INDECOMPOSABLE MODULES OF SOLVABLE LIE ALGEBRAS PAOLOCASATI,ANDREAPREVITALI,ANDFERNANDOSZECHTMAN 7 1 0 Abstract. We classify all uniserial modules of the solvable Lie algebra g = hxi⋉V, 2 where V is an abelian Liealgebra over an algebraically closed field ofcharacteristic 0 b andxisanarbitraryautomorphismofV. e F 8 1. Introduction ] T Let F be analgebraicallyclosedfieldofcharacteristic0. Allvectorspaces, including R allLiealgebrasandtheirmodules,areassumedtobefinitedimensionaloverF. . Recallthatamoduleissaidtobeindecomposableifitcannotbedecomposedasthedi- h rectsumoftwonon-trivialsubmodules. Naturally,knowingallindecomposablemodules t a of a given Lie algebra would provide a complete description of all its modules. Unfor- m tunately, the problem of classifying all indecomposable modules of a given Lie algebra [ -thatisnotsemisimpleorone-dimensional-isvirtuallyunsolvable,eveninthecaseofthe two-dimensional abelian Lie algebra, as observed in a celebrated paper by Gelfand and 2 v Ponomarev[GP]. 4 In spite of this fact, many types of indecomposable modules of non-semisimple Lie 8 algebras have been recently classified, see for example [CGS, CMS, CS, CS1, D, DdG, 0 DP,DR,P] 0 Inallthese papersthecentralidea isto considerparticularclasses ofindecomposable 0 . modulesforwhichacompleteclassificationcanbeachieved.Besidestheirreduciblemod- 2 ules,thesimplesttypeofindecomposablemoduleis,inacertainsense,theuniserialone. 0 Thisisa modulehavingauniquecompositionseries, i.e. anon-zeromodulewhosesub- 7 1 modulesformachain. Alternatively,suchmodulescanbedefinedasfollows. : LetgbeagivenLiealgebraandletU beanon-zerog–module.Thesocleseries v i 0=soc (U)⊂soc (U)⊂···⊂soc (U)=U X 0 1 k r ofU isinductivelydefinedbydeclaringsoci(U)/soci−1(U)tobethesocleofU/soci−1(U), a thatis, thesum ofallirreduciblesubmodulesofU/soc (U), for1 ≤ i ≤ k. ThenU is i−1 uniserialifandonlyifthesocleseriesofU hasirreduciblefactors. Inthelastyears,theclassificationoftheuniserialmodulesofimportantclassesofsolv- ableandperfectLiealgebrashasbeenachievedinvariousresearchpapers[CGS,CS,CS1, Pi, C]. In particular, [Pi] and [C] classify a wider class of modules, called cyclic in [Pi] andperfectcyclicin[C],overtheperfectLiealgebrassl(2)⋉F2 andsl(n+1)⋉Fn+1,for F =C,respectively. Theaimofthispaperistoproceedfurtherinthestudyofuniserialmodules. Weshall, indeed, classify the uniserial modules of a distinguished class of solvable Lie algebras, namely those of the form g = hxi ⋉ V, where V is an abelian Lie algebra and x is an arbitraryautomorphismofV. 2010MathematicsSubjectClassification. 17B10,17B30. Keywordsandphrases. uniserialmodule;indecomposablemodule;Clebsch-Gordanformula. 