Standard λ -lattices, rigid C ∗ tensor categories, and (bi)modules
SStandard λ -lattices, rigid C ∗ tensor categories, and (bi)modules Quan ChenSeptember 18, 2020
Abstract
In this article, we construct a 2-shaded rigid C ∗ multitensor category with canonicalunitary dual functor directly from a standard λ -lattice. We use the notions of tracelessMarkov towers and lattices to define the notion of module and bimodule over standard λ -lattice(s), and we explicitly construct the associated module category and bimodule categoryover the corresponding 2-shaded rigid C ∗ multitensor category.As an example, we compute the modules and bimodules for Temperley-Lieb-Jones stan-dard λ -lattices in terms of traceless Markov towers and lattices. Translating into the unitary2-category of bigraded Hilbert spaces, we recover DeCommer-Yamshita’s classification of T LJ modules in terms of edge weighted graphs, and a classification of
T LJ bimodules interms of biunitary connections on square-partite weighted graphs.As an application, we show that every (infinite depth) subfactor planar algebra embedsinto the bipartite graph planar algebra of its principal graph.
Introduction
Since Jones’ landmark article [Jo83], the modern theory of subfactors has developed deepconnections to numerous branches of mathematics, including representation theory, categorytheory, knot theory, topological quantum field theory, statistical mechanics, conformal fieldtheory, and free probability. The standard invariant of a type II subfactor was first defined asa standard λ -lattice [Po95]. Since, it has been reinterpreted as a planar algebra [Jo99] and aQ-system [Lo89], or unitary Frobenius algebra object, in a rigid C ∗ tensor category [M¨u03].The following theorem is a well known folklore result. It is for instance mentioned inthis form in [AV15, Remark 2.1]. A similar result with planar algebras in place of tensorcategories was announced in [Li14]. The folklore proof of this result makes use of Popa’ssubfactor reconstruction theorem [Po95, Thm. 3.1]. One primary motivation of this paper isto give a direct argument without making a detour via subfactors.
Theorem (Folklore) . There is a bijective correspondence between equivalence classes of thefollowing: (cid:26)
Standard λ -lattices A = ( A i,j ) ≤ i ≤ j (cid:27) ∼ = Pairs ( A , X ) with A a 2-shaded rigid C ∗ multitensorcategory with a generator X , i.e., 1 A = 1 + ⊕ − , 1 + , − are simple and X = 1 + ⊗ X ⊗ − Equivalence on the left hand side is unital ∗ -isomorphism of standard λ -lattices; equivalenceon the right hand side is unitary equivalence between their Cauchy completions which mapsgenerator to generator. Similarly, for a given standard λ -lattice, Jones proved in [Jo99, Thm. 4.2.1] that one can construct a subfactorplanar algebra by passing through Popa’s subfactor reconstruction theorem [Po95, Thm. 3.1]. a r X i v : . [ m a t h . OA ] O c t iven ( A , X ), it is well known that one can obtain a standard λ -lattice A by A i,j := (cid:40) id X alt ⊗ k ⊗ End (cid:0) X alt ⊗ ( j − k ) (cid:1) i = 2 k id X alt ⊗ (2 k +1) ⊗ End (cid:16) X alt ⊗ ( j − k − (cid:17) i = 2 k + 1where X is a dual of X and X alt ⊗ n := X ⊗ X ⊗ X ⊗ · · · (cid:124) (cid:123)(cid:122) (cid:125) n tensorands and similarly for X alt ⊗ n . The inclusion A i,j ⊂ A i,j +1 sends x to x ⊗ id, the inclusion A i +1 ,j ⊂ A i,j sends x to x . The Jones projections are defined using the canonical balanced evaluation andcoevaluation for X .Going the other way directly is harder. Using [CHPS18, Def. 3.1], we construct a skeletal(when d >
1) W ∗ category explicitly from A whose objects are [ n, ± ] for n ≥ A i,j . We endow it with a tensor structureusing the 2-shift map in the standard λ -lattice, which is a trace-preserving ∗ -isomorphism S i,j : A i,j → A i +2 ,j +2 [Bi97, Cor. 2.8]. We call this skeletal category a planar tensor category ,and we provide a string diagram calculus to perform computations. The Cauchy completion ofthis planar tensor category is the target 2-shaded rigid C ∗ multitensor category.Given a standard λ -lattice A , an A -module is a Markov tower as a standard A − module. Inmore detail, let A = ( A i,j ) ≤ i ≤ j< ∞ be a standard λ -lattice with Jones projection { e i } i ≥ andcompatible conditional expectations. An A − module is a Markov tower of finite dimensionalvon Neumann algebras ( M n ) n ≥ such that A ,n ⊂ M n together with conditional expectations E i : M i → M i − implemented by the Jones projections, which satisfy the appropriate commut-ing square conditions. M ⊂ M ⊂ M ⊂ · · · ⊂ M n ⊂ · · ·∪ ∪ ∪ ∪ A , ⊂ A , ⊂ A , ⊂ · · · ⊂ A ,n ⊂ · · ·∪ ∪ ∪ A , ⊂ A , ⊂ · · · ⊂ A ,n ⊂ · · · We refer the reader to Definition 1.1.3 below for the complete definition.We warn the reader that our definition is slightly different from the original one from[CHPS18, Def. 3.1]; our tower of algebras ( M n ) n ≥ does not necessarily have a Markov trace.An important difference in our construction is that we do not use the trace, but rather thecommuting square of conditional expectations. In § λ -lattice instead of merely pivotal modules.We call an A − module standard if [ M i , A k,l ] = 0 for i ≤ k ≤ l . By similar techniques usedin our new proof of the Folklore Theorem above, we obtain the following theorem. Theorem A.
There is a bijective correspondence between equivalence classes of the following:
Traceless Markov towers M =( M i ) i ≥ with dim( M ) = 1 asstandard right modules over astandard λ -lattice A ∼ = Pairs ( M , Z ) with M an indecomposablesemisimple right A− module C ∗ categorytogether with a choice of simple object Z = Z (cid:1) + A Equivalence on the left hand side is ∗ -isomorphism of traceless Markov towers as standard A − modules; equivalence on the right hand side is unitary A− module equivalence on Cauchycompletions which maps the simple base object to simple base object.Tracial Markov towers as standard A − modules correspond to pivotal A− module categories. §
3, we discuss bimodules. Given two standard λ -lattices A and B , we define an A − B bimodule as a standard Markov lattice , which consists of a doubly indexed sequence M =( M i,j ) i,j ≥ of finite dimensional von Neumann algebras with two sequences of Jones projections( e i ) i ≥ and ( f j ) j ≥ where the following conditions hold.(a) M i,j ⊂ M i,j +1 and M i,j ⊂ M i +1 ,j are unital inclusions.(b) M − ,j = ( M i,j , E M,li,j , e i +1 ) i ≥ are Markov towers with the same modulus d and e i ∈ M i +1 ,j for all i ; M i, − = ( M i,j , E M,ri,j , f j +1 ) j ≥ are Markov towers with the same modulus d and f j ∈ M i,j +1 for all j . We call M of modulus ( d , d ). M i +1 ,j ⊂ M i +1 ,j +1 ∪ ∪ M i,j ⊂ M i,j +1 (c) The commuting square condition: M i +1 ,jE M,li +1 ,j (cid:15) (cid:15) M i +1 ,j +1 E M,ri +1 ,j +1 (cid:111) (cid:111) E M,li +1 ,j +1 (cid:15) (cid:15) M i,j M i,j +1 E M,ri,j +1 (cid:111) (cid:111) is a commuting square, i.e., E M,ri,j +1 ◦ E M,li,j = E M,li,j +1 ◦ E M,ri +1 ,j +1 .We require A op i, ⊂ M i, and B ,j ⊂ M ,j for all i, j with conditional expectations satisfyingthe appropriate commuting square conditions. Here, we take the opposite λ -lattice A op of A ,where A op i,j is the opposite algebra of A i,j , so the indices for A and B are transposed. ∪ ∪ ∪ ∪ ∪ ∪ A , ⊂ A , ⊂ M , ⊂ M , ⊂ M , ⊂ M , ⊂∪ ∪ ∪ ∪ ∪ ∪ A , ⊂ A , ⊂ M , ⊂ M , ⊂ M , ⊂ M , ⊂∪ ∪ ∪ ∪ ∪ ∪ A , ⊂ A , ⊂ M , ⊂ M , ⊂ M , ⊂ M , ⊂∪ ∪ ∪ ∪ ∪ A , ⊂ M , ⊂ M , ⊂ M , ⊂ M , ⊂∪ ∪ ∪ ∪ B , ⊂ B , ⊂ B , ⊂ B , ⊂∪ ∪ ∪ B , ⊂ B , ⊂ B , ⊂ We call an A − B bimodule standard if [ M i,j , A p,q ] = 0 for i ≤ q ≤ p ; [ M i,j , B k,l ] = 0, for j ≤ k ≤ l . Similar to the proof of the Folklore Theorem and Theorem A above, we obtain thefollowing theorem. Theorem B.
There is a bijective correspondence between equivalence classes of the following:
Traceless Markov lattices M =( M i,j ) i,j ≥ with dim( M , ) = 1 asstandard A − B bimodules overstandard λ -lattices A, B ∼ = Pairs ( M , Z ) with M an indecompos-able semisimple C ∗ A−B bimodule cat-egory together with a choice of simpleobject Z = 1 + A (cid:3) Z (cid:1) + B quivalence on the left hand side is ∗ -isomorphism on the traceless Markov lattice as a standard A − B bimodule; equivalence on the right hand side is unitary A−B bimodule equivalence betweentheir Cauchy completions which maps the simple base object to simple base object.Tracial Markov lattices as standard A − B bimodules correspond to pivotal A − B bimodulecategories.
Examples
As a natural corollary from Theorem A, a Markov tower corresponds to a Temperley-Lieb-Jones(
T LJ ) module category. This result generalizes the pivotal module case from [CHPS18,Thm. A.]. To translate our classification into that of [DY15] which uses fair and balanced graphs,we obtain an elegant graphical version of a Markov tower using a W ∗ C (Λ , ω )of bigraded Hilbert spaces BigHilb which is built from a fair and balanced graph (Λ , ω ). Ourapproach is inspired by Ocneanu’s path algebras [Oc88] [EK98] [JS97, § § ∗ T LJ ( d ) − module category M C ( K, ev K )of BigHilb
Markov tower M with modulus d balanced d -fairbipartite graph (Λ , ω ) § § § § § Theorem C.
Every (infinite depth) subfactor planar algebra embeds in any bipartite graphplanar algebra of its fusion graph with respect to a module category. In particular, it embeds inthe bipartite graph planar algebra of its (dual) principal graph.
By Theorem B above, a Markov lattice corresponds to a
T LJ − T LJ bimodule cate-gory. By work-in-progress of Penneys-Peters-Snyder, pivotal
T LJ − T LJ bimodule categoriescorrespond to Ocneanu’s biunitary connections on associative square-partite graphs with ver-tex weightings. For the non-pivotal case, the weighting on the square-partite graph is theedge-weighting and we obtain the non-pivotal analog of a biunitary connection. To translatebetween these classifications, we use the well-known fact that a commuting square of finitedimensional von Neumann algebras gives a biunitary connection [JS97]. We then introduce agraphical version of a Markov lattice using a W ∗ C (Φ) of BigHilb obtained froma biunitary connection Φ. It turns out that the biunitary connection Φ corresponds to thebimodule associator of the bimodule category. The following diagram shows how these notionsare related to each other in § ∗ T LJ ( d ) − T LJ ( d ) bimodule category M C (Φ)of BigHilb
Markov lattice M with modulus ( d , d ) balanced ( d , d )-fairsquare-partite graph (Λ , ω )with biunitary connection Φ § § § § § cknowledgements I thank David Penneys and Corey Jones for providing this project andsome necessary techniques, including the idea of the 2-shift map in a standard λ -lattice, unitarydual functors and biunitary connections for the pivotal case. I thank Peter Huston for clarifyingLemma 1.5.4, and also providing good suggestion in the writing. I want to thank Jamie Vicaryfor clarifying the graphical calculus for 2-categories during the Summer Research Programon Quantum Symmetries at Ohio State University, 2019. The author is supported by theMathematics Department in Ohio State University as Graduate Teaching Associate and DavidPenneys’ NSF DMS grant 1654159. Contents λ -lattices and tensor category 6 λ -lattice and its properties . . . . . . . . . . . . . . . . . . . . . . . . . 81.3 The 2-shift map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111.4 String diagram explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151.5 Some useful lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191.6 From standard λ -lattice to pivotal planar tensor category . . . . . . . . . . . . . 211.6.1 Planar tensor category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 211.6.2 From standard λ -lattice to pivotal planar tensor category . . . . . . . . . 221.7 From 2-shaded rigid C ∗ multitensor category to standard λ -lattice . . . . . . . . 311.7.1 Rigid C ∗ multitensor category . . . . . . . . . . . . . . . . . . . . . . . . . 311.7.2 2-shaded rigid C ∗ multitensor category with a choice of generator andplanar tensor category . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331.7.3 From planar tensor category to standard λ -lattice . . . . . . . . . . . . . 34 λ -lattice and modulecategories 35 λ -lattice . . . . . . . . . 352.2 String diagram explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 362.3 From Markov tower as a standard module to planar module category . . . . . . . 372.3.1 Planar module category over planar tensor category . . . . . . . . . . . . 372.3.2 From Markov tower as a standard module to planar module category . . . 372.4 Indecomposable semisimple C ∗ A− module categories and planar A − module cat-egories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 392.4.1 Indecomposable semisimple C ∗ A− module category . . . . . . . . . . . . 392.4.2 From planar module category to Markov tower as a standard module overa standard λ -lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40 λ -lattices and bi-module categories 40 λ -lattices . . . . . . . . 423.3 String diagram explanation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 423.4 From Markov lattice as standard bimodule to planar bimodule category . . . . . 445.4.1 Planar bimodule category . . . . . . . . . . . . . . . . . . . . . . . . . . . 443.4.2 From Markov lattice as standard bimodule to planar bimodule category . 443.5 Indecomposable semisimple C ∗ A − B bimodules and planar A − B bimodulecategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 473.5.1 Indecomposable semisimple C ∗ A − B bimodule category . . . . . . . . . 473.5.2 From planar bimodule to Markov lattice as standard bimodule . . . . . . 47 d -fair bipartite graph . . . . . . . . . . . . . . . . . . . . . . . . . . . . 484.2 BigHilb and 2-subcategory C ( K, ev K ) . . . . . . . . . . . . . . . . . . . . . . . . . 494.3 The 2-subcategory of BigHilb generated by a balanced d -fair bipartite graph . . . 534.4 From C ( K, ev K ) to Markov tower . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.5 More properties of Markov tower . . . . . . . . . . . . . . . . . . . . . . . . . . . 564.6 From Markov tower to C (Λ , ω ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 574.7 C ( K, ev K ) and End † ( M , F ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62 d , d )-fair square-partite graph . . . . . . . . . . . . . . . . . . . . . . 665.2 2-subcategory C ( K , K , L , L , ev) of BigHilb and biunitary connection Φ . . . . 665.3 From C (Φ) to Markov lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 695.4 From Markov lattice to C (Γ , ω ; Φ) . . . . . . . . . . . . . . . . . . . . . . . . . . . 715.5 C (Φ) and End † ( M , F, G ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74 λ -lattices and tensor category Definition 1.1.1.
Let A ⊂ B be a unital inclusion of finite von Neumann algebras. A condi-tional expectation E : M → N is a positive linear map satisfying the following conditions:(a) E ( x ) = x for all x ∈ A ,(b) E ( axb ) = aE ( x ) b for all a, b ∈ A , x ∈ B . Definition 1.1.2.
Let C be a unital C ∗ -algebra. We call a linear functional tr : C → C a trace if it satisfies the following conditions:(a) (tracial) tr( xy ) = tr( yx ), for all x, y ∈ C .(b) (positive) tr( x ∗ x ) ≥
0, for all x ∈ C .(c) (faithful) tr( x ∗ x ) = 0 if and only if x = 0.In addition, we call tr unital if tr(1) = 1. Definition 1.1.3. A traceless Markov tower M = ( M n , E n , e n +1 ) n ≥ consists of a sequence( M n ) n ≥ of finite dimensional von Neumann algebras, such that M n is unitally included in6 n +1 . For each n , there is a faithful normal conditional expectation E n : M n → M n − togetherwith a sequence of Jones projections e n ∈ M n +1 for all n ≥
1, such that:(M1) The projections ( e n ) satisfy the Temperley-Lieb-Jones relations:(TLJ1) e n = e n = e ∗ n for all n .(TLJ2) [ e i , e j ] = 0 for | i − j | > modulus d > e n e n ± e n = d − e n for all n .(M2) For all x ∈ M n , e n xe n = E n ( x ) e n .(M3) E n +1 ( e n ) = d − · n ≥ M n +1 e n = M n e n for all n ≥ traceless unless stated otherwise. Proposition 1.1.4.
Some properties of a traceless Markov tower include: (1) [ x, e k ] = 0 , for x ∈ M n , k ≥ n + 1 . (2) The map M n (cid:51) x (cid:55)→ xe n ∈ M n +1 is injective. (3) For x ∈ M n +1 , d E n +1 ( xe n ) is the unique element y ∈ M n such that xe n = ye n . (4) Property (3) is equivalent to (M3).(5) If x ∈ M n and [ x, e n ] = 0, then x ∈ M n − . Together with (1), we have M n − = M n ∩ { e n } (cid:48) .(6) e n M n +1 e n = M n − e n . Proof. (1) For x ∈ M n and k ≥ n + 1, E k ( x ) = x, E k ( x ∗ ) = x ∗ , then xe k = E k ( x ) e k = e k xe k = ( e k x ∗ e k ) ∗ = ( E k ( x ∗ ) e k ) ∗ = ( x ∗ e k ) ∗ = e k x. (2) If x ∈ M n and xe n = 0, then by (M3),0 = E n +1 ( xe n ) = xE n +1 ( e n ) = d − x. Thus, x (cid:55)→ xe n is injective.(3) By (M4) and (2), the existence and uniqueness hold. Then by (M3), E n +1 ( xe n ) = E n +1 ( ye n ) = yE n +1 ( e n ) = d − y, so y = d E n +1 ( xe n ).(4) First, let’s prove that in this setting, M n ∈ x (cid:55)→ xe n ∈ M n +1 is injective. If xe n = 0, then0 = d E n +1 ( xe n ) = d xE n +1 ( e n ) . Note that E n +1 is faithful and E n +1 ( e n ) (cid:54) = 0, so x = 0.Let x = e n , then we have d E n +1 ( e n ) e n = e n . Since d E n +1 ( e n ) and 1 ∈ M n , we have d E n +1 ( e n ) = 1 by (2).(5) Since xe n = e n x , E n ( x ) e n = e n xe n = xe n e n = xe n . Then by (2), E n ( x ) = x , which implies x ∈ M n − .(6) By (M2) and (M4).We will explore more properties of traceless Markov towers in § emark 1.1.5. If there is a faithful normal trace on (cid:83) ∞ n =0 M n and E n is the canonical faithfulnormal trace-preserving conditional expectation for n = 1 , , · · · , then M is called a tracialMarkov tower . Thus, tracial Markov towers defined in [CHPS18] are traceless Markov towers. Example 1.1.6 (Markov tower without a trace) . Let d > d >
4. There is a unique λ ∈ (0 , ) such that d − = λ (1 − λ ). Then dλ + d (1 − λ ) = d and dλ + d (1 − λ ) = d . Let e ij denote the matrix units of M ( C ), i, j = 1 ,
2, and 1 = e + e ∈ M ( C ).Define E λ : M ( C ) → C by E λ ( e ) = λ , E λ ( e ) = 1 − λ and E λ ( e ) = E λ ( e ) = 0. It isclear that E λ is a normal faithful conditional expectation and not tracial.Define e λ ∈ M ( C ) ⊗ M ( C ) by e λ = (1 − λ ) e ⊗ e + λe ⊗ e + (cid:112) λ (1 − λ )( e ⊗ e + e ⊗ e ) , and one can check that:(a) e λ is a projection.(b) E λ ( e λ ) = d − ( e + e ) = d − · e λ ⊗ ⊗ e − λ )( e λ ⊗
1) = d − ( e λ ⊗
1) and ( e − λ ⊗ ⊗ e λ )( e − λ ⊗
1) = d − ( e − λ ⊗ M ( C ) → M ( C ) to be the identity map. Let M n := M ( C ) ⊗ n . The inclusion M n ⊂ M n +1 maps x to x ⊗ id. Jones projection e n +1 = 1 ⊗ n ⊗ e − λ ∈ M n +2 and e n +2 =1 ⊗ n +1 ⊗ e λ ∈ M n +3 , n = 0 , , , · · · The conditional expectation is defined as follows: E n +1 = id ⊗ n +1 ⊗ E λ E n +2 = id ⊗ n +2 ⊗ E − λ . Now we build a Markov tower with modulus d and without a trace:1 M ( C ) E λ (cid:111) (cid:111) M ( C ) ⊗ ⊗ E − λ (cid:111) (cid:111) M ( C ) ⊗ ⊗ ⊗ E λ (cid:111) (cid:111) M ( C ) ⊗ ⊗ ⊗ E − λ (cid:111) (cid:111) · · · (cid:111) (cid:111) λ -lattice and its properties Definition 1.2.1 ( [Po95]) . Let A = ( A i,j ) ≤ i ≤ j< ∞ be a system of finite dimensional C ∗ algebraswith A i,i = C with unital inclusions A i,j ⊂ A k,l , for k ≤ i, j ≤ l . A , ⊂ A , ⊂ A , ⊂ A , ⊂ A , ⊂ · · ·∪ ∪ ∪ ∪ A , ⊂ A , ⊂ A , ⊂ A , ⊂ · · ·∪ ∪ ∪ A , ⊂ A , ⊂ A , ⊂ · · ·∪ ∪ A , ⊂ A , ⊂ · · ·∪ A , ⊂ · · · . . .Let E ri,j : A i,j → A i,j − be the (horizontal) faithful normal conditional expectation, j =1 , · · · , i = 0 , · · · , j − E li,j : A i,j → A i +1 ,j be the (vertical) faithful normal conditionalexpectation i = 0 , , · · · , j = i + 1 , i + 2 , · · · . We also require that8a) (commuting square condition) A i,jE li,j (cid:15) (cid:15) A i,j +1 E ri,j +1 (cid:111) (cid:111) E li,j +1 (cid:15) (cid:15) A i +1 ,j A i +1 ,j +1 E ri +1 ,j +1 (cid:111) (cid:111) is a commuting square, i.e., E li,j ◦ E ri,j +1 = E ri +1 ,j +1 ◦ E li,j +1 .(b) (existence of Jones λ -projections)There exists a sequence of Jones projections { e i } i ≥ in (cid:83) n A ,n such that(b1) e j ∈ A i − ,k , for 1 ≤ i ≤ j + 1 ≤ k .(b2) The projections satisfy the Temperley-Lieb-Jones relations:(TLJ1) e i = e i = e ∗ i for all i .(TLJ2) e i e j = e j e i for | i − j | > d > e i e i ± e i = d − e i for all i .(b3) e j xe j = E ri,j ( x ) e j , for x ∈ A i,j , i + 1 ≤ j .(b4) e i xe i = E li,j ( x ) e i , for x ∈ A i,j , i + 1 ≤ j .(c) (Markov conditions)(c1) dim A i,j = dim A i,j +1 e j = dim A i +1 ,j +1 , for i ≤ j .(c2) E ri,j +1 ( e j ) = E lj − ,k ( e j ) = d −
1, for j ≥ i + 1 , k ≥ j + 1.Then A = ( A i,j ) ≤ i ≤ j< ∞ is called a λ -lattice of commuting squares. If there is a faithfulnormal trace tr on (cid:83) ∞ n =0 A ,n and E ri,j , E li,j are the canonical faithful normal trace-preservingconditional expectation, then A is called a tracial λ -lattice . Definition 1.2.2 ( [Po95]) . A λ -lattice ( A i,j ) ≤ i ≤ j is called a standard λ -lattice if [ A i,j , A k,l ] =0 for i ≤ j ≤ k ≤ l . This condition is called the standard condition . Remark 1.2.3.
In the definition of (standard) λ -lattice, we may not require a trace and theconditional expectations are trace-preserving. In fact, the reader can construct an example of(standard) λ -lattice without a trace from Example 1.1.6 easily. We will not further discuss thetraceless standard λ -lattices, though the following statements do NOT require the trace at all! Remark 1.2.4.
Each row A i = ( A i,j ) j ≥ i is a Markov tower, i = 0 , , , · · · ; each column A j = ( A i,j ) i = j is a Markov tower, j = 1 , , · · · . From Proposition 1.1.4, we have(1) If x ∈ A i,j , [ x, e k ] = 0 for k ≥ j + 1; [ x, e l ] = 0 for 1 ≤ l ≤ i − A i,j (cid:51) x (cid:55)→ xe j ∈ A i,j +1 is injective; the map A i,j (cid:51) x (cid:55)→ xe i ∈ A i − ,j is injective.(3) The Markov condition is equivalent to the pull-down condition:(c1)’ d E ri,j +1 ( xe j ) e j = xe j , for x ∈ A i,j +1 , j ≥ i ≥ d E li − ,j ( xe i ) e i = xe i , for x ∈ A i − ,j , j ≥ i ≥ Proposition 1.2.5.
