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Categories with projective functors [Elektronische Ressource] / Oleksandr Khomenko

48 pages
Ajouté le : 01 janvier 2004
Lecture(s) : 12
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Oleksandr Khomenko
DissertationzurErlangungdesDoktorgradesderFakult¨at fur Mathematik und Physik der ¨ Albert-Ludwigs-Universita¨tFreiburgimBreisgau
November 2003
Erster Referent:Prof. Dr. W.
Zweiter Referent: H. AndersenProf. Dr. H. Datum der Promotion:13 Februar 2004
ACKNOWLEDGEMENTS First of all I thank my scientific advisor Wolfgang Soergel for his friendly supervision, motivational discussions, many useful remarks, patience and for sharing his deep ideas and insights on mathematics. I thank Volodymyr Mazorchuk for useful and friendly discussions, sug-gestions, ideas, hospitality during my visits to Uppsala and for careful reading of this manuscript. I thank my colleagues Bijan Afshordel, Pe-terFiebig,SteenRyom-Hansen,GeraldH¨ohn,ViktorLevandovskyy, VladimirMatveev,OlafSchn¨urer,VsevolodShevchishinandCatharina Stroppel for creating friendly and motivating working atmosphere. Es-pecially I would like to thank my wife Nataliya Koval for emotional support. I Acknowledge the financial support of Freiburg University, Uppsala University, TMR program ”Algebraic Lie Representations”, and DFG project ”Projektive Funktoren und Hecke Algebra”.
Contents 1. Introduction 1 2. Full Projective Functors 5 3. Preliminaries from Representation Theory 13 4. Some subcategories ofg-mod 15 5. CategoryO19 5.1. Projective functors onO19 5.2. Zuckerman functors 20 5.3. Joseph’s version of completion functor 20 5.4. Coapproximation and functorV20 5.5. Arkhipov’s functor and Deodhar’s version of completion functor 21 5.6. Relations between functors onO24 5.7. Parabolic categoryO25 6. Categorical Realization of Arkhipov’s and Joseph’s Functors 26 7. Harish-Chandra bimodules 30 8. Kostant’s Problem 33 9. Structure of Induced Modules 36 References 43
1.Introduction In the representation theory of objects associated to Dynkin dia-grams (e.g. Lie algebras, Lie groups, Quantum groups) one often has translation functors. These functors encode certain symmetry of the representation category. In this paper we axiomatize some properties of these functors in the definition of acategory with full projective functors (see Definition 1) in order to understand better the common features of such categories. A categoryAwith full projective functors, is an abelian category with a distinguished object, called dominant object (an analog of the dominant Verma module inO), and a collection of endofunctors subject to the axioms from Definition 1. 1
2 O.KHOMENKO This approach leads to a better understanding of right exact functors which sufficiently naturally commute with projective functors (see Def-inition 2 for a more precise explanation of “sufficiently natural”). One of the first attempts of axiomatic characterization of some functors on categoryOgoes back to A. Joseph [J83], where he investigates a ver-sion of Enright’s completion functor, in particular he proves that these functors satisfy the braid relations. Later H. H. Andersen, C. Stroppel in [AnS] established some properties of Arkhipov’s functor and V. Ma-zorchuk and the author in [KM1] established some relations between different versions of Enright’s functor, Arkhipov’s functor (introduced in [Ar]) and certain categorically defined functors. One of the main results of the present paper is the following theorem Theorem 1.Right exact additive functors from a categoryAwith full projective functors to an abelian categoryBwhich naturally commute with projective functors are determined up to an isomorphism of func-tors by their values on the dominant object. Moreover one can provide a classification of such functors if one de-scribes the objects ofBwhich can appear as images of the dominant object inA Defi- objects will be called quasi-dominant objects.. Such nition 3 and Theorem 5 provide a description of these objects in terms of projective functors. Another important result of the paper is Theorem 2.Every morphism between the values on the dominant ob-ject of right-exact additive functors which naturally commute with pro-jective functors can be naturally lifted to a morphism of functors. Unfortunately the correspondence in the above theorem is, in gen-eral, only injective. In other words one looses some information about morphisms of functors when evaluating them on the dominant object. An attempt to explain the nature of this phenomena is made at the end of Section 2. Clearly, the dual notions can be developed to deal with left exact additive functors. The above ideas can also be generalized to work in derived categories with full projective functors. Going back to the original motivation, one can apply these results to Joseph’s and Deodhar’s versions of completion functor, Arkhipov’s functor, Zuckerman’s functor, slightly modified Soergel’s combinatorial functorVand various categorically defined functors. After proving or citing the fact that these functors naturally commute with projective functors on the categoryOfor semisimple finite dimensional complex Lie algebrag, one easily gets the braid relations for Joseph’s com-pletion functors and Arkhipov’s functors, a description of Arkhipov’s functor as the twist of Joseph’s completion functor by duality, the