1 2 PAOLOCASATI,ANDREAPREVITALI,ANDFERNANDOSZECHTMAN A properidealaof g isofthe forma = W, whereW isan x-invariantsubspaceof V. Thus, either W = V and g/a (cid:27) hxi is one-dimensional, or else g/a (cid:27) hxi ⋉ V, where V =V/W ,(0)andxistheautomorphismthatxinducesonV. Weknowfrom[CS]alluniserialmodulesoveranabelianLiealgebraaswellasalluni- serialg-moduleswhenxisdiagonalizable.Thus,itsufficestoclassifyallfaithfuluniserial g-moduleswhenxisnotdiagonalizable.Inthisregard,ourmainresultsareasfollows. In §2 we construct a family of non-isomorphic faithful uniserial representations of g whenxactsonV viaasingleJordanblockofsizen>1. Thisfamilyconsistsofallmatrix representations (1.1) R →gl(n+1), R →gl(n+1), R →gl(n+1), α,k,X α,n α,1 where α∈ F, 1<k <n, X ∈ M , k−1,n−k aswellasthematrixrepresentations (1.2) R →gl(n+2), α,a whichexistonlyforoddn,andwhere α∈F, a=(a ,...,a ), a =1, a =0foralleveni. 1 n 1 i In §3, we show that if x acts on V via a single Jordan block of size n > 1, then ev- ery faithful uniserial g-module is isomorphic to one and only one of the representations appearingin(1.1)and(1.2). In §4 we deal with the general case. By our results from §3, we may assume that x has e Jordan blocks, where e > 1. Moreover, as indicated above, we may also assume that x isnotdiagonalizable. Underthese assumptions. Theorem4.1givesnecessaryand sufficientconditionsforgtohaveafaithfuluniserialmoduleandclassifiesallsuchmodules whenevertheseconditionsaresatisfied. Indeed,let (1.3) V =V ⊕···⊕V 1 e beadecompositionofV intoindecomposableF[x]-submodules(thismeansthatxactson eachV viaaJordanblock)ofdimensions i n=n ≥···≥n , 1 e wheren > 1because xisnotdiagonalizable. Foreach1 ≤ i ≤ e,considerthesubalgebra g =hxi⋉V ofg,wherex = x| . i i i i Vi SupposethatghasafaithfuluniserialrepresentationR:g→gl(d).Thentherestriction R :g →gl(d)isalreadyuniserial.Inparticular, 1 1 d =n+1ord =n+2. In the first case, R is isomorphic to a unique R , the automorphism x has a single 1 α,k,X eigenvalueλ,andtheJordandecompositionofxis (1.4) Jn(λ)⊕Jn2(λ)⊕···⊕Jne(λ), where (1.5) n ≤n−2, n ≤n−4, n ≤n−6,..., n ≤n−2(e−1), 2 3 4 e and (1.6) e≤min{k,n+1−k}. INDECOMPOSABLEMODULESOFSOLVABLELIEALGEBRAS 3 In the second case, R is isomorphicto a uniqueR , n is odd, the automorphism x has 1 α,a twoeigenvaluesλand2λ,andtheJordandecompositionofxis (1.7) Jn(λ)⊕J1(2λ), sothate=2andn =1. 2 Conversely,if x hasa single eigenvalue,1 < k < n, and(1.5)-(1.6) are satisfied, then R can be extendedto a faithfuluniserial representationof g. In fact, let M be α,k,X k,n+1−k subspaceofgl(n+1)consistingofallmatrices 0 N , N ∈ M . 