GRADED SIMPLE MODULES AND LOOP MODULES ALBERTOELDUQUE⋆ ANDMIKHAILKOCHETOV† Abstract. Necessaryandsufficientconditions aregivenforaG-gradedsim- ple module over a unital associative algebra, graded by an abelian group G, tobeisomorphictoaloopmoduleofasimplemodule,aswellasfortwosuch 6 loop modules (associated to a subgroup H of G) to be isomorphic to each 1 other. Under some restrictions, these loop modules are completely reducible 0 (as ungraded modules), andsomeoftheir invariants —inertiagroup, graded 2 Brauer invariant and Schur index — which were previously defined for sim- p ple modules over graded finite-dimensional semisimple Lie algebras over an e algebraically closed field of characteristic zero, are now considered in a more S generalandnaturalsetting. 8 ] T 1. Introduction R . Finite-dimensional graded modules for semisimple Lie algebras over an alge- h braicallyclosedfieldF ofcharacteristiczerohavebeen studiedin[EK15a](see also t a [EK15b, Appendix A]). In particular, the graded simple modules are expressed in m terms of simple modules as follows. [ Let L be a (finite-dimensional) semisimple Lie algebra over F graded by an abelian group G, which can be assumed, without loss of generality, to be finitely 2 v generated. A G-grading V = g∈GVg on an L-module V is said to be compatible 8 ifLgVh Vgh for allg,h G,Lthatis, ifit makesVagradedL-module. Moreover, 0 V is said⊆to be graded si∈mple if it does not contain any proper nonzero graded 0 submodule. 3 The G-grading on L is given by a homomorphism of algebraic groups G 0 → 1. Aut(L), χ 7→ αχ, where G = Hom(G,F×) is the group of characters, so thatbthe 60 homFioxgeanCeoaurstacnomsupboanlgenebtsrbaaraengdivaensybstyemLgo=f s{imxp∈leLro:oαtχs(fxo)r=L,χa(ng)dxle∀tχΛ∈+Gd}e.note b 1 thesetofdominantintegralweights. ThegroupofcharactersG(aquasitorus)acts v: on the isomorphism classes of irreducible L-modules, and hence on Λ+. For any b i dominant integral weight λ Λ+, the inertia group is X ∈ r Kλ = χ G:χ fixes λ . a { ∈ } Then K is Zariski closed in G and [G:bK ] is finite. Therefore, λ λ H :=(K b)⊥ = gb G:χ(g)=1 χ K λ λ λ { ∈ ∀ ∈ } is a finite subgroup of G of size H = Gλ, and K is canonically isomorphic to λ λ | | | | the group of characters of G/H . λ b 2010 Mathematics Subject Classification. Primary16W50; Secondary17B70. Key words and phrases. Graded; simple; module; loop module; centralizer; central image; inertiagroup;gradedBrauerinvariant;gradedSchurindex. ⋆SupportedbytheSpanishMinisteriodeEconom´ıayCompetitividad—FondoEuropeodeDe- sarrolloRegional(FEDER)MTM2013-45588-C3-2-P,andbytheDiputaci´onGeneraldeArago´n— FondoSocialEuropeo(GrupodeInvestigacio´n deA´lgebra). †SupportedbyDiscoveryGrant341792-2013oftheNaturalSciencesandEngineeringResearch Council(NSERC)ofCanada. 1 2 A.ELDUQUEANDM.KOCHETOV LetVλbethesimpleL-modulewithhighestweightλand̺λ :U(L) EndF(Vλ) → the associated representation of the universal enveloping algebra. The G-grading on L induces naturally a G-grading on U(L). In general, we cannot expect V to λ admit a compatible G-grading. However, there is a homomorphism of algebraic groups Kλ Aut EndF(Vλ) −→ χ α˜ , (cid:0) (cid:1) χ 7→ where α˜ ̺ (x) = ̺ α (x) for all x L. This corresponds to a (G/H )- χ λ λ χ λ grading on(cid:0) EndF(cid:1)(Vλ) su(cid:0)ch tha(cid:1)t ̺λ becom∈es a homomorphism of (G/Hλ)-graded algebras, where the (G/H )-grading on L is given by L = L for all g G(acoarseningoftheλG-grading). Theclass[EndF(VgλH)λ]intLheh(∈GH/λHλg)h-graded ∈ Brauer group (see [EK15a, 2] or the