The module embedding theorem via towers of algebras

Jones and Penneys showed that a finite depth subfactor planar algebra embeds in the bipartite graph planar algebra of its principal graph, via a Markov towers of algebras approach. We relate several equivalent perspectives on the notion of module over a subfactor planar algebra, and show that a Markov tower is equivalent to a module over the Temperley-Lieb-Jones planar algebra. As a corollary, we obtain a classification of semisimple pivotal C* modules over Temperley-Lieb-Jones in terms of pointed graphs with a Frobenius-Perron vertex weighting. We then generalize the Markov towers of algebras approach to show that a finite depth subfactor planar algebra embeds in the bipartite graph planar algebra of the fusion graph of any of its cyclic modules.


Introduction
Jones' planar algebras [Jon99] are a powerful method to construct [BMPS12, GMP + 18] and classify [JMS14,Liu15,AMP15] finite index II 1 subfactors. Many exotic examples have been constructed via graph planar algebra embedding, i.e., by finding evaluable planar subalgebras of graph planar algebras. By [JP11], any finite depth subfactor planar algebra embeds in the graph planar algebra of its principal graph. This result also extends to infinite depth subfactor planar algebras by [MW10].
While none of the constructions above rely on the embedding theorem from [JP11], the embedding theorem gives us the motivation to do the hard work of looking for the embedding. However, the embedding theorem is necessary for Liu's important classification theorem for composites of A 3 and A 4 subfactor planar algebras [Liu15], in which he shows that higher quotients of A 3 * A 4 do not exist because the possible generator does not embed in the appropriate graph planar algebra. As noted in [JP11], it was rather surprising that the dual principal graph made no appearance in the embedding theorem. Adding to this mystery, certain examples above could be constructed by embedding into planar algebras of bipartite graphs which are completely different from the principal and dual principal graphs [Pet10,PP15,GMP + 18]. The answer to why this occurs is the following theorem: Theorem 1.1 ([GMP + 18]). Let P • be a finite depth subfactor planar algebra and C its unitary 2×2 multifusion category of projections with generator X ∈ P 1,+ , the unshaded-shaded strand. Endow C with the canonical spherical structure inherited from P • . There is a bijective correspondence between: • planar †-algebra embeddings P • → GPA(Γ) • , where Γ is a finite connected bipartite graph, and • indecomposable finitely semisimple pivotal left C-module C * categories M whose fusion graph with respect to X is Γ.
The proof in [GMP + 18] is mostly in the language of tensor and module categories. In this article, we provide an independent proof in the original towers of algebras approach to subfactor theory [Jon83,Wen88,GdlHJ89,Pop94] and the graph planar algebra embedding theorem [JP11].
In §2, we build on this correspondence by defining analogous notions of right modules for these algebraic objects. We briefly describe these objects here, and we refer the reader to §2.2 for more details.
A pivotal module category for C is a finitely semisimple C * category M which is an indecomposable right C-module category equipped with a faithful positive trace Tr M m on each endomorphism C * algebra M(m → m) which is compatible with the right C-action [Sch13, GMP + 18]. That is, for all m ∈ M, c ∈ C, and f ∈ End M (m c), A (connected) right planar module M • for a subfactor planar algebra P • is a sequence of finite dimensional von Neumann algebras (M k ) k≥0 with dim(M 0 ) = 1, 2 together with an action of the shaded planar module operad, which is a variation of Voronov's Swiss cheese operad [Vor99]. We refer the reader to Definition 2.7 for the details, but we include a representative tangle below which acts amongst the algebras M k and the box spaces P n,± : 3 Here, one can glue shaded planar module tangles into the module input semidisks, and one can glue shaded planar tangles into the circular input disks. In addition, the tower of algebras (M k ) k≥0 must satisfy that multiplication in the von Neumann algebra M k is given by the tangle The following theorem generalizes the correspondence between unitary 2 × 2 multitensor categories C with 1 C = 1 0 ⊕ 1 1 and generator X = 1 0 ⊗ X ⊗ 1 1 and subfactor planar algebras P • .
Theorem A. Let P • be a subfactor planar algebra corresponding to (C, X) as above. There is an equivalence between: (1) pivotal right C-module C * categories (M, Tr M ) with choice of simple basepoint m = m 1 0 , and (2) connected right planar modules M • for P • .
One passes from (2) to (1) in Theorem A by taking the category of projections, similar to the correspondence between P • and (C, X) [MPS10,BHP12]. One passes from (1) to (2) using the diagrammatic calculus for module categories, similar to how one gets a subfactor planar algebra from (C, X) via the diagrammatic calculus for pivotal categories [Gho11,Pen18].
From a pivotal semisimple right C-module C * category (M, Tr M ) together with a choice of simple basepoint m ∈ M with m = m 1 0 , we build a tower of finite dimensional von Neumann algebras by setting M n := End M (m X ⊗ X ⊗ · · · ⊗ X ? n tensorands ) where for our generator X ∈ C, we set X ? = X if n is even and X ? = X if n is odd. The trace Tr M endows each von Neumann algebra M n with a faithful tracial state tr n := Tr M (id n ) −1 Tr M together with canonical Jones projections e n ∈ M n+1 for all n ≥ 1. Based on the parity of n, the e n are defined for k ≥ 0 by Here, (X, ev X , coev X ) is the balanced dual of X, m is graphically represented by a red strand, and the left hand side of m is shaded red to denote the absence of a left C-action.
We call M • = (M n , tr n , e n+1 ) n≥0 a Markov tower as it satisfies the following axioms: (M1) The projections (e n ) satisfy the Temperley-Lieb-Jones relations with modulus d > 0 (our convention for d is e i e i±1 e i = d −2 e i .) (M2) For all x ∈ M n , e n xe n = E n (x)e n , where E n : M n → M n−1 is the canonical trace-preserving conditional expectation.
(M4) For all n ≥ 1, we have the Pimsner-Popa pull down property [PP86]: M n+1 e n = M n e n , which is equivalent to M n e n M n being a 2-sided ideal in M n+1 .
One should view a Markov tower as an analog of Popa's λ-lattices [Pop95] where we only have one tower of algebras rather than a tower/lattice of commuting squares. Indeed, one should compare (M1) and (M2) with (1. 3.2) and (M3) and (M4) with (1.3.3') from [Pop95] respectively. We expect that the notion of Markov tower with some compatibility axioms is the correct notion of a right module for Popa's λ-lattices (see Remark 3.34). We leave this exploration to a future article as it would take us too far afield. Markov towers satisfy many nice properties exhibited by standard invariants of finite index II 1 subfactors from [GdlHJ89, Ch. 4]; we mention a few here, and we refer the reader to §3 for more details. The traces satisfy the Markov property tr n+2 (xe n ) = d −2 tr n+1 (x) for every x ∈ M n+1 , and the Markov tower has a principal graph consisting of the non-reflected part of the Bratteli diagram at each step. The tower is called finite depth if the principal graph is finite.
From a Markov tower, we can form a semisimple C * projection category M, whose simple objects are in canonical bijection with the vertices of the principal graph. Moreover, the traces and Jones projections canonically equip M with the structure of a pivotal right T LJ (d)-module C * category. Now any pointed bipartite graph (Γ, v) with a quantum dimension function on vertices dim : gives us a Markov tower of modulus d, where we write w ∼ v to mean w is connected to v, and the sum is taken with multiplicity. We thus get the following corollary, which should be compared with the non-pivotal case in [DCY15].
Corollary B. Equivalence classes of pivotal T LJ (d)-module C * categories with simple basepoint are in bijection with pointed connected bipartite graphs (Γ, v) with a quantum dimension function.
We now specialize to the hypotheses of the module embedding theorem, i.e., Q • is a finite depth subfactor planar algebra, (C, X) is its corresponding spherical unitary multifusion category of projections with generator X = 1 0 ⊗ X ⊗ 1 1 the unshaded-shaded strand, and (M, Tr M , m) is a pivotal right C-module C * category with simple basepoint m = m 1 0 . In this case, the Markov tower M • constructed above has finite depth, and its principal graph Γ is the fusion graph of (M, m) with respect to X ∈ C. This means there is an r > 0 such that the inclusion M 2r ⊂ (M 2r+1 , tr 2r+1 ) is strongly Markov, meaning that there is a finite Pimsner-Popa basis {b} for M 2r+1 over M 2r By [JP11,§2.3], the inclusion A 0 := M 2r ⊂ (M 2r+1 , tr 2r+1 ) =: (A 1 , tr 1 ) has a canonical associated planar †-algebra P • , which is built from the tower of higher relative commutants. Moreover, by [JP11,Thm. 3.8], the planar algebra P • is non-canonically isomorphic to the bipartite graph planar algebra G • of the Bratteli diagram of the inclusion A 0 ⊂ A 1 , which is also the fusion graph Γ. (This isomorphism depends on the loop algebra representation for A 0 ⊂ A 1 from [JP11, §3.1], which amounts to choosing compatible bases for the algebras.) Theorem C (Module embedding). The unital †-algebra maps Φ n,± := id m id (X⊗X) ⊗r − : give a planar †-algebra embedding Q • → P • .
Choosing M = C 00 ⊕ C 10 and m = 1 0 corresponding to the unshaded empty diagram exactly recovers the embedding into the graph planar algebra of the principal graph of Q • from [JP11]. Similarly, we get an embedding into the graph planar algebra of the dual principal graph by choosing M = C 10 ⊕ C 11 and an arbitrary simple object m ∈ C 10 .
Notice we made three choices in our proof of the Module Embedding Theorem C; we picked a simple object m ∈ M with m = m 1 0 , an r ≥ 0 such that M 2r ⊂ (M 2r+1 , tr 2r+1 ) is strongly Markov, and a planar †-algebra isomorphism Q • ∼ = GPA(Γ) • . In our final Section 5.2, we explain that different choices still produce an equivalent planar †-algebra embedding Q • → GPA(Γ) • . Indeed, we show that the two corresponding strongly Markov inclusions are related by a shift and a compression by a projection with central support 1, and these processes yield planar †-isomorphisms on the associated canonical relative commutant planar algebras.
Acknowledgements. This article is the undergraduate research project of Desmond Coles and Srivatsa Srinivas, which was supervised by Peter Huston and David Penneys during Summer 2018, supported by Penneys' NSF DMS CAREER grant 1654159. The authors would like to thank Corey Jones for many helpful conversations. Additionally, David Penneys would like to thank Emily Peters and Noah Snyder for helpful conversations.

