Composed inclusions of A and A subfactors Advances in Mathematics

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Advances in Mathematics

www.elsevier.com/locate/aim

Composed inclusions of A

and A

subfactors

Zhengwei Liu

DepartmentofMathematics,VanderbiltUniversity,Nashville,TN37240, USA

a r t i c l e i n f o a b s t r a c t

Articlehistory:

Received18September2013 Accepted11March2015 Availableonline23April2015 CommunicatedbytheManaging EditorsofAIM

Keywords:

Subfactors Planaralgebras

Inthisarticle,weclassifyallstandardinvariantsthatcanarise froma composed inclusion of an A3 withan A4 subfactor.

Moreprecisely, ifN ⊂ P isanA3 subfactorandP ⊂ Mis anA4subfactor,thenonlyfourstandardinvariantscanarise fromthecomposedinclusionN ⊂ M.Weansweraquestion posedbyBischandHaagerupin1994.Thetechniquesofthis paperalsoshowthatthereareexactlyfourstandardinvariants forthecomposedinclusionoftwoA4subfactors.

© 2015ElsevierInc.All rights reserved.

1. Introduction

Jonesclassifiedtheindices ofsubfactorsoftypeII₁ in[13].Itisgivenby {4 cos²(π

n), n= 3,4,· · ·} ∪[4,∞].

Fora subfactorN ⊂ M of typeII₁ withfinite index,the Jonestower isa sequence of factors obtained by repeating the basic construction. The system of higher relative commutantsis called thestandardinvariantof thesubfactor[8,35].A subfactorissaid tobeoffinite depth,ifitsprincipalgraph isfinite.Thestandardinvariantisacomplete invariantofafinite depthsubfactor[35].

E-mailaddress:zhengwei.liu@vanderbilt.edu.

http://dx.doi.org/10.1016/j.aim.2015.03.017 0001-8708/© 2015ElsevierInc.All rights reserved.

SubfactorplanaralgebraswereintroducedbyJonesasadiagrammaticaxiomatization ofthestandardinvariant[17].OtheraxiomatizationsareknownasOcneanu’sparagroups [30]andPopa’sλ-lattices[37].EachsubfactorplanaralgebracontainsaTemperley–Lieb planar subalgebra which is generated by the sequence of Jones projections. When the index oftheTemperley–Liebsubfactorplanaralgebrais4cos²(_n+1^π ),itsprincipalgraph is theCoxeter–DynkindiagramAn.

Given two subfactors N ⊂ P and P ⊂ M, the composed inclusion N ⊂ P ⊂ M tells therelative position of these factors.The grouptype inclusion R^H ⊂ R⊂ RK for outer actionsof finite groupsH and K on thehyperfinite factor Rof typeII₁ was discussed byBischandHaagerup[5].

WeareinterestedinstudyingthecomposedinclusionoftwosubfactorsoftypeA,i.e., a subfactorN ⊂ M with anintermediate subfactor P, such thatthe principalgraphs of N ⊂ P andP ⊂ M aretypeA Coxeter–Dynkindiagrams. From theplanaralgebra point of view, the planaralgebra of N ⊂ M is a composition of two Temperley–Lieb subfactorplanaralgebras.Theirtensorproductiswellknown[17,23].Theirfreeproduct asaminimalcompositionwasdiscoveredbyBischandJones[6],calledtheFuss–Catalan subfactor planaralgebra.Ingeneral, thecompositionof two Temperley–Liebsubfactor planaralgebras isstillnotunderstood.

TheeasiestcaseisthecomposedinclusionoftwoA3subfactors.Inthiscase,theindex is 4,and such subfactorsare extended type D [8,36]. They also arise as agroup type inclusionR^H⊂ R⊂ RK,whereH ∼=Z2 andK∼=Z2.

Thefirstnon-group-likecaseisthecomposedinclusionofanA3withanA4subfactor.

ItsprincipalgraphiscomputedbyBischandHaagerupintheirunpublishedmanuscript in1994. Eitheritisafreecomposedinclusion,thenitsplanaralgebraisFuss–Catalan;

or itsprincipalgraphisaBisch–Haagerupfishgraphas

.

Thentheyaskedwhetherthissequenceofgraphsaretheprincipalgraphsofsubfactors.

The first Bisch–Haagerup fish graph is the principal graph of the tensor product of an A₃ and an A₄ subfactor. By considering the flip on R⊗ R, Bisch and Haagerup constructedasubfactorwhoseprincipalgraphisthesecondBisch–Haagerupfishgraph.

Later Izumi generalised the Haagerup factor [1] while considering endomorphisms of Cuntzalgebras[10],andheconstructedaHaagerup–IzumisubfactorforthegroupZ4in his unpublishednotes,alsocalled the3^Z4 subfactor[31].ThethirdBisch–Haagerupfish graphis theprincipalgraphof anintermediate subfactorofareducedsubfactorof the dual of3^Z4 [12].It turnsouttheevenhalfisMoritaequivalentto theeven halfof 3^Z4.

Inthispaper, weprovethefollowing classificationresult.

Theorem1.1.ThereareexactlyfoursubfactorplanaralgebrasasacompositionofanA₃ with an A4 planaralgebra.

This answers the question posed by Bisch and Haagerup. When n ≥ 4, the nth Bisch–Haagerupfishgraphisnottheprincipalgraphofasubfactor.Inthemeanwhile, Izumi, Morrisonand Penneyshaveruled outthe4th–10th Bisch–Haagerup fishgraphs usingadifferentmethod,see[12].

Three of the four subfactor planar algebras have finite depth which are complete invariantsofsubfactorsofthehyperfinite factorof typeII1 [35].TheFuss–Catalanone hasinfinite depth.Itcanalsoberealised fromahyperfinitesubfactor[38].

Bysimilar techniques,wealsoprovethefollowingclassificationresult.

Theorem1.2. Thereare exactlyfoursubfactors planaralgebrasasacomposition oftwo A₄ planaralgebras.

