (, r).)

    Hence these are the standard (resp. costandard) modules in G associated to (, r).

    2.5. Truncated subcategories. In this section, we recall the definition of certain Serresubcategories of G.

    Given , let

    G() be the full subcategory of

    G consisting of all M such that

    M ObG, [M : V(, r)] = 0 = (, r) .The subcategories G() and Gbdd() are defined in the obvious way. Observe that if (, r) ,

    then V(, r) G(), and we have the following trivial result.Lemma. The isomorphism classes of simple objects of G() are indexed by . Remark. Let C be one of the categories Gs, G, Gbdd, G(), Gbdd(), which are full subcate-

    gories of G. Then, we haveExt1

    G(M, N) = Ext1C(M, N) for all M, N C.

    2.6. A specific truncation. We now focus on of the form = P+ J, where J is aninterval in Z with one of the forms (, n], [m, n], [m, ) or Z, where n, m Z. We set

    a = infJ and b = supJ. Throughout this section, we assume that (, r) .Let P(, r)() be the maximal quotient of P(, r) which is an object of G() and let

    I(, r)() be the maximal submodule ofI(, r) which is an object ofG(). These are the inde-composable projective and injective modules associated to the simple module V(, r) G().

    For an object M G, let M be the subquotient

    M =M>aM>b

    .

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    where we understand M>a = M if a = and M>b = 0 ifb = .

    Remark.

    (1) IfM = s>pM[s] for some p, then M = M

    M>b.

    (2) IfM = sa.

    Clearly M G(), and because morphisms are graded, this assignment defines a functorfrom G to G(). It follows from Lemma 2.2 that is exact.

    If we define another subset = P+ {J}, then it follows from the definition of the gradedduality that if M Ob G() then M Ob G().

    Lemma. The module P(, r)() = P(, r) and I(, r)() = I(, r).

    We set

    (, r)() := (, r)

    , W(, r)() := W(, r)

    and (, r)() := (, r)

    .In light of the above remark, we can see that

    (, r)() =(, r)

    s>b(, r)[s], W(, r)() =

    W(, r)

    s>bW(, r)[s]

    and (, r)() = (, r)>a.

    Note that, with respect to the partial order , for each (, r) we have (, r)() themaximal quotient of P(, r)() such that

    [(, r)() : V(, s)] = 0 = (, s) (, r).

    Similarly, we see that (, r)() is the maximal submodule of I(, r)() satisfying

    [(, r)() : V(, s)] = 0 = (, s) (, r).

    These modules (, r)() and (, r)() are called, respectively, standard and co-standardmodules associated to (, r) .

    The following proposition summarizes the properties of (, r)() which are necessary forthis paper. They can easily be derived from the properties of the functor .

    Proposition.

    (i) The module (, r)() is generated as a g[t]-module by an element w,r with relations:

    n+[t]w,r = 0, (xi )

    (hi)+1w,r = 0,

    (h ts)w,r = s,0(h)w,r, U(g[t])[p]w,r = 0, if p > b r,

    for all i I, h h and s Z+, where if b = , then the final relation is empty relation.

    (ii) The module(, r)() is indecomposable and finite-dimensional and, hence, an object ofGbdd().

    (iii) dim(, r)() = dim (, r)()[r] = 1,(iv) wt(, r)() conv W ,(v) The module V(, r) is the unique irreducible quotient of (, r)().

    (vi) {chgr(, r)() : (, r) P+ Z} is a linearly independent subset ofZ[P][u, u1].

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    2.7. Here we collect the results on W(, r)() which we will need for this paper.

    Proposition. (i) The module W(, r)() is generated as a g[t]-module by an element w,rwith relations:

    n+[t]w,r = 0, (xi )

    (hi)+1w,r = 0,

    U(g[t])[p]w,r = 0, ifp > b r

    where if b = , then the final relation is empty relation. Here i I and h h .(ii) The module W(, r)() is indecomposable and an object of Gbdd().

    (iii) dim W(, r)()[r] = 1 and dim W(, r)()[s] = 0 if and only if r s b.(iv) wt W(, r)() conv W .(v) The module V(, r) is the unique irreducible quotient of W(, r)().

    (vi) {chgrW(, r)() : (, r) P+ Z} is a linearly independent subset ofZ[P][u, u1].

    2.8. The following proposition summarizes the main results on (, r)() that are neededfor this paper. All but the final result can be found by considering the properties of the functor and the paper [3].

    Proposition.

    (i) The module (, r)() is an indecomposable object of Gbdd().(ii) dim (, r)()[r] = 1 and dim (, r)()[s] = 0 a s r.

    (iii) wt (, r)() conv W ,(iv) Any submodule of (, r)() contains (, r)()[r] and the socle of (, r)() is the

    simple module V(, r).(v) {chgr(, r)() : (, r) P

    + Z} is a linearly independent subset ofZ[P][u, u1]].