Modules for subfactor planar algebras
The standard invariant of a finite index II 1 subfactor has many axiomatizations, including Popa's λ-lattices [Pop95] and Jones' subfactor planar algebras [Jon99]. Here, we use the language of subfactor planar algebras. We discuss the well-known correspondence between subfactor planar algebras and their projection unitary multitensor categories. We then introduce the notion of a planar module for a subfactor planar algebra, and we show it corresponds to a module category for the projection unitary multitensor category.

Unitary multitensor categories and subfactor planar algebras
In this section, we rapidly recall the definitions of a subfactor planar algebra [Jon99] and its unitary 2 × 2 multitensor category of projections [Gho11,Pen18].
Definition 2.1. The shaded planar operad consists of shaded planar tangles with the operation of composition. Shaded planar tangles have r ≥ 0 input disks each with 2k i boundary points, and an output disk with 2k 0 boundary points. Internal to the output disk are non-intersecting strings, which either attach 2 distinct boundary points, or are closed loops. There is also a checkerboard shading, and a distinguished interval marked for each input disk and the output disk. If the for the i-th disk is on an interval which meets an unshaded region, that disk has type (k i , +), and if it meets a shaded region, the disk has type (k i , −). A tangle with r input disks has type ((k 0 , ± 0 ); (k 1 , ± 1 ), . . . , (k r , ± r )) if the output disk has type (k 0 , ± 0 ) and the i-th input disk has type (k i , ± i ). There is a natural definition of the composite tangle T • i S when the output disk of a tangle T has the same type as the i-th input disk of a tangle T . We give a representative example below, and we refer the reader to [Pet10,Jon12] for a more precise definition. The shaded planar operad also has a †-structure, with the tangle T † obtained by reflecting T about a diameter.
A shaded planar algebra P • consists of a family P n,± of C-vector spaces together with an action of the shaded planar operad. That is, each shaded planar tangle T with input disks of type (k i , ± i ) for 1 ≤ i ≤ r and output disk of type (k 0 , ± 0 ) defines a multilinear map Z(T ) : r i=1 P k i ,± i → P k 0 ,± 0 , and tangle composition corresponds to composition of multilinear maps. Each P n,± should also have a dagger structure so that for every tangle T and tuple η 1 . . . r of inputs, Notation 2.2. We will try to shade our diagrams as much as possible for a shaded planar algebra. However, sometimes shading our diagrams requires us to split into many cases. In order to avoid this, we sometimes suppress the shading when it can be inferred from the indices. We also tend to suppress the external boundary disk of a shaded planar tangle; when we do so, the is always on the left. For explicit examples, compare (PA3) and (PA4) in the following definition.
Definition 2.4. A unitary 2 × 2 multitensor category C is an indecomposable rigid C * tensor category which is Karoubi complete such that 1 C has an orthogonal decomposition into simple objects as 1 C = 1 0 ⊕ 1 1 . We write C ij := 1 i ⊗ C ⊗ 1 j for i, j ∈ {0, 1}. By [LR97], such a C is automatically semisimple. When C is finitely semisimple, it is called a unitary 2 × 2 multifusion category [EGNO15].
We say X ∈ C 01 generates C if every object of C is isomorphic to a direct summand of an alternating tensor power of X and X X alt⊗n := X ⊗ X ⊗ · · · ⊗ X ? n tensorands where X ? = X if n is odd and X when n is even, and X ? = X when n is odd and X when n is even. Here, (X, ev X , coev X ) is the canonical balanced dual of X [BDH14, GL18,Pen18] which satisfies the zig-zag axioms and the balancing equation where ψ : C(1 C → 1 C ) → C is the linear functional such that ψ(id 1 0 ) = ψ(id 1 1 ) = 1.
The following theorem is well-known to experts.
Theorem 2.5. There is an equivalence of categories 4 Subfactor planar algebras P • ∼ = Pairs (C, X) with C a unitary 2×2 multitensor category together with a generator X ∈ C 01 .
Starting with a subfactor planar algebra P • , one may form its unitary 2×2 multitensor category of projections C [MPS10, BHP12,Pen18], which comes with a canonical generator corresponding to the unshaded-shaded strand in P 1,+ , and the canonical spherical unitary dual functor [BDH14,GL18,Pen18]. This unitary 2×2 multitensor category can also be thought of as a unitary 2-category called the paragroup; we refer the reader to [BP14] for more details.
Starting with a pair (C, X), we get a subfactor planar algebra by defining P n,+ := End C (X alt⊗n ) P n,− := End C (X alt⊗n ), and we define the action of the shaded planar operad via the diagrammatic calculus for pivotal tensor categories. We refer the reader to [Gho11,Pen18] for more details.