Ourclassificationresultis alsoimportant tothe classificationofsmall index subfac- tors. Theindex 3+√

5 isthenextfrontier after 5 where thesubfactorplanaralgebras werecompletedclassifiedrecently[18,28,26,11,32].Someinterestingexamplesand clas- sificationresultsareknownupto thisindex[27,25].

Now we sketchthe ideas of theproof. Following the spirit of [33,2], if theprincipal graphof asubfactorplanaralgebrais thenth Bisch–Haagerup fishgraph,then bythe embeddingtheorem[19],theplanaralgebraisembeddedinthegraph planaralgebra of itsprincipalgraph [14]. Bytheexistence ofa“normalizer” in theBisch–Haagerupfish graph, there will be a biprojection [3] inthe subfactor planar algebra, and the planar subalgebrageneratedbythebiprojectionisFuss–Catalan.Theimageofthebiprojection isdeterminedbytheuniquepossiblerefinedprincipalgraph,seeDefinition 3.14andThe- orem 3.26.Furthermore the planaralgebrais decomposed as anannular Fuss–Catalan module,similar to theTemperley–Lieb case, [15,20].Comparingthe principalgraphof this Fuss–Catalan subfactor planar algebra and the Bisch–Haagerup fishgraph, there isalowest weightvectorinthe orthogonalcomplementof Fuss–Catalan.It will satisfy somespecific relations, and there is a“unique” potential solution of these relations in thegraphplanaralgebra.

ThesimilarityofalltheBisch–Haagerupfishgraphsadmitsustocomputethecoeffi- cientsofloopsofthepotentialsolutionssimultaneously.Thecoefficientsoftwosequences ofloopshave periodicity5and20withrespectton.Comparingwiththecoefficientsof theothertwosequencesofloops,wewillruleoutthealltheBisch–Haagerupfishgraphs, exceptthefirstthree.

Theexistence ofthe firstthree follows from the constructionmentioned above.The uniquenessfollows fromthe“uniqueness”ofthepotentialsolution.

Furthermore we consider the composition of two A4 planar algebras in the same process.Inthislist,thereareexactlyfoursubfactorplanaralgebras.Theyallarisefrom reducedsubfactorsofthefourcompositionsofA3withA4.

Theskeintheoreticconstructionofthese subfactorplanaralgebras couldbe realised bytheFuss–CatalanJellyfishrelationsofageneratingvectorspace.

2. Background

Wereferthereaderto[16]forthedefinitionofplanaralgebras.

Notation 2.1. In aplanar tangle, we use a thickstring with a numberk to indicate k parallel strings.

A subfactor planar algebra S = {Sn,±}n∈N0 will be a spherical planar *-algebra over C, such that dim(Sn,±) < ∞, for all n, dim(S0,±) = 1, and the unnormal- ized Markovtrace induces apositivedefinite inner product of _Sn,± [16,17]. Note that dim(S0,±)= 1,thenS0,± isisomorphicto Casafield.Itissphericalmeans

=

as anumberinC, foranyx∈S1,±. Theinner productof Sn,± definedas

< y, z >=tr(z^∗y) = ,

the Markovtraceof z^∗y,for any y,z∈ Sn,±,is positivedefinite.(The diagram of the multiplication z^∗y is not the usual convention. The usual multiplication is from the bottomto thetop.)

A subfactor planaralgebrais alwaysunital, where unitalmeans any tangle without inner discscanbe identified as avector ofS. Note thatS0,± is isomorphic to C,the (shadedorunshaded)emptydiagramcanbeidentifiedasthenumber1 inC.Thevalue ofa(shadedorunshaded)closedstringisδ.Andδ⁻¹ in Sn,+,denotedbyen−1, for n≥2,is thesequenceofJones projections.ThefilteredalgebrageneratedbyJones projections isthesmallestsubfactorplanaralgebra,wellknownas theTemperley–Lieb algebra,denotedbyTL(δ). Itsvectorscanbe writtenaslinearsumsoftangleswithout inner discs.

Notation 2.2. Wemayidentify Sm,− asasubspaceof Sm+1,+ by addingonestringto theleft.

Definition 2.3. Let us define the (1-string) coproduct of x ∈ Si,± and y ∈ Sj,±, for i,j ≥1,tobe

x∗y= ,

whenevertheshadingmatched.

Let us recall some facts about the embedding theorem. Then we generalize these resultstoprovetheembeddingtheoremforanintermediatesubfactorinthenextsection.

2.1. Principal graphs

SupposeN ⊂ MisanirreduciblesubfactoroftypeII₁withfiniteindex.ThenL²(M) forms an irreducible (N,M) bimodule, denoted by X. Its conjugate X is an (M,N) bimodule.Thetensor productsX ⊗X⊗ · · · ⊗X, X⊗X⊗ · · · ⊗X, X⊗X⊗ · · · ⊗X andX⊗X⊗ · · · ⊗X aredecomposedintoirreduciblebimodulesover(N,N),(N,M), (M,N) and(M,M) respectively,where⊗isConnesfusionof bimodules.

Definition 2.4. The principal graph of the subfactor N ⊂ M is a bipartite graph. Its verticesareequivalenceclassesofirreduciblebimodulesover(N,N) and(N,M) inthe abovedecomposed inclusion.The numberof edges connecting two vertices, an (N,N) bimoduleY andan(N,M) bimoduleZ,isthemultiplicityoftheequivalenceclassofZ asasubbimoduleofY⊗X.Thevertexcorrespondingtothe(N,N) bimoduleL²(N) is markedbyastarsign.Thedimensionvectorofthebipartitegraphisafunctionλfrom thevertices of thegraphto R⁺. Its valueat avertex isdefinedto be thedimension of thecorrespondingbimodule.

Thedualprincipalgraphisdefinedsimilarlyfor(M,M) and(M,N) bimodules.