    (vi) Let

    = P+

    {J} and (, r) . Then (, r)()= W(0, r)(

    )

    .

    Proof. We prove the final item. As a vector space we have

    W(0, r)() = as=rW(0, r)[s].

    Since W(0, r)() is a quotient ofW(0, r), its dual must be a submodule of(, r).

    By the definition of the graded dual, we see that, as a vector space,

    W(0, r)() = rs=a(, r)[s].

    Hence, as vector spaces, we see that (, r)() = W(0, r)().

    The fact that W(0, r)() is a submodule completes the proof.

    3. The Main Theorem and some Homological Results

    Definition. We say that M Ob G() admits a () (resp. ())-filtration if there existsan increasing family of submodules

    0 M0 M1 with M =

    k

    Mk,

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    such that

    Mk/Mk1 = (,r) ()(, r)mk(,r) (resp., Mk/Mk1 = (,r) (, r)()

    mk(,r)),

    for some choice of mk(, r) Z+. We do not require

    (,r) mk(, r) < . If Mk = M for

    some k 0, then we say that M admits a finite () (resp. ())-filtration. Because ourmodules have finite-dimensional graded components, we can conclude that the multiplicity ofa fixed (, r)() (resp. (, r)()) in a () filtration (resp. () filtration) must be finite,and we denote this multiplicity by [M : (, r)()] (resp. [M : (, r)()]. Finally, we saythat M Ob G() is tilting if M has both a () and a ()-filtration.

    The main goal of this paper is to understand tilting modules in Gbdd(). (The case whereJ = Z was studied in [3].) In the case of algebraic groups (see [19, 23]) a crucial necessaryresult is to give a cohomological characterization of modules admitting a ()-filtration. The

    analogous result in our situation is to prove the following statement:

    An object M of Gbdd() admits a ()-filtration if and only if Ext1G

    (((, r)(), M) = 0

    for all (, r) .

    It is not hard to see that the forward implication is true. The converse statement howeverrequires one to prove that any object of Gbdd() can be embedded in a module which admitsa ()-filtration. This in turn requires Theorem 4. Summarizing, the first main result thatwe shall prove in this paper is:

    Proposition. LetM Ob Gbdd(). Then the following are equivalent

    (i) The module M admits a ()-filtration.

    (ii) M satisfies Ext1G((, r)(), M) = 0 for all (, r) .

    The second main result that we shall prove in this paper is the following:

    Theorem 3.

    (i) Given (, r) , there exists an indecomposable module T(, r)() Ob Gbdd() whichadmits a ()-filtration and a () filtration.

    T(, r)()[s] = 0 if s > r, T(, r)()[r] = 1, wtT(, r)() conv W,

    and T(, r)() = T(, s)() if and only if (, r) = (, s).(ii) Any indecomposable tilting module in Gbdd() is isomorphic to T(, r)() for some

    (, r) . Finally, any tilting module in Gbdd() is isomorphic to a direct sum of

    indecomposable tilting modules.

    3.1. We begin by proving part one of Proposition 3. To do so, we first establish somehomological properties of the standard and costandard modules which will be used throughoutthe paper.

    Proposition. Let, P+. Then we have the following

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    (i) Ext1G

    (W(, r)(), W(, s)()) = 0 = Ext1G

    ((, s)(), (, r)()) for all s, r Z if b r, then U(g[t])[p].x = 0 by grade considerations. It follows that thesequence splits. The proof for () is similar and omited.

    For part (ii), suppose we have a sequence 0 (, s)() X (, r)() 0 and . Then dim X[r] = 1, and ifx X[r] is a pre-image ofw,r then x satisfies the definingrelations of w,r and the sequence splits. If , then by taking duals we get a sequence

    0 (, r)()

    Y W(0, s)(

    ) 0. Again, if y Y0[s] is a pre-image ofw0,s, we see that y satisfies the defining relations of w0,s W(0, s)(), and

    the sequence splits.

    The proofs for parts (iii) and (iv) are similar to that for part (i) and are omitted.

    The proof of the following Lemma is standard (see for example [3]).

    Lemma. Suppose that M Gbdd() admits a (possibly infinite) ()-filtration. Then Madmits a finite ()-filtration

    0 M1 M2 Mk = M with Ms/Ms1 =rZ

    (s, r)()[M:(s,r)()]

    where i > j implies i > j. In particular if is maximal such that M = 0, then there existss Z and a surjective map M (, s)() such that the kernel has a ()-filtration.

    3.2. We now see that part one of Proposition 3 is a corollary of Lemma 3.1 and part (ii)of Proposition 3.1.

    Proposition. Suppose that N Ob G() is such that Ext1G()((, r)(), N) = 0 for all

    (, r) . If M has a ()-filtration then Ext1G()(M, N) = 0.