Modules for unitary multitensor categories and subfactor planar algebras
We now define the various notions of module for • a unitary 2 × 2 multitensor category C with its canonical unitary spherical structure and a generator X ∈ C 01 , and • a subfactor planar algebra Q • .
Definition 2.6. Let C be a unitary 2×2 multitensor category. A pivotal right C-module C * category is a pair (M, Tr M ) where M is a semisimple right C-module C * category, and Tr M is a family of positive traces Tr M n : M(n → n) → C on each endomorphism space for n ∈ M satisfying the following axioms: for all f ∈ M(m → n) and g ∈ M(n → m). A pivotal right module category is called pointed if it is indecomposable, we have a chosen simple object m ∈ M, and Tr M is normalized so that Tr M m (id m ) = 1 C . Generally, we choose m ∈ M 0 , but this choice is not essential.
When C is generated by a single X ∈ C 01 and (M, Tr M , m) is a pointed pivotal right module category with m ∈ M 0 , we define the cyclic pivotal right module category M m,X to be the (non Karoubi complete!) full subcategory of M whose objects are of the form m X alt⊗k for k ≥ 0, which is a pointed pivotal right module category over C X , the (non Karoubi complete!) full subcategory of C whose objects are of the form X alt⊗k and X alt⊗k for k ≥ 0.
We next give an appropriate definition of planar modules over a planar algebra as algebras over another operad. 5 Definition 2.7. The shaded planar module operad is a variant of the shaded planar operad, akin to a shaded, stranded version of the Swiss-cheese operad introduced in [Vor99]. In this operad, the starred region of the boundary of the output disk of a tangle is replaced by a vertical line on the left side of a tangle, and the adjacent region inside the tangle must be unshaded. In addition to the usual input disks, tangles may also have input semidisks, whose boundaries intersect the left wall. Similar to the definition of type for an input disk, a semidisk (input or output) has type k i if it has 2k i boundary points which meet 2k i strings. A tangle with r input semidisks and s input disks has type (k 0 ; k 1 , . . . , k r ; ( 1 , ± 1 ), . . . , ( s , ± s )) if the output disk has type k 0 , the i-th input semidisk has type k i , and the j-th input disk has type ( j , ± j ). The operadic composition comes from plugging tangles into semidisks when the types are compatible. A representative tangle appears below. Tangles of the shaded planar module operad can also be composed with shaded planar tangles, by plugging a shaded planar tangle into an input disk. One should think of the box spaces for semidisks as being endomorphisms of objects in a module category, while the involvement of full disks allows a planar algebra to act on the module. The input disks and semidisks are also numbered, with the numbering determining the order of the tensor factors in the domain of the action map as depicted above. Like the shaded planar operad, the shaded planar module operad is a symmetric operad, and vector spaces form a symmetric monoidal category, so we often suppress the numbering.
Definition 2.8. A right planar module M • for the subfactor planar algebra P • consists of a sequence of finite dimensional C-vector spaces (M k ) k≥0 and a conjugate-linear map † : M k → M k for all k ≥ 0, together with an action of the shaded planar module operad on the box spaces M • and P • compatible with the composition of tangles and the shaded planar algebra structure on P • , and the † operation. In other words, the box spaces M • and P • together must have the structure of an algebra over the shaded planar module operad, which must extend the original shaded planar operad algebra structure on P • .
Notice that each of the M k has a †-algebra structure with multiplication given by the tangle We require that each †-algebra M k is a finite dimensional C * /W * algebra. Moreover, we require that for each k, the following map M k → M 0 is positive, faithful, and tracial: We call M • connected if dim(M 0 ) = 1. In this case, we can canonically identify M 0 = C as a C * -algebra, and each Tr k is a scalar-valued. We define tr k := d −k Tr k , where d is the loop parameter of P • . Notice that the tr k are faithful tracial states. We will see that, under the correspondence of Theorem A, connected right planar modules correspond to cyclic pivotal right module C * -categories.
Example 2.9. Given a subfactor planar algebra P • , P + := (P k,+ ) k≥0 is a cyclic right planar module for P • , while P − := (P k,− ) k≥0 is a right planar module for the dual planar algebra of P • obtained by reversing the shading.
where T is obtained from T by first turning each half-open input semidisk into a closed input disk in the interior of the output disk with its on the left, rounding out the 90 • angles on the left boundary into a smooth curve, putting the external on the left hand side, and inserting one (0, +)-type input disk in the left-most region of the new tangle which is numbered first. We illustrate this procedure on the tangle above:

Equivalence of modules
In this section, P • will denote a subfactor planar algebra, and (C, X) will denote its unitary 2 × 2 multitensor category of projections, where X = 1 0 ⊗ X ⊗ 1 1 is the generating unshaded-shaded strand. We now sketch the proof of the following theorem.
Theorem (Theorem A). There is a canonical bijection between equivalence classes of (1) indecomposable pivotal right C-module C * categories (M, Tr M ) with simple basepoint m = m 1 0 , and (2) connected right planar modules M • for P • .
As an application, we get a classification of pivotal module C * categories for the 2-shaded Temperley-Lieb-Jones category with parameter d in Corollary B, whose proof appears in §3.5.
Definition 2.11. Suppose (M, Tr M , m) is an indecomposable pivotal right C-module C * category, and m ∈ M is a distinguished simple object with m = m 1 0 . We build a connected right planar P • -module M • by defining M k := End C (m X alt⊗k ) for k ≥ 0, and we define the action of the shaded planar module operad via the diagrammatic calculus for M. The process is similar to that We first define a standard form for tangles of the shaded planar module operad, such that every tangle is isotopic to one in standard form. We say a tangle is in standard form if (SF1) Each disk and semidisk, including the output semidisk, is rectangular in shape, with an equal number of strings emerging from the top and bottom, (SF2) in the case of a disk, the starred boundary interval includes the left side, and (SF3) a horizontal line through the tangle passes through a disk, semidisk, or extremum of a strand at most once.
Given a shaded planar module tangle T with type (k 0 ; k 1 , . . . , k r ; ( 1 , ± 1 ), . . . , ( s , ± s )) together with appropriate inputs (f 1 , . . . , f r , x 1 , . . . , x s ) with m i ∈ M k i and x j ∈ P j ,± j , we begin with the identity morphism of m X altk 0 in M k 0 , and we move an imaginary horizontal line upwards along the tangle. Each time the horizontal line passes a disk, semidisk, or local extrema (which can happen only one at a time!), we compose with a morphism from M. In more detail, when the horizontal line passes: • the i-th input semidisk with vertical stands to the right, we compose with f j id, where id is the appropriate identity morphism corresponding to the strands to the right of the input semidisk • the j-th input disk with vertical strands to the left and right, we compose with id m id l ⊗x ⊗ id r where id l , id r correspond to the appropriate identity morphisms corresponding to the strands to the left and right of the input disk • a local extrema with vertical strands to the left and right, we compose with id m id l ⊗v ⊗ id r where v stands for the following (co)evaluation or its dagger depending on the shading: and id l , id r are the appropriate identity morphisms as above.
The output is the composite morphism in M k 0 = End M (m X altk 0 ). One then checks that the resulting composite morphism is independent of the choice of standard form for the shaded planar module tangle T .
Example 2.12. Here is an explicit example of a tangle in standard form, together with the corresponding multi-linear map obtained by composing the associated morphisms in M from bottom to top: Definition 2.13. Given a connected right planar P • -module M • , we let M be its category of projections. The objects of M are the orthogonal projections in M k for k ≥ 0. The Hom-space M(p → q) for p ∈ M j and q ∈ M k is only nonzero if j ≡ k mod 2; in this case, we define Composition is given by a suitable version of the usual multiplication tangle, and the †-structure is given by † in M • . Given a projection p ∈ M k and projections q ∈ P n,+ and r ∈ P n,− , we define depending on parity For morphisms f ∈ M(p → q) with p ∈ M j and q ∈ M k and g ∈ C(r → s) with r ∈ P m,± and s ∈ P n,± such that p r and q s are well-defined, we define where shading depends on the parity of j and k. We leave the straightforward verification that M is a right C-module C * category to the reader. Finally, we replace M with its unitary Karoubi completion, which formally adds orthogonal direct sums and then takes the orthogonal projection completion.
The distinguished simple basepoint of M is given by the identity projection 1 M 0 ∈ M 0 . The trace Tr M p : M(p → p) → C for p ∈ M k is given by the non-normalized trace Tr k from (3) restricted to pM k p = M(p → p). By definition, we have assumed Tr k to be tracial and positive on endomorphisms. That the trace is also compatible with the action of C can be seen by composing the trace tangle with the action tangles.