Remark 2.5. By Frobenius reciprocity, the multiplicity of Z in Y ⊗X equals to the multiplicityofY inZ⊗X.

2.2. Standardinvariants

ForanirreduciblesubfactorN ⊂ MoftypeII1 withfinite index,theJones toweris asequenceoffactors N ⊂ M⊂ M1 ⊂ M2⊂ ·· ·obtainedby repeatingthe basic con- struction.Thesystemofhigherrelativecommutants

C=N∩ N ⊂ N∩ M ⊂ N∩ M1 ⊂ N∩ M2 ⊂ · · ·

∪ ∪ ∪

C =M∩ M ⊂ M∩ M1 ⊂ M∩ M2 ⊂ · · · iscalledthestandardinvariantofthesubfactor[8,35].

ThereisanaturalisomorphismbetweenhomomorphismsofbimodulesX⊗X⊗· · ·⊗X, X⊗X⊗· · ·⊗X,X⊗X⊗· · ·⊗X andX⊗X⊗· · ·⊗X andthestandardinvariantofthe subfactor[4].Theequivalenceclassofaminimalprojectioncorrespondstoanirreducible bimodule.Sotheprincipalgraphtellshowminimalprojectionsaredecomposedafterthe inclusion.Thenwecandefinetheprincipalgraphforasubfactorplanaralgebrawithout thepresumedsubfactor.Thefollowingtwopropositionsarewell knowntoexperts.

Proposition 2.6.Suppose S isasubfactorplanaralgebra.If P1,P2 are minimalprojec- tionsof _Sm,+,thenP₁e_m+1,+,P₂e_m+1,+ areminimalprojectionsof _Sm+2,+.Moreover P1 andP2 are equivalent in Sm,+ if andonly if P1em+1 and P2em+1 are equivalent in Sm+2,+.

Proposition 2.7 (Frobenius reciprocity). Suppose _S is a subfactor planar algebra. If P is a minimal projection of Sm,+ and Q is a minimal projection of Sm+1,+, then dim(P_Sm+1,+Q)= dim(P e_m+1Sm+2,+Q).

Bytheabovetwo propositions,theBrattelidiagram of_Sm,+⊂Sm+1,+ isidentified as asubgraphof theBrattelidiagramof Sm+1,+ ⊂Sm+2,+. Soitmakessense to take thelimitoftheBrattelidiagramof_Sm,+ ⊂Sm+1,+ asmapproaches infinity.

Definition 2.8. The principalgraphof asubfactorplanar algebra_S is thelimit of the Brattelidiagram of Sm,+ ⊂Sm+1,+.Thevertex correspondingto theidentityin S0,+

ismarkedbyastarsign.ThedimensionvectorλatavertexisdefinedtobetheMarkov traceoftheminimalprojectioncorresponding tothatvertex.

Similarlythedualprincipalgraphofasubfactorplanaralgebra_S isthelimitofthe Brattelidiagram of Sm,− ⊂Sm+1,−.Thevertex correspondingto theidentityin S0,−

ismarkedbyastarsign.ThedimensionvectorλatavertexisdefinedtobetheMarkov traceoftheminimalprojectioncorresponding tothatvertex.

The Brattelidiagram of Sm,+ ⊂Sm+1,+, as asubgraph ofthe Brattelidiagram of Sm+1,+⊂Sm+2,+,correspondstothetwo-sidedideal_Im+1,+ of_Sm+1,+ generatedby theJonesprojectionem.Sothetwographscoincideifandonlyif Sm+1,+=Im+1,+. Definition2.9.Forasubfactorplanaralgebra S,ifitsprincipalgraphisfinite,thenthe subfactorplanaralgebraissaidtobe finitedepth.Furthermoreitisofdepthm,ifmis thesmallestnumbersuchthat Sm+1,+=Sm+1,+emSm+1,+.

Definition 2.10.A vertex v intheprincipal graphhasdepthmifthe distance between v andthestar vertexis m.Thevertex hasmultiplicity nifthere arenlength-mpaths from thestarvertexto v.

A depth-m vertex in the principal graph corresponds to a central component in Sm,+/Im,+. The vertex has multiplicity n tells that the central component is an n bynmatrixalgebra.Thereforethedualofthevertex hasthesamemultiplicity.

2.3. Finite dimensionalinclusions

Wereferthereaderto Chapter3of[21]foradiscussionofinclusionsoffinitedimen- sionalvon Neumannalgebras.

Definition 2.11. Suppose A is a finite dimensional von Neumann algebra and τ is a trace on it. The dimension vector λ^τ_A is a function from the set of minimal central projections(orequivalence classesofminimal projectionsorirreducible representations upto unitaryequivalence) of A to C with following property, for any minimal central projectionz,λ^τ_A(z)=τ(x),wherex∈Aisaminimalprojectionwithcentralsupportz.

Thetraceofaminimalprojectiononlydependsonitsequivalenceclass,sothedimen- sionvectoriswelldefined.Ontheotherhand,givenafunctionfromtheset ofminimal centralprojectionsofAtoC,wecanconstructatraceofA,suchthatthecorresponding dimensionvectoristhegivenfunction.Sothemapλ→λ^τ_Aisabijection.

LetusrecallsomefactsabouttheinclusionoffinitedimensionalvonNeumannalge- brasB0⊂ B1.

TheBratteli diagram Br for theinclusionB0 ⊂ B1 isa bipartitegraph. Itseven or oddverticesareindexedbytheequivalenceclassesofirreduciblerepresentationsofB0or B1 respectively.Thenumberofedgesconnectsavertex correspondingto anirreducible representation U ofB0 toa vertex corresponding toan irreducible representationV of B1isgiven bythemultiplicity ofU intherestrictionofV onB0.

Let Br_± be the even/odd vertices of Br. The Bratteli diagram can be interpreted as the adjacency matrixΛ = Λ^B_B¹₀ : L²(Br₋)→ L²(Br+),where Λu,v is defined as the numberofedgesconnectsutov forany u∈Br₊,v∈Br₋.