end of Section 5 here) is called the graded § Brauer invariant of λ, and the degree of the graded division algebra representing [EndF(Vλ)] is called the graded Schur index of λ. For each G-orbit O in Λ+, select a representative λ. If k is the graded Schur index of λ, then the direct sum Vk of k copies of V is equipped with a compatible b λ λ (G/H )-grading. Finally, let W(O) be the induced module: λ b W(O)=IndGKλVkλ :=FG⊗FKλ Vkλ. b (HereFGdenotesthegroupalgebraofG,andsimilarlyforFK .) Then,by[EK15a, λ Theorem 8], up to isomorphism and shift of the grading, these W(O) are the G- b b graded simple finite-dimensional L-modules. In this way, the graded simple modules are obtained very explicitly in terms of simple modules. The inertia groups, graded Brauer invariants and Schur indices have been computed for the classical simple Lie algebras in [EK15a] and [EK15b, Appendix A]. A different, though less explicit, approachto describing the gradedsimple mod- ules is given by Billig and Lau in [BL07], based on the notion of thin coverings. It is proved in [BL07, Theorem 1.4] that any graded simple module that contains a maximal (ungraded) submodule can be obtained from a simple module endowed with a thin covering. Finally, in a recent work by Mazorchuk and Zhao, it is shown that if L is a Lie algebra over an algebraically closed field F, graded by a finitely generated abelian group G, and W is a graded simple L-module with dimW < F, then there is a subgroup H of G and a simple L-module V with a compati|bl|e G/H-grading satisfying some extra conditions such that W is the loop module of V, that is, the subspace VgH g V FFG, ⊗ ⊆ ⊗ gM∈G withtheG-gradinggivenbyW =V gandL-actiongivenbyx(v g)=xv g′g g gH for all g,g′ G, x Lg′ and v Vg (se⊗e [MZpr, Theorem 31]). ⊗ ⊗ ∈ ∈ ∈ The aim of this paper is twofold. First we will look at [BL07] and [MZpr] from a different perspective, based on the results by Allison, Berman, Faulkner and Pianzola [ABFP08] on the connection between graded simple algebras and loop algebras of simple, graded algebras. This will allow us to extend the results in [MZpr] and, we hope, put them in a more natural context. Second, we will relate this general (and less explicit) approach with our results in [EK15a] and [EK15b]. Since modules for a Lie algebra are just left modules for its universal enveloping algebra, in this paper, we will mostly work in the more general setting of left GRADED SIMPLE MODULES AND LOOP MODULES 3 modules for an associative algebra. The ground field F will be arbitrary unless indicated otherwise. It should also be noted that any (nonassociative) algebra A is a left module for its multiplication algebra, that is, the unital associative subalgebra of EndF(A) generated by the left and right multiplications by elements of A. (Recall that the centralizer of this module is called the centroid of A, which is commutative if A = A2.) Hence, our results in this paper will include the case of graded simple algebras, thus generalizing [ABFP08]. The paper is organized as follows. In Section 2, we recall the main definitions concerning gradedalgebras and modules. In particular, the centralizer of a graded simple module will play a key role throughout the paper. Section 3 is devoted to loop modules. Given an abelian group G, a G-graded unital associative algebra R and a subgroup H of G, R is naturally G/H-graded. Let π : G G/H be the natural homomorphism. For a left G/H-graded left R-module V→, the loop module L (V) is the subspace V g V FG, which