A Jacobson Radical Decomposition of the Fano-Snowflake Configuration

MetodSaniga1andPetrPracna2

AstronomicalInstitute,SlovakAcademyofSciencesSK-05960Tatransk´aLomnica,SlovakRepublic

(msaniga@astro.sk)

and

2

J.Heyrovsk´yInstituteofPhysicalChemistry,v.v.i.,AcademyofSciencesoftheCzechRepublic,Dolejˇskova3,CZ-18223Prague8,CzechRepublic

(pracna@jh-inst.cas.cz)

1

arXiv:0807.1790v1 [math-ph] 11 Jul 2008ProjectivelatticegeometriesoverunitalassociativeringsR(see,e.g.,[1]andreferencestherein)representaveryimportantgeneralizationofclassical(field)projectivespaces,beingendowedwithanumberofremarkablefeaturesnotexhibitedbythelatter.Oneofthemoststrikingdifferencesis,forcertainR,theexistenceoffreecyclicsubmodulesgeneratedbynon-unimodularvectorsofthefreeleftR-moduleRn,n≥1.Inacoupleofrecentpapers[2,3],anin-depthanalysishasbeenperformedofsuchnon-unimodularportionsofthelatticegeometriesforRbeingtheringofternions,i.e.,aringisomorphictothatofuppertriangular2×2matriceswithentriesfromanarbitrarycommutativefieldF.Ithasbeenfoundthatforanyn≥2thesenon-unimodularfreecyclicsubmodulesofRncanbeassociatedwiththelinesofPG(n,F),then-dimensionalprojectivespaceoverFsittinginthemiddleofsuchnon-unimodularworld.Inthefinitecase,F=GF(q),basiccombinatorialpropertiesofsuchconfigurationshavebeenderivedandillustratedinexhaustivedetailonthesimplest,n=q=2case—dubbedtheFano-Snowflakegeometry.InthepresentpaperweshallhaveanotherlookattheFano-SnowflakeandshowthatthisgeometryadmitsanintriguingdecompositionwithrespecttotheJacobsonradicaloftheringinquestion.

Tothisendinview,wefirstcollectthenecessarybackgroundinformationfrom[2,3].Weconsideranassociativeringwithunity1(=0),R,anddenotethefreeleftR-moduleonn+1generatorsoverRbyRn+1.ThesetR(r1,r2,···,rn+1),definedasfollows

󰀅󰀆

R(r1,r2,···,rn+1):=(αr1,αr2,···,αrn+1)|(r1,r2,···,rn+1)∈Rn+1,α∈R,(1)isaleftcyclicsubmoduleofRn+1.Anysuchsubmoduleiscalledfreeifthemappingα→(αr1,αr2,···,αrn+1)isinjective,i.e.,if(αr1,αr2,···,αrn+1)arealldistinct.Next,weshallcallavector(r1,r2,···,rn+1)∈Rn+1unimodularifthereexistelementsx1,x2,···,xn+1inRsuchthat

r1x1+r2x2+···+rn+1xn+1=1.

1

(2)

Table1:Addition(left)andmultiplication(right)inR♦.

+0

12

34

56

7

3

5

1

6

2

4

0

4

7

6

1

0

3

2

6

0

7

7

0

7

0

0

7

7

4

0

2

6

5

1

4

0

5

3

3

0

5

6

6

0

6

7

5

4

2

3

2

0

3

5

3

6

5

6

0

×0

0

1

2

3

4

5

6

7

Itisaverywell-knownfact(see,e.g.,[4]–[7])thatif(r1,r2,···,rn+1)isunimodular,thenR(r1,r2,···,rn+1)isfree;anysuchfreecyclicsubmodulerepresentsapointofthen-dimensionalprojectivespacedefinedoverR[5].Theconversestatement,however,isnotgenerallytrue.Thatis,thereexistringswhichalsogiverisetofreecyclicsubmodulesfeaturingexclusivelynon-unimodularvectors.Thefirstcasewhenthisoccursisthesmallest(non-commutative)ringofternions,R♦:

󰀁󰀃󰀄󰀂

ab

R♦≡|a,b,c∈GF(2).(3)

0cFromthisdefinitionitisreadilyseenthattheringcontainstwomaximal(two-sided)ideals,

󰀁󰀃󰀄󰀂

0b

I1=|b,c∈GF(2)(4)

0cand

I2=

󰀁󰀃

a0

b0󰀄

󰀂

|a,b∈GF(2),

(5)

whichgiverisetoanon-trivial(two-sided)JacobsonradicalJ,

󰀁󰀃󰀂󰀄

0b

J=I1∩I2=|b∈GF(2).

00AsforourfurtherpurposesitwillbemoreconvenienttoworkwithshallrelabeltheelementsofR♦asfollows

󰀃󰀄󰀃󰀄󰀃󰀄󰀃0010111

0≡,1≡,2≡,3≡

0001010󰀃󰀃󰀄󰀃󰀄󰀃󰀄0010010

4≡,5≡,6≡,7≡

0100000

(6)

numbersthanmatrices,we1

011󰀄

,.

(7)

󰀄

TheadditionandmultiplicationintheringisthatofmatricesoverGF(2),whichinourcompactnotationreadsasshowninTable1.Thetwomaximalidealsnowacquiretheform

I1:={0,4,6,7}and

I2:={0,3,5,6},

andtheJacobsonradicalreads,

J=I1∩I2={0,6}.

2

(10)(9)(8)

3

Thereexistaltogether21freecyclicsubmodulesofR♦whicharegeneratedbynon-unimodularvectors.Takingtheircompletelistfrom[2]oneseesthattheycanbeseparatedintothefollowingthreedisjointsets