Proposition 2.12. (See [13].) For the inclusion B0 ⊂ B1 and a trace τ on it, we have λ^τ_B0 = Λλ^τ_B1.

Ifthetraceτ is afaithfulstate,thenby GNSconstructionwe willobtainarightB1

module L²(B1). And L²(B0) is identified as a subspaceof L²(B1). Let e be the Jones projectiononto thesubspaceL²(B0).LetB2 bethevonNeumannalgebra(B1∪ {e}). ThenweobtainatowerB0⊂ B1⊂ B2whichiscalledthebasicconstruction.Furthermore if thetracial state τ satisfies the condition Λ^∗Λλ^τ_B1 =μλ^τ_B1 for some scalarμ, then it issaidto be aMarkovtrace.Inthis casethescalarμis||Λ||². Thenλ^τ =

λ^τ_B0 δλ^τ_B1

isa Perron–Frobeniuseigenvectorfor

0 Λ Λ^∗ 0

.

Definition2.13.Wecallλ^τ thePerron–FrobeniuseigenvectorwithrespecttotheMarkov traceτ.

Remark 2.14. Theexistence of a Markovtrace for theinclusion B0 ⊂ B1 follows from the Perron–Frobenius theorem.The Markov traceis unique ifand onlyif the Bratteli diagram fortheinclusionB0⊂ B1isconnected.

Wewill seetheimportanceof theMarkovtracefromthefollowingproposition.

Proposition 2.15. If τ is a Markov trace for the inclusion B0 ⊂ B1, then τ extends uniquely to a trace on B2, still denoted by τ. Moreover τ is a Markov trace for the inclusionB1⊂ B2.

In this case, we may repeat the basic construction to obtain a sequence of finite dimensional vonNeumann algebras B0⊂ B1⊂ B2 ⊂ B3 ⊂ · · ·and asequence ofJones projectionse1,e2,e3· · ·.

2.4. Graphplanaralgebras

Given afiniteconnectedbipartitegraphΓ,itcanberealised astheBrattelidiagram of theinclusion of finite dimensionalvon Neumann algebras B0 ⊂ B1 with a (unique) Markov trace. Applying the basic construction, we will obtain the sequence of finite dimensionalvonNeumannalgebrasB0⊂ B1⊂ B2⊂ B3⊂ · · ·.Take Sm,+ tobeB0∩ Bm

and S_m,−tobeB1∩Bm+1.Then{S_m,±}formsaplanaralgebra,calledthegraphplanar algebra ofthebipartitegraphΓ.Moreover_Sm,±hasanaturalbasisgivenbylength2m loopsofΓ.Wereferthereaderto[14,19]formoredetails.Wecite theconventionsused inSection3.4 of[19].

Definition 2.16. Let us define _G = {Gm,±} to be the graph planar algebra of a finite connectedbipartitegraphΓ.LetλbethePerron–Frobeniuseigenvectorwithrespectto theMarkovtrace.

A vertexof theΓ correspondstoanequivalence classofminimal projections,soλis alsodefinedasafunctionfromV±toR⁺.IfΓ istheprincipalgraphofasubfactor,then itsdimensionvectorisamultipleofthePerron–Frobeniuseigenvector.Inthispaper,we onlyneedtheproportionofvaluesofλatvertices. Wedonothavetodistinguishthese two vectors.

Let V± be the sets of even/odd vertices of Γ, and let E be the sets of all edges of Γ directed from even to odd vertices. Then we have the source and target functions s : E → V+ and t : E → V−. For adirected edge ε ∈ E, we define ε^∗ to be the same edge withanoppositedirection.Thesourcefunctions:E^∗={ε^∗|ε∈ E}→ V− and the target functiont:E^∗→ V+ aredefinedass(ε^∗)=t(ε) andt(ε^∗)=s(ε).

Alength2mloopinGm,+ isdenotedby[ε1ε^∗₂· · ·ε_2m−1ε^∗_2m] satisfying (i) t(εk)=s(ε^∗_k+1)=t(εk+1),foralloddk <2m;

(ii) t(ε^∗_k)=s(ε_k)=t(ε_k+1),forallevenk <2m;

(iii) t(ε^∗_2m)=s(ε_2m)=t(ε₁).

Thegraphplanaralgebraisalwaysunital.Theunshadedemptydiagram isgivenby

v∈V+v; and theshaded empty diagram is givenby

v∈V−v.It is worth mentioning thattheJonesprojectionisgivenby

e1=δ⁻¹ =δ⁻¹

s(ε1)=s(ε3)

λ(t(ε1))λ(t(ε3))

λ(s(ε₁))λ(s(ε₃))[ε1ε^∗₁ε3ε^∗₃].

Nowletusdescribesomeelementaryactionson_G.

Theadjointoperation isdefinedas theanti-linearextensionof [ε1ε^∗₂· · ·ε2m−1ε^∗_2m]^∗= [ε2mε^∗_2m−1· · ·ε2ε^∗₁].

ForGm,−,wehavesimilarconventions.

Forl1,l2∈Gm,+, l1= [ε1ε^∗₂· · ·ε2m−1ε^∗_2m],l2= [ξ1ξ₂^∗· · ·ξ2m−1ξ2m],wehave

= ^1≤k≤mδεm+k,ξ_m+1−k[ε1ε^∗₂· · ·ε^∗_mξm+1· · ·ξ2m−1ξ_2m^∗ ] whenmis even;

1≤k≤mδεm+k,ξ_m+1−k[ε1ε^∗₂· · ·εmξ^∗_m+1· · ·ξ_2m−1ξ_2m^∗ ] whenmis odd.

= ^s(ε)=s(εm)[ε₁ε^∗₂· · ·ε^∗_mεε^∗ε_m+1· · ·ε_2m−1ε^∗_2m] whenmis even;

t(ε)=t(εm)[ε1ε^∗₂· · ·εmε^∗εε^∗_m+1· · ·ε2m−1ε^∗_2m] whenmis odd.