is a G-graded left R-moduπle. It turns out thatLagG∈G-gragdHed⊗left⊆R-m⊗odule W is isomorphic to a loop module L (V) for a G/H-graded module V if and only π if the graded centralizer of W contains a graded subfield that is isomorphic, as a G-gradedalgebra,to the groupalgebraFH (Proposition3.4). Ifthe groundfieldF isalgebraicallyclosed,allfinite-dimensionalG-gradedsimpleleftR-modulessatisfy this condition and hence are isomorphic to loop modules (see Proposition 3.5). If, inaddition,H isfiniteandthecharacteristicofFdoesnotdivide H thentheloop | | module Lπ(V) is isomorphic to the induced module FG FK V (Proposition 3.8). In Section 4, we define two groupoids, M(π) and N(⊗π), for G, H, π and R as b above. The objects of M(π) are the simple, central, G/H-graded left R-modules with a certain tightness condition on the grading, while the objects of N(π) are the pairs (W,F) where W is a G-graded simple left R-module and F is a maximal gradedsubfieldofitscentralizerthatisisomorphictoFH asagradedalgebra. The loop functor L : M(π) N(π) is defined on objects as V L (V),L (F1) π π π → 7→ and shown to be faithful and essentially surjective (Theorem 4.14)(cid:0). The objects o(cid:1)f M(π)whoseimageinN(π)isisomorphicto(W,F)aredeterminedexplicitlyasthe central images of (W,F). A certain extension of the loop functor turns out to be an equivalence of categories (Theorem 4.16). Section 5 deals with the case of finite H. We will also assume that the ground field F is algebraically closed and its characteristic does not divide H . Then, for any object (W,F) of N(π), W is completely reducible as an ungrad|ed| R-module, and the simple submodules of W are, up to isomorphism, the central images of (W,F), so they are endowedwith the structure of G/H-gradedmodules (Theorem 5.1). Moreover, the isotypic components of W are actually G/Z-graded simple modules, where Z H is the support of the center of the centralizer C(W) of W (Theorem 5.7 and C≤orollary 5.8). For any G-graded simple left R-module W such that the dimension of C(W) is finite and not divisible by charF, we will define the inertia group, graded Brauer invariant, andgraded Schur index, thus extendingthe scope of the definitions given in [EK15a]. In Section 6, we consider a finite-dimensional G-gradedsimple left R-module W andassume that F is algebraicallyclosedandcharF does notdivide the dimension of C(W). We prove that if V is any simple (ungraded) submodule of W, then EndF(V) is endowed with a unique grading by G/Z compatible with the induced G/Z-grading on R, where Z is the support of the center of C(W), and the graded Brauer invariant of W is precisely the class [EndF(V)] in the G/Z-graded Brauer group(Theorem6.3andCorollary6.4). Moreover,Visendowedwithastructureof 4 A.ELDUQUEANDM.KOCHETOV G/H-gradedmoduleforanysubgroupH withZ H Gsuchthat C(W) is a maximal commutative graded subalgebra of≤C(W)≤. Lh∈H h Finally, in Section 7, assuming F algebraically closed of characteristic zero, we give a necessary and sufficient condition for a finite-dimensional simple left R- module to be a submodule of a G-graded simple module (Theorem 7.1). This condition is satisfied in the situation considered in [EK15a], which explains why every finite-dimensional simple module appears as a submodule of a gradedsimple module in that case. 2. Graded simple modules Throughoutthis work,Fdenotesa groundfield, whichis arbitraryunless stated otherwise. All vector spaces, algebras and modules are assumed to be defined over F. We start by reviewing