Let A , ⊂ A , ⊂ A , ⊂ A , ⊂ · · ·∪ ∪ ∪ A , ⊂ A , ⊂ A , ⊂ · · · e a λ -sequence of commuting squares, and define A i,j := A i − ,j ∩{ e i − } (cid:48) = A ,j ∩{ e , · · · , e i − } (cid:48) , ≤ i ≤ j . Then ( A i,j ) ≤ i ≤ j< ∞ is a λ -lattice of commuting squares.Proof. We construct A i,j and conditional expectation E li − ,j : A i − ,j → A i,j by induction on i , and show that Jones projections { e i +1 , · · · , e j − } ⊂ A i,j for i + 2 ≤ j . Suppose A i − ,j isconstructed (or given) and { e i , · · · , e j − } ⊂ A i − ,j , We define A i,j := A i − ,j ∩ { e i − } (cid:48) . Thenclearly, { e i +1 , · · · , e j − } ⊂ A i,j .According to Proposition 1.1.4(5) and (6), for each x ∈ A i − ,j ⊂ A i − ,j , there exists a y ∈ A i,j such that ye i − = e i − xe i − . By Proposition 1.1.4(2), A i − ,j (cid:51) y (cid:55)→ ye i − ∈ A i − ,j is injective, so y is unique for each given x . This technique is often used in this chapter. We define E li − ,j ( x ) := y . Now we show that E li − ,j is a faithful normal conditional expectation:(a) It is clear that E li − ,j is linear, and E li − ,j (1) = 1. The ultraweak continuity/normalityfollows from the definition.(b) E li − ,j ( x ∗ ) = E li − ,j ( x ) ∗ : E li − ,j ( x ) ∗ e i − = ( e i − E li − ,j ( x )) ∗ = ( e i − xe i − ) ∗ = e i − x ∗ e i − = E li − ,j ( x ∗ ) e i − . (c) E li − ,j ( axb ) = aE li − ,j ( x ) b for a, b ∈ A i,j : Note that [ a, e i − ] = [ b, e i − ] = 0, then E li − ,j ( axb ) e i − = e i − axbe i − = ae i − xe i − b = aE li − ,j ( x ) e i − b = aE li − ,j ( x ) be i − . (d) E li − ,j ( x ∗ x ) ≥ E li − ,j ( x ) ∗ E li − ,j ( x ), which follows that E li − ,j is positive: E li − ,j ( x ) ∗ E li − ,j ( x ) e i − = E li − ,j ( x ) ∗ e i − xe i − = e i − x ∗ e i − xe i − ≤ e i − x ∗ xe i − = E li − ,j ( x ∗ x ) e i − , so E li − ,j ( x ∗ x ) ≥ E li − ,j ( x ) ∗ E li − ,j ( x ) by applying the inductive hypothesis that E li − ,j is apositive conditional expectation and E li − ,j ( e i − ) = d − · E li − ,j ( x ∗ x ) = 0 if and only if x = 0, i.e., E li − ,j is faithful:0 = E li − ,j ( x ∗ x ) e i − = e i − x ∗ xe i − = ( xe i − ) ∗ ( xe i − ) , which follows that xe i − = 0. Note that A i − ,j (cid:51) x (cid:55)→ xe i − ∈ A i − ,j is an injection, so x = 0.Then define E ri,j +1 : A i,j +1 → A i,j as the restriction of E ri − ,j +1 on A i,j +1 , which is also aconditional expectation.Now we prove the commuting square condition E li − ,j ◦ E ri − ,j +1 = E ri,j +1 ◦ E li − ,j +1 : for x ∈ A i − ,j +1 , E li − ,j ( E ri − ,j +1 ( x )) e i − = e i − E ri − ,j +1 ( x ) e i − E ri,j +1 ( E li − ,j +1 ( x )) e i − = E ri − ,j +1 ( E li − ,j +1 ( x )) e i − = E ri − ,j +1 ( E li − ,j +1 ( x ) e i − )= E ri − ,j +1 ( e i − xe i − ) = e i − E ri − ,j +1 ( x ) e i − . Finally, we prove the Markov condition:(a) dim A i,j = dim A i − ,j ∩ { e i − } (cid:48) = dim A i − ,j ∩ { e j − } (cid:48) = dim A i − ,j − .(b) E ri,j +1 ( e j ) = E ri − ,j +1 ( e j ) = d − E li − ,j ( e i ) e i − = e i − e i e i − = d − e i − , so E li − ,j ( e i ) = d − · orollary 1.2.6. Let ( A i,j ) i ≤ j,i =0 , be a λ -sequence of commuting squares. If A i,j := { e , · · · , e i − } (cid:48) ∩ A i,j , for all ≤ i ≤ j , then ( A i,j ) ≤ i ≤ j is a standard λ -lattice if and only if ( A i,j ) i ≤ j,i =0 , satisfies [ A , , A ,j ] = 0 , ∀ ≤ j [ A ,i , A i,j ] = 0 , ∀ ≤ i ≤ j. Now we define the opposite standard λ -lattice, which will be used in Definition 3.2.1. Definition 1.2.7. A op = ( A i,j ) ≤ j ≤ i is the opposite of λ -lattice A if A op j,i = A i,j as oppositealgebras, E op ,lj,i = E ri,j , E op ,rj,i = E li,j for i ≤ j . Example 1.2.8.
The Temperley-Lieb-Jones algebra TLJ( d ) forms a standard λ -lattice withthe modulus d by letting A i,i = A i,i +1 = C and A i,j = (cid:104) e i +1 , · · · , e j − (cid:105) for j − i ≥
2, which iscalled a Temperley-Lieb-Jones standard λ -lattice. Example 1.2.9 ( [Po95]) . If A ⊂ A is a unital inclusion of type II subfactors with finiteindex and A ⊂ A ⊂ A ⊂ A ⊂ · · · is the Jones tower, then A i,j := A (cid:48) i ∩ A j forms a standard λ -lattice, which is called the standard invariant of A ⊂ A . In this section, we discuss an important type of ∗ -isomorphism in a standard λ -lattice,so-called the 2-shift map [Bi97]. Here we provide the definition by using the conditional expec-tations and Jones projections instead of trace and Pimsner-Popa basis.For i, k ≥
0, define the following element of A l,i +2 k , l + 1 ≤ i + 2 k : e ik := d k ( k − ( e k + i e k + i − · · · e i +1 )( e k + i +1 e k + i · · · e n − k +2 ) · · · ( e k + i − e k + i − · · · e k + i ) . For i, j, k ≥
0, define the following element of A l,i + j +2 k , l + 1 ≤ i + j + 2 k , e ij,k = d jk e ik e i +1 k · · · e i + jk . Clearly, e n = e n − = e n − , , e ik = e i ,k , ( e ik ) = ( e ik ) ∗ = e ik and e ij,k ( e ij,k ) ∗ = e i ,k , ( e ij,k ) ∗ e ij,k = e i + j ,k . Definition 1.3.1 (Multi-step condition expectation) . Define the k -step horizontal conditionalexpectation as E r,ki,j = E ri,j +1 − k ◦ E ri,j +2 − k ◦ · · · ◦ E ri,j : A i,j → A i,j − k for k ≤ j − i and we have E r, i,j = E ri,j ; the k -step vertical conditional expectation as E l,ki,j = E li + k − ,j ◦ E ri + k − ,j ◦ · · · ◦ E li,j : A i,j → A i + k for k ≤ j − i and we have E l, i,j = E li,j .In particular, the trace is made by the composition of conditional expectations, i.e., E l,i − j + ki,j − k ◦ E r,ki,j = tr = E r,j − i − ti + t,j ◦ E l,ti,j , for 0 ≤ k ≤ j − i , 0 ≤ t ≤ j − i Definition 1.3.2 (2-shift map) . Define the 2 -shift map S i,j : A i,j → A i +2 ,j +2 , i ≤ j by S i,j ( x ) := d j − i +2 E li,j +2 ( e i +1 e i +2 · · · e j xe j +1 e j · · · e i +1 ) . Proposition 1.3.3.
The followings are the properties of the 2-shift map. (1) S i,j is well defined, i.e., S i,j ( x ) ∈ A i +2 ,j +2 for x ∈ A i,j . S i,j is a unital ∗ -isomorphism. (3) ( commuting parallelogram ) S i,j − ◦ E ri,j = E ri +2 ,j +2 ◦ S i,j and S i +1 ,j ◦ E li,j = E li +2 ,j +2 ◦ S i,j . (4) S i,j +1 ( x ) = S i,j ( x ) for x ∈ A i,j and S i − ,j ( x ) = S i,j ( x ) for x ∈ A i,j . (5) ( shift ) e i +1 e i +2 · · · e j +1 x = S i,j ( x ) e i +1 e i +2 · · · e j +1 for x ∈ A i,j . Taking adjoints, xe j +1 e j · · · e i +1 = e j +1 e j · · · e i +1 S i,j ( x ) . In other word, e ij − i, x = S i,j ( x ) e ij − i, . (6) S i,j is trace-preserving. (7) S i,j ( e k ) = e k +2 , where i + 1 ≤ k ≤ j − .Proof. (1) Note that S i,j ( x ) ∈ A i +1 ,j +2 , we shall show that E li +1 ,j +2 ( S i,j ( x )) = S i,j ( x ). Since E li +1 ,j +2 ( S i,j ( x )) − S i,j ( x ) ∈ A i +1 ,j +2 and the map A i +1 ,j +2 (cid:51) y (cid:55)→ ye i +1 ∈ A i,j +2 is injective, we shall show that E li +1 ,j +2 ( S i,j ( x )) e i +1 = S i,j ( x ) e i +1 . E li +1 ,j +2 ( S i,j ( x )) e i +1 = e i +1 S i,j ( x ) e i +1 = d j − i +2 e i +1 E li,j +2 ( e i +1 e i +2 · · · e j xe j +1 e j · · · e i +1 ) e i +1 = d j − i e i +1 ( e i +1 e i +2 · · · e j xe j +1 e j · · · e i +1 ) (pull down)= d j − i e i +1 e i +2 · · · e j xe j +1 e j · · · e i +1 = d j − i +2 E li,j +2 ( e i +1 e i +2 · · · e j xe j +1 e j · · · e i +1 ) e i +1 (pull down)= S i,j ( x ) e i +1 . (2) For x ∈ A i,j , we have [ x, e j +1 ] = 0. First, we show that S i,j is a homomorphism, i.e., S i,j ( xy ) = S i,j ( x ) S i,j ( y ) for x, y ∈ A i,j . Note that the map A i +2 ,j +2 ⊂ A i +1 ,j +2 (cid:51) y (cid:55)→ ye i +1 ∈ A i,j +2 is injective, we shall prove that S i,j ( xy ) e i +1 = S i,j ( x ) S i,j ( y ) e i +1 . S i,j ( x ) S i,j ( y ) e i +1 = d j − i +2 S i,j ( x ) E li,j +2 ( e i +1 e i +2 · · · e j ye j +1 e j · · · e i +1 ) e i +1 = d j − i S i,j ( x ) e i +1 e i +2 · · · e j ye j +1 e j · · · e i +1 (pull down)= d j − i · d j − i ( e i +1 e i +2 · · · e j xe j +1 e j · · · e i +1 )( e i +1 e i +2 · · · e j ye j +1 e j · · · e i +1 )(pull down)= d j − i +2 e i +1 e i +2 · · · e j xe j +1 e j ye j +1 e j · · · e i +1 ( e k e k ± e k = d − e k )= d j − i +2 e i +1 e i +2 · · · e j xe j +1 e j e j +1 ye j · · · e i +1 ([ y, e j +1 ] = 0)= d j − i e i +1 e i +2 · · · e j xe j +1 ye j · · · e i +1 = d j − i e i +1 e i +2 · · · e j xye j +1 e j · · · e i +1 = d j − i +2 E li,j +2 ( e i +1 e i +2 · · · e j xye j +1 e j · · · e i +1 ) e i +1 (pull down)= S i,j ( xy ) e i +1 . Next, S i,j is a ∗ -homomorphism. Note that E li,j +2 is a ∗ -homomorphism, we have S i,j ( x ∗ ) = d j − i +2 E li,j +2 ( e i +1 e i +2 · · · e j x ∗ e j +1 e j · · · e i +1 )= d j − i +2 E li,j +2 (( e i +1 e i +2 · · · e j xe j +1 e j · · · e i +1 ) ∗ )= d j − i +2 E l, ∗ i,j +2 ( e i +1 e i +2 · · · e j xe j +1 e j · · · e i +1 )= S ∗ i,j ( x ) . When x = 1, e i +1 e i +2 · · · e j e j +1 e j · · · e i +1 = d − e i +1 e i +2 · · · e j − e j e j − · · · e i +1 = · · · = d i − j +2) e i +1 e i +2 e i +1 = d i − j ) e i +1 . S i,j (1) = d E li,j +2 ( e i +1 ) = 1, i.e., S i,j is unital.In order to prove that S i,j is an isomorphism, we shall show S i,j is injective and surjective.If S i,j ( x ) = 0, then0 = S i,j ( x ) e i +1 = d j − i e i +1 e i +2 · · · e j xe j +1 e j · · · e i +1 = d j − i ( e i +1 e i +2 · · · e j ) xe j +1 ( e i +1 e i +2 · · · e j ) ∗ , which follows that xe j +1 = 0. Since the map A i,j (cid:51) y (cid:55)→ ye j +1 ∈ A i,j +1 is injective, we have x = 0.Note that dim A i,j = dim A i +1 ,j +1 = dim A i +2 ,j +2 < ∞ , so the injectivity implies thesurjectivity. Thus, S i,j is a unital ∗ -isomorphism.(3) For x ∈ A i,j , E ri,j ( x ) ∈ A i,j − and [ E ri,j ( x ) , e j ] = 0, S i,j − ◦ E ri,j ( x ) = d j − i E li,j +1 ( e i +1 e i +2 · · · e j E ri,j ( x ) e j +1 e j · · · e i +1 )= d j − i E li,j +1 ( e i +1 e i +2 · · · E ri,j ( x ) e j e j +1 e j · · · e i +1 )= d j − i +2 E li,j +1 ( e i +1 e i +2 · · · E ri,j ( x ) e j · · · e i +1 )= d j − i +2 E li,j +1 ( e i +1 e i +2 · · · e j xe j · · · e i +1 ) ,E ri +2 ,j +2 ◦ S i,j ( x ) = E ri +2 ,j +2 ◦ E li +1 ,j +2 ◦ S i,j ( x )= E li +1 ,j +1 ◦ E ri +1 ,j +2 ◦ S i,j ( x ) (commuting square)= d j − i E li +1 ,j +1 ◦ E ri +1 ,j +2 ◦ E li,j +2 ( e i +1 e i +2 · · · e j xe j +1 e j · · · e i +1 )= d j − i E li +1 ,j +1 ◦ E li,j +1 ◦ E ri,j +2 ( e i +1 e i +2 · · · e j xe j +1 e j · · · e i +1 )= d j − i E li +1 ,j +1 ◦ E li,j +1 ( e i +1 e i +2 · · · e j xE ri,j +2 ( e j +1 ) e j · · · e i +1 )= d j − i +2 E li +1 ,j +1 ◦ E li,j +1 ( e i +1 e i +2 · · · e j xe j · · · e i +1 ) . = E li +1 ,j +1 ( S i,j − ◦ E ri,j ( x )) (since S i,j − ◦ E ri,j ( x ) ∈ A i +2 ,j +1 )= S i,j − ◦ E ri,j ( x ) . Thus, S i,j − ◦ E ri,j = E ri +2 ,j +2 ◦ S i,j .Note that { e i +1 , · · · , e j − } ⊂ A i,j , we have E li,j +2 ( e k xe n ) = e k E li,j +2 ( x ) e n for all k, n ∈ { i + 1 , · · · , j − } . ( ∗ )In order to prove that S i +1 ,j ◦ E li,j = E li +2 ◦ S i,j , by Remark 1.2.4 (2), we shall show that13 i +1 ,j ◦ E li,j ( x ) e i +2 = E li +2 ,j +2 ◦ S i,j ( x ) e i +2 for all x ∈ A i,j . S i +1 ,j ◦ E li,j ( x ) e i +2 = d j − i E li +1 ,j +2 ( e i +2 · · · e j E li,j ( x ) e j +1 · · · e i +2 ) e i +2 = d j − i − e i +2 · · · e j E li,j ( x ) e j +1 · · · e i +2 , (pull down) E li +2 ,j +2 ◦ S i,j ( x ) e i +2 = d j − i +2 E li +2 ,j +2 ( E li,j +2 ( e i +1 · · · e j xe j +1 · · · e i +1 )) e i +2 = d j − i +2 e i +2 E li,j +2 ( e i +1 e i +2 · · · e j xe j +1 · · · e i +2 e i +1 ) e i +2 (by ( ∗ ))= d j − i +2 E li,j +2 ( e i +2 e i +1 e i +2 · · · e j xe j +1 · · · e i +2 e i +1 e i +2 )= d j − i − E li,j +2 ( e i +2 · · · e j xe j +1 · · · e i +2 )= d j − i − e i +2 e i +1 · · · e j E li,j +2 ( x ) e j +1 · · · e i +1 e i +2 (by ( ∗ ))= d j − i − e i +2 e i +1 · · · e j E li,j ( x ) e j +1 · · · e i +1 e i +2 (commuting square)= S i +1 ,j ◦ E li,j ( x ) e i +2 . Thus, S i +1 ,j ◦ E li,j = E li +2 ◦ S i,j .(4) This is a particular case of (3) by the property of conditional expectation.(5) For x ∈ A i,j , [ x, e j +1 ] = 0, S i,j ( x ) e i +1 e i +2 · · · e j +1 = d j − i +2 E li,j +2 ( e i +1 e i +2 · · · e j xe j +1 e j · · · e i +2 e i +1 ) e i +1 e i +2 · · · e j +1 = d j − i ( e i +1 e i +2 · · · e j xe j +1 e j · · · e i +2 e i +1 ) e i +2 · · · e j +1 (pull down)= d j − i − ( e i +1 e i +2 · · · e j x ) · e j +1 · · · e i +2 · · · e j +1 = · · · = e i +1 e i +2 · · · e j xe j +1 = e i +1 e i +2 · · · e j e j +1 x. (6) By (3) and Definition 1.3.1.(7) Note that the map A i +2 ,j +2 ⊂ A i +1 ,j +2 (cid:51) y (cid:55)→ ye i +1 ∈ A i,j +2 is injective, we shall prove that S i,j ( e k ) e i +1 = e k +2 e i +1 . For i + 1 ≤ k ≤ j − S i,j ( e k ) e i +1 = d j − i +2 E li,j +2 ( e i +1 e i +2 · · · e j e k e j +1 e j · · · e i +1 ) e i +1 = d j − i e i +1 e i +2 · · · e j e k e j +1 e j · · · e i +1 (pull down)= d j − i e i +1 · · · e k − e k e k +1 e k e k +2 · · · e j e j +1 e j · · · e k +2 e k +1 · · · e i +1 ([ e i , e j ] = 0 for | i − j | ≥ d j − i e i +1 · · · e k − ( e k e k +1 e k )( e k +2 · · · e j e j +1 e j · · · e k +2 ) e k +1 · · · e i +1 = d k − i e i +1 · · · e k − e k e k +2 e k +1 e k · · · e i +1 ( e t e t ± e t = d − e t )= d k − i e k +2 e i +1 · · · e k − e k e k +1 e k · · · e i +1 = e k +2 e i +1 ( e t e t ± e t = d − e t ) Definition 1.3.4 (2 n -shift map) . Define S ( n ) i,j : A i,j → A i +2 n,j +2 n by S ( n ) i,j = S i +2( n − ,j +2( n − ◦ S ( n − i,j = S i +2( n − ,j +2( n − ◦ S i +2( n − ,j +2( n − ◦ · · · ◦ S i,j to be the 2 n -shift map . 14 roposition 1.3.5. The followings are the properties of the n -shift map. (1) S ( n ) i,j is a unital ∗ -isomorphism. (2) ( commuting parallelogram ) S ( n ) i,j − ◦ E r,ki,j = E r,ki +2 n,j +2 n ◦ S ( n ) i,j and S ( n ) i +1 ,j ◦ E l,ki,j = E l,ki +2 n,j +2 n ◦ S ( n ) i,j . (3) S ( n ) i,j + k ( x ) = S ( n ) i,j ( x ) for x ∈ A i,j and S ( n ) i − k,j ( x ) = S ( n ) i,j ( x ) for x ∈ A i,j . (4) ( shift ) For x ∈ A i,j , e ij − i,n x = S ( n ) i,j ( x ) e ij − i,n . By taking adjoint, xe i, ∗ j − i,n = e i, ∗ j − i,n S ( n ) i,j ( x ) . (5) S ( n ) i,j is trace-preserving.Proof. (1),(2),(3),(5) follow from Proposition 1.3.3.(4) First, we show that e i +2( n − j − i, e i +2( n − j − i, · · · e ij − i, x = S ni,j ( x ) e i +2( n − j − i, e i +2( n − j − i, · · · e ij − i, for x ∈ A i,j . S ni,j ( x ) e i +2( n − j − i, e i +2( n − j − i, · · · e ij − i, = S i +2( n − ,j +2( n − ( S ( n − i,j ( x )) e i +2( n − j − i, e i +2( n − j − i, · · · e ij − i, = e i +2( n − j − i, S ( n − i,j ( x ) e i +2( n − j − i, · · · e ij − i, = · · · = e i +2( n − j − i, e i +2( n − j − i, · · · e ij − i, x. Second, e ij − i,n = a ij − i,n e i +2( n − j − i, e i +2( n − j − i, · · · e ij − i, b ij − i,n with a ij − i,n ∈ A i,i +2 n and b ij − i,n ∈ A j,j +2 n , which will be showed below in Lemma 1.5.1 and 1.5.2. Then by the standard condition,since x ∈ A i,j and S ( n ) ( x ) ∈ A i +2 n,j +2 n , we have [ S ( n ) i,j ( x ) , a ij − i,n ] = 0 and [ x, b ij − i,n ] = 0, whichfollows that S ( n ) i,j ( x ) e ij − i,n = S ( n ) i,j ( x ) a ij − i,n e i +2( n − j − i, e i +2( n − j − i, · · · e ij − i, b ij − i,n = a ij − i,n S ( n ) i,j ( x ) e i +2( n − j − i, e i +2( n − j − i, · · · e ij − i, b ij − i,n = a ij − i,n e i +2( n − j − i, e i +2( n − j − i, · · · e ij − i, xb ij − i,n = a ij − i,n e i +2( n − j − i, e i +2( n − j − i, · · · e ij − i, b ij − i,n x = e ij − i,n x. In this section, we use Temperley-Lieb-Jones (TLJ) string diagram to explain the elementsin A i,j , horizontal (right) and vertical (left) conditional expectations, the Jones projections,2 n -shift maps and their properties.In the following sections, we will use these diagrams to do the algebraic computation andreaders may interpret these diagrams directly into the algebraic computations by looking at thedictionary here.( λ
1) Element x ∈ A i,j . A i,j is a (rectangular) box space with j shaded/unshaded strands wherethe left i strands are straight strands and together with a j − i box space. We set the leftpart of left most strand to be always unshaded; the shading on the left part of the j − i i : x j If 2 | i : = x i j − i = x j − ii x j If 2 (cid:45) i : = x i j − i = x j − ii Remark : The reader shall understand the meaning of rectangular box and round box ofan element. And the shading type of an element is the shading on the left of the roundbox.( λ
2) Horizontal inclusion x ∈ A i,j ⊂ A i,j +1 . The inclusion A i,j ⊂ A i,j +1 means adding onestraight strand on the right and regarding the j − i box space in A i,j as a part of the j − i + 1 box space in A i,j +1 together with the straight strand, which does not change theshading type of the box space: x j If 2 | i : = x j − i +1 i x j If 2 (cid:45) i : = x j − i +1 i ( λ
3) Vertical inclusion x ∈ A i,j ⊂ A i − ,j . The inclusion A i,j ⊂ A i − ,j means regarding the rightmost straight strand together with the original j − i box space in A i,j as a part of the j − i + 1 box space in A i − ,j , which changes the shading type of the box space: x j − i +1 i − If 2 | i : x j − i +1 i − If 2 | i :( λ
4) Jones projections: e k +1 =: d − k e k +2 = d − k +1 e ik = d − k i k e ij,k = d − k i j k e i, ∗ j,k = d − k i k j Remark : See the string diagram calculation of Jones projections in the Temperley-Lieb-Jones algebra.( λ
5) Horizontal (right) conditional expectation E ri,j : A i,j → A i,j − , x ∈ A i,j : E ri,j ( x ) = d − x j − = d − x i j − i − ( λ
6) Vertical (left) conditional expectation E li,j : A i,j → A i +1 ,j , x ∈ A i,j . The vertical (left)conditional expectation is the left conditional expectation acting on the left of the box16pace and then adding one straight strand on the left of the box space, which changes theshading type of box space:If 2 | i : E li,j ( x ) = d − x i j − i − = d − x j − i − i If 2 (cid:45) i : E li,j ( x ) = d − x i j − i − = d − x j − i − i ( λ e j xe j = E ri,j ( x ) e j , for x ∈ A i,j , i + 1 ≤ j : e i xe i = E li,j ( x ) e i , for x ∈ A i,j , i + 1 ≤ j : d − x j − = d − x j − d − x i − j − i − = d − x i − j − i − ( λ
8) Commuting square of conditional expectation: For x ∈ A i,j , E li,j ◦ E ri,j +1 ( x ) = E ri +1 ,j +1 ◦ E li,j +1 ( x ): E li,j ◦ E ri,j +1 ( x ) = x j − i − i = E ri +1 ,j +1 ◦ E li,j +1 ( x )( λ E ri,j +1 ( e j ) = E lj − ,k ( e j ) = d −
1, for j ≥ i + 1 , k ≥ j + 1. d − j − = d − j d − i −
11 1 = d − i +1 ( λ
10) Conditional expectation property E ri,j ( axb ) = aE ri,j ( x ) b , for x ∈ A i,j , a, b ∈ A i,j − ; E li,j ( axb ) = aE li,j ( x ) b , for x ∈ A i,j , a, b ∈ A i +1 ,j . d − axb j − = d − axb j − λ
11) Standard condition: For x ∈ A i,j , y ∈ A k,l with k ≥ j , then we regard x, y as elements in A i,l , xy = yx . x y j k − j l − k = x y j k − j l − k ( λ
12) Pull down condition d E ri,j +1 ( xe j ) e j = xe j , for x ∈ A i,j +1 , j ≥ i ≥ d E li − ,j ( xe i ) e i = xe i , for x ∈ A i − ,j , j ≥ i ≥ x j − j = x j − j y i j − ij − i − = y i j − i j − i − ( λ
13) 2-shift map S i,j : A i,j → A i +2 ,j +2 : For x ∈ A i,j , S i,j ( x ) = i j − i − x j − i − = x j − ii = x j ( λ
14) 2 n -shift map S ( n ) i,j : A i,j → A i +2 n,j +2 n : For x ∈ A i,j , S ( n ) i,j ( x ) = x j − i ni = x j n ( λ
15) Commuting parallelogram:For x ∈ A i,j , S ( n ) i,j − ◦ E r,ki,j ( x ) = E r,ki +2 n,j +2 n ◦ S ( n ) i,j ( x );For x ∈ A i,j , S ( n ) i +1 ,j ◦ E l,ki,j ( x ) = E l,ki +2 n,j +2 n ◦ S ( n ) i,j ( x ). S ( n ) i,j − ◦ E r,ki,j ( x ) = x j − k n = E r,ki +2 n,j +2 n ◦ S ( n ) i,j ( x )18 ( n ) i +1 ,j ◦ E l,ki,j ( x ) = x j − i − kki n = E l,ki +2 n,j +2 n ◦ S ( n ) i,j ( x )( λ
16) Shift property: For x ∈ A i,j , e ij,k x = S ki,j ( x ) e ij,k . x i jk k = x i jk k In this section, we are going to show some important lemmas. One can interpret the stringdiagram computation into algebraic computation by the above dictionary.