    Proof. Consider a short exact sequence

    0 N U M 0.

    Suppose that Mk

    Mk+1

    is a part of the ()-filtration of M and assume that

    Mk+1/Mk =

    (,s)

    ()(, s)ms .

    By assumption we have Ext1G()(Mk+1/Mk, N) = 0. Let Uk U be the pre-image of Mk and

    note that Uk+1/Uk = Mk+1/Mk. Consider the short exact sequence

    0 N Uk Mk 0.

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    12 MATTHEW BENNETT AND ANGELO BIANCHI

    This sequence defines an element of Ext1G()(Mk, N). Since Mk has a finite ()-filtration it

    follows that Ext1G()(Mk, N) = 0. Hence the sequence splits and we have a map k : Uk N.

    We want to prove that k+1 : Uk+1 N can be chosen to extend k. For this, applyingHomG()(, N) to

    0 Uk Uk+1 Uk+1/Uk 0,

    we get

    HomG()(Uk+1, N) HomG()(Uk, N) 0,

    which shows that we can choose k+1 to lift k. Now defining

    : U N, (u) = k(u), u Uk,

    we have the desired splitting of the original short exact sequence.

    Together with Proposition 3.1 above, we now have

    Corollary. LetM be a module with a ()-filtration. Then, Ext1G

    (M, (, r)()) = 0, for all

    (, r) .

    3.3. A Natural Embedding. In this section we show that every M Ob Gbdd() embedsinto an injective module I(M) Ob Gbdd(). Let soc M M be the maximal semi-simplesubmodule of M.

    Lemma. LetM Ob Gbdd().

    (i) If M = 0, then soc M = 0.(ii) Suppose soc M = V(, r)m,r . Then M I(, r)()m,r .

    Proof. For the first part, let M

    ObGbdd

    , suppose M= 0, and let s

    Z be minimal such

    that M Ob Gs. Then M[s] = 0 and M[s] soc M.

    For the second, let

    soc M = (,r)V(, r)m,r

    from which we get a natural injection soc M

    I(, r)m,r . By injectivity, we have a

    morphism Mf

    I(, r)m,r , which factors through . In fact, we can show that f is aninjection. If not, soc ker f = 0. On the other hand, soc ker f soc M, and f is injective onsoc M.

    If we assume that M Ob Gbdd(), then it is easy to conclude that imf I(, r)()m,r ,

    completing the proof.

    3.4. O-Canonical Filtration. In this section we shall establish a finite filtration on mod-ules M Ob Gbdd() where the successive quotients embed into direct sums of (). We thenuse the filtration to establish lower and upper bounds on the graded character of M. We usethe character bounds to prove Proposition 3.

    Now fix an ordering of P+ = {0, 1, } such that r > s implies that r > s. For M Ob G() we set Ms M as the maximal submodule whose weights lie in {conv W r | r s}.Evidently Ms1 Ms. We call this the o-canonical filtration, because it depends on the

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    order. This is a finite filtration for M Ob Gbdd() and we set k(M) to be minimal such thatM = Mk(M). Clearly

    Ms1 Ms, M =

    k(M)s=0

    Ms,

    Homg(V(, r), Ms/Ms1) = 0 = = s. (3.1)

    It follows from Lemma 3.3 and equation (3.1) that Ms/Ms1 embeds into a direct sum ofmodules of the form (s, r)(), r J. This gives,

    chgrM =s0

    chgrMs/Ms1 s0

    rJ

    dim HomG(V(s, r), Ms/Ms1)chgr(s, r)(), (3.2)

    i.e,

    [M : V(, )] s0

    rJ

    dim HomG(V(s, r), Ms/Ms1)[(s, r)() : V(, )],

    for all (, ) . We claim that this is equivalent to

    chgrM =

    chgrMs/Ms1 s0

    rJ

    dim HomG((s, r)(), M)chgr(s, r)().

    The claim follows from Lemma 3.4 below, and [3, 3.5], which states that

    HomG((, r), M) = HomG V(s, r),Ms

    Ms1Lemma. LetM Ob G() and (, r) . We have

    Hom G((, r)(), M)= Hom G((, r), M) and Ext

    1G

    ((, r)(), M) = Ext1G((, r), M).

    Proof. As M[s] = 0 for all s > b, we must have Hom G(s>b(, r)[s], M) = 0. Similarly, if

    0 M X s>b(, r)[s] 0

    is exact, by using again that M[s] = 0 for s > b, we have dim X[n] = dim(s>b(, r)[s])[n] forany n > b and so we have an injective map : s>b(, r)[s] X which splits the sequence.Thus, Ext1

    G

    (s>b(, r)[s], M) = 0. Now the other statements are easily deduced.

    Therefore, we get

    chgrM s0

    rJ

    dim HomG()((s, r)(), M)chgr(s, r)() (3.3)

    and the equality holds if, and only if, the o-canonical filtration is ()-filtration.