R♦(6,6,7)=R♦(6,6,4)={(0,0,0),(6,6,7),(6,6,4),(6,6,0),(0,0,4),(6,6,6),(0,0,6),(0,0,7)},R♦(6,7,6)=R♦(6,4,6)={(0,0,0),(6,7,6),(6,4,6),(6,0,6),(0,4,0),(6,6,6),(0,6,0),(0,7,0)},R♦(7,6,6)=R♦(4,6,6)={(0,0,0),(7,6,6),(4,6,6),(0,6,6),(4,0,0),(6,6,6),(6,0,0),(7,0,0)},R♦(0,6,7)=R♦(0,6,4)={(0,0,0),(0,6,7),(0,6,4),(0,6,0),(0,0,4),(0,6,6),(0,0,6),(0,0,7)},R♦(0,7,6)=R♦(0,4,6)={(0,0,0),(0,7,6),(0,4,6),(0,0,6),(0,4,0),(0,6,6),(0,6,0),(0,7,0)},R♦(6,0,7)=R♦(6,0,4)={(0,0,0),(6,0,7),(6,0,4),(6,0,0),(0,0,4),(6,0,6),(0,0,6),(0,0,7)},R♦(7,0,6)=R♦(4,0,6)={(0,0,0),(7,0,6),(4,0,6),(0,0,6),(4,0,0),(6,0,6),(6,0,0),(7,0,0)},R♦(6,7,0)=R♦(6,4,0)={(0,0,0),(6,7,0),(6,4,0),(6,0,0),(0,4,0),(6,6,0),(0,6,0),(0,7,0)},R♦(7,6,0)=R♦(4,6,0)={(0,0,0),(7,6,0),(4,6,0),(0,6,0),(4,0,0),(6,6,0),(6,0,0),(7,0,0)},R♦(4,6,7)=R♦(7,6,4)={(0,0,0),(4,6,7),(7,6,4),(6,6,0),(4,0,4),(0,6,6),(6,0,6),(7,0,7)},R♦(4,7,6)=R♦(7,4,6)={(0,0,0),(4,7,6),(7,4,6),(6,0,6),(4,4,0),(0,6,6),(6,6,0),(7,7,0)},R♦(6,4,7)=R♦(6,7,4)={(0,0,0),(6,4,7),(6,7,4),(6,6,0),(0,4,4),(6,0,6),(0,6,6),(0,7,7)},R♦(4,4,6)=R♦(7,7,6)={(0,0,0),(4,4,6),(7,7,6),(6,6,6),(4,4,0),(0,0,6),(6,6,0),(7,7,0)},R♦(4,6,4)=R♦(7,6,7)={(0,0,0),(4,6,4),(7,6,7),(6,6,6),(4,0,4),(0,6,0),(6,0,6),(7,0,7)},R♦(6,4,4)=R♦(6,7,7)={(0,0,0),(6,4,4),(6,7,7),(6,6,6),(0,4,4),(6,0,0),(0,6,6),(0,7,7)},R♦(0,4,7)=R♦(0,7,4)={(0,0,0),(0,4,7),(0,7,4),(0,6,0),(0,4,4),(0,0,6),(0,6,6),(0,7,7)},R♦(4,0,7)=R♦(7,0,4)={(0,0,0),(4,0,7),(7,0,4),(6,0,0),(4,0,4),(0,0,6),(6,0,6),(7,0,7)},R♦(4,7,0)=R♦(7,4,0)={(0,0,0),(4,7,0),(7,4,0),(6,0,0),(4,4,0),(0,6,0),(6,6,0),(7,7,0)},R♦(4,4,7)=R♦(7,7,4)={(0,0,0),(4,4,7),(7,7,4),(6,6,0),(4,4,4),(0,0,6),(6,6,6),(7,7,7)},R♦(4,7,4)=R♦(7,4,7)={(0,0,0),(4,7,4),(7,4,7),(6,0,6),(4,4,4),(0,6,0),(6,6,6),(7,7,7)},R♦(7,4,4)=R♦(4,7,7)={(0,0,0),(7,4,4),(4,7,7),(0,6,6),(4,4,4),(6,0,0),(6,6,6),(7,7,7)},accordingasthenumberofJacobsonradicalentriesinthegeneratingvector(s)istwo,oneorzero,respectively.EmployingthepictureoftheFano-Snowflakegivenin[2](reproduced,forconvenience,inFigure1),thestructureofandrelationbetweenthethreesetscanberepresenteddiagrammaticallyasshowninFigure2.AseachsubmoduleanswerstoasinglelineoftheassociatedcoreFanoplane,thedecompositionoftheFano-Snowflakeinducesanintriguingfactorizationofthelinesoftheplaneitself.ThisisdepictedinFigure3,bottompanel,anditisseentofundamentallydifferfromthecorrespondingpartitioningoftheFanoplanewithrespecttotheJacobsonradicalofitsgroundfieldGF(2)({0}),namelyGF(2)(1,0,0)={(0,0,0),(1,0,0)},GF(2)(0,1,0)={(0,0,0),(0,1,0)},GF(2)(0,0,1)={(0,0,0),(0,0,1)},GF(2)(1,1,0)={(0,0,0),(1,1,0)},GF(2)(1,0,1)={(0,0,0),(1,0,1)},GF(2)(0,1,1)={(0,0,0),(0,1,1)},GF(2)(1,1,1)={(0,0,0),(1,1,1)},

asdisplayedinFigure3,toppanel.

TheoriginofthefactorizationoftheFanoplanewhenrelatedtoitsgroundfieldGF(2)iseasytounderstand:thenumberoftheJacobsonradicalentries(i.e.,onlyzerosinthiscase)inthecoordinatesofaline(and,byduality,ofapointaswell)hasaclearmeaningwithrespecttothetriangleofbasepointsofthecoordinatesystem.SomethingsimilarholdsobviouslyforthefactorizationoftheFano-Snowflakewithrespecttoitsternioniccoordinates,butpassingtotheembeddedFanoplanethislinkseemstobelostorsubstantiallydistorted.Figure2illustratesthisfactquitenicely:inthetopfigureallthepolygonspassthroughthethreecorners/verticesofthebasictriangle,inthemiddlefigureallbranchesgothroughthepointsoneachsideofthe