Lemma 1.5.1. i j k − = i j i j k = i j k − Lemma 1.5.2.
For (cid:80) nl =1 k p l = (cid:80) mr =1 k q r , k p l , k q r ∈ Z ≥ , and x ∈ A i,j , we have: · · · · · · x i jk p k pn k q k qm = · · · · · · x i jk p k pn k q k qm Proof.
By the above lemma.These two lemmas are used in the proof of Proposition 1.3.5(4).
Lemma 1.5.3. a bi j i j = a bi j i Proof. bi j i j = d − i a b ii j i j = d − i a b ii j i j = d − i a bi ij i = a bi j i Lemma 1.5.4 ( [CHPS18]) . For x ∈ A m,n +2 i + j , m ≤ n + 2 i + j , we have: x n i + j ijii jn = x n i ji + j ii jn i j Proof. x n i + j ijii jn ○ = d − i x n ji + j ii in i j ○ = d − i x n ji + j ii in i j ○ = d − i x n ji + j ii in i j ○ = x n ji + j ii in i j ○ = x n i ji + ji in i j ○ = x n i ji + ji in i j ○ = x n i ji + j ii jn i j List of the formulas used in above equalities:1 ○ : top uses ( λ
8) and bottom uses Jones projection property; 2 ○ : uses ( λ ○ : middle uses ( λ
8) and bottom uses Jones projection property; 4 ○ : uses ( λ ○ : uses ( λ ○ : uses ( λ ○ : uses ( λ λ -lattice to pivotal planar tensor category A planar tensor category A has the following properties.(a) A is a 2-shaded category with objects [ n, +] , [ n, − ], n ∈ Z ≥ , where 1 + := [0 , +] , − :=[0 , − ] are simple and the tensor unit 1 A = 1 + ⊕ − , which means A is 2-shaded.(b) A is a strict tensor category. The tensor product of objects are[ m, ?] ⊗ [ n, ?] [2 i, +] [2 i + 1 , +] [2 i, − ] [2 i + 1 , − ][ n, +] [2 i + n, +] 0 0 [2 i + 1 + n, − ][ n, − ] 0 [2 i + 1 + n, +] [2 i + n, − ] 0There is an involution ( · ) such that [2 i, ± ] = [2 i, ± ], [2 i + 1 , +] = [2 i + 1 , − ] and ( · ) = id.(c) Only A ([ m, +] → [ m ± i, +]) and A ([ m, − ] → [ m ± i, − ]) are non-empty, m, i ∈ Z ≥ ,and A ([ m, +] → [ m, +]) and A ([ m, − ] → [ m, − ]) are finite-dimensional C ∗ -algebras. Thetensor product of morphisms should match the shading types.(d) A is a dagger category. There is a dagger structure † such that [ n, +] † = [ n, +] , [ n, − ] † =[ n, − ] and a anti-linear map A ([ m, ?] → [ n, ?]) → A ([ n, ?] → [ m, ?]) with † = id suchthat morphism ( x ◦ y ) † = y † ◦ x † and ( x ⊗ y ) † = x † ⊗ y † . In fact, A is a C ∗ category,see [CHPS18, § A is rigid. For X ∈ A , there exist(1) ev X : X ⊗ X → ? , where ? = + if X is unshaded on the right, i.e., X = 1 + ⊗ X , ? = − if X is shaded on the right, i.e., X = 1 − ⊗ X ;(2) coev X : 1 ? → X ⊗ X , where ? = + if X is unshaded on the left, ? = − if X is shadedon the left.such that • (id X ⊗ ev X ) ◦ (coev X ⊗ id X ) = id X (ev X ⊗ id X ) ◦ (id X ⊗ coev X ) = id X . • ev X := (coev X ) † and coev X = (coev X ) † .In other word, ( · ) is a unitary dual functor, which will be discussed in § Definition 1.6.2.
We call a planar tensor category A pivotal , if the left trace Tr L and righttrace Tr R defined as follows are faithful normal tracial and equal. For X = [2 k + 1 , +] and f ∈ A ( X → X ), since [2 k + 1 , +] = [2 k + 1 , − ], we defineev X ◦ (id X ⊗ f ) ◦ ev † X =: Tr L ( f )id + coev † X ◦ ( f ⊗ id X ) ◦ coev X =: Tr R ( f )id − We require that Tr R ( f ) = Tr L ( f ). Similar for other three cases [2 k, +] , [2 k, − ] and [2 k + 1 , − ].And there exists a d > [ n, ?] ◦ coev [ n, ?] = d n · ? , ? = + , − . Remark 1.6.3.
The traces Tr L , Tr R are defined in the sense of Definition 1.7.6. Definition 1.6.4.
The 2-shaded Temperley-Lieb-Jones multitensor category
T LJ ( d ) is a pla-nar tensor category with the endomorphism spaces being 2-shaded Temperley-Lieb-Jones alge-bras with modulus d , namely, End([ n, +]) is a 2-shaded Temperley-Lieb-Jones algebra with n points on one side and unshaded on the left; End([ n, − ]) is a 2-shaded Temperley-Lieb algebrawith n points on one side and shaded on the left. Remark 1.6.5.
The morphisms in A are determined by its representation in endomorphismand its domain and range.There is a canonical isomorphism φ : A ([ m, +] , [ m + 2 i, +]) → A ([ m + i, ?] → [ m + i, ?])by Frobenius reciprocity, where ? = + if i is even and ? = − if i is odd. φ : x mm + i i (cid:55)→ x mm + i i φ − : x m + im i (cid:55)→ x m + im i For morphism x ∈ A ([ m, ?] → [ n, ?]), we can write a triple ( φ ( x ); [ m, ?] , [ n, ?]) to represent x , where φ ( x ) ∈ End([ m + n , ?]), which is called the endomorphism representation part of x . In the following context, we simply write x instead of φ ( x ) in the triple ( x ; [ m, ?] , [ n, ?]). λ -lattice to pivotal planar tensor category We regard the elements in algebra A i,j as endomorphisms in the category and the idea inRemark 1.6.5 gives us the way to represent the morphism by using its corresponding endomor-phism, source and target, then we can construct a pivotal planar tensor category from a givenstandard λ -lattice. Definition 1.6.6.
Let A = ( A i,j ) ≤ i ≤ j be a standard λ -lattice. We define a planar tensorcategory A from A as follows.(a) The objects of A are the symbols [ n, +] , [ n, − ] for n ∈ Z ≥ .(b) Given n ≥
0, define A ([ n, +] → [ n, +]) := A ,n and A ([ n, − ] → [ n, − ]) := A ,n +1 . Define1 := [0 , +] ⊕ [0 , − ].(c) The identity morphism in A ([ n, +] → [ n, +]) is 1 A ,n and in A ([ n, − ] → [ n, − ]) is 1 A ,n +1 .22d) For ( x ; [ n, +] , [ n + 2 k, +]) (or ( x ; [ n + 2 k, +] , [ n, +])), we define the dagger structure as( x ; [ n, +] , [ n + 2 k, +]) † := ( x ∗ ; [ n + 2 k, +] , [ n, +]), where x, x ∗ ∈ A ,n + k ; for ( x ; [ n, − ] , [ n +2 k, − ]) (or ( x ; [ n +2 k, − ] , [ n, − ])), we define ( x ; [ n, − ] , [ n +2 k, − ]) † := ( x ∗ ; [ n +2 k, − ] , [ n, − ]),where x, x ∗ ∈ A ,n + k +1 .(e) We define composition in six cases.(C1) ( y ; [ n + 2 i, +] , [ n + 2 i + 2 j, +]) ◦ ( x ; [ n, +] , [ n + 2 i, +]) = ( d i E r,i ,n +2 i + j ( yxe nj,i ); [ n, +] , [ n +2 i + 2 j, +]), where x ∈ A ,n + i , y ∈ A ,n +2 i + j and d i E r,i ,n +2 i + j ( yxe nj,i ) ∈ A ,n + i + j .(C2) ( y ; [ n +2 i +2 j, +] , [ n +2 i, +]) ◦ ( x ; [ n, +] , [ n +2 i +2 j, +]) = ( d i E r,i + j ,n +2 i + j ( yxe n, ∗ j,i ); [ n, +] , [ n +2 i, +]), where x ∈ A ,n + i + j , y ∈ A ,n +2 i + j and d i E r,i + j ,n +2 i + j ( yxe n, ∗ j,i ) ∈ A ,n + i .(C3) ( y ; [ n, +] , [ n +2 i +2 j, +]) ◦ ( x ; [ n +2 i, +] , [ n, +]) = ( d i ye n, ∗ j,i x ; [ n +2 i, +] , [ n +2 i +2 j, +]),where x ∈ A ,n + i , y ∈ A ,n + i + j and d i ye n, ∗ j,i x ∈ A ,n +2 i + j .(C4) ( y ; [ n +2 i, − ] , [ n +2 i +2 j, − ]) ◦ ( x ; [ n, − ] , [ n +2 i, − ]) = ( d i E r,i ,n +2 i + j +1 ( yxe n +1 j,i ); [ n, +] , [ n +2 i +2 j, +]), where x ∈ A ,n + i +1 , y ∈ A ,n +2 i + j +1 and d i E r,i ,n +2 i + j +1 ( yxe n +1 j,i ) ∈ A ,n + i + j +1 .(C5) ( y ; [ n +2 i +2 j, − ] , [ n +2 i, − ]) ◦ ( x ; [ n, − ] , [ n +2 i +2 j, − ]) = ( d i E r,i + j ,n +2 i + j +1 ( yxe n +1 , ∗ j,i ); [ n, − ] , [ n +2 i, − ]), where x ∈ A ,n + i + j +1 , y ∈ A ,n +2 i + j +1 and d i E r,i + j ,n +2 i + j +1 ( yxe n +1 , ∗ j,i ) ∈ A ,n + i +1 .(C6) ( y ; [ n, − ] , [ n +2 i +2 j, − ]) ◦ ( x ; [ n +2 i, − ] , [ n, − ]) = ( d i ye n +1 , ∗ j,i x ; [ n +2 i, − ] , [ n +2 i +2 j, − ]),where x ∈ A ,n + i +1 , y ∈ A ,n + i + j +1 and d i ye n +1 , ∗ j,i x ∈ A ,n +2 i + j +1 .If x ∈ A ([ n + 2 i, − ] → [ n, − ]) and y ∈ A ([ n, − ] → [ n + 2 i + 2 j, − ]), we define y ◦ x := d i ye n +1 , ∗ j,i x ∈ A ,n +2 i + j +1 = A ([ n + 2 i, − ] → [ n + 2 i + 2 j, − ]) . We define the composition x † ◦ y † := ( y ◦ x ) † , which defines composition A ([ n + 2 i + 2 j, − ] → [ n, − ]) ⊗ A ([ n, − ] → [ n + 2 i, − ]) → A ([ n + 2 i + 2 j, − ] → [ n + 2 i, − ]) . It has been proved in [CHPS18, § A is a C ∗ category.Before we define the tensor product of morphisms, we use the string diagrams to explain thecomposition. The box space in the following diagram is always the endomorphism representationof corresponding morphism. yx n i + jinn i ij yx n ij + inn j ii yx n i + j jnn iii (C1) (C2) (C3)The string diagram of case (C4) comes from the string diagram of case (C1) by adding a straightstrand on the leftmost of the diagram and change the shading. In the same way, we obtain (C5)from (C2) and (C6) from (C3).Now we define the tensor product of morphisms.23 efinition 1.6.7. x ⊗ ⊗ y , x, y ∈ Hom( A ):First, we define x ⊗ x x ⊗ j ( x ; [ m, +] , [ m + 2 i, +]) , i ≤ j ( xe mj − i,i ; [ m + j, +] , [ m + 2 i + j, +])( x ; [ m, +] , [ m + 2 i, +]) , i > j ( xe m, ∗ i − j,j ; [ m + j, +] , [ m + 2 i + j, +])( x ; [ m, − ] , [ m + 2 i, − ]) , i ≤ j ( xe m +1 j − i,i ; [ m + j, − ] , [ m + 2 i + j, − ])( x ; [ m, − ] , [ m + 2 i, − ]) , i > j ( xe m +1 , ∗ i − j,j ; [ m + j, − ] , [ m + 2 i + j, − ]) Because of the shading, we define 1 ⊗ y as: y i ⊗ y i +1 ⊗ y ( y ; [ n, +] , [ n ± j, +]) ( S ( i )0 ,n ± j ( y ); [ n + 2 i, +] , [ n + 2 i ± j, +]) 0( y ; [ n, − ] , [ n ± j, − ]) 0 ( S ( i )1 ,n +1 ± j ( y ); [ n + 2 i, − ] , [ n + 2 i ± j, − ]) x n j − iin + i i x n j i − jn i j i ≤ j i > j y n i y n − i Proposition 1.6.8.
For x, y ∈ Hom( A ) , ( x ⊗ ◦ (1 ⊗ y ) = (1 ⊗ y ) ◦ ( x ⊗ .Proof. Here, we check the case ( x ; [ m, +] , [ m + 2 i, +]) and ( y ; [ n, +] , [ n + 2 j, +]), where 2 | m (or ( y ; [ n, − ] , [ n + 2 j, − ]) if 2 (cid:45) m ) and n + j ≥ i . We shall prove that(( x ; [ m, +] , [ m + 2 i, +]) ⊗ (1; [ n + 2 j, +] , [ n + 2 j, +])) ◦ ((1; [ m, +] , [ m, +]) ⊗ ( y ; [ n, +] , [ n + 2 j, +]))=((1; [ m + 2 i, +] , [ m + 2 i, +]) ⊗ ( y ; [ n, +] , [ n + 2 j, +])) ◦ (( x ; [ m, +] , [ m + 2 i, +]) ⊗ (1; [ n, +]; [ n, +]))First, they both in A ([ m + n, +] → [ m + n + 2 i + 2 j, +]).The right hand side:((1; [ m +2 i, +] , [ m +2 i, +]) ⊗ ( y ; [ n, +] , [ n +2 j, +])) ◦ (( x ; [ m, +] , [ m +2 i, +]) ⊗ (1; [ n, +]; [ n, +])): yx m i i n + j − imn i j i = yx m i i n + j − im n + ji The left hand side:(( x ; [ m, +] , [ m +2 i, +]) ⊗ (1; [ n +2 j, +] , [ n +2 j, +])) ◦ ((1; [ m, +] , [ m, +]) ⊗ ( y ; [ n, +] , [ n +2 j, +])):241) If i ≤ j , x y m i i n + j − ii j − im n i i ○ = x y m i i n + j − ii j − im n i i ○ = x y m i i n + j − ii j − im n i i ○ = x y m i i n + j − im n + ji ○ = x y m i i n + j − im n + ji ○ = x y mm i in + j n + j − ii List the formulas used in above equalities:1 ○ : uses ( λ ○ : uses ( λ
11) and ( λ ○ : Jones projection property; 4 ○ : uses ( λ ○ : uses Lemma 1.5.3. 252) If i > j , x y mm i inn + j − ij i − j j ○ = x y mm i i n + j − in j i − j j ○ = x y mm i i n + j − in j i − j j ○ = yx mm i i n + j − in j i − j j ○ = yx mm i i n + j − in j i − j j ○ = x y mm i in + j n + j − ii List of the formulas used in above equalities:1 ○ : uses Lemma 1.5.3; 2 ○ : uses ( λ ○ : uses ( λ
16) and Lemma 1.5.2; 4 ○ : uses ( λ
11) and ( λ ○ : Jones projection property.Therefore, ( x ⊗ ◦ (1 ⊗ y ) = (1 ⊗ y ) ◦ ( x ⊗
1) in this case. The remaining cases are left tothe reader.
Definition 1.6.9 (tensor product of morphisms) . Define x ⊗ y := ( x ⊗ ◦ (1 ⊗ y ).We need to prove that the tensor product defined above is functorial and associative. Proposition 1.6.10.
Tensor product is associative and strict, i.e., for x, y, z ∈ Hom( A ) , ( x ⊗ y ) ⊗ z = x ⊗ ( y ⊗ z ) .Proof. Here,we check the case ( x ; [ m, +] , [ m + 2 i, +]), ( y ; [ n, +] , [ n + 2 j, +]) and ( z ; [ l, − ] , [ l +2 k, − ]), where 2 | m, (cid:45) n and n + j ≥ i , l + k ≥ i + j . Then ( x ⊗ y ) ⊗ z , x ⊗ ( y ⊗ z ) ∈A ([ m + n + l, +] → [ m + n + l + 2 i + 2 j + 2 k, +]).By Proposition 1.6.8, the endomorphism representation parts of x ⊗ y and y ⊗ z are definedin this way: yx m i i n + j − im n + ji zy n j j l + k − jn l + kj x ⊗ y ) ⊗ z : yx z mn l + k i + jm i i n j j l + k − i − j ○ = d − i − j yx z mn l + k i + jm i i n j j l + k − i − ji j ○ = d − i − j yx z mn l + k i + jm i i n j j l + k − i − ji j ○ = yx z mn l + k i + jm i i n j j l + k − i − j List of the formulas used in above equalities:1 ○ : Jones projection property; 2 ○ : uses Lemma 1.5.4;3 ○ : Jones projection property.And x ⊗ ( y ⊗ z ): x y z mn l + k j i m i i n j j l + k − i − j ○ = x y z mn l + k j im i i n j j l + k − i − j ○ = x y z mn l + k j i m i i n j j l + k − i − j ○ = yx z mn l + k i + jm i i n j j l + k − i − j List of the formulas used in above equalities:1 ○ : uses ( λ ○ : uses ( λ ○ : Jones projection property.Therefore, ( x ⊗ y ) ⊗ z = x ⊗ ( y ⊗ z ) in this case. Readers can check the rest of the cases byusing the string diagram dictionary and the lemmas. Proposition 1.6.11.
For x, y ∈ Hom( A ) , ( x ◦ y ) ⊗ x ⊗ ◦ ( y ⊗ and ⊗ ( x ◦ y ) =(1 ⊗ x ) ◦ (1 ⊗ y ) . roof. By our construction, 1 ⊗ ( x ◦ y ) = (1 ⊗ x ) ◦ (1 ⊗ y ) only uses the fact that the shift mapis a ∗ -homomorphism.As for ( x ◦ y ) ⊗ x ⊗ ◦ ( y ⊗ x ; [ m, +] , [ m + 2 i, +]) and ( y ; [ m +2 i ] , [ m + 2 i + 2 j, +]), where n ≥ i + j . Then ( x ◦ y ) ⊗ n , ( x ⊗ n ) ◦ ( y ⊗ n ) ∈ A ([ m + n, +] → [ m + n + 2 i + 2 j, +]). Next, let us compare their endomorphism representation parts.( x ◦ y ) ⊗ n : y x m n − i − j i + jiim i j i i j ○ = d − i − j y x iii jm i j i i jm n − i − j i + j ○ = d − i − j y x m n − i − j i + jm i j i jiii j ○ = y x m n − i − j i + jm i j i jii List of the formulas used in above equalities:1 ○ : Jones projection property; 2 ○ : uses Lemma 1.5.4;3 ○ : Jones projection property.( x ⊗ n ) ◦ ( y ⊗ n ): y x m ↑ n − i − jj i j im i i j j ↓ n − i − jii ○ = y x m ↑ n − i − jj i j im i i j j ↓ n − i − jii ○ = y x m ↑ n − i − jj i j im i i j j ↓ n − i − jii ○ = y x m n − i − j i + jm i j i jii where only the straight strands are allowed in the blank.List of the formulas used in above equalities:1 ○ : uses ( λ ○ : uses ( λ ○ : uses Lemma 1.5.3 and Jones projection property.Therefore, ( x ◦ y ) ⊗ x ⊗ ◦ ( y ⊗
1) in this case. Readers can check the rest of the casesby using the string diagram dictionary and the lemmas.28 roposition 1.6.12.