Markov towers and their projection categories
So far, we have presented two versions of the concept of a module over a subfactor planar algebra. The algebraic data of each shares a common structure: that of a Markov tower of finite dimensional tracial von Neumann algebras. Studying elementary properties of Markov towers will therefore allow us to state many important results about planar modules in single, common language. The definition of a Markov tower can obtained from the definition of Popa's λ-sequences of commuting squares from [Pop95] by forgetting one of the towers, analogous to the way one defines a module for an algebraic object by replacing one argument of the algebraic operation with an element from the module. In short, Markov towers are the towers-of-algebras analog of a module category. In §3.5 below, we will see that Markov towers are exactly a λ-lattice approach to pivotal Temperley-Lieb-Jones module categories; this motivates the view of subfactor planar modules as simply Markov towers with an additional structure.

Markov towers and their elementary properties
(EP3) The traces tr n+1 satisfy the following Markov property with respect to M n and e n : for all x ∈ M n , tr n+1 (xe n ) = d −2 tr n (x).
(EP6) The map ae n b → ap n b gives a * -isomorphism from X n+1 = M n e n M n to M n , p n = M n p n M n , the Jones basic construction of M n−1 ⊆ M n acting on L 2 (M n , tr n ).
(EP7) Under the isomorphism X n+1 ∼ = M n p n M n , the canonical non-normalized trace Tr n+1 on the Jones basic construction algebra M n p n M n satisfying Tr n+1 (ap n b) = tr n (ab) for a, b ∈ M n equals d 2 tr n+1 | X n+1 .
("The new stuff comes only from the old new stuff" [GdlHJ89].) Proof.
(EP4) By (M4), e n M n+1 e n = e n M n e n . By (M2), e n M n e n = M n−1 e n .
(EP5) That M n e n M n is a 2-sided ideal is equivalent to (M4) as in Remark 3.3.
(EP6) It suffices to show the map is injective, which also shows it is well-defined. Suppose p n is injective by (EP1) applied to the Jones tower for M n−1 ⊂ (M n , tr n ), which is a Markov tower. Hence for all a, b ∈ M n , and thus a i e n b i = 0, so the map is injective.
(EP8) Since X 0 = (0) and X 1 = (0) by definition, we may assume n ≥ 2. As in the proof of [GdlHJ89, Thm. 4.6. 3.vi], we may assume y is a central projection in M n+1 such that ye n = 0. Then for all ae n−1 b ∈ X n , by (M1), yae n−1 b = d 2 yae n−1 e n e n−1 b = d 2 ae n−1 ye n e n−1 b = 0. The final claim follows from z n E n+1 (y) = E n+1 (z n y) = 0 where z n is the central support of e n−1 in M n .
Remark 3.5. The foregoing observations all hold in the case where the M n are arbitrary tracial von Neumann algebras. In this paper, we restrict our attention to the finite dimensional case because of the following, in which we obtain a principal graph for a Markov tower. To generalize to the infinite case, a measure-theoretic replacement for the principal graph would need to be introduced.
Notice that by (EP6), the Bratteli diagram for the inclusion M n ⊂ M n+1 consists of the reflection of the Bratteli diagram for the inclusion M n−1 ⊂ M n , together with possibly some new edges and vertices corresponding to simple summands of Y n+1 . By (EP8), the new vertices at level n + 1 only connect to the vertices that were new at level n. This leads to the following definition. It follows that a Markov tower has finite depth if and only if there is n ∈ N such that Y n = (0), as in (EP9). Let M • be a Markov tower with finite depth, and take the minimal integer n ∈ N such that Y n = (0). Notice that for k < n, the Bratteli diagram of M k ⊆ M k+1 contains the reflection of the Bratteli diagram of M k−1 ⊆ M k , along with additional vertices and edges which are part of the principal graph. At the base, all of the Bratteli diagram for M 0 ⊆ M 1 is part of the principal graph. We can therefore 'unravel' the Bratteli diagram for M n ⊆ M n+1 to obtain the principal graph for the Markov tower M • .
Fact 3.7. If a Markov tower M • has finite depth and n ∈ N is such that Y n = (0), then for k ≥ n, there is a canonical graph isomorphism between the principal graph of M • and the Bratteli diagram for M k ⊆ M k+1 .
Definition 3.8. The principal graph Γ of a Markov tower M • has a quantum dimension function dim : V (Γ) → R >0 given as follows. Let v ∈ V (Γ), and let p ∈ M k be a minimal projection with k minimal corresponding to the vertex v. We define dim(v) := d k tr k (p), and we note this dimension is independent of the choice of p ∈ M k representing v. Moreover, the quantum dimension function dim satisfies the Frobenius-Perron property where we write w ∼ v to mean w is connected to v, and the above sum is taken with multiplicity.

Examples of Markov towers
We discuss various examples of Markov towers in great detail.
The principal graph of M • is precisely the fusion graph of M with respect to X.
Example 3.11. We obtain the equivalent connected right planar module for the subfactor planar algebra T LJ (d) • to Example 3.10 under Theorem A as follows. We define M k := M k with its †-algebra structure and faithful tracial state tr k from Definition 2.8. Jones projections are defined depending on parity by e 2k+1 := 2k e 2k+2,+ := 2k + 1 . Proof. Let r be minimal such that P r+1,+ = P r,+ e r,+ P r,+ , and let {b} be a Pimsner-Popa basis for P r+1,+ over P r,+ so that b be r,+ b * = 1 P r+1,+ . Since we have that {b} is a Pimsner-Popa basis for M r+1 over M r . Hence M • has finite depth by (EP9). The last claim follows immediately.
Definition 3.13. Recall from [Pop94] that an inclusion of finite von Neumann algebras A 0 ⊂ A 1 with a faithful normal tracial state tr 1 on A 1 is called a Markov inclusion if the canonical faithful normal semifinite trace on the Jones basic construction A 2 = JA 0 J = A 1 , e 1 ⊂ B(L 2 (A 1 , tr 1 )) given by the extension of xe 1 y → tr 1 (xy) is finite and Tr 2 (1) −1 Tr 2 | A 1 = tr 1 . Following [JP11], we call such an inclusion strongly Markov if moreover there is a Pimsner-Popa basis for A 1 over A 0 , which is a finite subset {b} ⊂ A 1 such that 1 A 2 = b be 1 b * . This is equivalent to x = b bE A 0 (b * x) for all x ∈ A 1 , and also to A 2 = A 1 e 1 A 1 by [Con80,Prop. 3(b)] (see also [Wat90]).
Given a strongly Markov inclusion A 0 ⊂ (A 1 , tr 1 ), its Watatani index   Taking the relative commutant with A 0 , we get a Markov tower of finite dimensional von Neumann algebras is a Markov tower of finite dimensional von Neumann algebras.
We now classify all connected Markov towers in terms of pointed bipartite graphs and quantum dimension functions.
Example 3.16. Suppose (Γ, v) is a locally finite pointed bipartite graph with countably many vertices, and dim : V (Γ) → R >0 is a quantum dimension function satisfying (4). We construct a connected Markov tower M • by defining M 0 = C, and inductively constructing each M k as dictated by the principal graph Γ starting at v in the usual way [GdlHJ89,JS97]. We define the trace vector for M k by normalizing the vector obtained from dim applied to the minimal projections appearing at level k.
It is straightforward to check that M • has principal graph Γ with basepoint v corresponding to 1 M 0 . Moreover, by construction, the quantum dimension function of M • is exactly dim. Indeed, the above example can be easily generalized to the following result.