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    14 MATTHEW BENNETT AND ANGELO BIANCHI

    3.5. Note that even if N / Ob Gbdd, we can still define the submodules Ns, and by defini-tion Ns Ob Gbdd. The following result, or more precisely a dual statement about projective

    modules and global Weyl filtrations, was proved for g = sl2 in [4], for g = sln+1 in [5] and ingeneral for g in [11]. It is conjectured to hold in general.

    Theorem 4. For all (, r) and for all p Z+ the o-canonical filtration on I(, r)p is a-filtration. Moreover, for all (, s) we have

    [I(, r)p : (, s)] = dimHom G((, s), I(, r)p).

    As a consequence of the theorem and the exactness of the functor , we conclude thatI(, r)()p has a ()-filtration.

    For M Gbdd(), let p be minimal such that Mp = M. Then it is clear that we can refinethe embedding from Lemma 3.3 to M I(, r)p(). We can conclude the following:

    Corollary. For all M Ob Gbdd(), we have M I(M) Ob Gbdd() where I(M) admits a()-filtration.

    We now prove part (ii) of Proposition 3 following the argument in [3]. Let M G() andassume that Ext1

    G((, r)(), M) = 0. Let I(M) G() be as in Lemma 3.5, and consider

    the short exact sequence

    0 M I(M) Q 0

    where Q G(). The assumption on M implies that

    dim Hom((, r)(), M) = dimHom((, r)(), I(M)) dim Hom((, r)(), Q) (3.4)

    as objects of G(). As I(M) has a ()-filtration, we can conclude that Equation (3.3) is anequality. Hence we get that

    chgrM = chgrI(M) chgrQ

    rJs0

    (dimHomG()((s, r)(), I(M)) dimHomG()((s, r)(), Q))chgr(s, r).

    Combining this with Equation (3.4), we get that the character bound in (3.3) is an equalityfor M, and hence that M has a ()-filtration.

    Finally, we can prove the following.

    Proposition. The following are equivalent for a module M Ob Gbdd()

    (1) For all (, s) , we have Ext1G((, r)(), M) = 0.(2) M admits a () filtration.(3) The o-canonical filtration on M is a () filtration.

    Proof. The equivalence of (1) and (2) is Proposition 3. Clearly (3) implies (2), so it is enoughto show that (1) implies (3). Assuming (1), we have shown that the character bound in (3.3)is an equality, which is true if and only if the o-canonical filtration is a () filtration.

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    3.6. Our final result before constructing the tilting modules T(, r)() shows that thespace of extensions between standard modules is always finite-dimensional. The proof is anal-

    ogous to the proof in [3].Proposition. For all (, r), (, s) we have dim Ext1

    G((, r)(), (, s)()) < .

    Proof. Consider the short exact sequence

    0 K P(, r) (, r)() 0

    and apply the functor Hom G(, (, s)()). Because P(, r) is projective, we see that the

    result follows if dim Hom G(K, (, s)()) < . Let be such that (, s)()[p] = 0 for all

    p > . Then Hom G(K>, (, s)()) = 0, and hence we have an injection

    Hom G(K, (, s)()) Hom G(K

    K>, (, s)()) 0.

    The proposition follows because KK>

    is finite-dimensional.

    4. Construction of Tilting Modules

    4.1. In this section we construct a family of indecomposable modules in Gbdd(), the{T(, r)() : (, r) }, each of which admits a ()-filtration and satisfies

    Ext1G

    ((, s)(), T(, r)()) = 0, (, s) .

    It follows that the modules T(, r)() are tilting and we prove that any tilting module inGbdd() is a direct sum of copies ofT(, r)(), (, r) . The construction is a generalizationof 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(, r)(), using the algorithm given in [23], is to finda suitable subset (depending on (, r)) of which can be appropriately enumerated. Hencewe assume the following result, whose proof we postpone.

    Proposition. Fix (, r) and assume that under the enumeration we have = k. Thenthere exists a subset S(, r) such that

    (i) (, r) S(, r)(ii) if (, s) S(, r) then = i for some i k, and there exists an ri such that s ri.

    Further, the ri satisfy rk = r and ri rk.(iii) Ext1

    G((, s)(), (, s)()) = 0 for all (, s) S(, r) and (, s) / S(, r).

    Furthermore, there exists an injection : S(, r) Z0 such that for (i, pi) = 1(i) we

    have Ext1G((i, pi)(), (j , pj)()) = 0 if i < j, and (j, pj )()i [pi] = 0 for i < j .

    Without loss of generality we may assume that (, r) = 0 and the image of is an interval.We need the following elementary result.

    Lemma. Suppose that M, N ObG are such that0 < dim Ext1

    G(M, N) < and Ext1

    G(M, M) = 0.