some basic definitions and facts about gradings. For material not included here, the reader is referred to [EK13]. LetGbeanabeliangroup(writtenmultiplicatively,withneutralelemente)and let V be a vector space. A G-grading on V is a vector space decomposition Γ:V= V . (2.1) g gM∈G The support of Γ is the set Supp(Γ)= g G:V =0 . We will sometimes write g Supp(V) if Γ is fixed. If 0 = v V , v{is s∈aid to be6 hom}ogeneous of degree g, and g V is calledthe homogeneou6 s co∈mponent of degree g. If the gradingΓ is fixed, then g V will be referred to as a graded vector space. A subspace W V is said to be a graded subspace if W = (W V ). A graded homomorp⊆hism of G-graded g∈G ∩ g spaces is a linear map thatLpreserves degrees. A linear map f : V W of G-graded vector spaces is said to be homogeneous of degree g if f(V ) V→ for all h G. Thus the graded homomorphisms are the h gh ⊆ ∈ homogeneous linear maps of degree e. Denote the space of all homogeneous linear maps of degree g by Hom (V,W) and set Homgr(V,W) := Hom (V,W). If g g∈G g dimV is finite, then Homgr(V,W) = Hom(V,W), and thusLHom(V,W) becomes G-graded. For a group homomorphism α : G H and a G-grading Γ as in (2.1), the decomposition αΓ:V= V′, wher→e V′ := V , is an H-grading on h∈H h h g∈α−1(h) g V, called the grading indLuced by Γ by means of αL. In particular if Γ is a G-grading on V, H is a subgroupof G and π :G G/H is the natural homomorphism, then → πΓ will be called the G/H-grading induced by Γ. GivenanalgebraA(notnecessarilyassociative),aG-gradingonAisaG-grading as a vector space: A= A , that satisfies A A A for all g,h G. Then AwillbereferredtoasLaGg∈-gGradged algebra. Notetghathi⊆fAigshunitalthen∈itsidentity element 1 lies in A . e Let now R be a unital associative G-graded algebra. A G-graded left R-module is a left R-module V that is also a G-gradedvector space, V= V , such that g∈G g RgVh Vgh for all g,h G. The G-grading on V is then saLid to be compatible withth⊆eG-gradingonR.∈AG-graded right R-module isdefinedsimilarly. Onsome occasions, the action of R on V will not be denoted by juxtaposition, but as r v, r v, etc.; in such cases we will refer to the R-modules as (V, ), (V, ), etc.·A • · • graded submodule isagradedsubspacethatisalsoasubmodule. Ahomomorphism of G-graded modules (or G-graded homomorphism) is a linear map that is both a homomorphism of modules and of G-graded vector spaces. The vector space of graded homomorphisms between two G-graded left R-modules V and W will be denoted by HomGR(V,W). Note that HomGR(V,W) = HomR(V,W) Home(V,W). ∩ GRADED SIMPLE MODULES AND LOOP MODULES 5 WewilldenotebyRModG theabeliancategorywhoseobjectsaretheG-gradedleft R-modules and whose morphisms are the G-graded homomorphisms. We will follow the convention of writing endomorphisms of left modules on the right. Let V and W be G-graded left R-modules. The space Homgr(V,W) := R Homgr(V,W) HomR(V,W) is a graded subspace in Homgr(V,W). When V=W, we obtain a G∩-graded algebraCgr(V):=Homgr(V,V), called the graded centralizer R of V. Then V becomes a graded R,Cgr(V) -bimodule. A nonzero G-graded left R-m(cid:0)odule V is(cid:1)said to be G-graded simple if its only gradedsubmodules are 0 and V. If G or R are clear from the contextwe may omit them and refer to graded left modules, graded simple modules, etc. Proposition 2.1. Let G be an abelian group, let R be a unital associative G- graded algebra, and let V be a graded simple left R-module. Then its centralizer C(V):=HomR(V,V) coincides with its graded