3

Figure1:TheFano-Snowflake—adiagrammaticillustrationofaveryintricaterelationbetween

3

the21freeleftcyclicsubmodulesgeneratedbynon-unimodularvectorsofR♦.Eachcircle

33

representsavectorofR♦(infact,ofI1),itssizebeingroughlyproportionaltothenumberofsubmodulespassingthroughthegivenvector.Asthe(0,0,0)tripleisnotshown,eachsubmoduleisrepresentedbysevencircles(threebig,twomedium-sizedandtwosmall)lyingonacommonpolygon.Thesmallcirclesstandforthevectorsgeneratingthesubmodules.ThebigcirclesrepresentthevectorswhoseallthethreeentriesarefromJ;thesevectorscorrespondtothepointsoftheFanoplane.ThesevencolorswerechoseninsuchawaytomakealsothelinesoftheFanoplane,i.e.,theintersectionsofthesubmoduleswithJ3,readilydiscernible.See[2]and/or[3]formoredetails.

4

Figure2:Asketchyillustrationofthe9–9–3decompositionofthesetoffreecyclicsubmodulescomprisingtheFano-SnowflakewithrespecttotheJacobsonradicalofR♦accordingasthenumberofradicalentriesinthesubmodule’sgeneratingvector(s)istwo(top),one(middle)orzero(bottom),respectively.

5

Figure3:Acomparisonofthe“ternion-induced”6–7–3factorizationofthelinesoftheFanoplane(bottom)withtheordinary3–3–1one(top).Noteaprincipalqualitativedifferencebetweenthetwofactorizationsasthethreesets(factors)arepairwisedisjointinthelattercasebutnotintheformerone.

trianglewhicharenotvertices,whereasinthebottomfigureallthebranchessharetheonlypointwhichisoffthereferencetriangle.Itistheintersectionsofthebranches/polygonswiththecoreFanoplanewhichbehave“strange”andgiverisetothefundamentaldifferencebetweenthetwofactorizationsoftheFanoplaneshowninFigure3.

Theabove-describedobservationsclearlydemonstratethatthereismoretothealgebraicstructureoftheFanoplanethanmeetstheeye.Theplanewhenconsideredofitsownisfoundto“reveal”quitedifferentaspectscomparedwiththecasewhenbeingembeddedintoamoregen-eral,non-unimodularprojectivelatticesetting.Thisdifferenceislikelytogetmorepronounced,andmoreintricateaswell,aswepasstohigherorderringsgivingrisetomorecomplexformsofFano-Snowflakes.AkeyquestionistofindoutwhethertheSnowflakes’decompositionpatternsandtheirinducedfactorizationsofthelinesofthecoreFanoplanesremainqualitativelythesameasintheternioniccase;ourpreliminaryanalysisofsuchstructuresoveraparticularclassofnon-commutativeringsofordersixteenandhavingtwelvezero-divisorsindicatesthatthismightbeso.Anotherworth-pursuinglineofexplorationistostaywithternionsbutfocusonhigher-order(q>2)and/orhigher-dimensional(n>2)“Snowflake”geometriesandtheircoreprojectiveplanesand/orspaces.Finding,however,ageneralbuild-upprincipleforthesere-markablegeometricalstructureswithrespecttothepropertiesofdefiningringscurrentlyseemsatrulydifficult,yetextremelychallengingtaskdueto“ubiquity”oftheFanoplaneinvariousmathematicalandphysicalcontexts(see,e.g.,[8]).

Acknowledgments

TheworkwaspartiallysupportedbytheVEGAgrantagencyprojectsNos.6070and7012,theCNRS-SAVProjectNo.20246andbytheActionAustria-SlovakiaprojectNo.58s2.WethankHansHavlicek(ViennaUniversityofTechnology)forvaluablecommentsandsuggestions.

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