CATEGORIES WITH PROJECTIVE FUNCTORS 3 isomorphism of the (slight modification of) functorVand the square of Arkhipov’s functor, a description of Zuckerman’s functor as left de-rived of Arkhipov’s functor and various categorical realizations of above functors. These relations are proved just by computing corresponding functors on the dominant Verma module. Some of the above relations are already known (see [An, AnS, AnL, Ar, J83, KM1]) some of them seem to be new. A priori the definition of a category with full projective functors depends on the choice of the dominant object. However in many cases one gets equivalent constructions. More precisely we have Theorem 3.Let(A M{Pi i∈ I0})and(A0 M0{Pi0 i∈ I0})be two categories with full projective functors such that (1) there exist an additive functorF: Hom(AA)Hom(A0A0) which induces an equivalence of monoidal categories on cate-gories of projective functors and maps projective functors onA to right exact additive functors onA0 (2) for alli j∈ IandtHomF unc(PiPj)one hastM= 0if and only ifF(t)M0= 0. ThenAandA0are equivalent. Using this theorem one can establish for example the equivalences of various categories of modules which are presentable by certain class of projective modules. In this way one can deduce some properties of arbitraryg-modules from analogous properties of modules inO an. As application we obtain an answer to Kostant’s question (a description of the algebra of adjoint finite endomorphisms) for simple modules with a minimal annihilator. Theorem 4.LetMbe a simpleg-module with minimal annihila-tor. Then the algebra of adjoint finite endomorphisms ofMis a free U(g)/AnnU(g)M-module of finite rank. Moreover the rank of the module in the theorem can be described explicitly (see Section 8). The solution of this problem is well known for Verma modules, and in [Za] the problem was solved for Whittaker modules. We have to remark that the canonical morphism fromU(g) to the algebra of adjoint finite morphisms of a simple module given by multiplication is not surjective in general, (even for modules in category O) see [GJ, S89] for examples. Another application of the technique is a description of the struc-ture of parabolicaly induced modules. This problem was solved by D.Miliˇci´candW.Soergelin[MS]forWhittakermodules.Theirideas was adopted to work in the case of arbitrary simple moduleVwith
4 O.KHOMENKO minimal annihilator by V. Mazorchuk and the author in [KM2]. How-ever, if for some finite dimensional moduleEthe length ofEVdoes not equal to the dimension ofEone may loose some information about subquotients of induced module. The proofs presented here are slight modifications of those in [KM2]. The central idea of [MS] and [KM2] is that the (rough) structure of the module which is parabolicaly induced from a simple module with minimal annihilator depends only on its central character. The methods used in the investigation of Kostant’s problem and of the structure of induced modules can be adopted to work under some restrictions also for simple modules whose annihilator is an induced ideal. A more detailed discussion of the difficulties appearing in this case can be found at the end of Section 8. The discussed technique can also be applied to the category of Harish-Chandra bimodules with (generalized) central character from the right hand side (the last however need to be modified a bit). In this case one obtains two different structures of a category with full projective functors on it coming from the tensoring with finite dimensional mod-ules from the left hand side and from the right hand side. The second one provides another structure of category with full projective functors onOλbe used to deal with translation functors, (“classi-, which can cal”) projective functors, shuffling functors and Zuckerman functors. In particular, one can reprove the result of Bernstein and Gelfand that (“classical”) projective functors onOare determined by their values on the dominant Verma module. Let us describe the structure of the paper. In Section 2 we develop the main abstract constructions. In particular we present the definition of a category with full projective functors and prove the main results concerning description of right (left) exact additive functors which nat-urally commute with projective functors. After fixing notations and recalling some basic results from the representation theory in Section 3 we construct an example of a category with full projective functors and some functors on it in Section 4. The structure of a category with full projective functors onOis discussed in more details in Section 5. Here we also show that many “classical” functors onOnaturally commute with projective functors. Moreover it appears that there are not too much “essentially” different functors among them. In this section we also explain how the parabolic categoryOpcan be viewed as a category with full projective functors. We have to admit that the description of the homomorphism space between two projective functors on para-bolic categoryOseems to be rather difficult problem. Section 6 we In
CATEGORIES WITH PROJECTIVE FUNCTORS 5 present a categorical realization of Joseph’s version of Enright’s com-pletion functors and of Arkhipov’s functor. This result was obtained by V. Mazorchuk and the author in [KM1]. In Section 7 we apply the abstract nonsense to a certain completion of the category of Harish-Chandra bimodules. The Kostant’s problem is attacked in Section 8 and the structure of induced modules is investigated in Section 9. 2.Full Projective Functors LetA,BandCbe abelian categories. By Hom(AB) we denote the category of functorsfromAtoB. The objects of Hom(AB) are all functors fromAtoBand the morphisms are natural transformations of functors. The category Hom(AB) is an additive strict monoidal category with respect to compositionof functors. LetIdAdenote the identity functor onA. For two objectsF GObHom(AB)by HomF unc(F G) we denote the set of natural transformations (mor-phisms of functors) fromFtoG. LetH H0Ob(Hom(BB)) andHHomF unc(H H0 a morphism). ForfHomF unc(F G) andKOb(A) byfK:F(K)G(K) we denote the evaluation offonK, byHf we denote the natural transformation fromHFtoHGdefined by (Hf)N=H(fN) and byHFHomF unc(HF H0F) we denote the natural transformation defined by (HF)N=HF(N)forNOb(A). Definition 1.Acategory with full projective functorsis an abelian categoryAtogether with an objectMOb(A) (calleddominant ob-ject) and a collection of right exact and additive endofunctorsPi,i∈ I (calledprojective functors), closed under taking direct sums and com-positions of functors such that (PF1) The identity functorIdAis a projective functor. (PF2) For everyi∈ Ithe objectPi(M) is projective inA. (PF3) For everyNOb(A) there exist a projective functorPNand an epimorphismfN:PN(M)N. (PF4) For alli j∈ Ithe evaluation map evM: HomF unc(PiPj)HomA(Pi(M)Pj(M)) is surjective. The above definition immediately implies thatMis projective and any projective modulePis a direct summand ofPP(M). LetQbe a projective functor, thenQ(P) is a direct summand ofQ ◦ PP(M), thus it is projective. LetA=¯A M{Pi|i∈ I}be a category with full projective func-¯ tors. ByEA¯we will denote thecategory of projective functors ofA.