0 0 ! k×n+1−k LetvbeageneratoroftheF[x]-moduleV andset 1 Jk(α) 0 A=R (x)= ∈gl(n+1), E =R (v)∈M . α,k,X 0 Jn+1−k(α−λ) ! α,k,X k,n+1−k Then the extensions of R to a faithful uniserial representation of g are given by all α,k,X possibleF[t]-monomorphismsV →M suchthatv→E,wheretactsviaad x−λ1 k,n+1−k g g on V and via ad A − λ1 on M . Moreover, all such extensions produce gl(n+1) gl(n+1) k,n+1−k non-isomorphicrepresentationsofg. Illustrativeexamplesareprovidedin§5. Likewise, if x hasJordandecomposition(1.7), then R can be extendedto a faithful α,a uniserialrepresentationofg.Wedetermineallsuchextensionsandprovethattheyproduce non-isomorphic representations of g (this case is much simpler than the above and no examplesarerequired). Finally, a necessary and sufficient condition for g to have a faithful uniserial repre- sentation is that x has Jordan decomposition (1.4) and (1.5) holds, or that x has Jordan decomposition(1.7)andnisodd. Perhapssurprisingly, the representationtheory of sl(2), and in particularthe Clebsch- Gordanformula,playsadecisiveroleinourstudyandclassificationofuniserialg-modules. 2. Constructionofuniserialrepresentations Given p ≥ 1 and α ∈ F, we write J (α) (resp. Jp(α)) for the lower (resp. upper) p triangularJordanblockofsize p andeigenvalueα. We also let Ei,j ∈ gl(p)standforthe matrixwithentry(i, j)equalto1andallotherentriesequalto0. Wesupposethroughoutthissectionthatg=hxi⋉V,whereV isanabelianLiealgebra andx∈gl(V)actsonV viaasingle,lowertriangular,Jordanblock,say J (λ),relativetoa n basisv ,...,v ofV. Thecaseλ = 0isallowed. Themultiplicationtableforgrelative 0 n−1 toitsbasisx,v ,...,v is: 0 n−1 (2.1) [x,v ]=λv +v ,[x,v ]=λv +v ,...,[x,v ]=λv . 0 0 1 1 1 2 n−1 n−1 Wemaytranslate(2.1)into (2.2) (ad x−λ1 )kv =v , 0≤k≤n−1, g g 0 k and (2.3) (ad x−λ1 )nv =0. g g 0 Proposition2.1. Givenpositiveintegers p,q,letM besubspaceofgl(p+q)consisting p,q ofallmatrices 0 N N = , N ∈ M . 0 0 ! p×q b 4 PAOLOCASATI,ANDREAPREVITALI,ANDFERNANDOSZECHTMAN Givenα,λ∈F,set A= Jp(α)⊕Jq(α−λ). Letθstandfortheendomorphismad A−λ1 ofgl(p+q)restrictedtoitsinvariant gl(p+q) gl(p+q) subspaceM . Thenθisnilpotentwithelementarydivisors p,q tp+q−1,t(p+q−1)−2,t(p+q−1)−4,...,t(p+q−1)−2z, where z=min{p−1,q−1}. Moreover,givenanyN ∈ M ,theminimalpolynomialofN withrespecttoθistp+q−1 if p×q andonlyifN ,0. p,1 b Proof. For a ≥ 0, let V(a) stand for the irreduciblesl(2)-modulewith highest weight a. TheClebsch-Gordanformulastatesthat (2.4) V(a)⊗V(b)(cid:27)V(a+b)⊕V(a+b−2)⊕V(a+b−4)⊕···⊕V(a+b−2r), where r =min{a,b}. Let 0 1 1 0 0 0 e= ,h= , f = 0 0 ! 0 −1 ! 