Ad-nilpotent elements in algebras and superalgebras

Abstract

In thisthesiswewilldealwithad-nilpotentelementsinassociativealgebrasand superalgebraswithinvolutionandsuperinvolution,andad-nilpotentelementsinLie superalgebras.The rstaimofthiswork tswithHerstein'sbranchoftheorythat studies nilpotentinnerderivationsinalgebras.Therearemanystudiesonthisarea, highlightingforourworkthearticlesofW.S.MartindaleandC.R.Miers[55], [56] and T.K.Lee[54]. Later,inthesecondpart,westudyhowtoassociatesomeJordan structures toaLiesuperalgebra,motivatedbytheworkofA.Fern andez,E.Garc a and M.G omezLozano[24]. Objectives Three objectivesareaddressedthroughoutthisthesis.Inthe rstinstance,weseekto describeindetailthead-nilpotentelementsinsemiprimeassociativealgebraswithin- volution.Thesecondaimofthisthesisistocarryoverthedescriptionsofad-nilpotent elementsinsemiprimeassociativealgebrastoprimeassociativesuperalgebras,that is, togiveadetaileddescriptionofhomogeneousad-nilpotentelementsbelongingto prime associativesuperalgebras.Finally,motivatedbytheworkofA.Fern andez,E. Garc aandM.G omezLozanoin[24], toassociateaJordansuperstructuretoaLie superalgebrawithanad-nilpotentelementofacertainindex. Methodology Todevelopthe rsttwogoalswehaveworkedwithintheframeworkofsemiprime algebras withinvolutionandprimeassociativesuperalgebraswithsuperinvolution. V Moreover,theextendedcentroidwillbeanimportanttoolinthisthesis.Forthe last objective,wehaveworkedwithnonassociativesuperstructuressuchasLieand Jordan superalgebras,de nedbytheGrassmannenvelope,andJordansuperpairs. Wecanhighlightthehighcombinatorialcontentthroughouttheentirethesis. Results Wehavesuccessfullycoveredthethreeinitialgoals.First,wehavedescribedindetail ad-nilpotentelementsbelongingtoasemiprimeassociativealgebra.Moreover,we havesucceededinreducingthetorsionintheclassi cationofad-nilpotentelements in semiprimeassociativealgebraswithinvolutionduetothenewconceptofapure ad-nilpotentelement,introducedinthisthesisinChapter2.Theconditionsonthe scalar ringshasbeenweakenedtobefreeof 􀀀n s

and s torsion with s := [n+1 2 ] instead of beingfreeof n! torsion.Ontheotherhand,fortheskew-symmetricad-nilpotent elementsofasemiprimeassociativealgebra R with involution , wehavegivena description thatdependsontheirad-nilpotentindexmodulo4.Inthisdescription wecanemphasize:Ifaskew-symmetricelement a is ad-nilpotentsuchthatitsindex of ad-nilpotenceof K := Skew(R; ) and R do notcoincide,thatis,adn aK = 0but adn aR 6= 0,(itcanonlyoccurforad-nilpotentindicesof K congruentto0or3modulo 4) thenacertaincornerof R satis es aPI,hence R holds aGPI.Theseresultshave beendevelopedthroughoutChapter2whichhaveoriginatedanarticlethathasbeen published inthejournal BulletinoftheMalaysianMathematicalSciencesSociety ([12]). Thesecondaim,todescribeinprimeassociativesuperalgebraswithsuperin- volutionnilpotentinnerderivations,hasalsobeenpositivelysolvedduringChapter 3. Thisdescriptiondependsontheparityofthehomogeneouselement:iftheelement is even,whathasbeendevelopedinthepreviouschapterinalgebrasettings([12]), is largelyrescued.However,iftheelementisodd,wehaveworkedonitssquare, whichisanevenad-nilpotentelement,andwehaveappliedthedescriptionsforeven ad-nilpotentelementsstudiedabove.Theseresultshasbeenpublishedinthejournal LinearandMultilinearAlgebra ([28]). DuringChapter4,wehavegivenexamples for eachofthecasesappearinginthedescriptionsoftheelementsinbothalgebras VI and superalgebras,thusshowingthatthesedescriptionsarenottrivial.Finally,in Chapter 5,wehaveassociatedaJordansuperstructuretoaLiesuperalgebra L with a homogeneous ad-nilpotentelement a of index3or4,accordingtoitsparity.Further- more, theJordansuperpairwehaveconstructedfollowingthespiritofthepaperofA. Fern andez,E.Garc aandM.G omezLozano[24], coincideswiththesubquotientof a Liesuperalgebraassociatedwithanabelianinnerideal[a; [a; L]]. Thislastchapter has beenpublishedandcanbeconsultedinthejournal CommunicationsinAlgebra ([30]).