Operations on Markov towers to produce new Markov towers
In this section, we describe various operations on a Markov tower M • = (M n , tr n , e n+1 ) n≥0 which yield new Markov towers. We begin with shifting and compressing the tower. We then study the multistep tower. For each of these operations, we discuss how the principal graph changes.
We omit the proof of the following straightforward proposition. Remark 3.19. Notice that shifting a Markov tower simply truncates the Bratteli diagram, and by Fact 3.7, the principal graph is unchanged.
Given a Markov tower M • , we obtain another Markov tower by compression by a non-zero projection p ∈ P (M 0 ). First, for all n ≥ 0, we define a faithful trace tr p n on pM n p by tr p n (x) := tr n (p) −1 tr n (pxp).
It is straightforward to verify that the unique trace-preserving conditional expectation is given by Notice that since [e n , p] = 0 for all n ∈ N, for all pxp ∈ pM n p we have e n p(pxp)e n p = pe n xe n p = pE n (x)e n p = E p n (pxp)e n p, so the conditional expectation is implemented by e n p. Proof. First, it is easy to see that the projections (pe n ) n≥1 satisfy the Temperley-Lieb-Jones relations (M1), since [e n , p] = 0 for all n ≥ 0. That pe n implements the trace-preserving conditional expectation pM n p → pM n−1 p as in (M2) was shown above in (7). Using (6), this immediately implies that E p n+1 (pe n ) = pE n+1 (e n ) = d −2 p = d −2 1 Mn , so (M3) holds. Finally, for all n ≥ 1, pM n+1 p(pe n ) = pM n+1 e n p = pM n e n p = pM n pe n , so we have (M4).
Remark 3.21. We can determine the Bratteli diagram and principal graph for pM p • , as follows. If p has central support 1, then the Bratteli diagram is unchanged. In general, any vertices on the bottom row corresponding to simple summands of M 0 where p does not have support disappear, as well as those edges no longer supported from below. By proceeding up the tower and, at each level, removing those vertices and edges no longer supported from below, we obtain the Bratteli diagram for pM p • . Notation 3.22. We will make heavy use of the string diagrammatic representation of Temperley-Lieb-Jones diagrams. Ordinarily, for subfactors and planar algebras, Kauffman diagrams [Kau87] are drawn with strings going from bottom to top. We put the number k above or next to a strand to denote a bundle of k parallel strands, and the label is omitted for single strands. For example, the generators E i = de i are represented by Of particular importance will be the cabled/multi-step Jones projections from [PP88] which were of importance in [Bis97,JP11]: f j+k j := d k(k−1) (e j+k e j+k−1 · · · e j+1 )(e j+k+1 e j+k · · · e j+2 ) · · · (e j+2k−1 e j+2k−2 · · · e j+k ) We record the following relation for later use: Now suppose we fix j ≥ 0 and k ≥ 1. For n ∈ N, define the k-cabled Jones projections g n := f j+nk j+(n−1)k . It is straightforward to verify using Kauffman's diagrammatic calculus for Temperley-Lieb-Jones algebras that the projections (g n ) n∈N satisfy the Temperley-Lieb-Jones relations (M1) with d −2 replaced with d −2k .
We now show that taking every k-th algebra in a Markov tower gives us another Markov tower. Proof. We saw Condition (M1) holds from the diagrammatic calculus, and Conditions (M2) and (M3) are straightforward induction arguments.
We prove (M4) by strong induction on k. The base case k = 1 is exactly (M4) for the original Markov tower. Now suppose that (M4) holds for any multi-step towers with increment less than k. Consider the multi-step tower of algebras (M j+nk ) n≥0 , which has increment k. By Proposition 3.18, we may assume j = 0. Using (8) and (M4) for the original Markov tower, we have Since we may perform isotopy in the Temperley-Lieb-Jones subalgebra of M (n+1)k , we may decompose the diagram on the right hand side as follows: By the induction hypothesis, we have This completes the proof.
Remark 3.24. If M • is a Markov tower, then we know from Proposition 3.23 that M k• is also a Markov tower. The Bratteli diagram for M k• can be read off the original Bratteli diagram quite easily: the vertices of the level of the Bratteli diagram corresponding to M kn are the same in both towers, while the number of edges between two vertices in the new diagram is the number of upward paths between those vertices in the old diagram. Note that, since a vertex of the multistep principal graph may belong to the 'old stuff' in the original Bratteli diagram, the number of edges between adjacent vertices in the multistep principal graph is not simply the number of paths in the original principal graph. In the case where k is odd, taking the k-step basic construction therefore collapses the vertices of each k levels of the principal graph into one level; when k is even, we lose the odd part of the principal graph entirely, but aside from this, the situation is the same.

The projection category of a Markov tower
We now define the category of projections of a Markov tower.
We define the composite x † • y † := (y • x) † , which defines composition To show x † • y † is well-defined, we check that when i = j = 0, As above, we define the composite x † • y † := (y • x) † , which defines composition To show that x † • y † is well-defined, we check that when i = 0, As above, we define the composite x † • y † := (y • x) † , which defines composition To show that x † • y † is well-defined, we check that when j = 0, Showing that composition is associative directly from the definitions above is a highly non-trivial exercise using the axioms (M1) -(M4) of a Markov tower. A better way to prove associativity is to prove that each 4 × 4 (possibly non-associative) linking algebra [GLR85] L := is †/ * -isomorphic to a von Neumann algebra, which is necessarily associative! This technique also offers the advantage that it simultaneously proves M 0 is C * . 6 Notice we have an equality of sets We define the following map entry-wise; that is, for an element x ∈ L, we plug x ab into the input disk in the ab-th entry of the map π : L → p Mat 4 (M n+2i+2j+2k )p given by where p ∈ Mat 4 (M n+2i+2j+2k ) is the following projection: In Proposition 3.26 below, we verify the map π is an injective unital algebra map satisfying π(x † ) = π(x) * , and is thus an isomorphism onto its image. Thus im(π) is a unital * -subalgebra of the finite dimensional von Neumann algebra p Mat 4 (M n+2i+2j+2k )p, which means im(π) is a von Neumann algebra by the finite dimensional bicommutant theorem [Jon15, Thm. 3.2.1]. By looking at the 2 × 2 and 3 × 3 corners of the linking algebra L and (13), we immediately see: • † is a dagger structure on M 0 , • Hence M 0 is C * , and thus so is its unitary Karoubi completion M.
We begin with the following lemma.
7 Notice that in a C * category, these norms can be recovered from spectral radii together with the positivity and C * axioms. Thus these norms are not part of the data of the C * category.
Proof. We calculate Proof of Proposition 3.26. By inspection of the definition of π from (13), it is clear that π is injective, unital, C-linear, and respects the †-structure. The difficulty is in seeing that π is an algebra homomorphism. In the following, we suppress the rightmost 2k strings of entries of (13), as well as the factor d −k , since they are essentially inert when only three objects are considered. Because π respects †, as in Definition 3.25, we only need to consider 3 cases of composition.