=

δεm,εm+1

λ(s(εm))

λ(t(εm))[ε1ε^∗₂· · ·ε^∗_mεε^∗εm+1· · ·ε2m−1ε^∗_2m] whenmis even;

δεm,εm+1

λ(t(εm))

λ(s(εm))[ε1ε^∗₂· · ·εmε^∗εε^∗_m+1· · ·ε2m−1ε^∗_2m] whenmis odd.

Definition2.17.TheFouriertransformF:Gm,+→Gm,−,m>0 isdefinedasthelinear extensionof

F([ε1ε^∗₂· · ·ε2m−1ε^∗_2m]) =

⎧⎨

⎩

λ(s(ε2m)) λ(t(ε2m))

λ(s(εm))

λ(t(εm))[ε^∗_2mε₁ε^∗₂· · ·ε_2m−1] form even;

λ(s(ε2m)) λ(t(ε2m))

λ(t(εm))

λ(s(εm))[ε^∗_2mε₁ε^∗₂· · ·ε_2m−1] form odd.

Similarlyitisalsodefinedfrom Gm,− to Gm,+.

TheFouriertransformhasadiagrammatic interpretationas aone-clickrotation

.

Definition 2.18. Let us define ρ to be F². Then ρ is defined from _Gm,+ to _Gm,+ as a two-click rotationform>0,

ρ([ε1ε^∗₂· · ·ε2m−1ε^∗_2m]) =

λ(s(ε2m)) λ(s(ε_2m−1))

λ(s(εm))

λ(s(ε_m−1))[ε2m−1ε^∗_2mε1ε^∗₂· · ·ε2m−3ε^∗_2m−2].

It issimilarforGm,−.

Ingeneral,theactionofaplanartanglecouldbe realisedasacompositionofactions mentionedabove.Ithasaniceformula, seepage 11in[14].

2.5. Theembedding theorem

Foradepth2r(or2r+ 1) subfactorplanaralgebraS,wehave

Sm+1,+=Sm+1,+emSm+1,+=Sm,+em+1Sm,+, wheneverm≥2r+ 1.

So _Sm−1,+ ⊂ Sm,+ ⊂ Sm+1,+ forms a basic construction. Note that the Bratteli diagram of S2r,+ ⊂ S2r+1,+ is the principal graph. So the graph planar algebra G of theprincipalgraphisgivenby

Gk,+=S_2r,+ ∩S2r+k,+; Gk,−=S_2r+1,+ ∩S2r+k+1,+.

Moreover the map Φ : _S → G by adding 2r strings to the left preserves the planar algebrastructure.Itisnotobviousthattheleftconditionalexpectationispreserved.We havethefollowingembeddingtheorem,seeTheorem4.1 in[19].

Theorem 2.19. A finite depth subfactor planar algebra is naturally embedded into the graph planaralgebra of itsprincipalgraph.

Remark2.20.Theembeddingtheoremforgeneralcasesisprovedin[29].

2.6. Fuss–Catalan

TheFuss–CatalansubfactorplanaralgebraswerediscoveredbyBischandJonesasfree products of Temperley–Lieb subfactorplanar algebras while studying theintermediate

subfactorsof asubfactor[6].We referthe readerto [7,22] forthe definitionof thefree product of subfactor planar algebras. It has a nice diagrammatic interpretation. For two Temperley–Lieb subfactor planar algebras TL(δ_a) and TL(δ_b), their free product FC(δa,δb) is a subfactor planar algebra. A vector in FC(δa,δb)m,+ can be expressed as a linear sum of Fuss–Catalan diagrams, a diagram consisting of disjoint a,b-colour stringswhose boundarypointsareordered asabba abba · · ·abba

m

,mcopiesofabba,after the dollar sign. It is similar for a vector inFC(δa,δb)_m,−, while the boundary points are ordered as baab baab · · ·baab

m

. For the action of a planar tangle ona simple tensor of Fuss–Catalan diagrams, first we replace each string of the planar tangle by a pair ofparallel a-colourandb-colour stringswhichmatchesthea,b-colourboundarypoints, then the output is gluing the new tangle with the input diagrams. If there is an a or b-colourclosed circle,thenitcontributesto ascalarδ_a orδ_b respectively.

The Fuss–Catalan subfactor planar algebraFC(δa,δb) is naturallyderived from an intermediatesubfactorofasubfactor.SupposeN ⊂ M isanirreduciblesubfactorwith finite index,and P is anintermediate subfactor. Then there aretwo Jones projections e_N and e_P acting on L²(M), and we have the basic construction N ⊂ P ⊂ M ⊂ P1 ⊂ M1.Repeating this process, wewill obtainasequenceof factorsN ⊂ P ⊂ M⊂ P1 ⊂ M1 ⊂ P2 ⊂ M2· · · and a sequence of Jones projections e_N, e_P, e_M, e_P1· · ·. The algebra generated by these Jones projections forms a planar algebra, denoted by FC(δa,δb), where δa =

[P :N] and δb =

[M:P]. Moreover e_P ∈ FC(δa,δb)2,+

and e_P1 ∈ FC(δa,δb)2,− could be expressed as and respectively.Inparticular, F(e_P) isamultipleofe_P1.

Definition2.21.Foranirreduciblesubfactorplanaralgebra_S,aprojectionQ∈S2,+ is calledabiprojection,ifF(Q) isamultipleofaprojection.

If S istheplanaralgebrafor N ⊂ M, thene_P ∈S2,+ isabiprojection.Conversely allthebiprojections in_S2,+ arerealisedinthisway,see[6].

Proposition 2.22. If we identify _S2,− as a subspace of _S3,+ by adding a string to the left,thena biprojectionQ∈S2,+ willsatisfy QF(Q)=F(Q)Q,i.e.