6 O.KHOMENKO By definition, the objects ofEA¯are the functorsPi,i∈ Iand mor-phisms are the natural transformations of these functors. For any ob-jectNOb(A) the evaluation onNdefines a functor fromEA¯toA, which we denote by evN construction ev. ByN(Pi) =Pi(N) and for anytHomF unc(PiPj) holds evN(t) =tN. The Axiom (PF4) implies that the functor evMis full. A functorF:EA¯Hom(BB) is calledadmissibleif it is addi-tive,F(IdA) =IdB,F(Pi) is right exact, additive andF(Pi◦ Pj) = F(Pi)F(Pj) for alli j∈ I. For example, ifA=Bthen the identity functor on Hom(AA Fix a category with full) is always admissible. projective functorsA=¯A M{Pi|i∈ I}, an abelian categoryBand an admissible functorF:EA¯Hom(BB) for the rest of this section. Definition 2.A functorG:A → BnaturallyF-commutes with pro-jective functorsif there exist a collection of isomorphisms of functors CiG:F(Pi)GG◦ Pi,i∈ Isuch that for alli j∈ Iand tHomF unc(PiPj) the following diagram commutes:
F(Pi)GF(t)G//F(Pj)G CiGCjG G◦ PGit//G◦ Pj If the functorFis clear from the context we will sometimes omit it. Every morphism between values on the dominant objectMof right ex-act additive functors which naturally commute with projective functors comes from a natural transformation. More precisely we have Proposition 1.LetFandGbe two right exact additive functors from AtoBwhich naturallyF-commute with projective functors, then for every morphismφ:F(M)G(M)there exist a natural (with respect to the composition of morphisms) lift ofφto a morphismfφ:FG of functors such thatfφM=φ. Moreover,φis an isomorphism if and only iffφis. Proof.Givenφ:F(M)G(M) let us constructfφ For allas follows. i∈ Ithe morphismfPφi(M):F(Pi(M))G(Pi(M)) is defined to be 1 equal toCGiMF(Pi)(φ)CiFM. Note that ifφis an isomorphism, thenfφPi(M)are also isomorphisms.
CATEGORIES WITH PROJECTIVE FUNCTORS 7 Leti j∈ IandtHomF unc(PiPj). The top face of the following diagram is commutative by the definition of a natural transformation. (1) F(Pi)(G(M))F(t)G(M)//F(Pj)(G(M)) F(Pi)(φ)mmmmm66mmm66 mmmmmmmmmmmmmFm(mPmjm)m(φ) F(Pi)(F(M))F(t)F(M)//F(Pj)(F(M)) (CjG)M (CiG)M G(P//G(Pj M (CiF)MfmφPmim(mMm)mmmmmm66i(M))G(t)(jF)Mmmmfφmjm(mMmm)mm66(M)) C mmmmmP F(Pi(M))F(tM)//F(Pj(M)) The commutativity of front and back faces is given by Definition 2, and right and left faces commute by construction offφ. Since vertical arrows are isomorphisms one gets the commutativity of the bottom face. By the definition of a category with full projective functors, for every objectNOb(A) there exist projective functorsP1andP2onAand tHomF unc(P1P2) such thatNfits into the following exact sequence P1(M)tM//P2(M)p//N//0. The commutativity of the Diagram (1) implies the commutativity of the left square of the following diagram with exact rows F(P1(M))F(tM)//F(P2(M))F(p)//F(N)//0 fφP(M)fPφ(M)fNφ G(P1(M))G(tM)//G(P2(M))G(p)//G(N)//0 We definefNφto be the morphism making this diagram commutative. By standard argumentsfφNdoes not depend on the choice ofP1,P2and t that if. Noteφis an isomorphism, thenfφNis also an isomorphism. LetN1 N2Ob(A) andgHomB(N1 N2 standard argu-). By mentsgcan be lifted to a morphism between projective presentations ofN1andN2. By the definition of a category with full projective functors, it can be lifted to morphisms between corresponding projec-tive functors. All together we get: there exist projective functorsPji, i j= 12 onAand appropriate natural transformationst1,t2,g1,g2
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