1 0 ! standforthecanonicalbasisofsl(2). Itiswell-knownthateactsnilpotentlyonV(a)with asingleelementarydivisor,namelyta+1. ItfollowsfromtheClebsch-Gordanformulathat eactsnilpotentlyonV(a)⊗V(b)withelementarydivisors ta+b+1,t(a+b+1)−2,t(a+b+1)−4,...,t(a+b+1)−2r. Fora≥0,letR :sl(2)→gl(a+1)bethematrixrepresentationaffordedbyV(a)givenby a (2.5) R (h)=diag(a,a−2,...,−a+2,−a), a (2.6) R (e)= Ja+1(0), a (2.7) R (f)=diag(0,a,2(a−1),3(a−2),...,3(a−2),2(a−1),a)J (0). a a+1 Now V(a)⊗V(b)(cid:27)V(a)∗⊗V(b)(cid:27)Hom(V(a),V(b)), where (2.8) (y·φ)(v)=y·φ(v)−φ(y·v), y∈sl(2),φ∈Hom(V(a),V(b)),v∈V. Itfollowsthat (2.9) V(a)⊗V(b)(cid:27)M , a+1,b+1 where 0 N R (y) 0 0 N (2.10) y· = a , , y∈sl(2). 0 0 ! " 0 Rb(y) ! 0 0 !# Ontheotherhand,lettingB= Jp(0)⊕Jq(0),wereadilyverifythat (ad B)N =(ad A−λ1 )N, N ∈ M , gl(p+q) gl(p+q) gl(p+q) p×q whichmeansthatθistherestbrictionofadgl(p+q)BtoMp,qb. Settinga= p−1andb=q−1 andusing(2.6)aswellas(2.10),wededucethatθisnothingbuttheactionofeonM . a+1,b+1 ThestatedelementarydivisorsforθnowfollowfromthoseoftheactionofeonV(a)⊗V(b). INDECOMPOSABLEMODULESOFSOLVABLELIEALGEBRAS 5 Using (2.5) and (2.10) we find that, for 0 ≤ i ≤ r, the h-eigenspace of M with p,q eigenvalue−(a+b)+2i,sayS(i),consistsofallQsuchthattheentriesofQoutsideofits ithlowerdiagonalareequalto0. Herethe0thlowerdiagonalconsistsofposition(p,1), b the 1st lower diagonalof positions (p,2),(p−1,1), the 2nd lower diagonalof positions (p,3),(p−1,2),(p−2,1),andsoon. EachlowestweightvectorofM generatesanirreduciblesl(2)-submodule.Inviewof p,q themultiplicity-freedecomposition(2.4)andtheisomorphism(2.9),weseethatforeach 0≤i≤r,thereisoneandonlyone0, E(i)∈S(i),uptoscaling,suchthat (2.11) b fd·E(i)=0. LettingW(i)bethesl(2)-submodulegeneradtedbyE(i),wehave (2.12) Mp,q =W(0)⊕···d⊕W(r). GivenanarbitraryN ∈ M ,letuswriteN intermsof(2.12). Wehave p×q N =w(0)+w(1)··b·+w(r), w(i)∈W(i), where, b w(0)=α E(0)+α e·E(0)+···+α ea+b·E(0), α ∈ F. 0 1 a+b i FromthefirstpartoftheTheorem,weknowthat, relativetotheactionofe, theminimal d d d polynomialof E(0) is ta+b+1, while ta+b−1 annihilates all w(i), i > 0. It follows that the minimal polynomialof N is ta+b+1 if and only if α , 0. On the other hand, given that d 0 E(i)∈S(i),itfollowsthateveryE(i),i>1,hasentry(p,1)equalto0,whereasentry(p,1) b ofE(0)isnot0.Moreover,using(2.6)and(2.10)wefindthatifP=e·Q,thenentry(p,1) d ofPisequalto0foranyQ. Thus,N ,0ifandonlyifα ,0,asrequired. (cid:3) p,1 0 b b Proposition2.2. Givenα ∈ F, positiveintegers p,q,and N ∈ M suchthat N , 0, p×q p,1 considerthelinearmapR=R :g→gl(p+q)givenby α,p,q,N Jp(α) 0 x7→A= 0 Jq(α−λ) ! and 0 N v 7→(ad A−λ1 )k , 0≤k ≤n−1. k gl(p+q) gl(p+q) 0 0 ! ThenRisarepresentationofgifandonlyif p+q−1 ≤ n,inwhichcaseRisuniserial. Moreover,Risafaithfulrepresentationifandonlyif