On generators and relations for certain involutory subalge

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1、COMMUNICATIONSINALGEBRA,17(12),3165-3185(1989)OngeneratorsandrelationsforcertaininvolutorysubalgebrasofKac-MoodyLieAlgebras*S.BermanDepartmentofMathematicsUniversityofSaskatchewanSaskatoon,SaskatchewanS7NOW0CanadaDedicatedtoProfessorEiichiAbeAbstractWefindgeneratorsandrela

2、tionsforthosesubalgebrasofKac-MoodyLiealgebrasthatarethefixedpointalgebrasofcertaininvolutions.SpecificallytheinvolutionmustinvolvetheCartaninvolutionwhichinterchangesthepositiveandnegativegenerators.WegoontoapplytheseresultstotheG.I.M.algebras,whichwereintroducedasnatural

3、general-izationsofKac-MoodyalgebrasbyP.Slodowy.WeshowsuchalgebrasareisomorphictosubalgebrasofKac-Moodyalgebras.FromthisweareabletoderivesomeinterestinginterrelationsbetweencertainKac-Moodyalgebras.Introduction:EveryKac-MoodyLiealgebraGcanbedecomposedasG=G-$H~BG+whereHisaso

4、calledCartansubalgebraofGandwhereG+andG-areisomorphicsub-Downloadedby[SeoulNationalUniversity]at08:2724July2011algebras.Thestandardconventionistolettheelementsel,...,el(respectivelyfi,...,fr)denotethegeneratorsofG+(resp.G-)andwritehi=[ei,fi].ThenthereisaninvolutionqofGdefi

5、nedbytheequationsq(ei)=ji,q(ji)=ei,q(hi)=-hi,15i_

6、kker,Incofu=qy=yq.InthefirstsectionofthispaperwefindasystemofgeneratorsandrelationsforthesubalgebraS,offixedpointsofu.Itisimportantforusthatqisinvolvedinu.Aftercompletingthisworkwebecameawareoftherecentpaper[Borcherds]wherecertaingeneralizationsofKac-MoodyLiealgebrasarestu

7、died.ThesegeneralizationsofR.Borcherdsincludesubalgebrasoffixedpointsofdiagramautomorphisms(thesedonotinclude7)andsoourworkcanbeseenascomplementarytowhatheconsiders.Actually,wecameacrossthisproblembystudyinganothergeneralizationoftheno-tionofaKac-Moodyalgebra.Thisgeneraliz

8、ationwasintroducedbyP.SlodowyinhisHabiliationsschrift[Slodowy7,8]andarisesfromageometrica