= ,

called theexchangerelationof abiprojection.

Converselyifaself-adjointoperatorinS2,+ satisfiestheexchangerelation,thenitis amultipleofabiprojection.Thiscanbeprovedbyaddingcapsatproperpositions.We refer the reader to [23] for some other approaches to biprojections. The Fuss–Catalan subfactor planar algebra could also be viewed as the planar algebra generated by a biprojection withitsexchangerelation.

The planaralgebra of acomposed inclusion of an A3 with an A4 subfactoralways contains FC(δ_a,δ_b), where δ_a =√

2, δ_b = ^√5+1₂ , as aplanarsubalgebra. The principal graphanddual principalgraphofFC(δa,δb) aregivenas

.

3. Theembeddingtheoremforanintermediatesubfactor

A subfactor planaralgebrais embeddedin thegraphplanar algebraof itsprincipal graphbytheembeddingtheorem.Ifasubfactorplanaralgebracontainsabiprojection, then wehopetoknowtheimageofthebiprojection inthegraphplanaralgebra.Recall thattheimageoftheJonesprojectione1 isdeterminedbytheprincipalgraph,

δe1=

s(ε1)=s(ε3)

λ(t(ε1)) λ(s(ε₁))

λ(t(ε3))

λ(s(ε₃))[ε1ε^∗₁ε3ε^∗₃].

We willsee asimilarformula fortheimage ofthebiprojection.Itis determined bythe refined principalgraph. Therefined principalgraphis alreadyconsideredby Bisch and Haagerup for bimodules, by Bisch and Jones for planar algebras. For the embedding theorem,wewillusetheoneforplanaralgebras.

Thelopsided versionofembeddingtheorem foranintermediatesubfactorisinvolved in ageneralembedding theorem proved byMorrison in[29]. Toconsider thealgebraic structures,itisconvenienttoworkwiththesphericalversionoftheembeddingtheorem.

Their relationsaredescribed in[27].For convenience,we provethesphericalversion of embeddingtheorem,similarto theoneprovedbyJonesandPenneysin[19].

Inthissection,wealwaysassumeN ⊂ MisanirreduciblesubfactoroftypeII₁with finite index, and P is an intermediate subfactor. If the subfactorhas an intermediate subfactor,thenitsplanaralgebrabecomesanN −P −Mplanaralgebra.ForN −P −M planaralgebras,wereferthereadertoChapter4in[9].Inthiscase,thesubfactorplanar algebracontainsabiprojectionP,andaplanartanglelabelled byP canbe replacedby

aFuss–Catalan planar tangle. In this paper, we will use planar tangles labelled by P, insteadofFuss–Catalanplanartangles.

3.1. Principal graphs

For the embedding theorem,we will consider the principalgraph of N ⊂ P ⊂ M. It refines the principal graph of N ⊂ M. Instead of a bipartite graph, it will be an (N,P,M)colouredgraph.Thefollowingdefinitionsandpropositionsarewellknownto experts[6,7,22,9].

Definition3.1.An(N,P,M)colouredgraphΓ isalocallyfinitegraph,suchthattheset V of itsverticesisdividedinto threedisjoint subsetsVN,VP andVM,and thesetE of itsedgesisdividedintotwodisjointsubsetsE+,E−.MoreovereveryedgeinE+connects avertexinVN tooneinVP andeveryedgeinE− connectsavertexinVP tooneinVM. Thenwedefinethesourcefunctions:E → VN∪ VMandthetargetfunctiont:E → VP

intheobviousway. Theoperation∗reversesthedirectionofanedge.

Definition3.2.Froman(N,P,M)colouredgraphΓ,wewillobtainan(N,M) coloured bipartitegraphΓasfollows,theN/McolouredverticesofΓareidenticaltotheN/M coloured vertices of Γ; for two vertices vn in VN and vm ∈ VM, the number of edges betweenvnandvminΓ isgivenbythenumberoflengthtwopaths fromvn tovmin Γ. ThegraphΓ issaid tobe thebipartitegraphinduced fromthe graphΓ.ThegraphΓ issaidtobe arefinementofthegraphΓ.

Remark3.3.HereweabusethenotationsofΓ andΓ whichareusuallyreservedforthe principalgraphandthedualprincipalgraphrespectively.

For afactor M of typeII₁, ifN ⊂ P ⊂ M is a sequence of irreducible subfactors withfiniteindex,thenL²(P) formsanirreducible(N,P) bimodule,denotedbyX,and L²(M) formsanirreducible(P,M) bimodule,denotedbyY.TheirconjugatesX,Y are (P,N),(M,P) bimodulesrespectively.ThetensorproductsX⊗Y ⊗Y ⊗X⊗ · · · ⊗X, X⊗Y⊗Y⊗X⊗· · ·⊗X,X⊗Y⊗Y⊗X⊗· · ·⊗Y,X⊗Y⊗Y⊗X⊗· · ·⊗Y,aredecomposed intoirreducible bimodulesover(N,N),(N,P),(N,M) and(N,P) respectively.

Definition 3.4. The principal graph for the inclusion of factors N ⊂ P ⊂ M is an (N,P,M) coloured graph.Its vertices are equivalenceclasses of irreducible bimodules over (N,N), (N,P) and (N,M) in the above decomposed inclusion. The number of edgesconnectingtwovertices,an(N,N) (or(N,M))bimoduleU (orV)andan(N,P) bimoduleW,isthemultiplicityoftheequivalenceclassofU (orV)asasubbimoduleof W⊗X(orW⊗Y).Thevertexcorrespondingtotheirreducible(N,N) bimoduleL²(N) ismarked byastarsign∗.Thedimension vectoroftheprincipal graph isafunctionλ fromtheverticesofthegraphtoR⁺.Itsvalueatapointisdefinedtobethedimension ofthecorrespondingbimodule.