Temperley-Lieb-Jones module categories
We now show that the category of projections of a connected Markov tower of modulus d can be canonically endowed with the structure of a cyclic pivotal right Temperley-Lieb-Jones (T LJ (d)) module C * category. Moreover, all cyclic pivotal right T LJ (d)-module C * categories arise in this way. Combined with the classification of connected Markov towers of modulus d from Proposition 3.17, we get the following result, which should be compared with [DCY15] in the non-pivotal setting.
Corollary (Corollary B). Cyclic pivotal right module C * categories for T LJ (d) are classified by where we write w ∼ v to mean w is connected to v, and the above sum is taken with multiplicity.
Notice these formulas agree when j = k. Let ι denote the inclusion in M • . In the following, since g ∈ T LJ , we represent g by a ticket within Temperley-Lieb tangles, which we may freely move via isotopy. By definition, we have To show that is a well-defined bifunctor, it remains to show that for morphisms f and g in M and h and k in T LJ , we have (g 1) • (f 1) = (g • f ) 1 and (1 k) • (1 h) = 1 (k • h). Functoriality in the right variable comes directly from the definition, but functoriality in the left variable is more involved and done in cases. We illustrate a representative case. Suppose It is easier to check that the action is associative. Since we have already shown that is a bifunctor, it suffices to check associativity of triples of morphisms when two are the identity. That 1 (1 ⊗ h) = (1 1) h and that 1 (g ⊗ 1) = (1 g) 1 for all g, h ∈ T LJ follow directly from the definitions of and ⊗. Finally, in case f ∈ M([n] → [n + 2i]), we have The other cases are similar.
Remark 3.31. Notice that the principal graph of M • is exactly the fusion graph for the associated T LJ -module category M with respect to the unshaded-shaded strand X ∈ T LJ with basepoint the simple projection 1 M 0 ∈ M. By Remark 3.29, the operation of shifting the Markov tower does not change M (up to equivalence), but corresponds to replacing the basepoint 1 M 0 with [0] X alt⊗2n . Similarly, compressing by a minimal projection p ∈ M n corresponds to moving the basepoint to p. In contrast, the multistep basic construction, which may affect the principal graph, is analagous to replacing X ∈ T LJ with X alt⊗2n , without changing the basepoint of M. The T LJ -module structure on the category of projections M (n) of M n• comes from combining the action of the subcategory T LJ (n) of T LJ generated by subobjects of [kn] T LJ and the pivotal T LJ -right module structure of T LJ (n) .
Remark 3.32. Observe that we may identify the tensor category T LJ as the category of projections of the Markov tower T LJ • , where the tensor structure is given by the T LJ -module structure from Definition 3.30. One should think of the definitions for composition and tensor product in the category of projections as being obtained by isotoping the much simpler definitions for a planar algebra into a form that can be written down in terms of the data of a Markov tower. Notice that under this identification, ev Proof of Corollary B. We saw in Definitions 3.30 and 3.33 how a connected Markov tower of modulus d gives us a cyclic pivotal right T LJ (d)-module C * category. We saw in Example 3.10 that given a cyclic pivotal right T LJ (d)-module C * category (M, m, Tr M ), defining M n := End(m X alt⊗n ) and tr n := Tr M m X alt⊗n defines a connected Markov tower. One now shows these two processes are mutually inverse up to dagger equivalence.
Remark 3.34. While the process of defining a tensor structure on the category of projections P of a Markov tower P • obtained from a planar algebra P • is fairly straightforward and similar to Remark 3.32, it is far less obvious for a Markov tower coming from a standard λ-lattice as in [Pop95]. Given a standard λ-lattice A •• with λ = d −2 , we expect that a Markov tower M • of modulus d such that M n ⊃ A 0n for all n ≥ 0 satisfying certain compatibility conditions is the equivalent notion of a right module for A •• in the spirit of Theorems A and 2.5.
We leave this exploration to a future joint article, as it would take us too far afield.

The canonical planar algebra from a strongly Markov inclusion
We begin this section by recalling the construction of the canonical planar †-algebra from a strongly Markov inclusion. The reader is advised to review the definition of a strongly Markov inclusion from Definition 3.13 before proceeding. We then discuss various operations on the inclusion, and how such operations affect (or do not affect!) the planar algebra.

The canonical relative commutant planar algebra
There is a canonical planar algebra structure on the towers of relative commutants, called the canonical planar †-algebra of a strongly Markov inclusion corresponding to A 0 ⊆ (A 1 , tr 1 ). Denote by (A n , tr n ) n≥0 the Jones tower for inclusion A 0 ⊂ (A 1 , tr 1 ). The box spaces are defined by the relative commutants P n,+ := A 0 ∩ A n P n,− := A 1 ∩ A n+1 , which are finite dimensional by [Wat90,Prop. 2.7.3]. We refer the reader to [JP11,§2.3] for the action of tangles. The †-structure is given by * in the relative commutants. We remark that one important feature of this construction is that it depends on the existence of a Pimsner-Popa basis for A 1 over A 0 , but not on a choice of basis.
The following theorem uniquely characterizes the canonical relative commutant planar †-algebra.
Theorem 4.1 ([JP11, Thm. 2.50]). Given a strongly Markov inclusion A 0 ⊂ (A 1 , tr 1 ), there is a unique planar †-algebra P • of modulus d = [A 1 : A 0 ] 1/2 whose box spaces are given by such that (PA1) The †-structure of P n,± is given by x † = x * in the relative commutant, and stacking corresponds to multiplication in the relative commutant: n n n y x = xy ∈ P n,± .
(PA3) For x ∈ P n,+ = A 0 ∩ A n and {b} a Pimsner-Popa basis for A 1 over A 0 , We will proceed with the same general technique for defining the embedding of planar algebras as in [JP11], in that we will also initially embed into the canonical †-planar algebra, and then make use of the following theorem: Theorem 4.2 ([JP11, Theorem 3.28]). The canonical planar †-algebra associated to the strongly Markov inclusion of finite dimensional von Neumann algebras A 0 ⊆ (A 1 , tr 1 ) is isomorphic to the bipartite graph planar †-algebra of the Bratteli diagram for the inclusion A 0 ⊆ A 1 .
The following two subsections describe two isomorphisms between canonical †-planar algebras of related strongly Markov inclusions. Both are well known to experts, and will motivate constructions detailed in the later sections of this article.