p+q−1=n. Proof. Byconstruction,Rpreservesthefollowingrelationsofg: [v,v ]=0, (ad x−λ1 )kv =v , 0≤k≤n−1. i j g g 0 k On the other hand, due to Proposition 2.1, (ad A − λ1 )nN = 0 if and only if gl(p+q) gl(p+q) p+q−1≤n,whichmeansthatRpreservesthelastdefiningrelationofg,namely b (ad x−λ1 )nv =0, g g 0 if and only if p+q−1 ≤ n. Thus, condition p+q−1 ≤ n alone determines whether R is a representationor not. Suppose that indeed p+q−1 ≤ n. It is obviousthat R is uniserial. Moreover,R(v ),...,R(v )arelinearlyindependentifandonlyiftheminimal 0 n−1 polynomialofNwithrespecttoad A−λ1 hasdegreen. ByProposition2.1,this gl(p+q) gl(p+q) happensifandonlyif p+q−1=n. (cid:3) b 6 PAOLOCASATI,ANDREAPREVITALI,ANDFERNANDOSZECHTMAN Givenα∈F,1≤k≤n,andX ∈ M ,weset p=k,q=n+1−kand k−1,n−k 0 X N = ∈ M . 1 0 ! k,n+1−k ThenProposition2.2ensuresthat R =R :g→gl(n+1) α,k,X α,p,q,N is a faithfuluniserial representationof g. In the extreme cases k = n and k = 1 there is noX,andN isrespectivelyequalto 0 .  01.. ∈ Mn×1and(1,0,...,0)∈ M1×n. ThecorrespondingrepresentationswillberespectivelydenotedbyR andR α,n α,1 Givenanyα∈ Fanda=(a ,...,a )∈Fnsuchthata =1,weconsiderthelinearmap 1 n 1 R :g→gl(n+2)definedasfollows: α,a α 0 0 x7→A= 0 Jn(α−λ) 0 ,  0 0 α−2λ  0 a 0 vk 7→(adgl(n+2)A−λ1gl(n+2))k 0 0 en , 0≤k ≤n−1, where{e ,...,e }isthecanonicalbasisofthe0col0umn0spaceFn. 1 n Lemma2.3. R isarepresentationofgifandonlyifnisoddanda =0foralleveni,in α,a i whichcaseR isuniserial. α,a Proof. Bydefinition,R preservesrelations(2.2)and(2.3).WenextdeterminewhenR α,a α,a preservesrelations[v,v ]=0. LettingN = Jn(0),wehave i j 0 a 0 0 (−1)kaNk 0 (adgl(n+2)A−λ1gl(n+2))k 0 0 en = 0 0 Nken , 0≤k≤n−1. Thus[R(v ),R(v )]=0iff(−01)ka0Nk+0je=(−10)jaNk+j0e iffa 0 =0fork+ jodd.Since k j n n n−k−j a =1,thelastconditionisequivalenttonoddanda =0,foranys. Thisprovesthefirst 1 2s assertion.Asuniserialityisclear,theproofiscomplete. (cid:3) Finally, in the extreme case n = 1, given any α ∈ F and ℓ ≥ 2 we have the faithful uniserialrepresentationT :g→gl(ℓ)givenby α,ℓ x7→diag(α,α−λ,...,α−(ℓ−1)λ), v 7→ Jℓ(0). 0 Definition2.4. Givenpositiveintegersℓ,d,d ,...,d suchthatd +···+d =dandℓ>1, 1 ℓ 1 ℓ andamatrixA ∈ M ,weconsiderAaspartitionedintoℓ2 blocksA(i, j) ∈ M . Wesay d di×dj thatAisblockuppertriangularifA(i, j)=0foralli> j,andstrictlyblockuppertriangular ifA(i, j)=0foralli≥ j. If0≤i≤ℓ−1,bytheithblocksuperdiagonalofAwemeanthe ℓ−iblocksA(1,1+i),A(2,2+i),...,A(ℓ−i,ℓ). INDECOMPOSABLEMODULESOFSOLVABLELIEALGEBRAS 7 Lemma2.5. Givenpositiveintegersℓ,d1,...,dℓ,withℓ>1,setJi = Jdi(0),1≤i≤ℓ,and letGbethesubgroupofGL(d)consistingofallX ⊕···⊕X suchthatX ∈U(F[J]).Let 1 ℓ i i