7D dissident maps

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1、DissidentMapsontheSeven-DimensionalEuclideanSpaceErnstDieterichandLarsLindbergAbstractAdissidentmaponanite-dimensionaleuclideanvectorspaceVisunderstoodtobealinearmap:V^V!Vsuchthatv;w;(v^w)arelinearlyindependentwheneverv;w2Vare.Thisnotionofadissidentmapprovidesalinkbetweenseeminglydivers

2、easpectsofrealgeometricalgebra,therebyrevealingitsshiftingsignicance.Whileitgeneralizesontheonehandtheclassicalnotionofavectorproduct,itspecializesontheotherhandthestructureofarealdivisionalgebra.MoreoverityieldsnaturallyalargeclassofselfbijectionsoftheprojectivespaceP(V)manyofwhicharecol

3、lineations,butsomeofwhich,surprisingly,arenot.Dissidentmapsareknowntoexistinthedimensions0,1,3and7only.Inthedimensions0,1and3theyareclassiedcompletelyandirredundantly,butindimension7theyarestillfarfrombeingfullyunderstood.ThepresentarticlecontributestotheclassicationofdissidentmapsonR7wh

4、ichinturncontributestotheclassicationof8-dimensionalrealdivisionalgebras.WestudytwolargeclassesofdissidentmapsonR7.Therstclassisformedbyallcomposeddissidentmaps,obtainedfromavectorpro-ductonR7bycompositionwithadeniteendomorphism.Thesecondclassisformedbyalldoubleddissidentmaps,obtainedas

5、thepurelyimaginarypartsofthestructuresofthose8-dimensionalrealquadraticdivisionalgebraswhicharisefroma4-dimensionalrealquadraticdi-visionalgebrabydoubling.Foreachofthesetwoclassesweexhibitacompletebutredundantclassication,givenbya49-parameterfa-milyofcomposeddissidentmapsanda9-parameterfa

6、milyofdoubleddissidentmapsrespectively.Theproblemofrestrictingthesetwofa-miliessuchastoobtainacompleteandirredundantclassicationarisesnaturally.Regardingthesubproblemofcharacterizingwhentwocom-poseddissidentmapsbelongingtotheexhaustive49-parameterfamilyareisomorphic,wepresentanecessaryand

7、sucientcriterion.Regar-dingtheanalogoussubproblemfortheexhaustive9-parameterfamilyofdoubleddissidentmaps,wepresentasucientcriterionwhichisconjectured,andpartiallyproved,eventobenecessary.Finallywesolvethesubproblemofdescribingthosedissidentmapswhicharebothco