centralizer Cgr(V). Proof. Let 0 = v V be a homogeneous element. Since V is graded simple, g V = Rv. For6any 0∈= f C(V), we can write vf = v + +v for some natural number n an6d som∈e homogeneous elements v gg1 V···. Tghgenn, for any h H and r R , ggi ∈ ggi h h ∈ ∈ (r v)f =r (vf)=r v + +r v . h h h gg1 ··· h ggn But R v = V , so V f R V R V for all h G. Therefore, V f hV hg V hg fo⊆r allhhhggG1,⊕so··f·⊕ hn hgCgngr(V) C∈gr(V). (cid:3) h ⊆ hg1 ⊕···⊕ hgn ∈ ∈ i=1 gi ⊆ L A unital associative graded algebra is a graded division algebra if every nonzero homogeneouselementisinvertible. Commutativegradeddivisionalgebrasarecalled graded fields. Schur’s Lemma shows that the centralizer of a simple module is a division algebra. In the same vein, the graded Schur’s Lemma shows that the cen- tralizerofanygradedsimple module is a gradeddivisionalgebra(see,for instance, [EK13, Lemma 2.4]), and hence the module is free over its centralizer. A module V is called central (or Schurian) if its centralizerC(V) consists of the scalar multiples of the identity map. Similarly, a graded module V will be called graded central if C(V) consists of the scalar multiples of the identity map. e Theorem 2.2. Let G be an abelian group and let R be a unital commutative G- graded algebra. Suppose that V is a graded simple left R-module and let D=C(V). (i) (Graded Density) If v ,...,v V are homogeneous elements that are lin- 1 n early independent over D, the∈n for any w ,...,w V there exists an 1 n element r R such that rv =w for i=1,...,n. ∈ i i ∈ (ii) Vis graded centralif and only if V FK is a graded simple (R FK)-module ⊗ ⊗ for any field extension K/F. Proof. A proof of (i) appears in [EK13, Theorem 2.1]. Now,let V be gradedcentraland consider a nonzero homogeneouselement v 1 α1+ +vn αn inV FK,wherewemayassumev1, ,vn linearlyindependen⊗t over F···and 0⊗=α ,...,⊗α K. Since D =F1 and D i·s··a gradeddivision algebra, 1 n e eachnonzero6homogeneous∈component D is one-dimensional(for any 0=d D , g g the map D D , x xd, is a bijection). It follows that the elements6v ,.∈..,v e g 1 n are linearly i→ndepende7→nt over D. For any nonzero homogeneous element v V, by graded density there is an element r R such that rv = v and rv = 0∈for 1 i i = 2,...,n. Hence v α belongs to th∈e R-submodule generated by v α + 1 1 1 +vn αn. But V ⊗FK = (R FK)(v α1), so our arbitrarily chosen n⊗onzero h··o·mogen⊗eous element⊗generates ⊗V F K ⊗as a (R F K)-module, proving graded ⊗ ⊗ simplicity. 6 A.ELDUQUEANDM.KOCHETOV On the other hand, if V is not graded central, then D = F1, and for any d e De F1,K=F(d)isasubfieldofDe. ThenV FK=(V KK6) FK V K(K FK∈). But\K FKcontainsaproperidealI(thekern⊗elofthem⊗ultiplic⊗ation≃z1 ⊗z2 ⊗z1z2), and V⊗KI is a proper graded submodule of V FK. ⊗ 7→ (cid:3) ⊗ ⊗ Later on we will make use not only of G-gradings but also of G-pregradings, or G-coverings (see [Smi97] and [BL07]). Definition 2.3. Let G be an abelian group and let R be a unital associative G- graded algebra. Let V be a left R-module. A family of subspaces Σ = V : g G is called a G-pregrading on V if g • V= V and R V V{ for a∈ll g,}h G. g∈G g g h ⊆ gh ∈ GivePn two pregradings Σi = Vi : g G , i = 1,2, Σ1 is said to be a • { g ∈ } refinement of Σ2 (or Σ2 a coarsening of Σ1) if V1 V2 for all g G. If at g ⊆ g ∈ leastoneofthesecontainmentsisstrict,therefinementissaidtobeproper. A G-pregrading Σ is called thin if it admits no proper refinement. • Example 2.4. With G, R and V as in Definition 2.3, let H be a subgroup of G and consider the induced G/H-grading on R. Write G = G/H and g¯ = gH for g G. Assume that ∈ V= V (2.2) g¯ M g¯∈G is a G-grading on V making it a G-graded left R-module. Then the family Σ := V′ : g G , where V′ = V for all g G, is a G-pregrading of V. This is called { g ∈ } g g¯ ∈ the G-pregrading associated to the G-grading in (2.2). 