ThetheoryofgroupsandLiealgebrasisinterestingformanyreasons.Inthemathematicalviewpoint，itemploysatthesametimealgebra，analysisandgeometry.Ontheotherhand，itintervenesinotherareasofscience，inparticularindifferentbranchesofphysic8andchemistry.Itisanactivedomainofcurrentresearch.　　Thegeneraltheoryofalgebraicgroupsisstudiedinchapter821to28.TherelationsbetweenLiealgebrasandalgebraicgroups，whicharefundamentaltous，areestablishedinchapters23and24.Chapter29present8applicationsoftheserelationstotacklethesystematicstudyofLiealgebras.Thereaderwillobservethatthegeometricalaspectshaveanimportantpartinthisstudy.Inparticular，theorbitsofpointsundertheactionofanalgebraicgroupplaysacentralrole.