Lie_algebras_graded_by_finite_root_systems_and_the_intersection_matrix_algebras_of_Slodowy

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1、Invent.math.108,323-347(1992)Inventionesmathematicae9Springer-Verlag1992LiealgebrasgradedbyfiniterootsystemsandtheintersectionmatrixalgebrasofSlodowy*S.Berman1andR.V.Moody2'**1DepartmentofMathematics,UniversityofSaskatchewan,Saskatoon,SK,S7N0W0,Canada2DepartmentofMathematics,UniversityofA

2、lberta,Edmonton,AB,T6G2G1,CanadaOblatum21-I-1991&16-IX-19910IntroductionThispaperisabouttoroidalLiealgebras,certainintersectionmatrixLiealgebrasdefinedbySlodowy,andtheirrelationshiptooneanotherandtocertainLiealgebraanaloguesofSteinberggroups.Themainresultofthepaperistheidentificationofthei

3、ntersectionmatrixalgebrasarisingfrommultiply-affinizedCartanmatricesoftypesA,DandEwithcertainSteinbergLiealgebrasandtoroidalLiealgebras(Propositions5.9and5.10).AmajorpartofthepaperstudiesandclassifiesLiealgebrasgradedbyfiniterootsystems.Thesebecometheprinci-paltoolinouranalysisofintersecti

4、onmatrixalgebras.EachLiealgebragradedbyasimply-lacedfiniterootsystemofrank>2hasattachedtoitanalgebrawhich,accordingtothetypeandrank,iseithercommutativeandassociative,onlyassociative,oralternative.Allthesepossibilitiesoccurinourdescriptionofinter-sectionmatrixalgebras.LetRbeanyassociativeal

5、gebrawithidentity,notnecessarilyfinitedimen-sional,overafieldkofcharacteristic0.ForeachpositiveintegerntheassociativealgebraM,(R)ofnnmatriceswithcoefficientsinRformsaLiealgebraoverkunderthecommutatorproduct.WedenotethisLiealgebrabyol,(R).LetEisbethe(i,j)matrixunitofM.(R)andassumethatn>2.Th

6、esubalgebrae.(R)ofOI.(R)generatedbytheelementsrEis,r~R,i4=j,isanidealof91,(R)andisperfect,i.e.itisitsownderivedalgebra.NowanyperfectLiealgebraOhasauniversalcentralextension,alsoperfect,calledauniversalcoveringalgebra(u.c.a.)ofO[Ga],soinparticular,e.(R)hasau.c.a,thatwewilldenoteby~t,(R).Wed

7、efine112,.(R)bytheexactsequence(0.1)0~f2,,(R)~~t.(R)~e,(R)~0*DedicatedtoourteacherMariaJ.Wonenburger**BothauthorsgratefullyacknowledgethesupportoftheNaturalSciencesandEngineeringResearchCouncilofCanada324S.BermanandR.V.Moodywherenisthehomomorphismdefiningtheu.