3. Loop modules For the rest of the paper, G will denote an abelian group, H a subgroup of G, G will denote the quotient G/H, g¯= gH, and π : G G will denote the natural homomorphism: π(g)=g¯forallg G. Also,R willb→eaunitalassociativealgebra equipped with a fixed G-grading Γ∈: R = R . This G-grading on R induces g∈G g the G-grading πΓ. L Definition 3.1. Let V= V be a G-graded left R-module. The direct sum g¯∈G g¯ L Lπ(V):= Vg¯ g V FFG gM∈G ⊗ (cid:16)⊆ ⊗ (cid:17) with the left R-action given by rg′(vg¯ g)=(rg′vg¯) (g′g) ⊗ ⊗ for all g,g′ G, rg′ Rg′, and vg¯ Vg¯, is a G-graded left R-module, called the loop module∈of V rela∈tive to π. ∈ Remark 3.2. ThemoduleL (V)isdenotedbyM(G,H,V)in[MZpr]. Ournotation π is inspired by the notation for loop algebras in [ABFP08]. Therein, given a G- graded algebra A, the loop algebra is defined as L (A)= A g, which is a subalgebra of A FFG. This generalizes a well kπnown coLnsgt∈ruGctiog¯n⊗in the theory ⊗ of Kac–Moody Lie algebras. Remark 3.3. There is a natural ‘forgetful’ functor Fπ : RModG RModG. The imageofaG-gradedleftR-moduleWisWitselfwiththeinducedG→-grading(W := g¯ W for all g G). The loop construction gives us a functor in the reverse h∈H gh ∈ Ldirection: Lπ : RModG RModG. It is easy to see that Lπ is the right adjoint of → GRADED SIMPLE MODULES AND LOOP MODULES 7 F . Indeed, foranyG-gradedleft R-module WandanyG-gradedleft R-moduleV, π the map HomG F (W),V HomG W,L (V) R π R π −→ (cid:0) f(cid:1) [f˜:w (cid:0) f(w ) (cid:1)g], g g 7→ 7→ ⊗ for any g G and w W , is a bijection whose inverse is the map g g ∈ ∈ HomG W,L (V) HomG F (W),V R π R π −→ (cid:0) ϕ(cid:1) [ϕ¯:w (cid:0) (id ǫ)((cid:1)ϕ(w ))], g g 7→ 7→ ⊗ for any g G and w W , where ǫ:FG F is the augmentation map: ǫ(g)=1 g g ∈ ∈ → for all g G. ∈ A clue to understanding some of the main results in this paper is the following observation: Proposition 3.4. A G-graded left R-module W is graded isomorphic to a loop module L (V) for a G-graded left R-module V if and only if its graded centralizer π Cgr(W) contains a graded subfield isomorphic to the group algebra FH. Proof. If V is a G-graded left R-module, for any h H, the linear map δ : h L (V) L (V), given by (v g)δ =v (gh) for all g∈ G and v V =V , lies π → π ⊗ h ⊗ ∈ ∈ g¯ gh inCgr L (V) . Thelinearspanof δ :h H isagradedsubfieldofCgr L (V) , π h π { ∈ } isomor(cid:0)phic to(cid:1)the group algebra FH. (cid:0) (cid:1) Conversely,ifWisaG-gradedleftR-moduleandFisagradedsubfieldofCgr(W) isomorphic to FH, then F = Fc , with c c = c for all h ,h H. Let ρ : F F be the homomLorhp∈hHismhdefined bhy1 ρh(2ch) =h11h2for all h 1 H2 ∈(that is, the aug→mentation map if we identify F and FH), and let V = W∈/Wker(ρ). Then V is naturally G-graded, because so is W (with the G-grading induced by the G-grading) and Wker(ρ) is a G-graded submodule of W. The linear map φ:W L (V) given by φ(w) =(w+Wker(ρ)) g, for all g G and w W , is π g a G-gr→aded homomorphism. Also, for any g G⊗and h H, W∈ =W c ∈, hence gh g h ∈ ∈ W = W =W F =W (F1 ker(ρ)), g¯ gh g g ⊕ hM∈H so the map W W/Wker(ρ) g −→(cid:16) (cid:17)g¯ w w+Wker(ρ), 7→ is a linear isomorphism. Hence φ is an isomorphism. (cid:3) If F is algebraically closed then, for many