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1、AlgebraicGroupsandArithmeticGroupsJ.S.MilneVersion1.01June4,2006Thesenotesprovideanintroductoryoverviewofthetheoryofalgebraicgroups,Liealgebras,Liegroups,andarithmeticgroups.TheyarearevisionofthosepostedduringtheteachingofacourseatCMS,ZhejiangUniversity,HangzhouinSpring,2005.v0.00(February28–M
2、ay7,2005).Aspostedduringthecourse.v1.00May22,2005.Minorcorrectionsandrevisions;addedtableofcontentsandindex.v1.01June4,2006.Fixedproblemwiththediagrams.Pleasesendcommentsandcorrectionstomeatmath@jmilne.orgAvailableathttp://www.jmilne.org/math/ThephotoisofthefamouslaughingBuddhaonThePeakThatFle
3、wHere,Hangzhou.Copyrightc2005,2006J.S.Milne.ThiselectronicversionofthisworkislicensedunderaCreativeCommonsLicence(Attribution-NonCommercial-NoDerivs2.5).Thismeansthatyouarefreetocopy,distribute,display,andperformtheworkunderthefollowingconditions:Attribution.Youmustattributetheworkinthemanners
4、pecifiedbytheauthororlicensor.Noncommercial.Youmaynotusethisworkforcommercialpurposes.NoDerivativeWorks.Youmaynotalter,transform,orbuilduponthiswork.Foranyreuseordistribution,youmustmakecleartoothersthelicensetermsofthiswork.Formoredetails,seehttp://creativecommons.org/licenses/by-nc-nd/2.5/Con
5、tentsIntroduction1Notations2;Prerequisites2;References31Overviewandexamples4Thebuildingblocks4;Semisimplegroups5;Extensions6;Summary7;Exercises82Definitionofanaffinealgebraicgroup10Principleofpermanenceofidentities10;Affinealgebraicgroups10;Homomorphismsofal-gebraicgroups13;TheYonedalemma13;Theco
6、ordinateringofanalgebraicgroup14;Verybriefreviewoftensorproducts.14;Productsofalgebraicgroups15;Fibredproductsofal-gebraicgroups15;Extensionofthebasefield(extensionofscalars)15;Algebraicgroupsandbi-algebras16;Homogeneity18;Reducedalgebrasandtheirtensorproducts19;Reducedalge-braicgroupsandsmooth
7、algebraicgroups20;Smoothalgebraicgroupsandgroupvarieties20;Algebraicgroupsincharacteristiczeroaresmooth22;Cartierduality23;Exercises243Linearrepresentations25Linearrepresentationsandcomodules25;Stabilizersofsubspaces304MatrixGroups32Ane