Similarlythedualprincipalgraphfortheinclusionoffactorsisdefinedbyconsidering thedecomposedinclusionof(M,M),(M,P),(M,N) bimodules.

There is another principal graphgiven by decomposing (P,N), (P,P) and(P,M) bimodules underinclusions,butitisnotneededinthispaper.

Proposition 3.5. The (dual) principal graph fortheinclusionof factors N ⊂ P ⊂ M is a refinementofthe(dual) principalgraph of thesubfactorN ⊂ M.

Proof. If followsfrom thedefinition andthe factthat X⊗Y is the(N,M) bimodule L²(M). 2

Let δa be

[P :N],the dimension ofX, and δb be

[M:P], the dimensionof Y. Then byFrobeniusreciprocitytheorem,wehavethefollowing proposition.

Proposition 3.6. For the principal graph of factors N ⊂ P ⊂ M and the dimension vector λ,wehave

δaλ(u) =

ε∈E+,s(ε)=u

λ(t(ε)), ∀u∈ VN; δbλ(w) =

ε∈E−,s(ε)=w

λ(t(ε)), ∀w∈ VM; δ_aλ(v) =

ε∈E+,t(ε)=v

λ(s(ε)), ∀v∈ VP; δ_bλ(v) =

ε∈E−,t(ε)=v

λ(s(ε)), ∀v∈ VP.

Definition3.7. Foran(N,P,M) colouredgraphΓ,ifthereexits afunctionλ:V →R⁺ satisfying the proposition mentioned above, then we call it a graph with parame- ter (δ_a,δ_b).

Proposition 3.8. The principal graph of factors N ⊂ P ⊂ M isagraph with parameter (

[P:N],

[M:P]).ConsequentlyifN ⊂ Mhasfinitedepth,thentheprincipalgraph of N ⊂ P ⊂ M isfinite.

Proof. The first statement follows from the definition. Note that the dimension of a bimodule isatleast1.Bythisrestriction,N ⊂ Mhasfinitedepthimpliestheprincipal graphofN ⊂ P ⊂ Misfinite. 2

3.2. Standardinvariants

Wewilldefinetherefined(dual)principalgraphforasubfactorplanaralgebrawitha biprojection.Thisdefinitioncoincideswiththedefinitiongivenbybimodules,butwedo notneedthisfactinthispaper.GivenN ⊂ P ⊂ M,therearetwoJonesprojectionse_N ande_P actingonL²(M).ThenwehavethebasicconstructionN ⊂ P ⊂ M⊂ P1⊂ M1. Repeating thisprocess,wewillobtainasequenceoffactorsN ⊂ P ⊂ M⊂ P1⊂ M1⊂

P2⊂ M2· · ·andasequenceofJonesprojectionse_N,e_P,e_M,e_P1· · ·.Thenthestandard invariantisrefinedas

C=N∩ N ⊂ N∩ P ⊂ N∩ M ⊂ N∩ P1 ⊂ N∩ M1 ⊂ · · ·

∪ ∪ ∪

C=P∩ P ⊂ P∩ M ⊂ P∩ P1 ⊂ P∩ M1 ⊂ · · ·

∪ ∪ ∪

C =M∩ M ⊂ M∩ P1 ⊂ M∩ M1 ⊂ · · · For Fuss–Catalan, the corresponding Bratteli diagram is described by the middle patterns,see pages114–115in[6].

We hope to define the refined principal graph as the limit of the Bratteli diagram Brk of N∩ Mk−2 ⊂ N∩ Pk−1 ⊂ N∩ Mk−1. To show thelimit is well defined, we needtoprovethatBrk isidentifiedasasubgraphofBrk+1.Todefineitforasubfactor planar algebrawith a biprojection without the presumed factors, we need to do some translationsmotivatedbythefact

N∩ Pk=N∩(Mk∩ {e_Pk}) = (N∩ Mk)∩ {e_Pk}.

Definition 3.9. Let S = S_m,± be a subfactor planar algebra. Let e₁,e₂,· · · be the sequenceofJonesprojections.

Suppose p₁ is a biprojection in _S2,+. Then we obtain another sequence of Jones projections p₁,p₂,p₃,· · ·, corresponding to the intermediate subfactors, precisely p₂ in S2,− ⊂S3,+isamultipleofF(p1),andpk isobtainedbyaddingtwostringsontheleft sideofp_k−2.

Form≥1,letusdefine S_m,+ tobe Sm,+∩ {pm} and S_m,− tobe Sm,−∩ {pm+1}. Remark 3.10. If we interpret one string of a planar diagram as a pair of a/b-colour strings ,thensequenceofJonesprojectionsp1,p2,p3,· · ·,canbeinterpretedasfollow- inga/b-colour diagrams[6],

.

Thefollowingpropositionshowsthatthesubspace S_m,± of Sm,±consistsofdiagrams withana/b-colour throughstringontherightmost.

Proposition3.11.ForX ∈Sm,+,m≥1,wehave

Xpm=pmX ⇐⇒ F(X) =F(X)pm.

ThatmeansSm,+ istheinvariantsubspaceofSm,+ underthe“right action”of the biprojection.Diagrammaticallyitsconsistsofvectorswithonea/b-colourthroughstring ontherightmost.

Proof. IfpmX =Xpm, thentaketheaction givenbytheplanartangle ,we haveF(X)=F(X)pm.

Form odd,ifF(X)=F(X)p_m,thenX =X∗ F(p₁),i.e.

X = .

Bytheexchangerelationofthebiprojection,wehave

= .

Sop_mX =Xp_m.

Form even,theproofissimilar. 2

Note that Sm−1,+ is in the commutant of pm. So we have the inclusion of finite dimensionalvonNeumannalgebras

S0,+⊂S1,+ ⊂S1,+ ⊂S2,+ ⊂S2,+ ⊂ · · ·.