The shift isomorphism
In the rest of this article, we will make extensive use of the following lemma adapted from [JP11,Lem. 2.49], which provides sufficient conditions for a collection of maps to be a morphism of shaded planar †-algebras. . Suppose Φ n,± : P n,± → Q n,± is a collection of unital †-algebra maps such that (1) Φ maps Jones projections in P n,+ to Jones projections in Q n,+ , (2) Φ commutes with the action of the following tangles: Then Φ is a morphism of shaded planar †-algebras.
Suppose that A 0 ⊂ A 1 ⊂ (A 2 , tr 2 , e 1 ) is a strongly Markov inclusion of von Neumann algebras and (A n , tr n , e n ) n≥0 is the tower obtained by iterating the basic construction. We know from [JP11, Cor. 2.18] that for any 0 ≤ k ≤ n, the inclusion A k ⊂ A n is strongly Markov. Thus, we can find a Pimsner-Popa basis B for A n over A k .
By [JP11,Prop. 2.20], for 0 ≤ j ≤ 2n, we can represent A j on L 2 (A n , tr n ) via the multistep basic construction, and J n A 2n−j J n = A j ∩B(L 2 (A n , tr n )) where J n is the modular conjugation. We get a canonical trace on A j by tr j (x) := tr 2n−j (J n x * J n ) as discussed in Remark [JP11, Rem. 2.21]. : (A j ∩ B(L 2 (A n , tr n )), tr j ) → (A k ∩ B(L 2 (A n , tr n )), tr k ) is given by: and is independent of the choice of basis. Corollary 4.5. Adding k strings to the left of x gives a unital * -algebra isomorphisms P n,± → P n+k,± where ± = ± if k is even and ∓ if k is odd. Proof. We first prove the result for P n,+ = A 0 ∩ A n . The following implicitly uses a trick due to Vaughan Jones that can be found in [JP11,Theorem 4.1] along with the pictorial description of f n n−k in the Multistep Basic Construction [JP11, Remark 2.44] in the equality marked (!). Let B be a Pimsner-Popa basis for A k over A 0 . For x ∈ A k ∩ A n+k , the element y ∈ A 0 ∩ A n is uniquely determined by The proof is similar for P n,− .
Recall that the canonical planar †-algebra for the inclusion A 0 ⊆ (A 1 , tr 1 ) is denoted by P • . We denote the canonical planar †-algebra for A 2 ⊂ (A 3 , tr 3 ) by Q • . Proof. The map is an isomorphism between box spaces due to Corollary 4.5. In order to show that this map commutes with the action of tangles, we just have to show that it satisfies the requirements of Lemma 4.3. We draw the string diagrams of Q • in blue in order to increase clarity. Remark 4.7. By Remark 3.19, the categories of projections of P • and Q • as in Theorem 4.6 are equivalent.

The compression isomorphism
The following lemma is well known to experts. We provide a proof for convenience and completeness. (1) p(N ∩ M ) = pN ∩ pM p.
(2) Suppose the central support of p in N is z ∈ Z(N ). The map x → px is an isomorphism Proof.
Proof of (1): The proof of (1) is similar to the proof of the standard fact that (pN p) = N p.
Suppose u is a unitary in (N p) ∩ pM p. Let K be the closure of N pH. Let q ∈ B(H) be the projection onto K, which is clearly in N ∩ N = Z(N ). Define u 0 in B(K) by u 0 (npξ) := npuξ. One now verifies that u 0 is an isometry and thus is welldefined. Look at the operator u 0 q ∈ N ∩ B(H), and note that u = u 0 qp ∈ N p. We claim that u 0 q ∈ M , so that u = u 0 qp ∈ (N ∩ M )p. First, for any m ∈ M , n ∈ N , and ξ ∈ H, we have mu 0 npξ = mnupξ = nupmξ = u 0 npmξ = u 0 mnpξ. Thus u 0 ∈ qM q. Since q ∈ M , for all m ∈ M , we have u 0 qmξ = u 0 mqξ = mu 0 qξ. Hence u 0 q commutes with M on H, and u 0 q ∈ M .
Similar to the discussion in §3.3, given an inclusion of tracial von Neumann algebras A 0 ⊂ (A 1 , tr 1 ), we obtain another inclusion of tracial von Neumann algebras by compression by a nonzero projection p ∈ P (A 0 ). We define a faithful trace tr p 1 on pA 1 p by tr p 1 (x) := tr 1 (p) −1 tr 1 (pxp).
It is straightforward to verify that the unique trace-preserving conditional expectation is given by Notice that since [e 1 , p] = 0, we have for all pxp ∈ pA 1 p, we have so the conditional expectation is implemented by e 1 p. Suppose now that A 0 ⊂ (A 1 , tr 1 ) is strongly Markov. We would like to show that pA 2 p with trace tr p 2 (x) := tr 2 (p) −1 tr 2 (pxp) and Jones projection pe 1 is isomorphic to the basic construction of pA 0 p ⊂ (pA 1 p, tr p 1 ), but we will need an extra assumption on p. (This extra assumption will be automatic when A 0 ⊂ A 1 is a II 1 subfactor; see also [Bis94, , and (B, tr B , p) is a tracial von Neumann algebra containing A 1 together with a projection p ∈ P (B) such that (R3) B is algebraically spanned by A 1 and p, i.e., B = A 1 pA 1 := span {apb|a, b ∈ A 1 }.
Then the map A 2 → B given by ae 1 b → apb is a (normal) unital * -isomorphism of von Neumann algebras.
In this case, where (B, tr B , p) is isomorphic to the basic construction A 2 of A 0 ⊆ (A 1 , tr 1 ), we call the inclusion A 0 ⊆ (A 1 , tr 1 ) ⊆ (B, tr B , p) standard, after [JP11].
We now prove that compression by well-behaved projections of A 0 preserves the strong Markov structure. Here, 'well-behaved' is the condition A 0 = A 0 pA 0 := span {apb|a, b ∈ A 0 }, which implies that the central support of p in A 0 is 1. (1) Let {a} ⊂ A 0 be a finite set such that a apa * = 1 A 0 . 8 Then for any Pimsner-Popa basis {b} for A 1 over A 0 , {pbap} is a Pimsner-Popa basis for pA 1 p over pA 0 p.
Proof of (1): For all pxp ∈ pA 1 p, we have Proof of (2): Given that there exists a Pimsner-Popa basis for pA 1 p over pA 0 p by part (1) Proof of (3): We show the hypotheses of Lemma 4.9 hold. We already saw (R1) holds in (16). To see (R2) holds, note that E p 2 (pxp) = pE 2 (x)p for all x ∈ A 2 as in (15). Thus by part (2), Finally, (R3) follows immediately from the existence of a Pimsner-Popa basis for pA 1 p over pA 0 p by part (1).
By iterating Lemma 4.9 and Proposition 4.10, we immediately obtain the following.
Corollary 4.11. Assume the hypotheses of Proposition 4.10, and let A • = (A n , tr n , e n+1 ) n≥0 be the Jones tower for A 0 ⊂ (A 1 , tr 1 ). Then pAp • := (pA n p, tr p n , pe n+1 ) n≥0 is isomorphic to the Jones tower of pA 0 p ⊂ (pA 1 p, tr p 1 ).
8 Such a finite set necessarily exists by the same trick used in [Con80,Prop. 3(b)]. 9 The definition of tr p 2 is analogous to (14).
Suppose A 0 ⊂ (A 1 , tr 1 ) is a strongly Markov inclusion of tracial von Neumann algebras. Denote by P • the canonical planar †-algebra whose box spaces are given by P n,+ := A 0 ∩ A n P n,− := A 1 ∩ A n+1 .
Suppose p ∈ P (A 0 ) is a projection such that A 0 pA 0 = A 0 . By Corollary 4.11, the Jones tower for pA 0 p ⊂ (pA 1 p, tr p 1 ) is given by (pA n p, tr p n , pe n+1 ) n≥0 , and thus we get another canonical planar †-algebra Q • whose box spaces are given by Q n,+ := pA 0 ∩ pA n p Q n,− := pA 1 ∩ pA n+1 p.
Proof. We prove the unital * -algebra isomorphisms Φ n,± satisfy the conditions of Lemma 4.3. First, note that Φ n,± (e n ) = pe n , so Jones projections in P • map to Jones projections in Q • by Corollary 4.11. Hence Condition (1) of Lemma 4.3 is satisfied.
The only interesting part in checking Condition (2) of Lemma 4.3 holds is verifying that left capping commutes with Φ n,± . First, by Proposition 4.4, if {b} is a Pimsner-Popa basis for A 1 over A 0 , then for all x ∈ P n, This means that picking Pimsner-Popa bases {b} for A 1 over A 0 and {a} for pA 1 p over pA 0 p, we must show that The trick is to carefully choose our Pimsner-Popa basis for A 1 over A 0 . We take the Pimsner-Popa basis {a} for pA 1 p over pA 0 p and we take the disjoint union with {(1 − p)b}, where {b} was our Pimsner-Popa basis for A 1 over A 0 . We now claim {c} = {a} ∪ {(1 − p)b} is a Pimsner-Popa basis for A 1 over A 0 . Indeed, since a = pap ∈ pA 1 p for all a ∈ {a}, we have Thus for this special choice of Pimsner-Popa basis for A 1 over A 0 , we immediately obtain Hence (17) holds, and the result follows.