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    Then, there exists U Ob

    G, d Z+ and a non-split short exact sequence

    0 N U Md 0

    so that Ext1G

    (M, U) = 0.

    4.2. We now use to construct an infinite family of finite dimensional modules Mi, whosedirect limit will be T(, r)(). We note that the construction, at this point, will seem to bedependent on the ordering of P+ we have chosen, and on the set S(, r) and . We proveindependence at the end of this section.

    Set M0 = (0, p0)(). If Ext1G

    ((1, p1)(), (0, p0)()) = 0, then set M1 = M0. If not,

    then since dim(Ext1G

    ((1, p1)(), (0, p0)())) < by Prop 3.6, Lemma 4.1 gives us an

    object M1 ObG() and a non-split short exact sequence

    0 M0 M1 (1, p1)()

    d1 0.

    Let M1 M1 be an indecomposable submodule containing (M1)0 [p0]. By Proposition 4.1 wesee that (M1)0 [p0] = (M0)0 [p0]. Then we have M0

    0 M1. By examining the sequence

    . . . Ext1G()

    ((, p)(), M0) Ext1G()

    ((, p)(), M1)

    Ext1G()

    ((, p)(), (1, p1)()d1)

    we see that Ext1G()

    ((, p)(), M1) = 0 for (, p) / S(, r), and Ext1G()

    ((i, pi)(), M1) = 0

    for i = 0, 1.

    We use Lemma 4.1 again, with N = M1 and M = (2, p2)(), and we get

    0 M1 M2 (2, p2)()

    d2 0,

    a submodule M2 M

    2 containing (M

    2)i [pi], i = 0, 1, such thatM1

    1 M2, Ext

    1G()

    ((i, pi)(), M2) = 0 for i = 0, 1, 2,

    and Ext1G()

    ((, p)(), M2) = 0 for (, p) / S(, r).

    Repeating this procedure, and using Lemma 4.1 and the properties of, we have the followingproposition.

    Proposition. There exists a family {Ms}, s Z0, of indecomposable finite-dimensionalmodules and injective morphismss : Ms Ms+1 of objects ofGbdd() which have the followingproperties.

    (i) M0 = (k, r)() = (0, p0)(), and for s 1,

    Ms/s1(Ms1) = (s, ps)()ds

    , ds Z+,dim Ms[r]k = 1, wt Ms conv W k.

    (ii)Ms[p] = 0, for all s 0, p >> max{ri}. (4.1)

    (iii)

    Ext1G

    ((, p)(), Ms) = 0, 0 s, Ext1G

    ((, p)(), Ms) = 0, (, p) / S(, r).

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    (iv) Ms is generated as a g[t]-module by the spaces {Ms[p] : s}. Moreover, if we let

    r,s = s1 r : Mr Ms, r < s, r,r = id,

    then

    Ms[p] = ,s(M[p]), s . (4.2)

    4.3. Defining the tilting modules. Let T(k, r)() = T(, r)() be the direct limit of

    {Ms, r,s | r, s Z+, r s}. We have an injection Ms T(, r)(), and so Ms = Ms, whereMs is a submodule of T(, r)() such that Ms Ms+1, T(, r)() = Ms and MsMs1 = MsMs1 .In particular we see that T(, r)() has ()-filtration. We identify Ms with Ms.

    The argument that T(, r)() is indecomposable is identical to that from [3], which weinclude for completeness. We begin with an easy observation.

    T(, r)()[p] = M[p] , Ms =s

    U(g[t])T(k, r)[p] . (4.3)

    To prove that T(, r)() is indecomposable, suppose that

    T(, r)() = U1 U2.

    Since dim T(, r)()[r] = 1, we may assume without loss of generality that T(, r)()[r] U1and hence M0 U1. Assume that we have proved by induction that Ms1 U1. Since Msis generated as a g[t]-module by the spaces {Ms[p] : s}, it suffices to prove thatMs[ps]s U1. By (4.3), we have Ui[ps]s Ms and hence

    Ms = (Ms1 + U(g[t])U1[ps]s)U(g[t])U2[ps]s .Since Ms is indecomposable by construction, it follows that U2[ps]s = 0 and Ms U1 whichcompletes the inductive step.

    Proposition. Let(, r) .

    (i) Then there exists an indecomposable objectT(, r)() Gbdd() which admits a filtrationby finite-dimensional modules

    Ms =s

    U(g[t])T(, r)()[p] , s 0, (4.4)

    such that M0 = (, r)() and the successive quotients are isomorphic to a finite-direct

    sum of (, s)(), (, s) S(, r).(ii) We have wt T(, r)() conv W , dim T(, r)()[r] = 1 .(iii) For all (, s) , we have

    Ext1G()

    ((, s)()), T(, r)()) = 0. (4.5)

    Proof. Part (i) and (ii) are proved in the proceeding discussion. The proof for part (iii) isidentical to that found in [3].