G-graded simple modules, the cen- tralizer contains maximal graded subfields that are graded isomorphic to group algebras: Proposition 3.5. Let W be a G-graded simple left R-module. (i) C(W)=Cgr(W) contains maximal graded subfields. (ii) If W is graded central and F is algebraically closed, any graded subfield of C(W) is isomorphic to the group algebra of its support. (iii) IfFisalgebraicallyclosedanddimW< F (thesemaybeinfinitecardinals), then W is graded central. | | 8 A.ELDUQUEANDM.KOCHETOV Proof. RecallthatC(W)=Cgr(W)byProposition2.1,soC(W)isagradeddivision algebra. For(i),notethatF1isagradedsubfieldofC(W)andhenceZorn’sLemma guarantees the existence of maximal graded subfields. If W is graded central, C(W) = F1 and then dimC(W) = 1 for any g in e g the support of W. Then if F is algebraically closed and F is a graded subfield, we have F = C(W) and H := Supp(F) is a subgroup of G. But h∈Supp(F) h for all h H,LC(W) = Fx for some 0 = x . Then x x = σ(h ,h )x , where σ :∈H H hF× isha symmetric 62-cohcycle. Henhc1ehF2 is a co1mm2utaht1ihv2e twisted group×algebr→a of the abelian group H. Since F is algebraically closed, F is graded isomorphic to the group algebra FH (see e.g. [Pas85, Chapter 1, Lemma 2.9]; basically, this is a restatement of the fact that Ext(H,F×) = 0 since F× is a divisible abelian group). This proves (ii). Part (iii) is proved in [MZpr, Theorem 14]. (cid:3) The next result is transitivity of the loop construction (as expected in view of Remark 3.3): Proposition 3.6. Let K H G and let π′ : G G/K and π′′ : G/K G/H be the natural homomorph≤isms.≤Then, for any G/H→-graded module V, L →(V) and π Lπ′ Lπ′′(V) are isomorphic as G-graded modules. (cid:0) (cid:1) Proof. We haveLπ′ Lπ′′(V) = g∈GLπ′′(V)gK⊗g = g∈G VgH⊗gK ⊗g,and hence the map (cid:0) (cid:1) L L (cid:0) (cid:1) Lπ(V)= VgH g Lπ′ Lπ′′(V) , vgH g (vgH gK) g, ⊗ → ⊗ 7→ ⊗ ⊗ gM∈G (cid:0) (cid:1) is a G-graded isomorphism. (cid:3) Under certain conditions, the loop module construction is isomorphic to a well- knownconstructionofinducedmodules,whichweusedin[EK15a]. Assumefornow thatthesubgroupH isfiniteandthatFisalgebraicallyclosedanditscharacteristic does not divide n= H . Denote by G the group of characters of G, that is, group | | homomorphisms χ : G F×. The subgroup H⊥ = χ G : χ(h) = 1 h H is → b { ∈ ∀ ∈ } naturally isomorphic to the group of characters of G (recall that G=G/H). b Let V be a G-graded left R-module. Then V is a module for the group algebra F(H⊥) with χ v =χ(g)v g¯ g¯ · for χ H⊥, g G and v V . g¯ g¯ ∈ ∈ ∈ Since F is algebraically closed (and hence F× is a divisible abelian group), any character of H extends to a character of G, and since charF does not divide n, we have H = n. A transversal of H⊥ in G (that is, a set of coset representatives | | of H⊥ in G) is any subset χ ,...,χ in G such that the restrictions to H are b { 1 n} b distinct, so H = χ ,...,χ . Then FG=χ F(H⊥) χ F(H⊥). b { 1|H n|H} b 1 ⊕···⊕ n The induced FG-module b b b Iπ(V):b=IndGH⊥(V)=FG⊗F(H⊥)V=χ1⊗V⊕···⊕χn⊗V is a left R-module by means of b rg′(χj vg¯)=χj(g′)−1χj rg′vg¯ (3.1) ⊗ ⊗ for any j =1,...,n, g,g′ G, rg′ Rg′ and vg¯ Vg¯. ∈ ∈ ∈ Definition 3.7. Given an automorphism α of R and a left R-module V, we may define a new left R-module Vα =(V, ) which equals V as a vector space, but with the new action given by r v = α(r)v∗. This module Vα is referred to as V twisted ∗ by the automorphism α. GRADED SIMPLE MODULES AND LOOP MODULES 9 Equation (3.1) tells us that, as a left R-module, I (V) is the direct sum of its π