The tensor product is functorial. For x, y, z, w ∈ Hom( A ) , ( x ◦ y ) ⊗ ( z ◦ w ) = ( x ⊗ z ) ◦ ( y ⊗ w ) .Proof. Based on Proposition 1.6.8 and Proposition 1.6.11, we have( x ◦ y ) ⊗ ( z ◦ w ) = (( x ◦ y ) ⊗ ◦ (1 ⊗ ( z ◦ w ))= (( x ⊗ ◦ ( y ⊗ ◦ ((1 ⊗ z ) ◦ (1 ⊗ w ))= ( x ⊗ ◦ (( y ⊗ ◦ (1 ⊗ z )) ◦ (1 ⊗ w )= ( x ⊗ ◦ ((1 ⊗ z ) ◦ ( y ⊗ ◦ (1 ⊗ w )= (( x ⊗ ◦ (1 ⊗ z )) ◦ (( y ⊗ ◦ (1 ⊗ w ))= ( x ⊗ z ) ◦ ( y ⊗ w ) . Therefore, the tensor product in Definition 1.6.9 is well-defined.Next, we show that A has a pivotal structure. Definition 1.6.13 (ev and coev) . Note that [ n, ± ] ⊗ [ n, ± ] = [2 n ; ± ]; [ n, +] ⊗ [ n, +] = [2 n, +]if 2 | n and [2 n, − ] if 2 (cid:45) n ; [ n, − ] ⊗ [ n, − ] = [2 n, − ] if 2 | n and [2 n, +] if 2 (cid:45) n .Definecoev [ n, +] : 1 + → [2 n, +] = [ n, +] ⊗ [ n, +] as coev [ n, +] = ( d n ; [0 , +] , [2 n, +])ev [ n, +] : [ n, +] ⊗ [ n, +] = [2 n, ?] → ? as ev [ n, +] = ( d n ; [2 n, ?] , [0 , ?]) , ? = + , if 2 | n coev [ n, − ] : 1 − → [2 n, − ] = [ n, − ] ⊗ [ n, − ] as coev [ n, − ] = ( d n ; [0 , − ] , [2 n, − ])ev [ n, − ] : [ n, − ] ⊗ [ n, − ] = [2 n, ?] → ? as ev [ n, − ] = ( d n ; [2 n, ?] , [0 , ?]) , ? = − , if 2 | n Proposition 1.6.14. A is rigid.Proof. First we prove that(id [ n, +] ⊗ ev [ n, +] ) ◦ (coev [ n, +] ⊗ id [ n, +] ) = id [ n, +] . Note that id [ n, +] ⊗ ev [ n, +] = ( S ( n ) ( d n ); [2 n + n, +] , [0 + n, +]) = ( d n ; [3 n, +] , [ n, +]) andcoev [ n, +] ⊗ id [ n, +] = ( d n e n − n ) ,n ; [0 + n, +] , [2 n + n, +]) = ( d n e ,n ; [ n, +] , [3 n, +]).Then by the composition case (C2), where i = 0 , j = n ,(id [ n, +] ⊗ ev [ n, +] ) ◦ (coev [ n, +] ⊗ id [ n, +] ) = ( d n ; [3 n, +] , [ n, +]) ◦ ( d n e ,n ; [ n, +] , [3 n, +])= ( d E r, n ,n +2 n ( d n e ,n e n, ∗ j,i ); [ n, +] , [ n + 2 i, +])= ( d n E r,n , n ( e ,n ); [ n, +] , [ n, +])= (1; [ n, +] , [ n, +]) = id [ n, +] . The other three cases are left to the reader. Therefore, A is rigid. Proposition 1.6.15. A has a pivotal structure. roof. First, we prove that the right trace Tr R is a normal faithful trace. Let X = [ n, +]. Given( f ; [ n, +] , [ n, +]), f ⊗ id [ n, +] = ( f ; [2 n, +] , [2 n, +]), thenTr R ( f ) = coev † [ n, +] ◦ ( f ⊗ id [ n, +] ) ◦ coev [ n, +] = ( d n ; [2 n, +] , [0 , +]) ◦ ( f ; [2 n, +] , [2 n, +]) ◦ ( d n ; [0 , +] , [2 n, +])= ( d n ; [2 n, +] , [0 , +]) ◦ ( d n E r,n , n ( f · d n e ,n ); [0 , +] , [2 n, +])= ( d n ; [2 n, +] , [0 , +]) ◦ ( f ; [0 , +] , [2 n, +])= ( d E r,n ,n ( f e , ∗ n, ); [0 , +]; [0 , +])= (tr( f ); [0 , +] , [0 , +]) . The third equality uses (C1), where n = 0 , i = n, j = 0; the forth equality uses ( λ n = i = 0 , j = n .The case X = [ n, − ] is left to the reader.Next, we prove that the left trace Tr L is a normal faithful trace. Let X = [2 n, +]. Given( f ; [2 n, +] , [2 n, +]), id [2 n, +] ⊗ f = ( S ( n )0 , n ( f ); [4 n, +] , [4 n, +]), thenTr L ( f ) = ev [2 n, +] ◦ (id [2 n, +] ⊗ f ) ◦ ev † [2 n, +] = ( d n ; [4 n, +] , [0 , +]) ◦ ( S ( n )0 , n ( f ); [4 n, +] , [4 n, +]) ◦ ( d n ; [0 , +] , [4 n, +])= ( d n ; [4 n, +] , [0 , +]) ◦ ( d n E r, n , n ( S ( n )0 , n ( f ) · d n e , n ); [0 , +] , [4 n, +])= ( d n · d E r, n , n ( E r, n , n ( S ( n )0 , n ( f ) e , n ) e , ∗ , n ); [0 , +] , [0 , +])= (tr( f ); [0 , +] , [0 , +]) . The last equality: since e , ∗ , n = 1 and E r, n , n ◦ E r, n , n = tr = E r, n n, n ◦ E l, n , n , S ( n )0 , n ( f ) ∈ A n, n and S ( n )0 , n is trace-preserving, then d n · d E r, n , n ( E r, n , n ( S ( n )0 , n ( f ) e , n ) e , ∗ , n ) = d n tr( S ( n )0 , n ( f ) e , n )= d n E r, n n, n ◦ E l, n , n ( S ( n )0 , n ( f ) e , n ) (by ( λ E r, n n, n ( S ( n )0 , n ( f )) (by Prop 1.3.5(2))= E r, n , n ( f ) = tr( f ) . The cases X = [2 n + 1 , +] , [ n, − ] are left to the reader.Therefore, Tr R = Tr L are the trace, so A has a pivotal structure.Moreover, by the composition case (C2), where i = n = 0 , j = n ,ev [ n, +] ◦ coev [ n, +] = ( d n ; [2 n, +] , [0 , +]) ◦ ( d n ; [0 , +] , [2 n, +])= ( d E r,n , n ( d n e , ∗ n, ); [0 , +] , [0 , +])= ( d n ; [0 , +] , [0 , +]) = d n · + . Similarly, ev [ n, − ] ◦ coev [ n, − ] = d n · − .Combining above propositions, A constructed from a standard λ -lattice is a pivotal planartensor category. 30 .7 From 2-shaded rigid C ∗ multitensor category to standard λ -lattice In this section, we show the relation between the 2-shaded rigid C ∗ multitensor categoryand planar tensor category, and give the construction from the category to standard λ -lattice. C ∗ multitensor category In this subsection, we are going to review the unitary dual functors in a rigid C ∗ (multi)tensorcategory C [Pe18]. Definition 1.7.1.
Recall that every object c ∈ C is dualizable , i.e., there is an object c ∈ C together with morphisms ev c ∈ C ( c ⊗ c → C ) and coev c ∈ C (1 C → c ⊗ c ) satisfying the zig-zagcondition: (id c ⊗ ev c ) ◦ (coev c ⊗ id c ) = id c (ev c ⊗ id c ) ◦ (id c ⊗ coev c ) = id c . We also require that every object c ∈ C admits a predual object c such that ( c ) ∼ = c . Definition 1.7.2.
A choice of dual for every object in C assembles into a dual functor ( · ) : C → C mop , which is a tensor functor with a canonical tensorator ν a,b . To be precise, for amorphism f ∈ C ( a → b ), define f := (ev b ⊗ id a ) ◦ (id b ⊗ f ⊗ id a ) ◦ (id b ⊗ coev a ) : b → a.f := fb aba The tensorator ν a,b : a ⊗ b → b ⊗ a is defined as ν a,b := (ev a ⊗ id b ⊗ a ) ◦ (id a ⊗ ev b ⊗ id a ⊗ id b ⊗ a ) ◦ (id a ⊗ b ⊗ coev b ⊗ a ) . Note that ν is completely determined by ev and coev. Proposition 1.7.3.
Any two dual functors ( · ) and ( · ) are equivalent up to a unique naturalisomorphism. Define ζ : ( · ) → ( · ) as follows: for c ∈ C , ζ c := (ev c ⊗ id c ) ◦ (id c ⊗ coev c ) .ζ c = coev c ev c cc c Then we have ζ ( f ) = ζ a ◦ f ◦ ζ − b = ζ ( f ) for all f ∈ C ( a → b ) . efinition 1.7.4. [EGNO15] A pivotal structure on a rigid monoidal category C is a pair(( · ) , ϕ ), where ( · ) is a dual functor and ϕ : id ⇒ ( · ) is a monoidal natural isomorphism. To beprecise, for all a, b ∈ C , the following diagram commutes: a ⊗ b ϕ a ⊗ ϕ b (cid:47) (cid:47) ϕ a ⊗ b (cid:15) (cid:15) a ⊗ b ν b,a (cid:15) (cid:15) a ⊗ b ν a,b (cid:47) (cid:47) b ⊗ a Definition 1.7.5 (Pivotal trace) . Let 1 C = (cid:76) ri =1 i be a decomposition into simples. For c ∈ C and f ∈ C ( c → c ), define the left/right pivotal traces tr ϕL and tr ϕR : C ( c → c ) → C (1 C → C ) ∼ = M r ( C ) by tr ϕL ( f ) := ev c ◦ (id c ⊗ f ) ◦ (id c ⊗ ϕ − c ) ◦ coev c tr ϕR ( f ) := ev c ◦ ( ϕ c ⊗ id c ) ◦ ( f ⊗ id c ) ◦ coev c . tr ϕL ( f ) = f ϕ − c cccc tr ϕR ( f ) = fϕ c ccc c The traces are tracial and non-degenerate.
Definition 1.7.6.
Let p i ∈ C (1 C → C ) be the projection onto 1 i , i = 1 , , · · · , r . We define the M r ( C )-valued traces Tr ϕL and Tr ϕR by the formulas:(Tr ϕL ( f )) i,j id j := tr ϕL ( p i ⊗ f ⊗ p j )(Tr ϕR ( f )) i,j id i := tr ϕR ( p i ⊗ f ⊗ p j ) . Note that Tr ϕL and Tr ϕR are tracial, and Tr ϕL ( f ) = Tr ϕR ( f ) T for all f ∈ C ( c → c ).We call the pivotal structure (( · ) , ϕ ) spherical , if Tr ϕL ( f ) = Tr ϕR ( f ), for all c ∈ C , f ∈C ( c → c ). Definition 1.7.7.
For each c ∈ C , define Dim ϕL , Dim ϕR ∈ M r ( C ) byDim ϕL ( c ) := Tr ϕL (id c ) Dim ϕR ( c ) := Tr ϕR (id c ) . If c is simple, then Dim ϕL ( c ) , Dim ϕR ( c ) have only one non-zero entry, which we denotedim ϕL ( c ) , dim ϕR ( c ) respectively.If the pivotal structure (( · ) , ϕ ) is spherical, Dim ϕL ( c ) = Dim ϕR ( c ) := Dim( c ) for all object c . Definition 1.7.8. A dagger structure on a C -linear category is a collection of anti-linearmaps † : C ( c → d ) → C ( d → c ) for all c, d ∈ C such that ( f ◦ g ) † = g † ◦ f † and ( f † ) † = f . Amorphism f : C ( a → b ) is called unitary if f † = f − .A dagger (multi)tensor category is a (multi)tensor category equipped with a dagger structureso that ( f ⊗ g ) † = f † ⊗ g † for all morphisms f, g , and all associator and unitors are unitary. Definition 1.7.9.
A functor between dagger categories F : C → D is called a dagger functor if F ( f † ) = F ( f ) † for all f ∈ Hom( C ). 32 efinition 1.7.10 (Rigid C ∗ (multi)tensor category) . A C ∗ category is a dagger category whichis Cauchy complete and each endomorphism algebra is a C ∗ -algebra, where the dagger structureis compatible with the ∗ -structure.A C ∗ (multi)tensor category is a dagger (multi)tensor category whose underlying daggercategory is C ∗ .A rigid C ∗ (multi)tensor category is a C ∗ (multi)tensor category equipped with a dualfunctor. It is known that a rigid C ∗ multitensor category is Cauchy complete if and only if it issemisimple [LR96]. Proposition 1.7.11 (Unitary dual functor) . Fix a dual functor ( · ) on a rigid C ∗ (multi)tensorcategory C , the followings are equivalent:(1) ( · ) is a unitary dual functor , i.e., for all a, b ∈ C , f ∈ C ( a → b ) , the tensorator ν a,b isunitary and f † = f † .(2) Defining ϕ c := (coev † c ⊗ id c ) ◦ (id c ⊗ coev c ) is a pivotal structure ϕ : id ⇒ ( · ) .Proof. [Se11], see also [Pe18, Prop. 3.9]. Definition 1.7.12.
Two unitary dual functors are called unitary equivalent , if the canonicalnatural transformation ζ from Proposition 1.7.3 is unitary, i.e., ζ c is unitary for all c ∈ C . Proposition 1.7.13.
For a unitary dual functor ( · ) , the left/right pivotal traces have alternateformulas: tr ϕL ( f ) = ev c ◦ (id c ⊗ f ) ◦ ev † c tr ϕR ( f ) = coev † c ◦ ( f ⊗ id c ) ◦ coev c . Theorem 1.7.14 ( [BDH14] [Pe18, Prop. 3.24]) . For a rigid C ∗ (multi)tensor category C ,there exists a unique unitary dual functor whose induced pivotal structure is spherical up tounitary equivalence. In other words, the pivotal structure can be trivial, so that ev c = coev † c and coev c = ev † c for all c ∈ C . C ∗ multitensor category with a choice of generator and planartensor category Let A be a 2-shaded rigid C ∗ multitensor category together with 1 = 1 + ⊕ − , where 1 + , − are simple, and a generator X = 1 + ⊗ X ⊗ − . Here, the generating means for any simple object P , it is a direct summand of X alt ⊗ n or X alt ⊗ n (defined below) for some n ∈ Z ≥ .Let ( · ) be a unitary dual functor that induced a spherical pivotal structure ϕ . Note thatonly (+ , − ) entry of Dim( X ) is non-zero and we denote this number as d X to be the modulusof category C . Construction 1.7.15.
We construct a planar tensor category A from ( A , X ). By MacLane’scoherence theorem, A is unitary equivalent to a strict tensor category with the above propertiesand the dual functor is strict, WLOG, we also denote it as A . Construct the pivotal planartensor category A as follows:(a) Objects: Define [0 , +] := 1 + , [0 , − ] := 1 − , and[ n, +] := [ n − , +] ⊗ X ? = ( · · · ( X ⊗ X ) ⊗ X ) ⊗ · · · ) ⊗ X ? (cid:124) (cid:123)(cid:122) (cid:125) n tensorands =: X alt ⊗ n , X ? = X if n is even and X if n is odd, and[ n, − ] := [ n − , − ] ⊗ X ? = ( · · · ( X ⊗ X ) ⊗ X ) ⊗ · · · ) ⊗ X ? (cid:124) (cid:123)(cid:122) (cid:125) n tensorands =: X alt ⊗ n , where X ? = X if n is even and X if n is odd, for n ∈ Z ≥ .(b) Morphisms: A is the full subcategory of A with above objects.(c) Duality: The dual functor is unitary as a dual functor on the subcategory, which alsoinduces a spherical pivotal structure on the subcategory.Given A to be a pivotal planar tensor category, then its Cauchy completion (cid:99) A is a Cauchycompleted 2-shaded rigid C ∗ multitensor category with a generator [1 , +] and a canonical unitarydual functor ( · ) . Proposition 1.7.16.
Suppose A is a pivotal planar tensor category constructed from ( A , X ) ,then there is a unitary equivalence between ( (cid:99) A , [1 , +]) and the Cauchy completion of ( A , X ) with respect to their unitary dual functors. Remark 1.7.17.
Suppose A , B are two 2-shaded rigid C ∗ multitensor categories with generator X and Y respectively and A , B are corresponding pivotal planar tensor categories. Then A and B are unitary equivalent if and only if the Cauchy completions of A and B are unitaryequivalent which maps generator to generator. Remark 1.7.18.
The planar tensor category A is not Cauchy complete, i.e., additive completeand idempotent complete. In fact, as for skeletalness, strictness and Cauchy complete, mosttensor categories can require at most two of them. Vec( G ) is an exception. λ -latticeConstruction 1.7.19. Let A be a pivotal planar tensor category with modulus d . Define A ,j = End([ j, +]), A ,j = id [1 , +] ⊗ End([ j − , − ]), j ∈ Z ≥ , so that A , = A , = C . Ingeneral, for i ≤ j , define A i,j = (cid:40) id [ i, +] ⊗ End([ j − i, +]) 2 | i id [ i, +] ⊗ End([ j − i, − ]) 2 (cid:45) i. Then we check A = ( A i,j ) i,j ≥ to be a standard λ -lattice.(a) The vertical inclusion A i +1 ,j ⊂ A i,j is clear. The right inclusion: the right inclusion send x ∈ A i,j to x ⊗ id [1 , ?] ∈ A i,j +1 , where ? = + if 2 | j and ? = − if 2 (cid:45) j .(b) Horizontal conditional expectation: Define E ri,j : A i,j → A i,j − by E ri, k ( x ) = d − (id [2 k − , +] ⊗ ev [1 , +] ) ◦ ( x ⊗ [1 , +]) ◦ (id [2 k − , +] ⊗ coev [1 , +] ) E ri, k +1 ( x ) = d − (id [2 k, +] ⊗ ev [1 , − ] ) ◦ ( x ⊗ [1 , − ]) ◦ (id [2 k, +] ⊗ coev [1 , − ] ) . (c) Vertical conditional expectation: Define E li,j : A i,j → A i +1 ,j by E l k,j = d − (id [2 k +2 , +] ⊗ ev [1 , +] ) ◦ (id [2 , +] ⊗ x ) ◦ (id [2 k +2 , +] ⊗ coev [1 , +] ) E l k +1 ,j = d − (id [2 k +3 , +] ⊗ ev [1 , − ] ) ◦ (id [2 , +] ⊗ x ) ◦ (id [2 k +3 , +] ⊗ coev [1 , − ] ) . n -th Jones projection is defined as e k +1 = d − · id [2 k, +] ⊗ (coev [1 , +] ◦ ev [1 , +] ) ∈ A i, k +2 e k +2 = d − · id [2 k +1 , +] ⊗ (coev [1 , − ] ◦ ev [1 , − ] ) ∈ A i, k +3 . The check that A = ( A i,j ) j ≥ i ≥ satisfies Definition 1.2.1(a), (b), (c) and standard condition isleft to the reader. In particular, e n e n ± e n = d − e n , E ri,j +1 ( e j ) = E lj − ,k ( e j ) = d − A . Remark 1.7.20.
The idea of drawing the string diagram explanation in § A , X ) with A a 2-shaded rigid C ∗ multitensor category and X a generator induces the class of isomorphic pivotal planar ten-sor categories; in § λ -lattices.Combining above discussion, we can deduce the equivalence between standard λ -lattice A and pair 2-shaded rigid C ∗ multitensor category with a generator ( A , X ). Theorem 1.7.21.
There is a bijective correspondence between equivalence classes of the fol-lowing: (cid:26)
Standard λ -lattices A = ( A i,j ) ≤ i ≤ j (cid:27) ∼ = Pairs ( A , X ) with A a 2-shaded rigid C ∗ multitensorcategory with a generator X , i.e., 1 A = 1 + ⊕ − , 1 + , − are simple and X = 1 + ⊗ X ⊗ − Equivalence on the left hand side is unital ∗ -isomorphism of standard λ -lattices; equivalenceon the right hand side is unitary equivalence between their Cauchy completions which mapsgenerator to generator. λ -lattice and module categories Now we move to the module case. One motivation that regards a Markov tower as a rightmodule over a standard λ -lattice is to answer the question in [CHPS18, Rmk. 3.34]. λ -lattice Definition 2.1.1. M ⊂ M ⊂ M ⊂ · · · ⊂ M n ⊂ · · ·∪ ∪ ∪ ∪ A , ⊂ A , ⊂ A , ⊂ · · · ⊂ A ,n ⊂ · · ·∪ ∪ ∪ A , ⊂ A , ⊂ · · · ⊂ A ,n ⊂ · · · Let A = ( A i,j ) ≤ i ≤ j< ∞ be a standard λ -lattice with Jones projection { e i } i ≥ and compatibleconditional expectations. Let M = ( M n , e n ) n ≥ be a Markov tower with conditional expectation E i : M i → M i − , i ≥
1. ( M and A share the same Jones projections) We call a Markov tower M a standard right A − module , if it satisfies the following three conditions.35a) A ,i ⊂ M i is a unital inclusion, i = 0 , , , · · · .(b) E i | A ,i = E r ,i , i = 1 , , · · · .(c) (standard condition) [ M i , A k,l ] = 0 for i ≤ k ≤ l .In the rest of this Chapter, we only consider the Markov tower with dim( M ) = 1 unlessstated. We now introduce the diagrammatic explanation of the element, conditional expectation,Jones projection and their relations in a Markov tower with the same spirit in § x ∈ M n : x n := x n (MT2) Vertical inclusion x ∈ A ,n ⊂ M n : x n (MT3) Horizontal inclusion x ∈ M n ⊂ M n +1 : x n (MT4) Jones projections: e i +1 = d − i ∈ M i +2 e i +2 = d − i +1 ∈ M i +3 (MT5) Conditional expectation E n : M n → M n − and E n | A ,n = E r ,n : E n ( x ) = d − x n − , x ∈ M n E n ( x ) = E r ,n ( x ) = d − x n − , x ∈ A ,n (MT6) Pull down condition: For x ∈ M n +1 , xe n = dE n +1 ( xe n ) e n . x j − j = x j − j f ∈ M i , x ∈ A k,l with k ≥ i , then we regard φ, x as elementsin M l , f x = xf . f x i k − i l − k = f x i k − i l − k Let A be a planar tensor category defined in § M be an inde-composable semisimple C ∗ right A − module category with following properties:(a) Object: The objects of M are [ n ] = [ n ] M , n ∈ Z ≥ , where [0] is simple.(b) The tensor product of objects are[ m ] M (cid:1) [ n, +] A = [ m + n ] M , [ m ] M (cid:1) [ n, − ] A = 0 . (c) Only M ([ n ] → [ n ± i ]) is non-empty, n, i ∈ Z ≥ . The module product of morphism inHom( M ) and Hom( A ) should match the shading types.(d) M is a strict right A − module category, i.e., the module associator is identity. For x , x ∈A and f ∈ M , ( f (cid:1) x ) (cid:1) x = f (cid:1) ( x ⊗ x ) . (e) M is a C ∗ category with a natural dagger structure such that (cid:1) is a dagger functor, i.e.,for x ∈ Hom( A ) and f ∈ Hom( M ),( f (cid:1) x ) † = f † (cid:1) x † . Such module category is called a planar module category . Remark 2.3.2.
Similar to Remark 1.6.5, the morphisms in M is determined by its represen-tation as an endomorphism and its domain and range.There is a canonical isomorphism φ : M ([ m ] → [ m + 2 i ]) → M ([ m + i ] → [ m + i ]) byusing the rigid structure on A . φ : x mm + i i (cid:55)→ x mm + i i φ − : x m + im i (cid:55)→ x m + im i For morphism x ∈ M ([ m ] , [ n ]), we can write a triple ( φ ( x ); [ m ] , [ n ]) to represent x , where φ ( x ) ∈ End([ m + n ]), which is called the endormophism representation part of x . In thefollowing context, we simply write x instead of φ ( x ) in the triple ( x ; [ m ] , [ n ]). Define the multi-step conditional expectation E mn = E n − m +1 ◦ · · · ◦ E n , for m ≤ n . Similarto Definition 1.6.6, we may regard the elements in M n as endomorphisms in the category, wecan construct a planar module category from a given Markov tower as a standard module overa standard λ -lattice. 37 efinition 2.3.3. Let M = ( M n ) n ≥ be a Markov tower as a standard right module overstandard λ -lattice A = ( A i,j ) with dim( M ) = 1. We define a planar module category M from M as follows.(a) The objects of M are the symbols [ n ] for n ∈ Z ≥ .(b) Given n ≥
0, define M ([ n ] → [ n ]) := M n .(c) The identity morphism in M ([ n ] → [ n ]) is 1 M n .(d) For ( f ; [ m ] , [ n ]) with 2 | m + n , we define ( f ; [ m ] , [ n ]) † := ( f ∗ ; [ n ] , [ m ]), where f, f ∗ ∈ M m + n .(e) We define composition in three cases.(C1) ( g ; [ n + 2 i ] , [ n + 2 i + 2 j ]) ◦ ( f ; [ n ] , [ n + 2 i ]) = ( d i E in +2 i + j ( gf e nj,i ); [ n ] , [ n + 2 i + 2 j ]), where f ∈ M n + i , g ∈ M n +2 i + j and d i E in +2 i + j ( gf e nj,i ) ∈ M n + i + j .(C2) ( g ; [ n + 2 i + 2 j ] , [ n + 2 i ]) ◦ ( f ; [ n ] , [ n + 2 i + 2 j ]) = ( d i E i + jn +2 i + j ( gf e n, ∗ j,i ); [ n ] , [ n + 2 i ]),where f ∈ M n + i + j , g ∈ M n +2 i + j and d i E i + jn +2 i + j ( gf e n, ∗ j,i ) ∈ M n + i .(C3) ( g ; [ n ] , [ n + 2 i + 2 j ]) ◦ ( f ; [ n + 2 i ] , [ n ]) = ( d i ge n, ∗ j,i f ; [ n + 2 i ] , [ n + 2 i + 2 j ]), where f ∈ M n + i , g ∈ M n + i + j and d i ge n, ∗ j,i f ∈ M n +2 i + j .For the other cases, we can use the dagger structure f † ◦ g † := ( g ◦ f ) † to define.Similarly, the composition and the dagger structure are well defined, and M is C ∗ [CHPS18, § gf n i + jinn i ij gf n ij + inn j ii gf n i + j jnn iii (C1) (C2) (C3) Remark 2.3.4.