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    4.4. The next result is an analog of Fittings Lemma for the infinite-dimensional modulesT(, r)(), whose proof is identical to that in [3].

    Lemma. Let : T(, r)() T(, r)() be any morphism of objects of G. Then(Ms) Msfor all s 0 and is either an isomorphism or locally nilpotent, i.e., given m M, thereexists 0 (depending on m) such that (m) = 0.

    In the rest of the section we shall complete the proof of the main Theorem by showingthat any indecomposable tilting module is isomorphic to some T(, r)() and that any tiltingmodule in Gbdd() is isomorphic to a direct sum of indecomposable tilting modules. LetT Gbdd() be a fixed tilting module. Then we have

    Ext1G

    (T, (, r)()) = 0 = Ext1G

    ((, r)(), T)), (, r) . (4.6)

    Lemma. Suppose that T1 is any summand of T. Then T1 admits a ()-filtration and

    Ext1G

    (T1, (, r)()) = 0, (, r) .

    Proof. Since Ext1 commutes with finite direct sums, for all (, r) we have

    Ext1G

    (T1, (, r)()) = 0, Ext1G

    ((, r)(), T1)) = 0.

    Under the assumption that Proposition 3 is true, the second equality implies that T1 has a()-filtration and the proof of the Lemma is complete.

    4.5. The preceding lemma illustrates one of the difficulties we face in our situation.Namely, we cannot directly conclude that T1 has a ()-filtration from the vanishing Ext-condition by using the dual of Proposition 3. However, we have the following, whose proof isgiven in [3].

    Proposition. Suppose that N Gbdd() has a ()-filtration and satisfiesExt1

    G(N, (, r)()) = 0, for all (, r) .

    Let(, s) be such that N (, s)() 0. Then T(, s)() is a summand of N.

    The following is immediate. Note that this also shows that our construction of the inde-composable tilting modules is independent of the choice of enumeration of P+, the set S(, r)and .

    Corollary. Any indecomposable tilting module is isomorphic to T(, r)() for some(, r) .Further if T Ob Gbdd() is tilting there exists (, r) such that T(, r)() is isomorphicto a direct summand of T.

    Proof. Since T and T(, r)() are tilting they satisfy (4.6) and the corollary follows.

    We can now prove the following theorem.

    Theorem 5. Let T Ob Gbdd(). The following are equivalent.

    (i) T is tilting.(ii) Ext1

    G((, r)(), T) = 0 = Ext1

    G(T, (, r)()), (, r) .

    (iii) T is isomorphic to a direct sum of objects T(, s)(), (, s) ().

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    Proof. Because the other implications follow easily, we show that (ii) implies (iii). Recall thatby Lemma 3.1, we can assume that if is maximal such that T = 0, then T (, r)() 0

    for some r. Choose integers r1 r2 . . . such that[T : (, s)()] = 0 = s = ri for some i.

    By Proposition 4.5 we have T = T(, r1)() T1 and we see that T1 has a () filtration,and that T1 maps onto (, r2)(), and hence T(, r2)() is isomorphic to a summand of T1.Continuing, we find that for j 1, there exists a summand Tj of T with

    T = Tj js=1 T(, rs)().

    Let j : T js=1T(, rs)() be the canonical projections. Because T has finite dimensional

    graded components, and the ri are decreasing, it follows that for m T we have j (m) = 0for all but finitely many j. Hence we have a surjection

    : T j0T(, rj )() 0 and ker =Tj.In particular, we have T = T(, ri)() ker and ker = 0. We now apply to argumentto ker , and the result follows.

    4.6. The set S(, r) and enumeration . In this section we prove Proposition 4.1.

    The set S(, r). Recall that = P+ J, and that a = infJ and b = supJ. Using theenumeration of P+, let = k and define integers recursively by rk = r,

    rs = max{r | (s+1, rs+1)()[r] = 0}.

    Note that (i, ri)()[p] = 0 if p > rj for any j < i, and hence that (i, s)()[p] = 0 forp > rj, for any j < i and for any s ri.

    We set S(, r) = {(i, s)| i k, s ri}, and note that it satisfies conditions 1 and 2 ofProposition 4.1. For condition 3, there are two cases; either = i for i > k, or

    = i forsome 0 i k and s > ri. The first case is covered by Proposition 3.1. It is enough to provethat Ext1

    G((i, s

    )(), (j , s)()) = 0 for i j, and s > ri, and s rj . By the total

    order, it follows that i < j. It follows that s rj ri < s. It follows that the Ext1 group is

    zero, and the third property is established.

    The enumeration . It remains to define the enumeration . The case where J = Z is donein [3], and the case where J is a finite or of the form [b, ) (in which case S(, r) is in factfinite), we use the enumeration defined by

    (1) (i, s) < (j , s) if i > j

    (2) (i, s) < (i, s 1)

    We are left with the case where J = (, a], and note that it is enough to define a settheoretic inverse, which by abuse of notation we also denote by .