submodules χ V, j = 1,...,n, and each χ V is isomorphic to the module V j j ⊗ ⊗ twisted by α−1, where the automorphism α , for any χ G, is given by χj χ ∈ α :R R, r χ(g)r , b (3.2) χ g g → 7→ for all g G and r R . g g ∈ ∈ Since V and its twists are G-graded, I (V) has a natural G-grading, with ho- π mogeneous component of degree g¯ being χ V χ V . Clearly, any 1 g¯ n g¯ ⊗ ⊕···⊕ ⊗ χ H⊥ acts on this component as the scalar χ(g¯), and each of the χ restricts to j ∈ a diagonalizableoperator. It follows that the G-gradingonI (V) canbe refined to π a G-grading: I (V) := x χ V χ V :χ x=χ(g)x χ G . (3.3) π g 1 g¯ n g¯ { ∈ ⊗ ⊕···⊕ ⊗ · ∀ ∈ } b (The sum of these subspaces is direct because G separates points of H, that is, for any e=h H, there exists χ G such that χ(h)=1.) 6 ∈ ∈ b 6 Proposition 3.8. Assume that Hb is finite, F algebraically closed and charF does not divide n = H . Choose a transversal χ ,...,χ of H⊥ in G. Let V be a 1 n G-graded left R-|mo|dule and consider the lin{ear maps } b ϕ:L (V) I (V) π π −→ n v g χ (g)−1χ v , g¯ j j g¯ ⊗ 7→ ⊗ Xj=1 for any g G and v V , and g¯ g¯ ∈ ∈ ψ :I (V) L (V) π π −→ 1 χ v χ (gh)v gh, j g¯ j g¯ ⊗ 7→ n ⊗ hX∈H for any j =1,...,n, g G and v V . Then we have: g¯ g¯ ∈ ∈ (i) ϕ does not depend on the choice of transversal χ ,...,χ ; 1 n { } (ii) ϕandψ areaG-gradedmapswithrespecttotheG-gradingonI (V)defined π by (3.3); (iii) ϕ and ψ are homomorphisms of R-modules; (iv) ϕ and ψ are inverses of each other; (v) I (V) is a G-graded R-module isomorphic to L (V). π π Proof. Let χ˜ = χ ̟ where ̟ H⊥ for any j = 1,...,n. Then, for any g G j j j j and v V , we have ∈ ∈ g¯ g¯ ∈ n n χ˜ (g)−1χ˜ v = χ (g)−1̟ (g)−1χ ̟ v j j g¯ j j j j g¯ ⊗ ⊗ · Xj=1 Xj=1 n n = χ (g)−1̟ (g)−1χ ̟ (g)v = χ (g)−1χ v . j j j j g¯ j j g¯ ⊗ ⊗ Xj=1 Xj=1 This proves (i). 10 A.ELDUQUEANDM.KOCHETOV For(ii), weclearlyhaveϕ L (V) χ V χ V . Letus verifythat π g 1 g¯ n g¯ ⊆ ⊗ ⊕···⊕ ⊗ χ ϕ(v g)=χ(g)ϕ(v g(cid:0)) for any(cid:1)g G, χ G and v V . Indeed, g¯ g¯ g¯ g · ⊗ ⊗ ∈ ∈ ∈ n b χ ϕ(v g)= χ (g)−1χχ v g¯ j j g¯ · ⊗ ⊗ Xj=1 n =χ(g) (χχ )(g)−1(χχ ) v =χ(g)ϕ(v g), j j g¯ g¯ ⊗ ⊗ Xj=1 where we have used (i). Therefore, ϕ is a G-graded map. The result for ψ will follow from (iv). Now consider (iii). For j = 1,...,n, g ,g G, r R and v V , we 1 2 ∈ g1 ∈ g1 g¯2 ∈ g¯2 obtain: ϕ r (v g ) =ϕ(r v g g ) g1 g¯2 ⊗ 2 g1 g¯2 ⊗ 1 2 (cid:0) (cid:1) n = χ (g g )−1χ r v j 1 2 j ⊗ g1 g¯2 Xj=1 n = χ (g )−1χ (g )−1χ r v j 1 j 2 j ⊗ g1 g¯2 Xj=1 n =r χ (g )−1χ v g1(cid:16)Xj=1 j 2 j ⊗ g¯2(cid:17) =r ϕ(v g ), g1 g¯2 ⊗ 2 so ϕ is a homomorphism of R-modules. The result for ψ will follow from (iv). Finally, for j =1,...,n, g G and v V , g¯ g¯ ∈ ∈ n 1 ϕψ(χ v )= χ (gh)−1χ (gh)χ v j g¯ i j i g¯ ⊗ n ⊗ hX∈HXi=1 n 1 = χ (g)−1χ (g) χ (h)−1χ (h) χ v i j i j i g¯ Xi=1 (cid:16)nhX∈H (cid:17) ⊗ =χ v , j g¯ ⊗ where the last equality follows from the first orthogonality relation for characters of a finite group, since H = χ ,...,χ . Also, 1 H n H { | | } n b 1 ψϕ(v g)= χ (g)−1χ (gh)v gh g¯ j j g¯ ⊗ n ⊗ hX∈HXj=1 n 1 = χ (h) v gh j g¯ hX∈H(cid:16)nXj=1 (cid:17) ⊗ =v g, g¯ ⊗ by the second orthogonality relation for characters of a finite group. This proves (iv) and hence (v). (cid:3) We return to the general setting. Our next result rephrases [MZpr, Lemma 27]. Proposition 3.9. Let V be a G-graded left R-module. (i) If L (V) is G-graded simple, then V is G-graded simple. π (ii) IfVis G-graded simple, thenL (V)is G-graded simple if andonlyif theG- π pregradingonVassociated toitsG-grading(seeDefinition2.3andExample 2.4) is thin.