The module embedding theorem via towers of algebras
We have finally developed the tools necessary to prove Theorem C, which turns a finite depth cyclic right pivotal module (M, m) over the category of projections C of a subfactor planar algebra Q • (or equivalently, a finite depth connected right planar module over Q • ) into an embedding of Q • into the graph planar algebra of the fusion graph of M with respect to the unshaded-shaded strand X ∈ Q 1,+ . As a special case, we recover the embedding of a subfactor planar algebra into the graph planar algebra of its own principal graph, described in [JP11], along with an embedding into the graph planar algebra of the dual principal graph. We then verify that, up to an automorphism of the graph planar algebra, the resulting embedding does not depend on the choice of generating object for the module category.

The Embedding Theorem
Suppose M • is a a connected right planar module over Q • . Since Q • has finite depth, so does M • , by Lemma 3.12. The tangle n gives a map Q • → M • , which is injective since M • is non-zero and connected. Let M • be the Markov tower obtained from M • from Example 3.11. Choose r ≥ 0 such that the inclusion M 2r ⊆ (M 2r+1 , tr 2r+1 ) ⊆ (M 2r+2 , tr 2r+2 , e 2r+1 ) is standard. Then setting (A n , tr n ) := (M 2r+n , tr 2r+n ) for n ≥ 0 as described in §4.2, A 0 ⊆ (A 1 , tr 1 ) is a strongly Markov inclusion, and the tower (A n , tr 2r+n ) is a strongly Markov tower. Let P • be the canonical relative commutant planar †-algebra of the inclusion A 0 ⊆ (A 1 , tr 1 ) described in §4.1. Therefore, the tangle 2r n gives an embedding Φ : Q n → P n on the level of Markov towers. In terms of the string calculus of pivotal modules over tensor categories, Φ places 2r strings to the left of elements in Q n,+ and 2r + 1 strings to the left of elements in Q n,− : Here, the unshaded-shaded strand is a generator of the category of projections C of Q • , while the shaded-unshaded red strand is a simple generator of a cyclic pivotal right module category over C.
Theorem 5.1. The map Φ is a †-planar algebra embedding.
Proof. For clarity, let us denote the conditional expectations and inclusions in Q • by E and ι, reserving the plain symbols for their counterparts in P • . Similarly, let us denote the n-th Jones projection in Q • by ε n , reserving e n for the Jones projection in P • .
We need only check the hypotheses of Lemma 4.3: that Φ commutes with the action of several tangles. • (Left Capping) In §4.1, we discussed that the left-capping tangle in the canonical planar * -algebra is given for n ≥ 1 by where {b} is a Pimnser Popa basis of A 1 over A 0 . This means that, for any x ∈ Q n,+ , we have • (Left Inclusion) The left inclusion l n : P n,− → P n+1,+ is just the inclusion A 1 ∩ A n+1 → A 0 ∩ A n+1 . Graphically, n : Q n,− → Q n+1,+ is equivalent to adding a string on the left. Thus, for x ∈ Q n,− , we have that: = Φ( n (x)).
We have checked that Φ : Q • → P • is a planar †-algebra inclusion. Let G • be the bipartite graph planar algebra of the fusion graph of X acting on M. Then by [JP11,Th,. 3.33], we know that P • is a planar †-algebra isomorphic to G • . Thus, we have an embedding of Q • into G • .
Corollary 5.2 (The Embedding Theorem). A finite depth subfactor planar algebra Q • can be embedded into the bipartite graph planar algebra of the fusion graph of a connected right planar Q • -module.
In particular, by considering (Q • , 1 0,+ ) as a connected right planar Q • -module, we recover the main result of [JP11]. By instead considering (Q • , Y ) for Y a simple summand of the shadedunshaded strand X ∈ Q 1,− , we obtain an embedding of Q • into the graph planar algebra for the dual principal graph.

Invariance of the embedding
As the observant reader may have noted, we made three choices in defining the embedding map from Q • → GPA(Γ) • . First, we chose a simple object m ∈ M to get our Markov tower M n := End M (m X alt⊗n ), and second, we chose r ≥ 0 such that the inclusion M 2r ⊆ (M 2r+1 , tr 2r+1 ) ⊆ (M 2r+2 , tr 2r+2 , e 2r+1 ) is standard. Third, we chose a basis for the strongly Markov inclusion M 2r ⊆ (M 2r+1 , tr 2r+1 ) to obtain a planar †-algebra isomorphism from the canonical relative commutant planar algebra P • of the inclusion to the graph planar algebra GPA(Γ) • of the fusion graph Γ. In this section, we show that the embedding does not depend on these choices up to a †-automorphism of GPA(Γ) • .
Definition 5.3. Suppose Q • is a subfactor planar algebra and P • , P • are two unitary shaded planar algebras together with planar algebra embeddings Φ : Q • → P • and Φ : Q • → P • . We say the embeddings Φ and Φ are equivalent if there is a planar †-algebra isomorphism Ψ : P • → P • such that the following diagram commutes: We now treat our three choices for our embedding in reverse order. First, note that choosing a different basis for the inclusion just alters the isomorphism P • ∼ = GPA(Γ) • by a †-automorphism of GPA(Γ) • , resulting in equivalent embeddings.
Second, suppose we chose a different r ≥ 0 such that the inclusion M 2r ⊆ (M 2r +1 , tr 2r +1 ) ⊆ (M 2r +2 , tr 2r +2 , e 2r +1 ) is standard. Without loss of generality, we may assume r = r + k for k ∈ N. Denoting the canonical relative commutant planar algebra for the strongly Markov inclusion M 2r+2k ⊆ (M 2r+2k+1 , tr 2r+2k+1 ) by P • , we see that P • ∼ = P • by iteratively applying the shift-by-2 planar algebra isomorphism from Theorem 4.6. Hence, replacing r with r results in an equivalent embedding. Third, suppose we chose the simple object n ∈ M 0 ⊂ M instead of m, where M 0 = M 1 0 . For i ≥ 0, define N i := End M (n X alt⊗i ). Since M is indecomposable as a right C-module category, there is a j > 0 such that n is a subobject of m X alt⊗2j . Fix an orthogonal projection p ∈ M 2j = End M (m X alt⊗2j ) with image isomorphic to n. Notice that the compressed shifted Markov tower (pM 2j+k p, tr p 2j+k , pe 2j+k+1 ) k≥0 is * -isomorphic to the Markov tower (N k , tr k , f k+1 ) k≥0 , where we denote by e i the Jones projections in M • and by f i the Jones projections in N • . Again, since M is indecomposable, we may fix k > 0 sufficiently large such that the following three conditions hold: (1) The projection p id X alt⊗2k has central support 1 in the finite dimensional von Neumann algebra M 2(j+k) = End M (m X alt⊗2(j+k) ), which is equivalent to M 2(j+k) = M 2(j+k) pM 2(j+k) by finite dimensionality.