Readers can observe the similarity between the diagrammatic explanation ofelements in M n and A i,n , difference only appears on the leftmost. Moreover, the similar versionof Lemma 1.5.3 and Lemma 1.5.4 is also true for Markov tower case.Now we define the module action of morphisms. Definition 2.3.5. f (cid:1) (cid:1) x , f ∈ Hom( M ) and x ∈ Hom( A ). The idea is the same asin Definition 1.6.7.First, we define f (cid:1) f f (cid:1) j ( f ; [ m ] , [ m + 2 i ]) , i ≤ j ( f e mj − i,i ; [ m + j ] , [ m + 2 i + j ])( f ; [ m ] , [ m + 2 i ]) , i > j ( f e m, ∗ i − j,j ; [ m + j ] , [ m + 2 i + j ]) The definition of 1 (cid:1) x will be the same as 1 ⊗ x by using the 2-shift maps in Definition1.6.7. 38 n j − iin + i i f n j i − jn i j i ≤ j i > j The proof of following propositions are the same as in Proposition 1.6.8, 1.6.10 and 1.6.11.
Proposition 2.3.6.
For f ∈ Hom( M ) , x ∈ Hom( A ) , ( f (cid:1) ◦ (1 (cid:1) x ) = (1 (cid:1) x ) ◦ ( f (cid:1) . Definition 2.3.7.
Define f (cid:1) x := ( f (cid:1) ◦ (1 (cid:1) x ).The following propositions guarantee the module action defined above is well-defined. Proposition 2.3.8.
For f ∈ Hom( M ) , x, y ∈ Hom( A ) , ( f (cid:1) x ) (cid:1) y = f (cid:1) ( x ⊗ y ) . Proposition 2.3.9.
For f, g ∈ Hom( M ) , ( f ◦ g ) (cid:1) f (cid:1) ◦ ( g (cid:1) and (cid:1) ( x ⊗ y ) =(1 (cid:1) x ) ◦ (1 (cid:1) y ) . C ∗ A− module categories and planar A − modulecategories C ∗ A− module category Let A be a 2-shaded rigid C ∗ multitensor category with a generator X = 1 + ⊗ X ⊗ − witha canonical unitary dual functor ( · ). Let M be a Cauchy complete indecomposable semisimpleC ∗ A− module category. Note that there is a natural dagger structure on M , and the moduleaction (cid:1) is a dagger functor, namely, for morphism f ∈ Hom( M ) and x ∈ Hom( A ),( f (cid:1) x ) † = f † (cid:1) x † . We call a module category M indecomposable if for any two simple objects P, Q ∈ M , Q is a direct summand of P (cid:1) X alt ⊗ n if P = P (cid:1) + ( P (cid:1) X alt ⊗ n if P = P (cid:1) − ) for some n ∈ Z ≥ . Construction 2.4.1.
Let A be a planar tensor category obtained from ( A , X ) via the con-struction in § M A is unitary equivalent to a strict one,i.e., M and A are strict and the right module associator is trivial. Then M is also a strict right A − module category.We construct the planar A − module category M as follows:(a) Objects: Pick a simple object Z = Z (cid:1) + ∈ M , define [0] := Z , and[ n + 1] := [ n ] (cid:1) [1 , ?] , where [1 , ?] = [1 , +] if 2 | n and [1 , ?] = [1 , − ] if 2 (cid:45) n .(b) Morphisms: M is a full subcategory of M with above objects.Given M to be a planar A − module category, then its Cauchy completion (cid:99) M is an (cid:99) A − module, compatible with the dagger structure. The proof is left to the reader as an exercise. Remark 2.4.2.
Suppose M is a planar A − module category constructed from ( M , Z ) over( A , X ), then there is a unitary equivalence between M as A− module and (cid:100) M as (cid:99) A − module,which sends base object to base object. 39 .4.2 From planar module category to Markov tower as a standard module overa standard λ -latticeConstruction 2.4.3. Let M be a planar A − module category with modulus d and A is astandard λ -lattice constructed from A as in § M j = End([ j ]), j ∈ Z ≥ . Then wecheck M = ( M j ) j ≥ to be a Markov tower as a standard A − module.(a) The horizontal inclusion M j ⊂ M j +1 sends x ∈ M j to x (cid:1) id [1 , ?] ∈ M j +1 , where ? = + if2 | j and ? = − if 2 (cid:45) j . The vertical inclusion A ,j ⊂ M j sends x ∈ A ,j to id [0] (cid:1) x ∈ M j .(b) Conditional expectation: Define E Mj : M j → M j − by E M k ( x ) = d − (id [2 k − ⊗ ev [1 , +] ) ◦ ( x (cid:1) [1 , +]) ◦ (id [2 k − ⊗ coev [1 , +] ) ,E M k +1 ( x ) = d − (id [2 k ] ⊗ ev [1 , − ] ) ◦ ( x (cid:1) [1 , − ]) ◦ (id [2 k ] ⊗ coev [1 , − ] ) . (c) Jones projections: the same Jones projections in A and identify e n ∈ A ,n +1 with 1 (cid:1) e n ∈ M n +1 .The check that M is a Markov tower and a standard A − module is left to the reader. Inparticular, we have E n +1 ( e n ) = d − · M , Z ) with M anindecomposable right A− module category and Z a simple base point induces the equivalentclass of planar module categories; according to § λ -lattices.Combining above discussion, we can deduce the equivalence between ( M , Z ) as A− modulecategory and Markov tower M as standard A − module. Theorem 2.4.4.
There is a bijective correspondence between equivalence classes of the follow-ing:
Traceless Markov tower M =( M i ) i ≥ with dim( M ) = 1 asa standard right module over astandard λ -lattice A ∼ = Pairs ( M , Z ) with M an indecomposablesemisimple C ∗ right A− module categorytogether with a choice of simple object Z = Z (cid:1) + A Equivalence on the left hand side is ∗ -isomorphism of traceless Markov towers as standard A − modules; equivalence on the right hand side is unitary A− module equivalence on their Cauchycompletions which maps the simple base object to simple base object. Corollary 2.4.5.
Any Markov tower M with modulus d and dim( M ) = 1 is naturally astandard right TLJ( d ) − module, where TLJ( d ) is a Temperley-Lieb-Jones standard λ -lattice asin Example 1.2.8, which corresponds to an indecomposable semisimple C ∗ right T LJ ( d ) − modulecategory with a simple base object. Remark 2.4.6.
The tracial case will be discussed in § λ -lattices and bimodule categories In this chapter, we extend the discussion into the bimodule case. We give the notionMarkov lattices and Markov lattices as bimodule over two standard λ -lattices, by using thesimilar method, which correspond to bimodule categories.40 .1 Markov lattice and basic properties Definition 3.1.1 (Markov lattice) . A tuple M = ( M i,j , E M,li,j , E
M,ri,j , e i , f j ) i,j ≥ is called aMarkov lattice if the following conditions hold. M i +1 ,j ⊂ M i +1 ,j +1 ∪ ∪ M i,j ⊂ M i,j +1 (a) M i,j ⊂ M i,j +1 and M i,j ⊂ M i +1 ,j are unital inclusions.(b) M − ,j = ( M i,j , E M,li,j , e i ) i ≥ are Markov towers with the same modulus d and e i ∈ M i +1 ,j for all j ; M i, − = ( M i,j , E M,ri,j , f j ) j ≥ are Markov towers with the same modulus d and f j ∈ M i,j +1 for all i . We call M of modulus ( d , d ).(c) The commuting square condition: M i +1 ,jE M,li +1 ,j (cid:15) (cid:15) M i +1 ,j +1 E M,ri +1 ,j +1 (cid:111) (cid:111) E M,li +1 ,j +1 (cid:15) (cid:15) M i,j M i,j +1 E M,ri,j +1 (cid:111) (cid:111) is a commuting square, i.e., E M,ri,j +1 ◦ E M,li,j = E M,li,j +1 ◦ E M,ri +1 ,j +1 .Here are some properties of Markov lattice. Proposition 3.1.2.
Let M = ( M i,j , E M,li,j , E
M,ri,j , e i , f j ) i,j ≥ be a Markov lattice.(1) E M,ri +1 ,j +1 ( e i ) = e i and E M,li +1 ,j +1 ( f j ) = f j for each i, j = 1 , , · · · .(2) [ f j , e i ] = 0 for each i, j = 1 , , , · · · .Proof. (1) Note that e i ∈ M i +1 ,j ⊂ M i +1 ,j +1 and E ri +1 ,j +1 : M i +1 ,j +1 → M i +1 ,j is a conditional expec-tation, we have E ri +1 ,j +1 ( e i ) = e i . Similarly, E M,li +1 ,j +1 ( f j ) = f j .(2) By Proposition 1.1.4(1). Remark 3.1.3.
If there is a faithful normal trace on (cid:83) i,j ≥ M i,j and E M,ri,j , E
M,li,j are the canon-ical faithful normal trace-preserving conditional expectations for i, j = 0 , , , · · · , then M iscalled a tracial Markov lattice .In the rest of this Chapter, we only consider the traceless Markov lattice with dim( M , ) = 1unless stated. 41 .2 Markov lattice as a standard bimodule over two standard λ -lattices Definition 3.2.1 (Markov lattice as a standard bimodule over two standard λ -lattices) . ∪ ∪ ∪ ∪ ∪ ∪ A , ⊂ A , ⊂ M , ⊂ M , ⊂ M , ⊂ M , ⊂∪ ∪ ∪ ∪ ∪ ∪ A , ⊂ A , ⊂ M , ⊂ M , ⊂ M , ⊂ M , ⊂∪ ∪ ∪ ∪ ∪ ∪ A , ⊂ A , ⊂ M , ⊂ M , ⊂ M , ⊂ M , ⊂∪ ∪ ∪ ∪ ∪ A , ⊂ M , ⊂ M , ⊂ M , ⊂ M , ⊂∪ ∪ ∪ ∪ B , ⊂ B , ⊂ B , ⊂ B , ⊂∪ ∪ ∪ B , ⊂ B , ⊂ B , ⊂ Let A op = ( A i,j ) ≤ j ≤ i< ∞ B = ( B i,j ) ≤ i ≤ j< ∞ be two standard λ -lattices with Jones pro-jection e i ∈ A i +1 ,j , f j ∈ B i,j +1 respectively and compatible conditional expectations. Here, A and M share the same Jones projections e i ; B and M share the same Jones projections f j .(Warning: here we use the opposite λ -lattice A op , see Definition 1.2.7)Let M = ( M i,j , e i , f j ) i,j ≥ be a Markov lattice with conditional expectation E M,r , E
M,l . Wecall a Markov lattice M a standard A − B bimodule where the left action is the opposite action,if it satisfies the following three conditions.(a) A i, ⊂ M i, , B ,j ⊂ M ,j are unital inclusions, i, j = 0 , , , · · · .(b) E M,li, | A i, = E A,li, , E M,r ,j | B ,j = E B,r ,j i = 1 , , · · · .(c) (standard condition) [ M i,j , A p,q ] = 0 for i ≤ q ≤ p ; [ M i,j , B k,l ] = 0, for j ≤ k ≤ l . Remark 3.2.2.
The standard condition implies that [ A p,q , B k,l ] = 0 for all q ≤ p, k ≤ l since A p,q ⊂ A p, ⊂ M p, and B k,l ⊂ B ,l ⊂ M ,l . Moreover, E M,ri,j | A k,l = id, E M,li,j | B k,l = id. Inparticular, we have E M,ri,j ( e k ) = e k , E M,li,j ( f l ) = f l for Jones projections. We now provide the string diagram explanation of the element, conditional expectation,Jones projection and their relations in a Markov lattice with the same spirit in § x ∈ M i,j : x i j = x i j (ML2) Horizontal inclusion x ∈ M i,j ⊂ M i,j +1 and x ∈ A i, ⊂ M i,j : x i j x i j x ∈ M i,j ⊂ M i +1 ,j and x ∈ B ,j ⊂ M i,j : x i j x i j (ML4) Horizontal conditional expectation E M,ri,j : M i,j → M i,j − and E M,ri,j | B ,j = E B,r ,j : E M,ri,j ( x ) = d − x i j − , x ∈ M i,j E M,ri,j ( x ) = E B,r ,j ( x ) = d − x i j − , x ∈ B ,j (ML5) Vertical conditional expectation E M,li,j : M i,j → M i − ,j and E M,li,j | A i, = E A,li, : E M,li,j ( x ) = d − x i − j , x ∈ M i,j E M,li,j ( x ) = E A,li, ( x ) = d − x i − j , x ∈ A i, (ML6) Commuting square of conditional expectation E M,ri,j +1 ◦ E M,li,j = E M,li,j +1 ◦ E M,ri +1 ,j +1 : M i +1 ,j +1 → M i,j , x ∈ M i +1 ,j +1 : E M,ri,j +1 ◦ E M,li,j ( x ) = E M,li,j +1 ◦ E M,ri +1 ,j +1 ( x ) = d − d − x i j (ML7) Horizontal Jones projections f j ∈ M i,j +1 and vertical Jones projections e i ∈ M i +1 ,j : f j +1 = d − ji f j +2 = d − j +1 i e i +1 = d − i j e i +2 = d − i +1 j (ML8) Standard condition: • [ M i,j , A p,q ] = 0 for i ≤ q ≤ p . For g ∈ M i,j , x ∈ A p,q , regard them as elements in M p,j ,then gx = xg ; • [ M i,j , B k,l ] = 0, for j ≤ k ≤ l . For g ∈ M i,j , y ∈ B k,l , regard them as elements in M i,l ,then gy = yg : gx i jq − ip − q = gx i jq − ip − q g y i jk − j l − k = g y i jk − j l − k .4 From Markov lattice as standard bimodule to planar bimodule category Let A and B be planar tensor categories. Let M be a C ∗ A − B bimodule categorywith following properties:(a) Object: The objects of M are [ m, n ] = [ m, n ] M , m, n ∈ Z ≥ , where [0 ,
0] := 1 M issimple.(b) The module tensor product of objects are[ i, +] A (cid:3) [ m, n ] M = [ m + i, n ] M , [ i, − ] A (cid:3) [ m, n ] M = none[ m, n ] M (cid:1) [ j, +] B = [ i, n + j ] M , [ m, n ] M (cid:1) [ j, − ] B = none([ i, +] A (cid:3) [ m, n ] M ) (cid:1) [ j, +] B = [ m + i,n + j ] M = [ i, +] A (cid:3) ([ m, n ] M (cid:1) [ j, +] B )(c) Only M ([ m, n ] → [ m ± i, n ± j ]) is non-empty, m, n, i, j ∈ Z ≥ . The module tensorproduct of morphisms in Hom( A ), Hom( M ) and Hom( M ) should match the shadingtypes.(d) M is a strict A −B bimodule category, i.e., the left/right module associator and bimoduleassociator are trivial. For x, x , x ∈ Hom( A ), g ∈ Hom( M ) and y, y , y ∈ Hom( B ), x (cid:3) ( x (cid:3) g ) = ( x ⊗ x ) (cid:3) g ( g (cid:1) y ) (cid:1) y = g (cid:1) ( y ⊗ y )( x (cid:3) g ) (cid:1) y = x (cid:3) ( g (cid:1) y ) . (e) M is a C ∗ category with a natural dagger structure such that (cid:1) and (cid:3) are dagger functors,i.e., for x ∈ Hom( A ) , g ∈ Hom( M ) and y ∈ Hom( B ),( x (cid:3) g (cid:1) y ) † = x † (cid:3) g † (cid:1) y † . Such bimodule category is called a planar bimodule category . Remark 3.4.1.
As in Remark 2.3.2, the morphisms in M is determined by its representationas an endomorphism and its domain and range.There is a canonical isomorphism φ : M ([ m, n ] → [ m + 2 i, n + 2 j ]) → M ([ m + i, n + j ] → [ m + i, n + j ]) by using the rigid structure on A and B . xφ : m + i n + jm ni j (cid:55)→ x m + i n + jmi jn Remark 3.4.2.
Let M and N be planar bimodule categories over the same planar tensorcategory. If they are unitary monoidal equivalent, then they are unitary isomorphic. Use the similar notion as we define the planar module category in Definition 2.3.3.Define the multi-step conditional expectations E l,im,n := E M,lm − i +1 ,n ◦ · · · ◦ E M,lm,n and E r,km,n := E M,rm,n − k +1 ◦ · · · ◦ E M,rm,n . Definition 3.4.3.
Let
A, B be standard λ -lattices and M = ( M m,n ) m,n ≥ be a Markov latticeas a standard A − B bimodule with dim( M , ) = 1. We define a planar bimodule category M from M as follows. 44a) The objects of M are the symbols [ m, n ] for m, n ∈ Z ≥ .(b) Given m, n ≥
0, define M ([ m, n ] → [ m, n ]) := M m,n .(c) The identity morphism in M ([ m, n ] → [ m, n ]) is 1 M m,n .(d) For ( f ; [ m , n ] , [ m , n ]) with 2 | m + m and 2 | n + n , define ( f ; [ m , n ] , [ m , n ]) † =( f ∗ [ m , n ] , [ m , n ]), where f, f ∗ ∈ M m m , n n .(e) Define the composition in nine cases.(C11) ( h ; [ m + 2 i, n + 2 k ] , [ m + 2 i + 2 j, n + 2 k + 2 t ]) ◦ ( g ; [ m, n ] , [ m + 2 i, n + 2 k ])= ( d i d k E l,im +2 i + j,n + k + t ( E r,km +2 i + j,n +2 k + t ( hgf nt,k e mj,i )); [ m, n ] , [ m + 2 i + 2 j, n + 2 k + 2 t ]),where g ∈ M m + i,n + k , h ∈ M m +2 i + j,n +2 k + t and d i d k E l,im +2 i + j,n + k + t ( E r,km + i + j,n +2 k + t ( hgf nt,k e mj,i )) ∈ M m + i + j,n + k + t .(C12) ( h ; [ m + 2 i, n + 2 k + 2 t ] , [ m + 2 i + 2 j, n + 2 k ]) ◦ ( g ; [ m, n ] , [ m + 2 i, n + 2 k + 2 t ])= ( d i d k E l,im +2 i + j,n + k ( E r,k + tm +2 i + j,n +2 k + t ( hgf n, ∗ t,k e mj,i )); [ m, n ] , [ m + 2 i + 2 j, n + 2 k ]), where g ∈ M m + i,n + k + t , h ∈ M m +2 i + j,n +2 k + t and d i d k E l,im +2 i + j,n + k ( E r,k + tm +2 i + j,n +2 k + t ( hgf n, ∗ t,k e mj,i )) ∈ M m + i + j,n + k .(C13) ( h ; [ m + 2 i, n ][ m + 2 i + 2 j, n + 2 k + 2 t ]) ◦ ( g ; [ m, n + 2 k ] , [ m + 2 i, n ])= ( d i d k E l,im +2 i + j,n +2 k + l ( hf n, ∗ l,k ge mj,i ); [ m, n + 2 k ] , [ m + 2 i + 2 j, n + 2 k + 2 t ]), where g ∈ M m + i,n + k , h ∈ M m +2 i + j,n + k + t and d i d k E l,im +2 i + j,n +2 k + l ( hf n, ∗ l,k ge mj,i ) ∈ M m + i + j,n +2 k + t .(C21) ( h ; [ m + 2 i + 2 j, n + 2 k ] , [ m + 2 i, n + 2 k + 2 t ]) ◦ ( g ; [ m, n ] , [ m + 2 i + 2 j, n + 2 k ])= ( d i d k E l,i + jm +2 i + j,n + k + t ( E r,km +2 i + j,n +2 k + t ( hgf nt,k e m, ∗ j,i )); [ m, n ] , [ m +2 i, n +2 k +2 t ]), where g ∈ M m + i + j,n + k , h ∈ M m +2 i + j,n +2 k + t and d i d k E l,i + jm +2 i + j,n + k + t ( E r,km +2 i + j,n +2 k + t ( hgf nt,k e m, ∗ j,i )) ∈ M m + i,n +2 k + t .(C22) ( h ; [ m + 2 i + 2 j, n + 2 k + 2 t ] , [ m + 2 i, n + 2 k ]) ◦ ( g ; [ m, n ] , [ m + 2 i + 2 j, n + 2 k + 2 t ])= ( d i d k E l,i + jm +2 i + j,n + k ( E r,k + tm +2 i + j,n +2 k + t ( hgf n, ∗ t,k e m, ∗ j,i )); [ m, n ] , [ m + 2 i, n + 2 k ]), where g ∈ M m + i + j,n + k + t , h ∈ M m +2 i + j,n +2 k + t and d i d k E l,i + jm +2 i + j,n + k ( E r,k + tm +2 i + j,n +2 k + t ( hgf n, ∗ t,k e m, ∗ j,i )) ∈ M m + i,n + k .(C23) ( h ; [ m + 2 i + 2 j, n ] , [ m + 2 i, n + 2 k + 2 t ]) ◦ ( g ; [ m, n + 2 k ] , [ m + 2 i + 2 j, n ])= ( d i f k E l,i + jm +2 i + j,n +2 k + t ( hf n, ∗ t,k ge m, ∗ j,i ); [ m, n + 2 k ] , [ m + 2 i, n + 2 k + 2 t ]), where g ∈ M m + i + j,n + k , h ∈ M m +2 i + j,n + k + t and d i f k E l,i + jm +2 i + j,n +2 k + t ( hf n, ∗ t,k ge m, ∗ j,i ) ∈ M m + i,n +2 k + t .(C31) ( h ; [ m, n + 2 k ] , [ m + 2 i + 2 j, n + 2 k + 2 t ]) ◦ ( g ; [ m + 2 i, n ] , [ m, n + 2 k ])= ( d i d k E r,km +2 i + j,n +2 k + t ( he m, ∗ j,i gf nt,k ); [ m + 2 i, n ] , [ m + 2 i + 2 j, n + 2 k + 2 t ]), where g ∈ M m + i,n + k , h ∈ M m + i + j,n +2 k + t and d i d k E r,km +2 i + j,n +2 k + t ( he m, ∗ j,i gf nt,k ) ∈ M m +2 i + j,n + k + t .(C32) ( h ; [ m, n + 2 k + 2 t ] , [ m + 2 i + 2 j, n + 2 k ]) ◦ ( g ; [ m + 2 i, n ] , [ m, n + 2 k + 2 t ])= ( d i d k E r,k + tm +2 i + j,n +2 k + t ( he m, ∗ j,i gf n, ∗ t,k ); [ m + 2 i, n ] , [ m + 2 i + 2 j, n + 2 k ]), where g ∈ M m + i,n + k + t , h ∈ M m + i + j,n +2 k + t and d i d k E r,k + tm +2 i + j,n +2 k + t ( he m, ∗ j,i gf n, ∗ t,k ) ∈ M m +2 i + j,n + k .(C33) ( h ; [ m, n ] , [ m + 2 i + 2 j, n + 2 k + 2 t ]) ◦ ( g ; [ m + 2 i, n + 2 k ] , [ m, n ])= ( d i d k hf n, ∗ t,k e m, ∗ j,i g ; [ m +2 i, n +2 k ] , [ m +2 i +2 j, n +2 k +2 t ]), where g ∈ M m + i,n + k , h ∈ M m + i + j,n + k + t and d i d k hf n, ∗ t,k e m, ∗ j,i g ∈ M m +2 i + j,n +2 k + t .For the other cases, we can use the dagger structure g † ◦ h † := ( h ◦ g ) † to define.Similarly, we use the string diagrams to explain the composition.45 f mi + ji mmii j n kt + knn t kk gf mij + i mmji i n k + t tnn kkk gf mi + jj mmi i i n k + tknn k kt (C12) (C23) (C31)The composition is well-defined and M is a C ∗ category as before. Remark 3.4.4.
The composition is well-defined, because of the commuting square of left/rightconditional expectation condition and Proposition 3.1.2.The definition of x (cid:3) (cid:1) y for x ∈ Hom( A ) and y ∈ Hom( B ) are the same as in Definition 3.4.5. (cid:3) g (cid:1) x (cid:3) (cid:1) y , g ∈ Hom( M ), x ∈ Hom( A ) and y ∈ Hom( B ).The idea is the same as in Definition 2.3.5. First, we define 1 (cid:3) g (cid:1) g j (cid:3) g (cid:1) t ( g ; [ m, n ] , [ m + 2 i, n + 2 k ]) , i ≤ j, k ≤ t ( ge mj − i,i f nt − k,k ; [ m + j, n + t ] , [ m + 2 i + j, m + 2 k + t ])( g ; [ m, n ] , [ m + 2 i, n + 2 k ]) , i > j, k ≤ t ( ge m, ∗ i − j,j f nt − k,k ; [ m + j, n + t ] , [ m + 2 i + j, m + 2 k + t ])( g ; [ m, n ] , [ m + 2 i, n + 2 k ]) , i ≤ j, k > t ( ge mj − i,i f n, ∗ k − t,t ; [ m + j, n + t ] , [ m + 2 i + j, m + 2 k + t ])( g ; [ m, n ] , [ m + 2 i, n + 2 k ]) , i > j, k > t ( ge m, ∗ i − j,j f n, ∗ k − t,t ; [ m + j, n + t ] , [ m + 2 i + j, m + 2 k + t ]) Note that here we use the fact that the Jones projection [ e i , f k ] = 0 for all i, k ≥ (cid:3) g ) (cid:1) (cid:3) ( g (cid:1)
1) =: 1 (cid:3) g (cid:1) x (cid:3) (cid:1) y will be the same as x ⊗ ⊗ y in Definition 1.6.7 byusing the shift maps. g n t − kkn + k kmji − j mij i ≥ j, k ≤ t The proof of the following propositions are the same as in the Markov tower case with thefact in Remark 3.2.2. To be precise, the diagrammatic proof can be split as left-hand-side andright-hand-side independently, and the proof on each side is the same as the Markov tower case.