E ∈ M bestrictlyblockuppertriangular,withdiagonalblocksofsizesd ×d ,...,d ×d . d 1 1 ℓ ℓ Let E ∈ M ,...,E ∈ M be the blocksin the first block superdiagonalof E, 1 d1×d2 ℓ−1 dℓ−1×dℓ andsupposethatthebottomleftcornerentryofeachE isnon-zero. i Then E isG-conjugateto a matrix H suchthateachofthe blocks H ,...,H in the 1 ℓ−1 first block superdiagonal of H has first column equal to the last canonical vector, and H haslastrowequaltothefirstcanonicalvector. Likewise, E isalsoG-conjugatetoa ℓ−1 matrix H such thateach ofthe blocks H ,...,H in the first block superdiagonalof H 1 ℓ−1 haslastrowequaltothefirstcanonicalvector,andthefirstcolumnof H isequaltothe 1 lastcanonicalvector. Proof. RecallthatA ∈ F[Jp(0)]ifandonlyifAisuppertriangularandallitssuperdiago- nalshaveconstantvalue.Forinstance,atypicalelementofA∈F[J4(0)]hastheform α β γ δ 0 α β γ (2.13) A= . It is clear that the unit groupof F[Jp(000)] ac00ts trα0ansαβitivelyfrom the left (right) on the set ofcolumn(row)vectorsofFp thathavenon-zerolast(first)entry. Moreover,itisequally clearthatifU ∈ U(F[Jp(0)])and B ∈ M (resp. B ∈ M )thenthefirstcolumn(resp. q×p p×q lastrow)of BU (resp. UB)isthatof Bscaledbya non-zeroconstant. Itfollowsatonce fromtheseconsiderationsthatwecanfindX ,...,X (resp. X ,...,X )sothatforanyX 1 ℓ−1 2 ℓ ℓ (resp. X )theresultingX ∈GwillconjugateE intoamatrixH suchthatthefirstcolumn 1 (resp. last row) of every H is equal to a non-zeroscalar multiple, say by α, of the last i i (resp. first)canonicalvector. MakingasecondselectionofscalarmatricesY ,...,Y and 1 ℓ conjugatingHbytheresultingY =Y ⊕···⊕Y ,wecanmakeallα =1above.Finally,by 1 ℓ i suitablychoosingX (resp. X )withwith1’sonthediagonalandtakingallotherX =1 , ℓ 1 i di we can makethe last row (resp. firstcolumn)of H (resp. H ) equalto the first (resp. ℓ−1 1 last)canonicalvector. (cid:3) Proposition2.6. Supposeλ,0andn>1. ThentherepresentationsR ,R ,R and α,k,X α,n α,1 R arenon-isomorphictoeachother. α,a Proof. Considering the eigenvalues of the image of x as well as their multiplicities, the only possible isomorphisms are easily seen to be between R and R , or R and α,k,X α,k,Y α,a R . α,b SupposefirstT ∈GL(n+1)satisfies TR (y)T−1 =R (y), y∈g. α,k,X α,k,Y ThenT commuteswithR (x)= Jk(α)⊕Jn+1−k(α−λ),andthereforeT =T ⊕T ,where α,k,X 1 2 T (resp. T )isapolynomialinJk(0)(resp. Jn+1−k(0))withnon-zeroconstantterm. Thus 1 2 TR (v )T−1 =R (v ) α,k,X 0 α,k,Y 0 translatesinto 0 X 0 Y (2.14) T = T 1 1 0 ! 1 0 ! 