    We recursively define another set of integers {ri} by setting rk = r and letting r

    i =

    max{r|(i+1, ri+1)[r] = 0}. It is easy to see that ri r

    i. Set as := r

    s r

    s+1. Then it

    is easy to see that

    Ext1G

    ((i, c)(), (s, d)()) = 0 ifc d as + 1 and i < s

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    We define : Z0 S(, r) in the following way: Set (0) = (k, rk). If is definedon {0, . . . , m 1} and (m 1) = (i, pi), we define (m) as follows. Suppose that i > 0,

    and (i1, pi + ai1) S(, r), then we set (m) = (i1, pi + ai1). Otherwise, we let(m) = (k, pm 1), where pm is the minimal integer such that (k, p) has a pre-image under.

    To prove that Ext1G

    ((i, pi)(), (j , pj)()) = 0 if i < j, we assume that i j. If

    i = j then this follows from Proposition 3.1. So we assume that i < j, and lets say thatj = . In this case we must have pi pj > a and so the Ext group is zero by the observationabove.

    5. Some different considerations on the categories G()

    Throughout this section we discuss some trivial tilting theories for the category G() byconsidering different type of orders on the set . These categories with the described order

    below already appeared in the literature as in [6, 8, 9, 10, 12] and references therein.

    We keep considering the categories G and G in the same way introduced before.5.1. Truncated categories with the covering relation. Consider a strict partial order

    on in the following way. Given (, r), (, s) , say that (, s) covers (, r) if and onlyif s = r + 1 and R {0}. It follows immediately that for any (, s) the setof (, r) such that (, s) covers (, r) is finite. Let be the unique partial order on generated by this cover relation. Then {(, s) : (, s) (, r)} is finite for all (, r) .

    One of the main inspirations to consider this relation comes from the following proposition

    Proposition. [8, Proposition 2.5] For (, r), (, s) , we have

    Ext1G(V(, r), V(, s)) =0, if s = r + 1,

    Homg(V(), g V()), if s = r + 1.

    In other words, Ext1G(V(, r), V(, s)) = 0 unless (, s) covers (, r).

    5.2. Given , set

    V+ = {v V[s] : n

    +v = 0, (, s) },

    V = U(g)V+, V = V /V\.

    Proposition. [8, Propositions 2.1 and 2.7] Let be finite and convex and assume that(, r), (, s) .

    (i) The object I(, r) is the injective envelope of V(, r) in G[] while P(, r) is the pro-jective cover of V(, r) in G[]. In particular, G[] has enough projectives.

    (ii) [P(, r) : V(, s)] = [P(, r) : V(, s)] = [I(, s) : V(, r)] = [I(, s) : V(, r)].(iii) HomG(P(, r), P(, s)) = HomG[](P(, r)

    , P(, s)).(iv) LetK(, r) be the kernel of the canonical projection P(, r) V(, r), and let(, s) .

    Then [K(, r) : V(, s)] = 0 only if (, r) (, s).(v) Let (, s) . Then [I(, r)/V(, r) : V(, s)] = 0 only if (, s) (, r).

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    5.3. Following Section 2.6 but using the partial order defined above, for each (, r) we denote by (, r)() the maximal quotient of P(, r) such that

    [(, r)() : V(, s)] = 0 = (, s) (, r).

    Similarly, we denoted by (, r)() the maximal submodule of I(, r) satisfying

    [(, r)() : V(, s)] = 0 = (, s) (, r).

    These modules (, r)() and (, r)() are called, respectively, standard and co-standard

    modules associated to (, r). Further, any module in G with a ()-filtration and a ()-filtration is called tilting.

    Proposition. Let finite and convex.

    (i) For all (, r) , there exists indecomposable tilting module T(, r)().(ii) For al l indecomposable tilting module T, we have T = T(, r)() for some (, r) .

    (iii) Every tilting module is isomorphic to a direct sum of indecomposable tilting modules.

    Before proving this proposition, we have some remarks to do:

    Remark.

    (1) It follows from Proposition 2.3 (iii) and Proposition 5.2, parts (ii) and (v), that the

    costandard modules in G associated to (, r) is I(, r) and similarly it follows fromPropositon 5.2, parts (ii) and (iv), that the standard module in

    G associated to (, r)

    is the simple module V(, r).(2) For any M G let k(M) the such that M Gk(M). Thus M admits a filtration {Mi}

    where Mi = i

    j=0M[k(M) j] which can be refined into a Jordan-Holder series sinceeach quotient Mi+1/Mi is a finite-dimensional g-module.

    Proof. of Proposition 5.3 Part (i) follows from the Remark (1) since we have T(, r)() :=I(, r). Part (ii) and (ii) are direct consequences of the injectivity of I(, r).