Proposition 3.4.6. M is a left A − module. That is, (1) For g ∈ Hom( M ) , x ∈ Hom( A ) , (1 (cid:1) g ) ◦ ( x (cid:1)
1) = ( x (cid:1) ◦ (1 (cid:1) g ) . (2) For g ∈ Hom( M ) , x , x ∈ Hom( A ) , x (cid:3) ( x (cid:3) g ) = ( x ⊗ x ) (cid:3) g . (3) For g , g ∈ Hom( M ) , x , x ∈ Hom( A ) , (cid:3) ( g ◦ g ) = (1 (cid:3) g ) ◦ (1 (cid:3) g ) and ( x ◦ x ) (cid:3) x (cid:3) ◦ ( x (cid:3) . Proposition 3.4.7.
Similarly, M is a right B − module. That is, For g ∈ Hom( M ) , y ∈ Hom( B ) , ( g (cid:1) ◦ (1 (cid:1) y ) = (1 (cid:1) y ) ◦ ( g (cid:1) . (2) For g ∈ Hom( M ) , y , y ∈ Hom( B ) , ( g (cid:1) y ) (cid:1) y = g (cid:1) ( y ⊗ y ) . (3) For g , g ∈ Hom( M ) , y , y ∈ Hom( B ) , ( g ◦ g ) (cid:1) g (cid:1) ◦ ( g (cid:1) and (cid:1) ( x ◦ x ) =(1 (cid:1) x ) ◦ (1 (cid:1) x ) . Proposition 3.4.8. M is a A − B bimodule. That is, for g ∈ Hom( M ) , x ∈ Hom( A ) , y ∈ Hom( B ) , ( x (cid:3) ◦ (1 (cid:1) y ) ◦ (1 (cid:3) g (cid:1)
1) = (1 (cid:1) y ) ◦ ( x (cid:3) ◦ (1 (cid:3) g (cid:1) .Proof. By Remark 3.2.2.
Definition 3.4.9.
Define x (cid:3) g (cid:1) y := ( x (cid:3) ◦ (1 (cid:1) y ) ◦ (1 (cid:3) g (cid:1) C ∗ A − B bimodules and planar A − B bi-module categories C ∗ A − B bimodule category
Let A and B be 2-shaded rigid C ∗ multitensor categories with generators X = 1 + A ⊗ X ⊗ −A and Y = 1 + B ⊗ Y ⊗ −B . Let M be a Cauchy complete indecomposable semisimple C ∗ A − B bimodule category. Note that there is a natural dagger structure on M , and the left/rightmodule actions are dagger functors, i.e., for morphism g ∈ Hom( M ), x ∈ Hom( A ) and y ∈ Hom( B ), ( x (cid:3) g ) † = x † (cid:3) g † , ( f (cid:1) y ) † = f † (cid:1) y † . We call M indecomposable if for any two simple objects P, Q ∈ M (WLOG, P = 1 + A (cid:3) P (cid:1) + B ), Q is a direct summand of ( X alt ⊗ m (cid:3) P ) (cid:1) Y alt ⊗ n for some m, n ∈ Z ≥ .Let A , B be planar tensor categories constructed from ( A , X ) and ( B , Y ) respectively. ByMacLane’s coherence theorem, A M B is unitary equivalent to a strict one, i.e., A , B are strict,the right/left module associators and the bimodule associator are trivial. This strict categoryis also a strict A − B bimodule category. WLOG, we also denote it as M .Pick a simple object Z = 1 + A (cid:3) Z (cid:1) + B ∈ M , then we construct a planar A − B bimodulecategory M as follows:(a) Objects: Define [0 ,
0] := Z , and[ m + 1 ,
0] := [1 , ?] A (cid:3) [ m, , [ m, n + 1] := [ m, n ] (cid:1) [1 , ?] B , where [1 , ?] A = [1 , +] A if 2 | m and [1 , ?] A = [1 , − ] A if 2 (cid:45) m ; [1 , ?] B = [1 , +] B if 2 (cid:45) n and [1 , ?] B = [1 , − ] B if 2 | n .(b) M is a full subcategory of M with above objects.Given M to be a planar A − B bimodule category, for the similar reason, its Cauchycompletion (cid:100) M is a (cid:99) A − (cid:99) B bimodule category, compatible with the dagger structure. Remark 3.5.1.
Suppose M is a planar A − B bimodule category constructed from M over( A , X ) and ( B , Y ), then there is a unitary equivalence between M as A − B bimodule categoryand (cid:100) M as (cid:99) A − (cid:99) B bimodule category, which maps base object to base object. Now let M i,j = End([ i, j ]), i, j ∈ Z ≥ . After identifying f ∈ M i,j withid [1 , ?] (cid:3) f ∈ M i +1 ,j and f (cid:1) id [1 , ?] ∈ M i,j +1 and identifying x ∈ A i, = End([ i, +] A ) with x (cid:1) id [0 ,j ] ∈ M i,j and y ∈ B ,j = End([ j, +] B ) with id [ i, (cid:1) y ∈ M i,j . It is easy to show that M = ( M i,j ) i,j ≥ is a Markov lattice as a standard A − B bimodule with modulus ( d , d ).47imilar to the module case, combining above discussion, we have the following theorem. Theorem 3.5.3.
There is a bijective correspondence between equivalence classes of the follow-ing:
Traceless Markov lattice M =( M i,j ) i,j ≥ with dim( M , ) = 1 asa standard A − B bimodule overstandard λ -lattices A, B ∼ = Pairs ( M , Z ) with M an indecompos-able semisimple C ∗ A−B bimodule cat-egory together with a choice of simpleobject Z = 1 + A (cid:3) Z (cid:1) + B Equivalence on the left hand side is the ∗ -isomorphism on the traceless Markov lattice as standard A − B bimodule; the equivalence on the right hand side is the unitary A−B bimodule equivalencebetween their Cauchy completions which maps the simple base object to simple base object.
Corollary 3.5.4.
Any Markov lattice M with modulus ( d , d ) and dim( M ) = 1 is naturallya standard TLJ( d ) − TLJ( d ) bimodule, which corresponds to an indecomposable semisimple C ∗ T LJ ( d ) − T LJ ( d ) bimodule category with a simple base object. Remark 3.5.5.
The tracial case will be discussed in § In this Chapter, as an application, we are going to classify all indecomposable semisimple
T LJ − modules (see Corollary 2.4.5) to get Markov tower, which are also the same as balanced d -fair bipartite graphs [DY15]. We will explain exactly how these two classifications agreeby directly constructing the correspondence passing through the 2-category BigHilb [FP19].Although this is known [DY15, FP19], we explain in detail here so that we are able to do thebimodules in § d -fair bipartite graph In [DY15], the authors classify unshaded unoriented
T LJ ( d ) − modules in terms of the com-binatorial data of fair and balanced graphs. This classification was generalized to T LJ (Γ) − modulesin [FP19], where T LJ (Γ) is a generalized Temperley-Lieb-Jones category associated to aweighted bidirected graph Γ. We will be interested in the special case of 2-shaded
T LJ ( d ) − modules. Notation 4.1.1.
Let Λ be a graph where V (Λ) is the set of vertices and E (Λ) is the set ofedges. Let s, t : E (Λ) → V (Λ) be the source and target functions respectively. Definition 4.1.2.
Let Λ be a bipartite graph with vertices V (Λ) = V (cid:116) V and { e | s ( e ) , t ( e ) ∈ V i } = ∅ , i = 0 ,
1. Let ω : E (Λ) → (0 , ∞ ) be the weighting on the edges of graph [FP19].We call (Λ , ω ) a d -fair graph if for each P ∈ V , Q ∈ V (cid:88) { e | s ( e )= P } w ( e ) = (cid:88) { e | s ( e )= Q } w ( e ) = d. We call (Λ , ω ) a balanced graph if there exists an involution ( · ) on E (Λ) that switchessources and targets for each e ∈ E (Λ) and ω ( e ) ω ( e ) = 1 . roposition 4.1.3. Suppose (Λ , ω ) is a balanced d -fair bipartite graph. Then the graph islocally finite, i.e., the number of edges coming in or out of any vertex is uniformly bounded: { e : s ( e ) = P } = { e : t ( e ) = P } ≤ d for any vertex P. Proof.
Suppose P has N edges, then there exists an edge e : P → Q such that ω ( e ) ≤ dN andhence ω ( e ) = ω ( e ) ≥ Nd . Note that d = (cid:88) { e | s ( e )= Q } ω ( e ) ≥ ω ( e ) ≥ Nd , which follows that N ≤ d < ∞ . Definition 4.1.4.
We call θ : (Λ , ω ) → (Λ (cid:48) , ω (cid:48) ) an isomorphism of edge-weighted graphs if θ isa graph isomorphism and ω (cid:48) ( θ ( e )) = ω ( e ) for each e ∈ E (Λ). BigHilb and 2-subcategory C ( K, ev K ) Definition 4.2.1.
Let
U, V be countable sets. Define a category
Hilb U × Vf as follows:(a) Object: U × V − bigraded Hilbert spaces H = (cid:77) u ∈ Uv ∈ V H uv , where H uv is finite dimensional for each pair ( u, v ), and only finite many H uv is non-trivialfor each fixed u ∈ U or each fixed v ∈ V .(b) Morphism: The morphisms are defined as uniformly bounded operators f = (cid:77) u ∈ Uv ∈ V f uv : H → G, where f uv : H uv → G uv are morphisms in Hilb f , the category of finitely dimensional Hilbertspaces. Uniformly boundedness meanssup u ∈ Uv ∈ V (cid:107) f uv (cid:107) < ∞ . (c) The composition: For morphisms f, g , define the composition entry-wisely as g ◦ f := (cid:77) u ∈ Uv ∈ V g uv ◦ f uv . (d) The identity morphism: Define the identity morphism id H : H → H asid H := (cid:77) u ∈ Uv ∈ V id H uv , where id H,uv = id H uv is the identity map on H uv . Definition 4.2.2.
Let
BigHilb be a dagger 2-category defined as follows:49a) Object: Countable sets.(b) For objects
U, V , Hom(
U, V ) =
Hilb U × Vf .(c) The composition of 1-morphisms: For 1-morphisms H : U → V , G : V → W , the composi-tion of U, V denoted by ⊗ is defined as G ◦ H = H ⊗ G := (cid:77) u ∈ Uw ∈ W (cid:77) v ∈ V H uv ⊗ G vw : U → W, where the ⊗ on the right hand side is the tensor product of Hilbert spaces. The operator isanalogous to matrix multiplication, the product is replaced by tensor product and the sumis replaced by direct sum. Clearly, ( H ⊗ G ) ⊗ L = H ⊗ ( G ⊗ L ) . (d) The identity 1-morphism: For an object U , the identity 1-morphism C | U | ∈ Hom(
U, U ) isdefined as C | U | := (cid:77) u,v ∈ U δ u = v · C . (e) The dual 1-morphism: For 1-morphism H = (cid:76) u ∈ Uv ∈ V H uv : U → V , define its dual as H := (cid:77) v ∈ Vu ∈ U H vu : V → U, where H vu := H uv and H uv is the complex conjugate Hilbert space of H uv .(f) Tensor product of 2-morphisms. Let H , H : U → V , G , G : V → W , and f : H → H , g : G → G , define f ⊗ g as( f ⊗ g ) uw := (cid:77) v ∈ V f uv ⊗ g vw : (cid:77) v ∈ V H ,uv ⊗ G ,vw → (cid:77) v ∈ V H ,uv ⊗ G ,vw . Clearly, ( f ⊗ g ) ⊗ h = f ⊗ ( g ⊗ h ).(g) Dagger structure: For a 2-morphism f = (cid:76) u,v f uv : H → G , define its adjoint f † := (cid:76) u,v f ∗ uv : G → H , where f ∗ uv is the adjoint of f uv as a bounded linear map. Clearly,( f † ) † = f . Definition 4.2.3.
We call a 1-morphism H : U → V dualizable , if there exist evaluation andcoevaluation 2-morphisms ev H : H ⊗ H → C | V | and coev H : C | U | → H ⊗ H meeting the zigzagcondition : (id H ⊗ ev H ) ◦ (coev H ⊗ id H ) = id H (ev H ⊗ id H ) ◦ (id H ⊗ coev H ) = id H . We are going to discuss the evaluation and coevaluation ev H and coev H in more details. Definition 4.2.4.
Note that ev
H,uv : (cid:76) w H uw ⊗ H wv = ( H ⊗ H ) uv → ( C | V | ) uv = δ u = v · C ,only ev H,vv is nonzero for v ∈ V . Let C H,vu : H vu ⊗ H uv = H uv ⊗ H uv → C such thatev H,vv = (cid:76) u ∈ U C H,vu . Similarly, only coev
H,uu : C → ( H ⊗ H ) uu = (cid:76) v ∈ V H uv ⊗ H vu is nonzerofor u ∈ U . Let D H,uv : C → H uv ⊗ H vu = H uv ⊗ H uv such that coev H,uu = (cid:76) v ∈ V D H,uv .50hen id
H,uv = ((id H ⊗ ev H ) ◦ (coev H ⊗ id H )) uv = (id H ⊗ ev H ) uv ◦ (coev H ⊗ id H ) uv = (cid:32) (cid:77) w ∈ V id H,uw ⊗ ev H,wv (cid:33) ◦ (cid:32)(cid:77) t ∈ U coev H,ut ⊗ id H,tv (cid:33) = (id
H,uv ⊗ ev H,vv ) ◦ (coev H,uu ⊗ id H,uv )= (id
H,uv ⊗ C H,vu ) ◦ ( D H,uv ⊗ id H,uv )for u ∈ U, v ∈ V . Similarly,id H,vu = (ev
H,vv ⊗ id H,vu ) ◦ (id H,vu ⊗ coev H,uu ) = ( C H,vu ⊗ id H,vu ) ◦ (id H,vu ⊗ D H,uv ) , for v ∈ V, u ∈ U . Remark 4.2.5. ev H and coev H are completely determined by C H,uv and D H,uv . Definition 4.2.6.
Let C ( K, ev K ) = C ( K, ev K , coev K ) be a 2-subcategory of BigHilb with a1-morphism generator K : V → V and distinguished 2-morphisms evaluation and coevaluationev K , coev K . We require that(a) K is dualizable.(b) The evaluation and coevaluation for the dual K :ev K := (coev K ) † and coev K := (ev K ) † . (c) They satisfy the d − fairness condition , namely,ev K ◦ coev K = d · id C | V | ev K ◦ coev K = d · id C | V | . In other words, C K,uv = ( D K,uv ) † D K,vu = ( C K,vu ) † , and For each P ∈ V , (cid:88) Q ∈ V C K,P Q ◦ D K,P Q = d · id C For each Q ∈ V , (cid:88) P ∈ V C K,QP ◦ D K,QP = d · id C , Here, the 1-morphism generator means all the 1-morphism is Cauchy generated by K and K . Remark 4.2.7. coev K , ev K and coev K are determined by ev K in C ( K, ev K ). Proposition 4.2.8.
The followings are some properties of C ( K, ev K ) . (1) Let V = V (cid:116) V , then all the 1-morphisms in C ( K, ev K ) , including K, K , can be regardedas V × V − bigraded Hilbert spaces. So we can regard C ( K, ev K ) as a 2-category with oneobject V . Then all the 2-morphisms can be regarded as V × V − bigraded uniformly boundedoperators.If ( P, Q ) (cid:54)∈ V × V , then K P Q = K QP = 0 , which follows that C K,QP = D K,P Q = 0 . Thezigzag condition between them still hold.
All the 1-morphisms in C ( K, ev K ) are dualizable. (3) sup P ∈ V ,Q ∈ V dim( K P Q ) < ∞ . In fact, we will see sup P ∈ V ,Q ∈ V dim( K P Q ) ≤ d in the nextsection § (4) There exist standard spherical evaluation and coevaluation in 2-morphisms: ev st K : K ⊗ K → C | V | coev st K : C | V | → K ⊗ K ev st K := (coev K ) † coev st K := (ev K ) † . In more details, Let { (cid:15) i } ki =1 be the orthonormal basis ( ONB ) of K P Q and { (cid:15) ∗ i } be the dualbasis of K P Q , P ∈ V , Q ∈ V then C st K,QP : K QP ⊗ K P Q = K P Q ⊗ K P Q → C D st K,ab : C → K P Q ⊗ K QP = K P Q ⊗ K P Q C st K,P Q := ( D st K,P Q ) † D st K,QP := ( C st K,QP ) † are defined as C st K,QP : (cid:15) ∗ i ⊗ (cid:15) j (cid:55)→ δ i = j D st K,P Q : 1 (cid:55)→ k (cid:88) i =1 (cid:15) i ⊗ (cid:15) ∗ i . Note that ev st K and coev st K are well-defined 2-morphisms because of (3) , and the definitionsof ev st K and coev st K do not depend on the choice of ONB on each K P Q and they also meetthe zigzag condition..
Notation 4.2.9.
Now, we use the graphic calculus to describe C ( K, ev K ). The idea is fromthe graphical calculus for 2- Hilb [RV16]. However, in their paper, they only care about the casewhen ev = ev st and coev = coev st , which is not necessarily true in our context.First we provide the single object version:(1) For P ∈ V , Q ∈ V , C K,P Q , D K,QP , C st K,P Q and D st K,QP . K PQ K QP P Q C K,PQ : K PQ ⊗ K QP → C K QP K PQ Q P D K,QP : C → K QP ⊗ K PQ K PQ K QP P Q C st K,PQ : K PQ ⊗ K QP → C K QP K PQ Q P D st K,QP : C → K QP ⊗ K PQ (2) Rigidity: P Q P Q P Q = =
P Q P Q P Q = =(3) d -fairness. For P ∈ V , (cid:80) Q ∈ V P Q = d · P Then the graphical calculus version: In the n -category setting, n -morphisms are n-morphismsare used to label codimension n cells of an n -manifold. So here, 0-morphisms in BigHilb labelregions of the plane, 1-morphisms label strings from left to right, and 2-morphisms label tickets(including ev and coev) from bottom to top. Shading is just shorthand for the labelling. Theunshaded region indicates the object V and the shaded region indicates V .521) coev K , ev K , coev st K and ev st K . coev K : C | V | → K ⊗ K ev K : K ⊗ K → C | V | coev st K : C | V | → K ⊗ K ev st K : K ⊗ K → C | V | (2) Rigidity: = = = =(3) d -fairness: = d · = d · (4) Dagger structure on ev and ev st . (cid:18) (cid:19) † = (cid:18) (cid:19) † = BigHilb generated by a balanced d -fair bipartitegraph In this section, we show the relation between 2-categories C ( K, ev K ) and d -fair bipartitegraphs (Λ , ω ). Then we may regard the generator K as a Hilb -enriched graph, and the edge-weighting ω giving the interesting dual pair. Construction 4.3.1.
First, we construct a W ∗ C (Λ , ω ) of BigHilb from a bal-anced d -fair bipartite graph (Λ , ω ) as follows:(a) Object is V = V (Λ) = V (cid:116) V , which is a countable set.(b) The 1-morphism generator K = K Λ : At ( P, Q ) ∈ V × V , K P Q is the Hilbert space withONB {| e (cid:105) : e ∈ E (Λ) , s ( e ) = P, t ( e ) = Q } and other entries are 0. The uniform boundednesscondition follows from Proposition 4.1.3.As for the dual 1-morphism K , at entry ( Q, P ) ∈ V × V , K QP is the Hilbert space withONB {| e (cid:105) : e ∈ E (Λ) , s ( e ) = Q, t ( e ) = P } = {| e (cid:105) : e ∈ E (Λ) , s ( e ) = P, t ( e ) = Q } , where ( · )is the involution of edge.So we may regard K as a Hilb -enriched graph.(c) All the 1-morphisms are Cauchy generated by K and K .(d) 2-morphisms are V × V -bigraded uniformly bounded operators between those 1-morphisms.(e) The edge-weighting gives the distinguished evaluation and coevaluation ev and coev. Notethat K P Q is a Hilbert space with orthonormal basis {| e (cid:105) : e ∈ E (Λ) , s ( e ) = P, t ( e ) = Q } ,53hen {| ¯ e (cid:105) : e ∈ E (Λ) , s ( e ) = P, t ( e ) = Q } is an orthonormal basis for K QP . Define C K,P Q : K P Q ⊗ K QP → C by | e (cid:105) ⊗ | e (cid:48) (cid:105) (cid:55)→ δ e = e (cid:48) w ( e ) , e : P → QD K,P Q : C → K P Q ⊗ K QP by 1 (cid:55)→ (cid:88) e : P → Q w ( e ) | e (cid:105) ⊗ | e (cid:105) = (cid:88) e : Q → P w ( e ) | e (cid:105) ⊗ | e (cid:105) .C K,QP : K QP ⊗ K P Q → C by | e (cid:105) ⊗ | e (cid:48) (cid:105) (cid:55)→ δ e = e (cid:48) w ( e ) , e : Q → PD K,QP : C → K QP ⊗ K P Q by 1 (cid:55)→ (cid:88) e : Q → P w ( e ) | e (cid:105) ⊗ | e (cid:105) = (cid:88) e : P → Q w ( e ) | e (cid:105) ⊗ | e (cid:105) . Proposition 4.3.2. C (Λ , ω ) satisfies the condition in Definition 4.2.6.Proof. We shall prove that C (Λ , ω ) is rigid and d -fair.(a) Rigidity: For each P, Q ∈ V , e : P → Q ,( C K,P Q ⊗ id K,P Q ) ◦ (id K,P Q ⊗ D K,QP )( | e (cid:105) ⊗
1) = ( C K,P Q ⊗ id K,P Q ) | e (cid:105) ⊗ (cid:88) e : P → Q w (¯ e ) | ¯ e (cid:105) ⊗ | e (cid:105) = w ( e ) w (¯ e ) | e (cid:105) = | e (cid:105) , (id K,P Q ⊗ C K,QP ) ◦ ( D K,P Q ⊗ id K,QP )(1 ⊗ | e (cid:105) ) = (id K,P Q ⊗ C K,QP ) (cid:88) e : P → Q w ( e ) | e (cid:105) ⊗ | ¯ e (cid:105) ⊗ | e (cid:105) = w ( e ) w (¯ e ) | e (cid:105) = | e (cid:105) . (b) d -fairness: (cid:88) Q ∈ V C K,P Q ◦ D K,P Q (1) = (cid:88) Q ∈ V C K,P Q (cid:88) e : P → Q w ( e ) | e (cid:105) ⊗ | ¯ e (cid:105) = (cid:88) { e | s ( e )= P } w ( e ) w ( e ) = d, ; (cid:88) P ∈ V C K,QP ◦ D K,QP (1) = (cid:88) a ∈ V C K,QP (cid:88) e : Q → P w ( e ) | e (cid:105) ⊗ | ¯ e (cid:105) = (cid:88) { e | s ( e )= Q } w ( e ) w ( e ) = d. Remark 4.3.3.
Suppose θ : (Λ , ω ) → (Λ (cid:48) , ω (cid:48) ) is an isomorphism of edge-weighted graphs (seeDefinition 4.1.4). We construct a unitary equivalence between C (Λ , ω ) and C (Λ (cid:48) , ω (cid:48) ). For the1-morphism generators K Λ and K Λ (cid:48) , we have K Λ ,P Q ∼ = K Λ (cid:48) ,θ ( P ) θ ( Q ) as finite dimensional Hilbert spaces, via the bijection of ONBs given by | e (cid:105) (cid:55)→ | θ ( e ) (cid:105) . Denoteby u θ : K Λ → K Λ (cid:48) this unitary isomorphism.As for the evaluation ev K Λ and ev K Λ (cid:48) , we look at C K Λ ,P Q and C K Λ (cid:48) ,θ ( P ) θ ( Q ) (see Definition4.2.4). Note that C K Λ (cid:48) ,θ ( P ) θ ( Q ) : K Λ (cid:48) ,θ ( Q ) θ ( P ) ⊗ K Λ (cid:48) ,θ ( P ) θ ( Q ) → C by | θ ( e ) (cid:105) ⊗ | θ ( e (cid:48) ) (cid:105) (cid:55)→ δ θ ( e )= θ ( e (cid:48) ) ω (cid:48) ( θ ( e )) = δ e = e (cid:48) ω ( e ) , ∀ e : Q → P ∈ E (Λ) . We have C K Λ (cid:48) ,θ ( P ) θ ( Q ) = C K Λ ,P Q ◦ ( u θ † QP ⊗ u θ † P Q ) .
54n other words, ev K Λ (cid:48) = ev K Λ ◦ ( u θ † ⊗ u † θ ) . Therefore, C (Λ , ω ) and C (Λ (cid:48) , ω (cid:48) ) are unitary equivalent up to the unitary 2-morphism u θ .Next, start with a 2-category C ( K, ev K ), we construct a balanced d -fair bipartite graph(Λ , ω ). Definition 4.3.4.