2 Explicitly writing T and T , as in (2.13), we infer from (2.14) that T = α1 = T , 1 2 1 n+1 2 whenceX =Y. 8 PAOLOCASATI,ANDREAPREVITALI,ANDFERNANDOSZECHTMAN SupposenextS ∈GL(n+2)satisfies SR (y)S−1 =R (y), y∈g. α,a α,b Asabove,S = J1(β)⊕S ⊕J1(γ),whereS isapolynomialinJn(0)withnon-zeroconstant 2 2 termandβ,γarenon-zero,butthen SR (v )=R (v )S α,a 0 α,b 0 forcesS tobeanon-zeroscalarmatrix,whencea=b. (cid:3) 3. Classificationofuniserialrepresentations Lemma3.1. LetT :h→kbeahomomorphismofLiealgebras.Then T((ad y−µ1 )kz)=(adT(y)−µ1)kT(z), y,z∈h,µ∈ F,k≥0. h h k k Proof. Thisfollowseasilybyinduction. (cid:3) Theorem3.2. ConsidertheLiealgebrag=hxi⋉V,whereVisanabelianLiealgebraand x ∈ GL(V)actsonV viaasingleJordanblock J (λ),λ , 0. LetRbeafaithfuluniserial n representationofg. Then (a)Ifn>1thenRisisomorphictooneandonlyoneoftherepresentationsR ,R , α,k,X α,n R ,R . α,1 α,a (b)Ifn=1thenRisisomorphictooneandonlyoneoftherepresentationsT . α,ℓ Proof. LetU beafaithfuluniserialg-module,sayofdimensiond. Lie’stheoremensures theexistenceofabasisB={u ,...,u }ofU suchthatthecorrespondingmatrixrepresen- 1 d tationR:g→gl(d)consistsofuppertriangularmatrices. Since x ∈GL(V),wehave[g,g]=V,whenceR(v)isstrictlyuppertriangularforevery v∈V. Set A=R(x)andE =R(v ), 0≤k ≤n−1. k k Inviewof[CS,Lemma2.2],wemayassumethatAsatisfies: (3.1) A =0wheneverA , A . ij ii jj Moreover,from[CS,Lemma2.1],weknowthatforevery1 ≤ i < d thereissomey ∈ g i suchthat (3.2) R(y) ,0. i i,i+1 Step1. IfA , A then(E ) ,0andA −A =λ. i,i i+1,i+1 0 i,i+1 i,i i+1,i+1 Indeed,(2.2),(2.3)andLemma3.1imply (3.3) (ad A−λ1 )kE =E , 0≤k ≤n−1, gl(d) gl(d) 0 k and (3.4) (ad A−λ1 )nE =0. gl(d) gl(d) 0 SinceAisuppertriangularandE isstrictlyuppertriangular,(3.3)and(3.4)give 0 (3.5) (A −A −λ)k(E ) =(E ) , 0≤k≤n−1,1≤i<d. i,i i+1,i+1 0 i,i+1 k i,i+1 and (3.6) (A −A −λ)n(E ) =0, 1≤i<d. i,i i+1,i+1 0 i,i+1 INDECOMPOSABLEMODULESOFSOLVABLELIEALGEBRAS 9 Fixisuchthat1≤i<dandA ,A . By(3.1),wehave i,i i+1,i+1 (3.7) A =0. i,i+1 Combining(3.2),(3.5)and(3.7)weobtain (3.8) (E ) ,0. 0 i,i+1 From(3.6)and(3.8)wededuce A −A =λ. i,i i+1,i+1 Step2. Wehave (3.9) A= A ⊕···⊕A , A ∈gl(d), 1 ℓ i i whereeachA hasscalardiagonal,sayofscalarα,and,settingα=α ,wehave i i 1 α =α−(i−1)λ. i Thisfollowsatoncefrom(3.1)andStep1. Step 3. Let us write each E in block form compatiblewith (3.9), that is, with diagonal k blocksofsizesd ×d ,...,d ×d . ThenalldiagonalblocksofeveryE areequalto0. 1 1 ℓ ℓ k Indeed,supposei≤ jand A =···=A . i,i j,j SettingUr =span{u ,...,u },weseethatthesection 1 r Ui,j =Uj/Ui−1 of U is a (uniserial) g-module of dimension e = j − i + 1. Let T : g → gl(e) be the corresponding matrix representation relative to the basis u + Ui−1,...,u + Ui−1 of U. i j ThenT(x)isuppertriangularwithscalardiagonal,soad T(x)isnilpotent.Ontheother gl(e) hand,sinceT isaLiehomomorphism,Lemma3.1gives (ad