    5.4. Truncated categories related to restricted Kirillov-Reshetikhin modules.

    One of the goals of [6, 10] was to study the modules P(, r) under certain very specificconditions on . In these papers it was shown that the modules P(, r) are giving in termsof generators and relations which allows us to regard these modules as specializations of thefamous Kirillov-Reshetikhin modules (in the sense of [14, 15]). These papers develop a generaltheory over a Z+-graded Lie algebra a = iZg[i] where g0 is a finite-dimensional complexsimple Lie algebra and its non-zero graded components g[i], i > 0, are finite-dimensional g0-modules. By focusing in these algebras with g[i] = 0 for i > 1, we have a = g V, where V isa g-module, and in this context a very simple and particular tilting theory can be consideredas follows.

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    22 MATTHEW BENNETT AND ANGELO BIANCHI

    Assume that wt(V) = {0} and fix a subset wt(V) satisfying

    m = wt(V) n (m, n Z+) = m wt(V) nand (5.1)

    m =

    wt(V)

    n only if n = 0 for all / .

    Remark. It is proved in [21] that such subsets are precisely those lying on a proper face ofthe convex polytope determined by wt(V).

    Consider the reflexive and transitive binary relation on P given by

    if Z+,

    where Z+ is the Z+-span of . Set also

    d(, ) = min

    m : =

    m, m Z+

    .

    By [12, Proposition 5.2], is in fact a partial order on P and

    d(, ) + d(, ) = d(, ) whenever .

    Moreover, it induces a refinement of the partial order on by setting

    (, r) (, s) if , s r Z+, and d(, ) = s r.

    Finally, if is finite and convex with respect to and there exists (, r) suchthat (, r) (, s) for all (, s) . It was shown [12, Lemma 5.5] that

    HomG[](P(, s)

    , P(, t)

    ) = 0 only if (, s) (, t) (5.2)and

    ExtjG[](V(, s), V(, t)) = 0 only if (, t) (, s) and j = d(, ). (5.3)

    In particular, it follows from Proposition 5.2, parts (ii) and (iii), and (5.2) that the standardmodules in G() are the simple modules and the costandard modules are the injective hulls ofthe simple modules and, hence, T(, r)() = P(, r).

    References

    [1] E. Ardonne and R. Kedem, Fusion products of Kirillov-Reshetikhin modules and fermionic multiplicityformulas, J.Algebra 308 (2007), 270-294.

    [2] M. Bennett, A realization of tilting modules for the current algebra associated to A1, work in progress.

    [3] M. Bennett and V. Chari, Tilting modules for the current algebra of a simple Lie algebra, Proceedingsof Symposia in Pure Mathematics 86: Recent Developments in Lie Algebras, Groups and RepresentationTheory 2012, 75-97.

    [4] M. Bennett, V. Chari, and N. Manning, BGG reciprocity for current algebras, Adv. Math. 231 (2012), no.1, 276305.

    [5] M. Bennett, A. Berenstein, V. Chari, A. Khoroshkin, and S. Loktev, Macdonald Polynomials and BGGreciprocity for current algebras, to appear in Selecta Mathematica

    [6] A. Bianchi, V. Chari, G. Fourier, and A. Moura, On multigraded generalizations of Kiril lov-Reshetikhinmodules, to appear in Algebr. Represent. Theory (2013).

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    TILTING MODULES IN TRUNCATED CATEGORIES 23

    [7] V. Chari, G. Fourier, and T. Khandai, A categorical approach to Weyl modules, Transform. Groups 15(2010), no. 3, 517549.

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    simple Lie algebras, Adv. in Math. 220 (2009), 11931221.[10] V. Chari and J. Greenstein, Minimal affinizations as projective objects, J. Geom. Phys. 61 (2011), 594609.[11] V. Chari, B. Ion, BGG Reciprocity for Current Algebras arXiv:1307.1440[12] V. Chari, A. Khare, and T. Ridenour, Faces of polytopes and Koszul algebras, J. Pure Appl. Algebra 216

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    algebras, Comm. Math. Phys. 266 (2006), no. 2, 431454[15] V. Chari and A. Moura, KirillovReshetikhin modules associated to G2, Contemp. Math. 442 (2007) 4159.[16] V. Chari and A.Pressley, Weyl modules for classical and quantum affine algebras, Represent. Theory 5

    (2001), 191223[17] E. Cline, B. Parshall and L. Scott, Finite-dimensional algebras and highest weight categories, J. Reine

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    Department of Mathematics, State University of Campinas, Brazil

    E-mail address: [email protected]

    E-mail address: [email protected]

    http://arxiv.org/abs/1307.1440http://arxiv.org/abs/1012.5480http://arxiv.org/abs/1012.5480http://arxiv.org/abs/1307.1440