For P ∈ V , Q ∈ V , let v P Q : K P Q → K P Q = K QP be the canonical dualmap that ξ (cid:55)→ ξ ∗ and v † P Q : K QP → K P Q defined by ξ ∗ → ξ ∗∗ = ξ . Then v † P Q ◦ v P Q = id
K,P Q and v P Q ◦ v † P Q = id
K,QP . Define ϕ K,P Q : K QP → K P Q by ϕ K,P Q = (id
K,P Q ⊗ C st K,QP ) ◦ ( D K,P Q ⊗ v † P Q ) ϕ K,QP : K P Q → K QP by ϕ K,QP = (id
K,QP ⊗ C st K,P Q ) ◦ ( D K,QP ⊗ v † P Q ) . Proposition 4.3.5.
Here are some properties for ϕ K and ϕ K . (1) ϕ K,P Q ◦ ϕ K,QP = id
K,P Q . (2) (cid:80) Q ∈ V Tr( ϕ † K,P Q ◦ ϕ K,P Q ) = (cid:80) P ∈ V Tr( ϕ † K,QP ◦ ϕ K,QP ) = d .Proof. See [DY15, Prop. 1.8], [FP19, Prop. 3.10].
Construction 4.3.6.
Define the graph Λ to be V (Λ) := V and the number of edges from P ∈ V to Q ∈ V to be dim K P Q . Define edge-weighting function ω : E (Λ) → (0 , ∞ ) as themultiset { ω ( e ) } e : P → Q := { eigenvalues of ϕ K,P Q ◦ ϕ † K,P Q }{ ω ( e ) } e : Q → P := { eigenvalues of ϕ K,QP ◦ ϕ † K,QP } . From above Proposition 4.3.5, (Λ , ω ) is a d -fair and balanced bipartite graph. To be precise,(1) gives the balance condition and (2) gives the d -fairness. In fact, ϕ K,P Q ◦ ϕ † K,P Q =(id
K,P Q ⊗ C st K,QP ) ◦ ( D K,P Q ⊗ id K,P Q ) ◦ ( C K,P Q ⊗ id K,P Q ) ◦ (id K,P Q ⊗ D st K,QP ) ϕ K,QP ◦ ϕ † K,QP =(id
K,QP ⊗ C st K,P Q ) ◦ ( D K,QP ⊗ id K,QP ) ◦ ( C K,QP ⊗ id K,QP ) ◦ (id K,QP ⊗ D st K,P Q ) .P Q Q P Remark 4.3.7.
For a given 2-category C ( K, ev K ), let (Λ , ω ) be the balanced d -fair bipartitegraph obtained from Construction 4.3.6. When we construct the 1-morphism generator K = K Λ in C (Λ , ω ) from the bipartite graph Λ, we secretly make a choice of ONB for each ( K Λ ) P Q , sothere is a unitary 2-morphism α : K → K Λ such that ev K = ev K Λ ◦ ( α ⊗ α ). Therefore, C ( K, ev K ) and C (Λ , ω ) are unitary equivalent up to a unitary 2-morphism α .55 .4 From C ( K, ev K ) to Markov tower Construction 4.4.1.
Here, we are going to build a tower of algebra from the 2-category C ( K, ev K ) discussed above with a chosen point, say P ∈ V . Let C | P | be a 1-morphism withall the entry being 0 except ( C | P | ) P P = C .Note that C | P | ⊗ K alt ⊗ n is a 1-morphism for each n ∈ Z ≥ .Let M n = End (cid:0) C | P | ⊗ K alt ⊗ n (cid:1) and identify M n (cid:51) x with x ⊗ id K ? ∈ M n +1 , where K ? = K if 2 | n , K ? = K if 2 (cid:45) n . We use the graphical calculus to show M = ( M n ) n ≥ is a Markovtower.(1) Element x ∈ M n : xP · · ·· · · C | P | K K K n th (2) Inclusion x ∈ M n ⊂ M n +1 : xP · · ·· · · C | P | K K K n th ( n +1) th (3) Conditional expectation E n +1 : M n +1 → M n , x ∈ M n : E n +1 ( x ) = d − xP · · ·· · · C | P | K K K n th ( n +1) th Here, the choice of the duality pair (coev K , (coev K ) † ) or (ev K , (ev K ) † ) depends on the shad-ing.(4) Jones projection e n ∈ M n +1 : e n = d − P · · ·· · · C | P | st n th (5) The pull down property is true automatically in this setting. See the diagram 2.2(MT6). Here, we are going to explore more properties of Markov tower. The tracial version hasbeen proved in [GHJ89, Thm. 4.1.4, Thm. 4.6.3] [CHPS18, Prop. 3.4]. For convenience, herewe will prove those properties for the traceless case.56 emma 4.5.1.
Suppose A ⊂ B is a unital inclusion of finite dimensional C ∗ -algebras and E : B → A is a faithful conditional expectation. Then there is an orthonormal basis { u i } i ∈ I such that (cid:80) i ∈ I u i E ( u ∗ i x ) = x for all x ∈ B , where | I | < ∞ .Proof. Regard B as a right A -module equipped with an A -valued inner product (cid:104) x | y (cid:105) A := E ( x ∗ y ). Note that A and B are finite dimensional, so B is a finitely generated projectiveHilbert A -module. By [FL02, Thm. 4.1] [KW00, Lemma. 1.7], there exists an orthonormal basis { u i } i ∈ I ⊂ B such that x = (cid:80) i ∈ I u i (cid:104) u i | x (cid:105) A = (cid:80) i ∈ I u i E ( u ∗ i x ) for all x ∈ B and | I | < ∞ . Proposition 4.5.2. (1) X n +1 := M n e n M n is a 2-sided ideal of M n +1 and hence M n +1 splits as a direct sum of vonNeumann algebras X n +1 ⊕ Y n +1 . We also define Y = M , Y = M so that X = X = 0 . X n +1 is called the old stuff and Y n +1 is called the new stuff. (2) X n +1 is isomorphic to M n ⊗ M n − M n , which is the basic construction from E n : M n → M n − .Denote this isomorphism as φ . Here, M n ⊗ M n − M n is a ∗ -algebra with multiplication ( x ⊗ y )( x ⊗ y ) = x E n ( y x ) ⊗ y and adjoint ( x ⊗ y ) ∗ = y ∗ ⊗ x ∗ . (3) If y ∈ Y n +1 and x ∈ X n , then yx = 0 in M n +1 . Hence E n +1 ( Y n +1 ) ⊂ Y n , which means thenew stuff comes from the old new stuff. (4) If Y n = 0 , then Y k = 0 for all k ≥ n .Proof. (1) Note that M n +1 e n = M n e n , then M n +1 M n e n M n ⊂ M n +1 e n M n = M n e n M n and M n e n M n M n +1 =( M n +1 M n e n M n ) ∗ ⊂ ( M n e n M n ) ∗ = M n e n M n .(2) See Watatani index theory [Wa90, §
1] with Lemma 4.5.1.(3) Note that as a finite dimensional von Neumann algebra, M n +1 = (cid:76) i M n +1 p i , where p i arethe minimum central projections. So if y ∈ Y n +1 , then y = (cid:80) j m j p j , where [ p j , e n ] = 0.For ae n − b ∈ X n and m j p j ∈ Y n +1 , by Jones projection property, m j p j ae n − b = d − m j p j ae n − e n e n − b = d − m j ae n − p j e n e n − b = 0 , so yx = 0 for any x ∈ X n , y ∈ Y n +1 .Let X n = (cid:76) k M n q k , where q k are the minimum central projections. For any y ∈ Y n +1 , q k E n +1 ( y ) = E n +1 ( q k y ) = 0 for all k , which implies that E n +1 ( y ) ∈ Y n .(4) By (3) and faithfulness of E n . C (Λ , ω ) Now we are able to extract the so-called principal graph data from the Markov tower, whichis similar to the classical tracial Markov tower [Oc88] [JS97, § A is a finite dimensional C ∗ -algebra, we write π ( A ) to be the set of minimal centralprojections of A . If A ⊂ B is a unital inclusion of finite dimensional C ∗ -algebras, then theinclusion matrix is the π ( A ) × π ( B ) matrix, with ( p, q )-th entry being (dim C ( pqA (cid:48) pq ∩ pqBpq )) .If A ⊂ B ⊂ B is a basic construction, then the inclusion matrix of B ⊂ B is the transpose ofthe inclusion matrix of A ⊂ B [GHJ89, §
2] [JS97].The inclusion matrix of A ⊂ B can be described as the Bratteli diagram of A ⊂ B , whosevertices are the minimal central projections and the number of edges between p and q is the( p, q )-th entry.The Bratteli diagram ∆ of the Markov tower M = ( M n ) n ≥ contains all the Bratteli dia-gram ∆ n of M n ⊂ M n +1 . Then by the property of inclusion matrix of basic construction and57roposition 4.5.2(2), the Bratteli diagram for M n ⊂ M n +1 contains the reflection of the Brattelidiagram of M n − ⊂ M n and new part, which is called the principal part . A vertex in the newpart is called a new vertex , otherwise, called an old vertex . The reflected vertex from a newvertex is called a new old vertex . Moreover, for a new vertex p ∈ Y n , denote p (cid:48) to be the newold vertex of p in M n +2 .The principal graph Λ contains the new part in the Bratteli diagram ∆, so its verticesare new vertices. To be precise, V (Λ) contains all the minimal central projections p in the newstuff. By Proposition 4.5.2(4), the new stuff comes from the old new stuff, then for p, q ∈ Λ, E (Λ) contains all the edges between p and q .It is clear that both the Bratteli diagram and the principal graph are bipartite. We can alsouse the principal graph to construct the Bratteli diagram by doing the reflection at each level. p qp (cid:48) p, q are new vertices p (cid:48) is the new old vertex of p The red part is principal partLet us then compute the edge weighting w : E (Λ) → (0 , ∞ ). Before that, we first give alemma: Lemma 4.6.1.
The follows are some properties for the relative commutant in
BigHilb : (1) Let H , H , · · · , H n , G , G , · · · , G n be finite dimensional Hilbert spaces. We identify B ( H i ) with B ( H i ) ⊗ id G i and B ( G i ) with id H i ⊗ B ( G i ) as subalgebras in B ( (cid:76) ni =1 H i ⊗ G i ) for each i = 1 , · · · , n , then the relative commutant n (cid:92) i =1 (cid:32) B ( H i ) (cid:48) ∩ B (cid:32) n (cid:77) i =1 H i ⊗ G i (cid:33)(cid:33) = n (cid:77) i =1 B ( G i ) . ( ∗ )(2) Let H be a 1-morphism in BigHilb , then the center Z (End( H )) is the linear span of all thedirect summands of id H . (3) Let G be another 1-morphism in BigHilb such that H ⊗ G is nondegenerate, i.e., for eachnonzero H pq , there is a nonzero G qr and vice versa. We identify End( H ) with End( H ) ⊗ id G and End( G ) with id H ⊗ End( G ) as subalgebras in End( H ⊗ G ) . Then the relative commutant End( H ) (cid:48) ∩ End( H ⊗ G ) = Z (End( H )) ⊗ End( G ) . (4) Moreover, if H pq is nonzero only when p = p ∈ V , then the relative commutant can berepresented as End( H ) (cid:48) ∩ End( H ⊗ G ) = id H ⊗ End( G ) . Warning : the tensor product in (1) is the tensor product of Hilbert spaces and bounded oper-ators; the tensor product in (3) and (4) is the tensor product of 1-morphisms/2-morphisms in
BigHilb , see Definition 4.2.2.Proof. ⊃ is clear. We show ⊂ .For f ∈ B ( (cid:76) ni =1 H i ⊗ G i ), f = (cid:76) ni,j =1 f i,j , where f i,j ∈ B ( H i ⊗ G i , H j ⊗ G j ). We shall provethat f i,j = 0 for i (cid:54) = j and f i,i ∈ id H i ⊗ B ( G i ) if f ∈ LHS of equation ( ∗ ). Let x i ∈ B ( H i ), then f ( x i ⊗ id G i ) = n (cid:77) j =1 f i,j ( x i ⊗ id G i ) = n (cid:77) k =1 ( x i ⊗ id G i ) f k,i = ( x i ⊗ id G i ) f, which implies that f i,j ( x i ⊗ id G i ) = ( x i ⊗ id G i ) f k,i = 0 for k (cid:54) = i, j (cid:54) = i and f i,i ( x i ⊗ id G i ) =( x i ⊗ id G i ) f i,i .From the first half, if we choose x i = id H i , we obtain f i,j = f k,i = 0, j (cid:54) = i, k (cid:54) = i ; fromthe second half, from a well-known statement that B ( H i ) (cid:48) ∩ B ( H i ⊗ G i ) = B ( G i ), so that f i,i ∈ id H i ⊗ G i .(2) Clear, see Definition 4.2.1(d).(3) ⊃ is clear. We show ⊂ .For f ∈ End( H ) (cid:48) ∩ End( H ⊗ G ), we shall prove that f pq ∈ (cid:76) r ∈ V id H pr ⊗ B ( H rq ).Note that (End( H ⊗ G )) pq = End(( H ⊗ G ) pq ) = B (cid:32)(cid:77) r ∈ V H pr ⊗ G rq (cid:33) For f ∈ End( H ) (cid:48) ∩ End( H ⊗ G ), f pq commute with B ( H pr ) ⊗ id G rq for all r ∈ V . By (1),we have f pq ∈ (cid:76) r ∈ V id H pr ⊗ B ( H rq ). Together with (2), we prove this statement.(4) From (3), for f ∈ End( H ) (cid:48) ∩ End( H ⊗ G ), f = (cid:77) q ∈ V id H p q ⊗ g ( q ) , where g ( q ) ∈ End( G ).Now we define g ∈ End( G ) by g ij := g ( i ) ij . Then f = id H ⊗ g .By § ∗ C (Λ) without providing the dis-tinguished evaluation and coevaluation given by the edge weighting, though we still havethe canonical evaluation and coevaluation denoted by ev st and coev st , which are drawn ingreen below. We denote the generators by K = K Λ and K . From Construction 4.4.1, let N n := End( C | p | ⊗ K alt ⊗ n ). Notation 4.6.2. and Observation
Denote Λ n to be the subgraph of Λ with vertices depth ≤ n and the corresponding Hilb -enriched graph to be K n := K Λ n and K n the dual space in thesense of Construction 4.3.1. As a convention, p is of depth 0. Observe that N n = End( K ⊗ K ⊗ K ⊗ K ⊗ · · · ⊗ K ? n ) . where K ? n = K n if 2 (cid:45) n , K ? n = K n if 2 | n . 59 xample 4.6.3. Let us take A graph for example. We label the vertices as follows. p p p p p p p p p p p p p p Then K = C
00 0 0 0 00 0 0 0 00 0 0 0 00 0 0 0 0 K = C C K = C
00 0 0
C C K = C C
C C = K k , K = C
00 0 0
C C C = K k , k = 0 , , , · · · K ⊗ K ⊗ K = C C K ⊗ K ⊗ K ⊗ K = C C C For this example, observe that End( K ⊗ K ⊗· · ·⊗ K ? n ) is the semisimple quotient of T LJ n ( √ n as the subgraph of the Bratteli diagram between depth n − n ,and K n is the Hilb -enriched graph of Λ n . The entry ( i, j ) in K ⊗ K ⊗ · · · ⊗ K ? n indicates thenumber of paths from the vertex p i at depth 0 to the vertex p j at depth n . Note that the basepoint is a single vertex p , so entry only at (1 , j ) can be nonzero. Proposition 4.6.4. N (cid:48) n − ∩ N n +1 = (cid:40) id K ⊗ K ⊗···⊗ K k − ⊗ End( K k ⊗ K k +1 ) n = 2 k id K ⊗ K ⊗···⊗ K k ⊗ End( K k +1 ⊗ K k +2 ) n = 2 k + 1 . Proof.
Note that K ⊗ K ⊗ · · · ⊗ K ? n satisfies the condition in Lemma 4.6.1(3) and (4).The idea is to transport the Jones projections from the Markov tower ( M n ) to the endomor-phism algebras ( N n ) in order to obtain the edge weighting ω . Let ψ n : M n → N n be a ∗ -algebraisomorphism for each n ≥ ψ n +1 | M n = ψ n .Let us consider the image of Jones projection ψ ( e n ) ∈ N n +1 . Note that e n ∈ M (cid:48) n − ∩ M n +1 ,so ψ ( e n ) ∈ N (cid:48) n − ∩ N n +1 . 60 roposition 4.6.5. WLOG, let n = 2 k . There exists a projection ε k ∈ End( K k ⊗ K k +1 ) such that ψ ( e k ) = id K ⊗ K ⊗···⊗ K k − ⊗ ε k .Proof. By proposition 4.6.4, there exists ε k ∈ End( K k ⊗ K k +1 ) such that ψ ( e k ) = id K ⊗ K ⊗···⊗ K k − ⊗ ε k . Note that e k is a projection, so is ε k . Lemma 4.6.6.
Let H be a Hilbert space and p (cid:54) = 0 be a projection on H . Suppose pf p ∈ C p for all f ∈ B ( H ) , then p = r ∗ r , where r : H → C and rr ∗ = 1 .Similarly, let H be a 1-morphism in BigHilb and p (cid:54) = 0 be a projection on H . Suppose pf p ∈ C p for all f ∈ End( H ) , then p = r ∗ r , where r : H → C | V | and rr ∗ = C | V | .Proof. For the Hilbert space case: Note that Im( f p ) can be any subspace of H and Im( p ( f p )) =Im( p ), so Im( p ) does not depend on the input, i.e., p facts through C . Let r : H → C and p = r ∗ r with rr ∗ = 1, since p ∗ = p = p ∗ p .The similar argument on 1-morphisms in BigHilb .As we see the construction of Jones projection in Construction 4.4.1(4), we shall prove thatthe Jones projection splits into two pieces.By Proposition 1.1.4(6), e n M n +1 e n = M n − e n , so ψ ( e n ) N n +1 ψ ( e n ) = N n − e n . WLOG, let n = 2 k . For each f ∈ End( K k ⊗ K k +1 ), id K ⊗ K ⊗···⊗ K k − ⊗ f ∈ N k +1 , there exists x ∈ N k − such thatid K ⊗ K ⊗···⊗ K k − ⊗ ( ε k f ε k ) = ( x ⊗ id K k ⊗ K k +1 )(id K ⊗ K ⊗···⊗ K k − ⊗ ε k ) = x ⊗ ε k , which follows that ε k f ε k ∈ C ε k .By Lemma 4.6.6, there exists r k : K k ⊗ K k +1 → C | V , k − | such that ε k = r † k r k and r k r † k = C | V , k − | , where V , k +1 contains all the simple objects in Λ k +1 with odd depth.Similarly, we can define ε k +1 ∈ End( K ⊗ K ) corresponding to Jones projection e k +1 andthere exists r k +1 : K k +1 ⊗ K k +2 → C | V , k | such that ε k +1 = r † k +1 r k +1 and r k +1 r † k +1 = C | V , k | , where V , k contains all the simple objects in Λ k with even depth.Now consider u k := d (id K ⊗ r k +1 ) ◦ ( r † k ⊗ id K ) ∈ End( K ). Note that e k e k +1 e k = d − e k and e k +1 e k e k +1 = d − e k +1 , we have u † k u k = id K k and u k u † k = id K k +2 , so u k is a unitary. d r † k r k r † k +1 r k +1 r † k +1 r k +1 · · ·· · ·· · · p C | p | K K K K K = r † k +1 r k +1 · · · p C | p | K K K K K d r † k r k r † k r k r † k +1 r k +1 · · ·· · ·· · · p C | p | K K K K K = r † k r k · · · p C | p | K K K K K p, q ∈ Λ with p at depth n and q at depth n + 1, we shallcompute the edge weighting on the edges e : p → q and e : q → p . WLOG, n = 2 k .Define ϕ k and ϕ k +1 as follows: ϕ k = d r k · · · p C | p | K K K K K ϕ k +1 = d r k +1 u † k · · · p C | p | K K K K K K and we have following properties:(1) ϕ k +1 ◦ ϕ † k = id.(2) Tr( ϕ † k ◦ ϕ k ) = d Tr( r † k r k ) = d Tr( r k r † k ) = d .(3) Tr( ϕ † k +1 ◦ ϕ k +1 ) = d Tr( u k r † k +1 r k +1 u † k ) = d Tr( r k +1 r † k +1 ) = d . Definition 4.6.7.
Define the edge-weighting function ω as the multiset: { ω ( e ) } e : p → q := { eigenvalues of ( ϕ † k ◦ ϕ k ) pq }{ ω ( e ) } e : q → p := { eigenvalues of ( ϕ † k +1 ◦ ϕ k +1 ) pq } Combining Construction 4.3.6 and our definition with properties for ϕ k , ϕ k +1 , the edgeweighting ω we obtained for bipartite graph Λ is d -fair and balanced. C ( K, ev K ) and End † ( M , F ) In this section,
T LJ ( d ) means the 2-shaded pivotal rigid C ∗ multitensor category fromDefinition 1.6.4 with endomorphism spaces the Temperley-Lieb algebras and simple generator X = 1 + ⊗ X ⊗ − .We have already seen the ways to construct a Markov tower from C ( K, ev K ) in this chapteror from M in § Z , where M is an indecomposable semisimple C ∗ T LJ ( d ) − module category. In this section, we will show their relation to each other. Definition 4.7.1 (Endofunctor monoidal category) . Define End † ( M ) to be a C ∗ tensor categoryas follows:(a) Objects: The objects are all the dagger endofunctors of M .(b) Morphisms: The morphisms are the uniformly bounded natural transformations betweenthese dagger endofunctors which compatible with the dagger structure.(c) Tensor structure: The tensor product is given by the composition of endofunctors, i.e., F ⊗ F := F ◦ F for endofunctors F , F . Definition 4.7.2.
Define F := − (cid:1) X, F := − (cid:1) X , which are endofunctors of M . Note that F and F are adjoint functors, with unit ev F and counit coev F induced by ev X and coev X .Define End † ( M , F ) to be the full category Cauchy generated by F and F . Since thegenerators are dualizable, the category is rigid.We warn the reader that End † ( M , F ) will only be multitensor (dim(End(id M )) < ∞ ) when M is finitely semisimple. Moreover, the dual functor on End † ( M , F ) given by ev F and coev F is not a unitary dual functor.We can give an alternative description of End † ( M , F ) using the following remark.62 emark 4.7.3. Let A be a 2-shaded rigid C ∗ multitensor category with generator X . Thefollows are equivalent [GMPPS18]:(1) M is an indecomposable semisimple C ∗ right A -module category;(2) there is a faithful dagger tensor functor φ : A →
End † ( M ), where End † ( M ) is a tensorcategory with all the dagger endofunctors being objects and uniformly bounded naturaltransformations being morphisms.We see that under this equivalence, End † ( M , F ) := φ ( A ) is the C ∗ category Cauchy tensorgenerated by the image of the tensor functor T LJ →
End † ( M ), where F = − (cid:1) X . ThenEnd † ( M , F ) is clearly a rigid C ∗ tensor category.As the end of this chapter, we are going to show that the tensor category End † ( M , F ) and2-category C ( K, ev K ) are unitarily equivalent. Construction 4.7.4.
We construct C ( K, ev K ) from End † ( M , F ) functorially.(a) Object: Let V be a set of representatives of all isomorphism classes of simple objects P ∈ M such that P = P (cid:1) + and V a set of representatives of all isomorphism classes ofsimple objects Q ∈ M such that Q = Q (cid:1) − . Then the object is the set V = V (cid:116) V .(b) 1-morphism: Let G ∈ End † ( M , F ) be an object with adjoint G . Define the V × V − bigradedHilbert space H G by H G,P Q := Hom(
Q, G ( P )) , with inner product (cid:104) f | g (cid:105) G,P Q for f, g ∈ Hom(
Q, G ( P )) defined by f † ◦ g = (cid:104) f | g (cid:105) G,P Q · id Q , since Q is simple and f † ◦ g ∈ End( Q ) ∼ = C · id Q . Note that Hom( Q, G ( P )) ∼ = Hom( G ( Q ) , P )is a natural isomorphism, so H G,QP and H G,P Q are dual Hilbert spaces.(c) Composition of 1-morphisms:
Proposition 4.7.5.
For G , G ∈ End † ( M , F ) , we have H G ◦ G ∼ = H G ◦ H G as V × V − bigraded Hilbert spaces, i.e., H G ◦ G ,P Q ∼ = ( H G ◦ H G ) P Q = ( H G ⊗ H G ) P Q = (cid:77) R H G ,P R ⊗ H G ,RQ . is a unitary isomorphism between Hilbert spaces for each pair ( P, Q ) ∈ V × V .Proof. Note that the direct sum contains finite many components. For each nonzero com-ponent with respect to R , define θ R : H G ,P R ⊗ H G ,RP → H G ◦ G ,P Q by θ R ( f ⊗ f ) := G ( f ) ◦ f . First, we prove that θ R is an isometry, i.e., (cid:104) θ ( f ⊗ f ) | θ ( g ⊗ g ) (cid:105) G ◦ G ,P Q = (cid:104) f ⊗ f | g ⊗ g (cid:105) = (cid:104) f | g (cid:105) G ,P R · (cid:104) f | g (cid:105) G ,RQ f , g ∈ H G ,P R , f , g ∈ H G ,RQ . LHS = (cid:104) G ( f ) ◦ f | G ( g ) ◦ g (cid:105) G ◦ G ,P Q = ( G ( f ) ◦ f ) † ◦ ( G ( g ) ◦ g )= f † ◦ G ( f † ◦ g ) ◦ g ( G is a dagger functor)= f † ◦ G ( (cid:104) f | g (cid:105) G ,P R · id R ) ◦ g = (cid:104) f | g (cid:105) G ,P R · f † ◦ id G ( R ) ◦ g ( G is a functor)= (cid:104) f | g (cid:105) G ,P R · f † ◦ g = RHS.