T(x)−λ1 )nT(v )=0, 0≤k≤n−1. gl(e) gl(e) k ItfollowsthateveryT(v )isageneralizedeigenvectorofad T(x)forthedistincteigen- k gl(e) valuesλand0. WeinferthateveryT(v ) = 0. Itfollowsthatalldiagonalblocksofevery k E areequalto0. k Step4. ReferringtotheblockdecompositionofE usedinStep3,ifi < jand j , i+1, k thenblock(i, j)ofE is0forall0≤k ≤n−1. k Indeed,recallingDefinition2.4,welet S(1),S(2),...,S(ℓ−1) bethesubspacesofgl(d)correspondingtotheblocksuperdiagonals1,2,...,ℓ−1,andset S =S(1)⊕···⊕S(ℓ−1). ThenS(i)isthe generalizedeigenspaceofad AactingonS forthe eigenvalueiλ, for gl(d) all1 ≤ i ≤ ℓ−1. Ontheotherhand,everyE ∈ S byStep3,while(3.3)and(3.4)imply k thatevery E belongsto the generalizedeigenspaceof ad A forthe eigenvalueλ. We k gl(d) concludethateveryE isinS(1). k 10 PAOLOCASATI,ANDREAPREVITALI,ANDFERNANDOSZECHTMAN Step5. WemayassumewithoutlossofgeneralitythatAisinJordanform (3.10) Jd1(α)⊕Jd2(α−λ)⊕···⊕Jdℓ(α−(ℓ−1)λ). Indeed,by(3.2)andStep3,thefirstsuperdiagonalofeveryA appearingin(3.9)con- i sists entirely of non-zero entries. Thus, for each 1 ≤ i ≤ ℓ there is X ∈ GL(d) such i i that XiAiXi−1 = Jdi(α−(i−1)λ). Set X =X ⊕···⊕X ∈GL(d). 1 ℓ Then XAX−1 is equal to (3.10) and XE X−1 is strictly block upper triangular with each k block(i, j), j,i+1,equalto0. Step6. Ahasatleast2Jordanblocks. Ifnot,V isannihilatedbyR,bySteps3and4,contradictingthefactthatRisfaithful. Step7. d +d ≤n+1forall1≤i<ℓ. i i+1 ApplyProposition2.2tosuitablesectionsofU. Step8.Withoutlossofgeneralitywemayassumethatthefirstcolumnofeachblockalong the first block superdiagonalof E is equal to the last canonicalvector, and that the last 0 rowofthelastoftheseblocksisequaltothefirstcanonicalvector. ThisfollowsfromLemma2.5. Final Stepwhen n = 1. Supposen = 1. Thenalld = 1byStep 7,so Steps5, 6and8 i yield that R is isomorphic to a representation T . As these representations are clearly α,ℓ non-isomorphictoeachother,theTheoremisproveninthiscase. Weassumefortheremainderoftheproofthatn>1. Step9. AhasatleastoneJordanblockofsize>1. Ifnot,d +d < n+1forallibyStep7. Sincethe x-invariantsubspacesofV forma i i+1 chain,itfollowsfromProposition2.2thatR(v )=0,contradictingthefaithfulnessofR. n−1 Step10.LetJdi(α−(i−1)λ)beaJordanblockofAofsize>1ofA,asensuredbyStep9. Theni≥ℓ−1andi≤2. Letusfirstseethati≥ℓ−1. Suppose,ifpossible,thatAhasconsecutiveJordanblocks Ja(β),Jb(β−λ),Jc(β−2λ)witha>1. Case1. b = 1. ConcentratingonasuitablesectionofU,asintheproofofStep3,wesee thatghasamatrixrepresentationP:g→gl(4)suchthat β 1 0 0 0 0 0 0 0 β 0 0 0 0 1 0 P(x)= , P(v )= . HeretheshapeofP(E )00ise00nsuβre−0dλbySβte−0p2sλ3,4and80. Now00 00 00 10  0 P(v )=[P(x),P(v )]−λP(v )=E1,3, 1 0 0