ThenweobtaintheBrattelidiagramBr_mfortheinclusion_Sm−1,+⊂Sm,+ ⊂Sm,+.To takethelimitofBrm,weneedtoprovethatBrmisidentified asasubgraphofBrm+1. Proposition 3.12. If P1, P2 are minimal projections of Sm,+ . Then P1pm, P2pm are minimalprojectionsof S_m+1,+ .MoreoverP1andP2 areequivalentin S_m,+ ifandonly if P1pm andP2pm areequivalent in S_m+1,+ .

This propositionisprovedsimilar toProposition 2.6.

Proposition 3.13(Frobenius reciprocity).

(1) For a minimal projection P ∈ Sm−1,+ and a minimal projection Q ∈ Sm,+ , we have that Qpm is a minimal projection of S_m+1,+ , P em is a minimal projection

of Sm+1,+,and

dim(P(_Sm,+ )Q) = dim(P e_m(_Sm+1,+)Qp_m).

(2) Foraminimalprojection P ∈Sm,+ andaminimalprojectionQ ∈Sm,+,wehave Ppm isaminimalprojectionof S_m+1,+ ,and

dim(P(Sm,+)Q) = dim(Ppm(Sm+1,+ )Q).

Proof. (1)Considerthemaps

φ1= :Sm,+→Sm+1,+, φ2= :Sm+1,+→Sm,+.

Formodd,ifX∈P(S_m,+ )Q,thenbyProposition 3.11,wehaveX=P(X∗ F(p1))Q for some X ∈ Sm,+. So φ1(X) ∈ P em(Sm+1,+)Qpm. On the other hand, if Y ∈ P e_m(_Sm+1,+)Qp_m, then φ₂(Y) ∈ P(_Sm,+ )Q. While φ₁ ◦φ₂ is the identity map on P e_m(_Sm+1,+)Qp_mandφ₂◦φ₁istheidentitymaponP(_Sm,+ )Q.Sodim(P(_Sm,+)Q)= dim(Pp_m(_Sm+1,+ )Q).

Formeven,theproofissimilar.

(2)ThisisthesameasProposition 2.7. 2

ByProposition (2.6)(3.13), the Brattelidiagram Br_m is identified as a subgraph of Br_m+1.

Definition3.14.Letusdefinetherefinedprincipalgraphof S withrespecttothebipro- jectionp₁ to be thelimit of theBratteli diagram of Sm,+ ⊂S_m+1,+ ⊂ Sm+1,+. The vertexcorresponding totheidentityin_S0,+ ismarked byastarsign.

Similarly let us define the refined dual principal graph of _S with respect to the biprojection p₁ to be thelimitof the Brattelidiagram of_Sm,− ⊂Sm+1, − ⊂Sm+1,−. Thevertexcorresponding totheidentityinS0,− ismarked byastarsign.

Therefinedprincipalgraphisan(N,P,M) colouredgraph.TheN,P, Mcoloured vertices are given by equivalence classes of minimal projections of _S2m,−, _{S2m+1, −}, S2m+1,−respectivelyasmapproachesinfinity.Similarlytherefineddualprincipalgraph isan(M,P,N) colouredgraph.

Definition3.15.Thedimensionvector λoftheprincipalgraphisdefinedasfollows, for an N or M coloured vertex, its value is the Markov trace of the minimal projection correspondingtothatvertex;foraP colouredvertexv,supposeQ∈Sm,+ isaminimal projection corresponding to v. Then λ(v) = δ⁻_a¹tr(Q), when m is even, where δa = tr(p1);λ(v)=δ⁻_b¹tr(Q),whenmisodd,whereδb=δδ⁻_a¹.

Remark 3.16.Anelement in Sm,+ has ana/b-colour throughstringon therightmost.

When wecompute thedimension vectorfor aminimal projectionin S_m,+ , thatstring shouldbe omitted.Sothere isafactorδ_a⁻¹ orδ⁻¹_b .

NotethatthedimensionvectorsatisfiesProposition 3.6.Sotherefinedprincipalgraph is a graph with parameter (δ_a,δ_b). If the Bratteli diagram of _Sm,+ ⊂ Sm+1,+ is the same as thatof Sm+1,+ ⊂Sm+2,+, i.e.S has finite depth,then Brm+1 =Brm+2 by therestrictionofthedimensionvector.Inparticular,theBrattelidiagramof S_m+1,+ ⊂ Sm+1,+isthesameasthatof Sm+1,+⊂S_m+2,+ .So S_m+1 ⊂Sm+1,+⊂S_m+2,+ forms a basicconstruction,and p_m+1 isthe Jonesprojection. Whileapplying the embedding theorem, the image of theJones projection canbe expressed as alinear sum ofloops.

Wewill seetheformulalater.

ThesubfactorplanaralgebraFC(√

2,¹⁺₂^√5) containsatrace-2 biprojection.Consid- eringthemiddlepatternofitsminimalprojections,seepages114–115in[6],wehaveits refined principalgraph,as

; and itsrefineddualprincipalgraphas

,

where the black, mixed, white points are N,P,M coloured vertices. We will discuss moreaboutthesegraphsinSection4.1.

3.3. Finitedimensional inclusions

Now given an inclusionof finite dimensional von Neumannalgebras B0 ⊂ B1 ⊂ B2, similarlywemayconsideritsBrattelidiagram,adjacencymatrixes,Markovtraceifthere exists one,andthebasicconstruction.

Definition3.17.TheBrattelidiagramBrfortheinclusionB0⊂ B1⊂ B2isa(B0,B1,B2) coloured graph.ItsBi colouredvertices areindexed bytheminimal centralprojections (or equivalently the irreducible representations) of Bi, for i = 0,1,2.The subgraph of Br consistingof B0, B1 colouredvertices andtheedges connectingthemisthesameas theBrattelidiagramfortheinclusionB0⊂ B1.ThesubgraphofBr consistingofB1,B2

colouredverticesandtheedgesconnectingthemisthesameas theBrattelidiagramfor theinclusionB1⊂ B2.