Growth of polynilpotent varieties of Lie algebras

GrowthofpolynilpotentvarietiesofLiealgebrasandfastincreasingentirefunctionsPetrogradskyV.M.February17,1997FacultyofMathematicsUlyanovskStateUniversityLevTolstoy42,Ulyanovsk,432700Russiae-mail:[e-mail]:FacultyofMathematics,UniversityofBielefeldPostfach100131,33501Bielefeld,Germanye-mail:[e-mail]ncn(V)forvarietiesofLiealgebrasisstudied;wherecn(V)isthedimensionofalinearspanofallmultilinearwordsonndistinctlettersinafreealgebraF(V;X)ofthevarietyV.ToanynontrivialvarietyofLiealgebrasVcorrespondsthecomplexityfunctionC(V;z),whichisanentirefunctionofacomplexvariable.IncaseofapolynilpotentvarietyofLiealgebrasVestimatesforcomplexityfunctionarestudied,inmostcasesitisofinniteorder.AconnectionbetweenagrowthoffastincreasingentirefunctionandanasymptoticofitsTaylorcoecientsisstudied.Themainresultisanasymptoticforafunctioncn(V)incaseofthepolynilpotentvarietyofLiealgebrasV.AlsoweproveananalogofRegev’stheoremonanupperboundforanarbitraryvarietyofLiealgebras.Asaresultweobtainabetterboundthananestimate,obtainedbytheauthorearlier.1Introduction:growthofvarietiesofLiealgebrasLetHbeaLiealgebraLieoveraeldK.LetF=F(X)beafreeLiealgebrawithacountablesetoffreegeneratorsX=fxiji2Ng.Then06=f2Fissaidanidentity(oridenticalrelation)inH,iforanyai2H;i2Nandanyhomomorphism:F!H;(xi)=ai;i2Nwehave(f)=0;alsointhiscasewesaythatHisaPI-algebra.SupposethatIF,thenaclassofallLiealgebrasV,satisfyingidentitiesI,issaidavariety.AsetofallidentitiesV(X),whicharetrueinagivenvarietyV,isanidealinF(X),sowecanconsiderF(V;X)=F(X)=V(X),whichiscalledfreealgebraofthevarietyV.Ithasapropertythat8H2V;8ai2H;i2N;9!:F(V;X)!H:(xi)=ai;i2N.Avarietyiscallednontrivialifitsatisessomenonzeroidentity.ForthetheoryofvarietiesofLiealgebrasseeamonograph[1].LetPn(V)F(V;X)beasetofmultilinearelementsonlettersx1;:::;xn.Namely,Pn(V)isalinearspanofallmonomialsonx1;:::;xn(moreexactly,theirimagesinF(V;X))suchthateachpartiallysupportedbygrantRFFI96-01-00146;theauthorisgratefulltotheUniversityofBielefeldforhospitality,wherehewasDAAD-fellow1letterxi;i=1;:::;nentersmonomialsexactlyonetime.ForagivenvarietyVappearsafunctionofcodimensiongrowthcn(V)=cn(F(V;X);X)=dimKPn(V);n2N:Itisevidentthatforothervariablesxi1;:::;xin2Xwewillgetthesamenumber.Abehaviourofthesequencecn(V)isanimportantcharacteristicofthevarietyV.Thesameobservationsholdalsoforvarietiesofassociativealgebras.Aboutthecodimensiongrowthofassociativealgebrasthefollowingfactisknown.Theorem1.1(Regev,[2],seealso[1])LetVbeavarietyofassociativealgebraswhichsatisessomenontrivialidentityofdegreed.Thencn(V)Cn;n2N;whereC=(d1)2:RecallthedenitionofapolynilpotentvarietyofLiealgebrasNsqNs2Ns1,correspondingtothetuple(sq;:::;s2;s1).Ifq=1,thenonehasavarietyofnilpotentLiealgebrasNs1,denedbyanidentityT1(x1;:::;xs1)=[x1;:::;xs1]0:(Liebracketsweconsidertobeleftnormed,thatis[a1;:::;an]=[:::[a1;a2];:::;an]).SupposethatwehaveconstructedanidentityTq1(x1;:::;xt),whichdenesNsq1Ns1,thenNsqNs1isdenedbyanidentityTq(x11;:::;xtsq)=[Tq1(x11;:::;xt1);Tq1(x12;:::;xt2);:::;Tq1(x1sq;:::;xtsq)]0;wherexij;1it;1jsqaredistinctelementsfromX.ItiseasytoseethatNsqNs1consistsofallLiealgebrasH,suchthatthereexistsachainofideals0=Hq+1Hq:::H1=H,Hi=Hi+12Nsi.Ifsq=:::=s2=s1thenonehasavarietyofsolvableLiealgebrasAq.AgrowthofLiealgebrasnotexceedinganexponentialgrowthhasbeenstudiedextensively,seereview[3].Unlikeassociativecase,(seeTheorem1.1)agrowthofarathersmallvarietyAN2isoverexponential[4](i.e.itcannotbeboundedbyanyexponent).Moreover,insomesense,thissituationistypicalforLiealgebras.So,wehaveavastareaoftheoverexponentialgrowthforLiealgebras.In[5]therewassuggestedascaletomeasuretheoverexponentialgrowthofLiealgebras.IntermsofthisscaleagrowthofpolynilpotentvarietiesofLiealgebraswasdescribed(seeThe-orem2.1).Thegoalofthepresentpaperistogivemoreprecisedescriptionofthegrowthofpolynilpotentvarieties(Theorem2.2,seealsoLemma3.3).Asaresultappearsmorenescalefortheoverexponentialgrowth(Section2).Ourmaininstrumentisacomplexityfunctionofthevariety,westudyestimatesonitsgrowth(Section3).OfindependentinterestisanasymptoticforthecomplexityfunctionforthepolynilpotentLievariety,obtainedinTheorem3.2.InSection4westudyanasymptoticsforcoecientsoffastincreasingentirefunction,thisisanimportantstepinaproofofTheorem2.2.InSection5wedirectlyprovethelowerestimatesforTheorem2.2.InSection6weproveananalogofTheorem1.1forLiealgebras(Theorem2.3).2MorenescaleforgrowthofLiePI-algebrasandgrowthofpolynilpotentvarietiesIn[5]appearedthefollowingtypesofoverexponentialgrowths.Denoteln(1)x=lnx;ln(s+1)x=ln(ln(s)x);exp(1)x=expx;exp(s+1)x=exp(exp(s)x);s2N:2Consideraseriesoffunctionsofanaturalargumentq(n);q=2;3;:::witharealparameter:q(n)=8:(n!)1;1;q=2;n!(ln(q2)n)n=;0;q=3;4:::Letf(n)beafunctionofanaturalargument,wewillcompareitwithouretalonfunctionsinthefollowingway.Supposethat(n)isafunctionofanaturalargumentincreasingwithrespecttoarealparameter.Letc2Rbexed.Thenwedenotef(n)cc(n)()inff0jf(n)a(n)g=c;