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1、ANOTEONDERIVATIONSOFLIEALGEBRASM.SHAHRYARIAbstract.Inthisnote,wewillprovethatafinitedimensionalLiealge-braLofcharacteristiczero,admittinganabelianalgebraofderivationsD≤Der(L)withthepropertyXnL⊆d(L)d∈Dforsomen>1,isnecessarilysolvable.Asaresult,ifLhasaderivationnd:L→L,s
2、uchthatL⊆d(L),forsomen>1,thenLissolvable.In[2],F.LadischprovedthatafinitegroupG,admittinganelementawiththepropertyG′=[a,G],issolvable.Usingthisresult,onecanprovethatafinitegroupissolvable,ifithasafixedpointfreeautomorphism.Inthisnote,weproveasimilarresultforLiealgebrasi
3、namoregeneralframework;weshowthatafinitedimensionalLiealgebraLofcharacter-isticzero,issolvableifithasanabeliansubalgebraAwiththepropertyLn⊆[A,L],forsomen>1.Next,weusethisresulttoprovethatafi-nitedimensionalLiealgebraLofcharacteristiczero,admittinganabelianalgebraofderi
4、vationsD≤Der(L)withthepropertyXLn⊆d(L)d∈Dforsomen>1,isnecessarilysolvable.Asaspecialcase,weconcludethatiftheLiealgebraLadmitsaderivationd:L→L,suchthatLn⊆d(L),forsomen>1,thenLissolvable.NotethatasimilarresultwasobtainedbyN.Jacobsonin[1]:afinitedimensionalLiealgebraofch
5、aracteristiczero,arXiv:1011.1609v1[math.RT]7Nov2010admittinganinvertiblederivation,isnilpotent.Ourmaintheorem(Theorem1bellow)isalsotrueforconnectedcompactLiegroupsandso,itmaybealsotrueforfinitegroups.Therefore,weaskthefollowingquestion;LetGbeafinitegroupadmittinganabel
6、iansubgroupAwiththepropertyGn⊆[A,G],forsomen>1.IsittruethatGissolvable?Duringthisnote,LisafinitedimensionalLiealgebraoverafieldKofcharacteristiczero.ByLnandL(n),wewilldenotethen-thtermsoftheDate:November9,2010.MSC(2010):17B40Keywords:Liealgebras;Derivations;SolvableLie
7、algebras;CompactLiegroups.12M.SHAHRYARIlowercentralseriesandderivedseriesofL,respectively.Also,Der(L)willdenotethealgebraofderivationsofL.Theorem1.SupposethereexistsanabeliansubalgebraA≤Landanintegern>1,suchthatLn⊆[A,L].ThenLissolvable.Proof.LetS=Ln−1.First,weshowtha
8、tSissolvable.Todothis,weuseCartancriterion.Letx∈Sandy∈S′.SinceS′=[Ln−1,Ln−1]⊆Ln⊆[A,L],soXy=[ai,ui],iforsomea1,...,ak∈Aandu1,...,uk∈