It follows that (cid:76) R θ R : (cid:76) R H G ,P R ⊗ H G ,RQ → H G ◦ G ,P Q is an isometry.Note that for a semisimple rigid C ∗ category,dim H G ◦ G ,P Q = dim Hom( Q, G ◦ G ( P ))= dim Hom( G ( Q ) , G ( P ))= dim (cid:77) R Hom( G ( Q ) , R ) ⊗ Hom(
R, G ( P ))= dim (cid:77) R Hom(
Q, G ( R )) ⊗ Hom(
R, G ( P ))= dim (cid:77) R H G ,RQ ⊗ H G ,P R = dim (cid:77) R H G ,P R ⊗ H G ,RQ . Note that (cid:76) R θ R is an isometry and hence injective, so (cid:76) R θ R : (cid:76) R H G ,P R ⊗ H G ,RQ → H G ◦ G ,P Q is a bijection and hence a unitary.It follows that H G ◦ G ◦ H G ∼ = H G ◦ G ◦ G ∼ = H G ◦ H G ◦ G as V × V − bigraded Hilbert space.(d) 1-morphism generator: Define K := H F and K := H F . It is clear that C | V | = H I + and C | V | = H I − .(e) 2-morphism: The 2-morphism of C ( K ) is the morphism of End † ( M , F ). Let α : G → G be a uniformly bounded natural transformation. Then α ( P ) : G ( P ) → G ( P ) and hence α P Q := α P ◦ − : H G ,P Q = Hom( Q, G ( P )) → Hom(
Q, G ( P )) = H G ,P Q is a uniformly bounded linear map.(f) Composition of 2-morphisms: Let α : G → G , α : G → G be uniformly boundednatural transformations. Then G ( P ) α ( P ) −−−−→ G ( P ) α −→ G ( P ), then( α ◦ α ) P Q = ( α ◦ α ) P ◦− = α ,P ◦ α ,P ◦− = α ,P Q ◦ α ,P Q : H G ,P Q → H G ,P Q → H G ,P Q . (g) Tensor product of 2-morphisms: Let α : G → G , α : G → G be uniformly boundednatural transformation. Then α ⊗ α : G ◦ G = G ⊗ G → G ⊗ G = G ◦ G defined as G ◦ G G ◦ G G ◦ G G ◦ G = ⇒ H G ⊗ H G H G ⊗ H G H G ⊗ H G H G ⊗ H G K and coev K : Define ev K to be the unit of adjoint pair ( F, F ) and coev K to be the counitof ( F, F ). Note that the duality is a property, not an extra structure. The dual functor isgenerated by the duality of generator, which is not necessarily a unitary dual functor.
Definition 4.7.6 ( [EGNO15, Def. 7.2.1]) . Let M and N be two semisimple C ∗ module categorycategories over a semisimple rigid C ∗ (multi)tensor category C . A C -module functor from M to N consists of a functor ψ : M → N and a natural isomorphism s X,M : ψ ( M (cid:1) X ) → ψ ( M ) (cid:1) X for all X ∈ C , M ∈ M which satisfies the pentagon equation.We call that M and N are C -module equivalent if ψ is an equivalence of categories.Let C = T LJ ( d ). Now we discuss the relation between the equivalence on T LJ ( d ) − modulecategory and the equivalence on End † ( M , F ), where F = − (cid:1) X , and the corresponding 2-category C ( K, ev K ). Remark 4.7.7.
Let M be an indecomposable semisimple T LJ ( d ) − module C ∗ categories and( ψ, s ) : M → M is an
T LJ ( d ) − module equivalence. Then ψ ∈ End † ( M ) is an object. Since T LJ ( d ) is generated by X , s − , − in above Definition 4.7.6 is determined by s X, − . Note that s X, − : ψ ( F ( − )) = ψ ( − (cid:1) X ) → ψ ( − ) (cid:1) X = F ( ψ ( − ))is a unitary natural isomorphism. Note that as an equivalence, ψ maps simple objects in M tosimple objects. Then we have H F,ψ ( P ) ψ ( Q ) = Hom( ψ ( Q ) , F ( ψ ( P ))) ∼ −−−−→ −◦ s − Hom( ψ ( Q ) , ψ ( F ( P ))) ∼ = Hom( Q, F ( P )) = H F,P Q . It follows that the 1-morphism generator K = H F indexed by V and H F indexed by ψ ( V ) areunitary equivalent.Comparing the discussion here with Remark 4.3.3, the T LJ ( d ) − module equivalence cor-responds to the unitary equivalence on C ( K, ev K ), which corresponds to isomorphism of edge-weighted graphs (Λ , ω ). Theorem 4.7.8.
There is a bijective correspondence between equivalence classes of the follow-ing: (cid:26)
Indecomposable semisimple C ∗ T LJ ( d ) − module categories M (cid:27) ∼ = W ∗ C (Λ , ω ) of BigHilb ,where Λ is a balanced d -fair bipartitegraph with edge-weighting ω Equivalence on the left hand side is unitary equivalence; equivalence on the right hand side isisomorphism of edge-weighted graphs.Proof.
We can prove this correspondence for the version with base point by passing throughthe Markov tower. According to Construction 4.7.4, the correspondence holds without fixingthe base point. As for the equivalence, see Remark 4.7.7.
Remark 4.7.9.
Given a semisimple C ∗ category C , similar to Construction 4.7.4, we get adagger tensor functor from End † ( C ) to the tensor category Hilb
Irr( C ) × Irr( C ) f , which is the endo-morphism tensor category of the object Irr( C ) in BigHilb . One should view this as a concreteversion of End † ( C ). Note that dualizable endofunctors always map to dualizable 1-morphisms.65 Markov lattices and biunitary connections ( d , d ) -fair square-partite graph Definition 5.1.1.
Let Γ be an oriented square-partite graph with vertices V (Γ) = V (cid:116) V (cid:116) V (cid:116) V . V V Λ V V We call that Γ associative if for any two vertices on opposite corners of Γ, there are the samenumber of length 2 paths going either way around Γ. In more details, • for any P ∈ V and R ∈ V , there are the same number of length 2 paths from P to R (or R to P ) through vertices Q ∈ V and through vertices S ∈ V ; • for any Q ∈ V and S ∈ V , there are the same number of length 2 paths from Q to S (or S to Q ) through vertices P ∈ V and through vertices R ∈ V .Let ω : E (Γ) → (0 , ∞ ) be a weighting on the edges of graph Γ.Let Λ i denote the full subgraph of Γ on V i (cid:116) V i , i = 0 ,
1; let Ω i denote the full subgraphof Γ on V i (cid:116) V i , i = 0 ,
1. Then Λ , Λ , Ω , Ω are oriented bipartite graphs.We call (Γ , ω ) a balanced ( d , d ) -fair square-partite graph if Λ , Λ are balanced d -fairbipartite graphs and Ω , Ω are balanced d -fair bipartite graphs. Remark 5.1.2.
We can define the edge-weighting preserving graph isomorphism literally thesame as in Definition 4.1.4 for balanced ( d , d )-fair square partite graph. C ( K , K , L , L , ev) of BigHilb and biunitary connection Φ Definition 5.2.1.
Let C ( K , K , L , L , ev) be a W ∗ BigHilb with four 1-morphism generators K i : V i → V i , L i : V i → V i , i = 0 , K i , L i are dualizable, i = 0 , ? := (coev ? ) † and coev ? := (ev ? ) † , where ? = K i , L i , i = 0 , d , d ) − fairness condition, namely,ev K ◦ coev K = d · id C | V | ev K ◦ coev K = d · id C | V | ev K ◦ coev K = d · id C | V | ev K ◦ coev K = d · id C | V | ev L ◦ coev L = d · id C | V | ev L ◦ coev L = d · id C | V | ev L ◦ coev L = d · id C | V | ev L ◦ coev L = d · id C | V | Notation 5.2.2.
Now, we provide the graphical calculus to describe C ( K , K , L , L , ev). Thewhite region indicates the object V , the lightest gray for V , the medium gray for V and thedarkest gray for V ; the black edge indicates K , K and red for L , L , so white and mediumgray, lightest gray and darkest gray will not be adjacent.66 oev K : C | V | → K ⊗ K ev K : K ⊗ K → C | V | coev L : C | V | → L ⊗ L ev L : L ⊗ L → C | V | Remark 5.2.3.
Similar to the discussion in § d , d )-fair square-partite graph (Γ , ω ), we can construct a 2-subcategory C (Γ , ω ) of BigHilb ; on the other hand,if we start with C ( K , K , L , L , ev), we can obtain the (Γ , ω ). Moreover, C ( K , K , L , L , ev)and C (Γ , ω ) are unitary equivalent.Similar to the discussion in Remark 4.3.3, the edge-weighting preserving graph automor-phism will result in the unitary equivalence on C (Γ , ω ).In the rest of this section, we define a special 2-morphism Φ in C ( K , K , L , L , ev), called biunitary connection . Definition 5.2.4 (Biunitary connection) . A biunitary connection Φ : K ⊗ L → L ⊗ K is a 2-morphism which is a vertical unitary and a horizontal unitary, as defined as follows. Hereis the graphical calculus of Φ.(1) The biunitary connection Φ: Φ(2) Vertical unitary: Φ † ◦ Φ = id K ⊗ id L and Φ ◦ Φ † = id L ⊗ id K .ΦΦ † = ΦΦ † =(3) Horizontal unitary:(id L ⊗ ev K ⊗ id L ) ◦ (Φ ⊗ Φ † ) ◦ (id K ⊗ coev L ⊗ id K ) = coev L ◦ ev K (id K ⊗ ev L ⊗ id K ) ◦ (Φ † ⊗ Φ) ◦ (id L ⊗ coev K ⊗ id L ) = coev K ◦ ev L . Φ Φ † = ΦΦ † =Here Φ is defined as the dual of Φ in the sense of Definition 1.7.2.67 efinition 5.2.5. C ( K , K , L , L , ev) equipped with a biunitary connection Φ is written as C ( K , K , L , L , ev; Φ) or simply C (Φ). Remark 5.2.6.
The existence of Φ implies thatdim( K ⊗ L ) uv = dim( L ⊗ K ) uv dim( K ⊗ L ) uv = dim( L ⊗ K ) uv , for each pair ( u, v ) ∈ V × V . In other word, the corresponding square-partite graph is associative.We are going to discuss some properties of biunitary connection. Definition 5.2.7 (Rotation by 90 ◦ ) . Define the rotation by 90 ◦ to beΦ r := (id K ⊗ id L ⊗ ev K ) ◦ (id K ⊗ Φ ⊗ id K ) ◦ (coev K ⊗ id L ⊗ id K ) . Similarly,Φ r := (id L ⊗ id K ⊗ ev L ) ◦ (id L ⊗ Φ r ⊗ id L ) ◦ (coev L ⊗ id K ⊗ id L ) = Φ . Φ r := Φ Φ r := Φ r Remark 5.2.8.
Here are some properties for biunitary connections and rotation.(1) The group (cid:104) r, †(cid:105) = (cid:104) r, †| r = † = id , r † = † r (cid:105) for the biunitary connection is isomorphicto the dihedral group D .(2) Φ is a biunitary connection if and only if Φ g is both vertical unitary and horizontal unitary,where g ∈ (cid:104) r, †(cid:105) . Definition 5.2.9 ( [RV16, § . We call biunitary connections Φ : K ⊗ L → L ⊗ K andΦ (cid:48) : K (cid:48) ⊗ L (cid:48) → L (cid:48) ⊗ K (cid:48) gauge equivalent , if there exist unitaries u : K (cid:48) → K , u : L → L (cid:48) , u : K → K (cid:48) and u : L (cid:48) → L such that Φ = ( u ⊗ u ) ◦ Φ ◦ ( u ⊗ u ).Φ (cid:48) = Φ u u u u Notation 5.2.10. and Observation
Observe that once we know the color of region and the color of edge, the biunitary connectionin the circle is determined. So we can simplify the graphical calculus of biunitary connectionas follows. Φ = ⇒ g , g ∈ (cid:104) r, †(cid:105) .Here are the simplified graphical calculus of vertical unitarity and horizontal unitarity. Inthe following context, We require that the leftmost regions in the uncolored equality have thesame color. = == = Proposition 5.2.11.
Here are some properties that will be used in the next section and theproof is left to the reader. (1) =(2)
For 2-morphism x ∈ End( F ⊗ K ⊗ L ) , where F is a proper -morphism, we have x F K L = x F K L L K C (Φ) to Markov lattice Construction 5.3.1.
Here we are going to construct a Markov lattice from the 2-category C (Φ) discussed above with a chosen point, say P ∈ V . Let C | P | be a 1-morphism with all theentry being 0 except ( C | P | ) P P = C .Note that C | P | ⊗ K alt ⊗ i ⊗ L alt ⊗ j ? is a 1-morphism for each i, j ∈ Z ≥ .Let M i,j = End (cid:16) C | P | ⊗ K alt ⊗ i ⊗ L alt ⊗ j ? (cid:17) , where L ? = L if 2 | i and L ? = L if 2 (cid:45) j . Weuse the graphical calculus to show M = ( M i,j ) i,j ≥ is a Markov lattice.691) Element x ∈ M i,j : · · ·· · ·· · ·· · · x C | P | st i th st j th P (2) Horizontal inclusion x ∈ M i,j ⊂ M i,j +1 : · · ·· · ·· · ·· · · x C | P | st i th st j th ( j +1) th P (3) Vertical inclusion x ∈ M i,j ⊂ M i +1 ,j : · · ·· · ·· · ·· · · x C | P | st i th st j th ( i +1) th P (4) Horizontal conditional expectation E M,ri,j : M i,j → M i,j − , , x ∈ M i,j : E M,ri,j ( x ) = d − · · ·· · ·· · ·· · · x C | P | st i th st ( j − th j th P (5) Vertical conditional expectation E M,li,j : M i,j → M i − ,j , x ∈ M i,j : E M,li,j ( x ) = d − · · ·· · ·· · ·· · · x C | P | st ( i − th st j th i th P E M,ri − ,j ◦ E M,li − ,j − = E M,li − ,j ◦ E M,ri,j : M i,j → M i − ,j − , x ∈ M i,j : E M,ri − ,j ◦ E M,li − ,j − ( x ) = d − d − · · ·· · ·· · ·· · · x C | P | st ( i − th st ( j − th j th i th P = d − d − · · ·· · ·· · ·· · · x C | P | st ( i − th st ( j − th j th i th P = E M,li − ,j ◦ E M,ri,j ( x )(7) Vertical Jones projections e i ∈ M i +1 ,j and horizontal Jones projection f j ∈ M i,j +1 : e i = d − · · ·· · ·· · ·· · · C | P | st i th st j th P f j = d − · · ·· · ·· · ·· · · C | P | st i th st j th P (8) It is clear that M j = ( M i,j , E M,li,j , e i ) i ≥ are Markov towers with the same modulus d and e i ∈ M i +1 ,j for all i , i, j = 0 , , , · · · ; M i = ( M i,j , E M,ri,j , f j ) j ≥ are Markov towers with thesame modulus d and f j ∈ M i,j +1 for all j . Remark 5.3.2.
A gauge equivalence Φ ∼ Φ (cid:48) will result in an isomorphism of the correspondingMarkov lattices. C (Γ , ω ; Φ) First, we are going to explore more properties of Markov lattice.
Proposition 5.4.1. (a) X i +1 ,j +1 := (cid:104) e i , f j (cid:105) is a 2-sided ideal of M i +1 ,j +1 and hence M i +1 ,j +1 can split as a directsum of von Neumann algebras X i +1 ,j +1 ⊕ Y i +1 ,j +1 . We also define Y , = M , , Y , = M , , Y , = M , , Y , = M , so that X , = X , = X , = X , = 0 . X i +1 ,j +1 is called theold stuff and Y i +1 ,j +1 is called the new stuff. (b) If y ∈ Y i +1 ,j +1 and x ∈ X i +1 ,j or x ∈ X i,j +1 , then yx = 0 in M i +1 ,j +1 . Hence E ri +1 ,j +1 ( Y i +1 ,j +1 ) ⊂ Y i +1 ,j and E li +1 ,j +1 ( Y i +1 ,j +1 ) ⊂ Y i,j +1 , which means the new stuff comes from the old newstuff. (c) If Y i,j = 0 , then Y k,l = 0 for all k ≥ i, l ≥ j .Proof. Similar to Proposition 4.5.2.Now we are going to construct C (Γ , ω ; Φ) from a given Markov lattice M . Construction 5.4.2.
The square partite graph and the edge weighting (Γ , ω ):From Markov lattice M , since each row and column is a Markov tower, we can obtain aBratteli diagram ∆ as in § ∩ Y i,j and the edges between them, we obtain the principal graph Γ is not necessary a square-partite graph, so we have todo some identification.For the new vertices p ∈ Γ ∩ Y i,j and p ∈ Γ ∩ Y i +2 ,j − , as in § p (cid:48) be the new oldvertex of p in M i +2 ,j and p (cid:48) be the new old vertex of p in M i +2 ,j . We identify p with p if p (cid:48) ∈ M i +2 ,j p (cid:48) (or equivalently p (cid:48) ∈ M i +2 ,j p (cid:48) ).For the pairs of new vertices p ∈ Γ ∩ Y i,j and q ∈ Γ ∩ Y i +1 ,j , and the pairs of new vertices p ∈ Γ ∩ Y i +2 ,j − and q ∈ Γ ∩ Y i +3 ,j − , suppose p and p are identified in M i +2 ,j , q and q are identified in M i +3 ,j on above sense, then the numbers of edges between p , q and p , q areequal, since they both equal to(dim C ( p (cid:48) q (cid:48) M (cid:48) i +2 ,j p (cid:48) q (cid:48) ∩ p (cid:48) q (cid:48) M i +3 ,j p (cid:48) q (cid:48) )) , see the discussion in § p , q and p , q . Similarstatement for p ∈ Γ ∩ Y i,j and r ∈ Γ ∩ Y i,j +1 , and the pairs of new vertices p ∈ Γ ∩ Y i +2 ,j − and r ∈ Γ ∩ Y i +2 ,j − . After above identification as well as the edges between those identifiedvertices (see following example), we obtain a graph Γ, which is a square-partite graph.Then V ij ⊂ V (Γ) contains all the vertices in V (Γ ) ∩ M i +2 m,j +2 n , i, j = 0 , m, n ∈ Z ≥ .The edge-weighting ω can be obtained the same way as in § Example 5.4.3.
Here we provide an example to see the difference between the square-partitegraph and the principal graph of a Markov lattice. In the diagram below, if p is at depth zero,then p is at depth 2 of the principal graph. Therefore, as a new vertex, p will appear in twoplaces M , and M , , but their reflections/new old vertices coincide in M , . p p p p p p square-partite graph = ⇒ p p p p p p p p p p p p p p p p p p p p p p p principal graph with base point p and Bratteli diagram Remark 5.4.4.
Suppose vertex q ∈ V is at depth 2 n of the principal graph, then q will firstappear in M i, n − i , i = 0 , , · · · , n ; if q ∈ V is at depth 2 n + 1, then q will first appearin M i +1 , n − i , i = 0 , , · · · , n ; if q ∈ V is at depth 2 n + 1, then q will first appear in M i, n +1 − i , i = 0 , , · · · , n ; if q ∈ V is at depth 2 n +2, then q will first appear in M i +1 , n +1 − i , i = 0 , , · · · , n .Next, we compute the biunitary connection Φ. Notation 5.4.5. and Observation
We choose p ∈ V as the base point, which is at depth0. Similar to Observation 4.6.2, denote Λ ,n to be the subgraph of Λ with vertices depth ≤ n ,similar definition for Ω ,n , Λ ,n and Ω ,n , see Definition 5.1.1. The corresponding Hilb -enriched72raphs are K i,n := K Λ i,n , L i,n := L Ω i,n . From Construction 5.3.1, N i,j := End( C | p | ⊗ K alt ⊗ i ⊗ L alt ⊗ j ? ). WLOG, let 2 (cid:45) i . Observe that N i,j = End( K , ⊗ K , ⊗ · · · K ,i ⊗ L ,i +1 ⊗ L ,i +2 ⊗ · · · ⊗ L ?1 ,i + j ) , where L ?1 ,j = L ,j if 2 (cid:45) j , L ?1 ,j = L ,j if 2 | j . Example 5.4.6.
Following Example 5.4.3, K , K , K , K , K , K , K , K , K , K , K , K , L , L , L , L , L , L , L , L , L , L , L , L , N , N , N , we have K , = C K , = C C K , = C C K , = C C L , = C L , = C C K , ⊗ K , ⊗ L , = C ∼ = K , ⊗ L , ⊗ K , ∼ = L , ⊗ K , ⊗ K , Similar to Example 4.6.3, the entry ( i, j ) in N m,n indicates number of paths from the vertex p i at depth 0 to the vertex p j at depth m + n . Note that the base point is a single vertex p ,so only at entry (1 , j ) can be nonzero. 73 emark 5.4.7. Any automorphism of M n ( C ) is inner. To be precise, if α ∈ Aut( M n ( C )), thenthere exists a unitary u ∈ M n ( C ), such that α ( x ) = uxu ∗ = Ad( u )( x ), for any x ∈ M n ( C ).Moreover, this unitary u is unique up to a unit scalar. Indeed, if uxu ∗ = u xu ∗ for all x ∈ M n ( C ),then x ( u ∗ u ) = ( u ∗ u ) x , which implies that u ∗ u is in the center of M n ( C ). Thus, u ∗ u = a ∈ C with | a | = 1 and hence u = au .As a corollary, for 1-morphisms H, G , if α : End( H ) ∼ = End( G ) is a ∗ -isomorphism, thenthere exists a unitary 2-morphism u : H → G such that α = Ad( u ). Warning : the unitary u is obtained by taking a unitary u i,j in each entry. Thus any two choicesof implementing unitary u = ( u i,j ) and v = ( v i,j ) differ by a matrix of scalars ( a i,j ) which maybe distinct. Hence the unitary u is unique up to a matrix of scalars. Construction 5.4.8.
The biunitary connection Φ: The construction (for the tracial case) hasbeen written in [JS97, § C (Γ , ω ) can be constructed.In order to obtain the biunitary connection Φ, we shall compute it componentwise, whichis similar to the idea to compute the edge-weighting in § pr :( K ⊗ L ) pr = (cid:76) q ∈ V K ,pq ⊗ L ,qr → (cid:76) s ∈ V L ,ps ⊗ K ,sr = ( L ⊗ K ) pr for each pair( p, r ) ∈ V × V .Suppose p is at depth 2 n of the principal graph and r is at depth 2 n + 2. By Remark 5.4.4, p first appear in M , n and r first appears in M , n +1 .Consider two path models M , ⊂ M , ⊂ · · · ⊂ M , n ⊂ M , n +1 ⊂ M , n +1 and M , ⊂ M , ⊂ · · · ⊂ M , n ⊂ M , n ⊂ M , n +1 .Similar to Proposition 4.6.4, we have N (cid:48) , n ∩ N , n +1 = id K , ⊗ K , ⊗···⊗ K , n ⊗ End( K , n +1 ⊗ L , n +1 ) for the first model N (cid:48) , n ∩ N , n +1 = id K , ⊗ K , ⊗···⊗ K , n − ⊗ End( L , n ⊗ K , n +1 ) for the second model . Let ψ : M , n +1 → N , n +1 denote the ∗ -isomorphism onto the first model and ψ (cid:48) : M , n +1 → N , n +1 denote the ∗ -isomorphism onto the second model, then ψ : M (cid:48) , n ∩ M , n +1 → N (cid:48) , n ∩ N , n +1 ∼ = End( K , n +1 ⊗ L , n +1 ) ψ (cid:48) : M (cid:48) , n ∩ M , n +1 → N (cid:48) , n ∩ N , n +1 ∼ = End( L , n ⊗ K , n +1 ) . are ∗ -isomorphisms. Then ψ (cid:48) ◦ ψ − : End( K , n +1 ⊗ L , n +1 ) → End( L , n ⊗ K , n +1 ) is a ∗ isomorphism between two 1-morphisms. By Remark 5.4.7, their exists a unique unitary u upto a matrix of scalars such that ψ (cid:48) ◦ ψ − = Ad( u ). We define Φ pr := u pr .Similar to Remark 4.3.7, we secretly make a choice of ONB when we construct the generators K i , L j from the square-partite graph Γ, i, j = 0 ,
1. Different choice results in multiplying a uni-tary on each generator. Combining Definition 5.2.9 of gauge equivalence and above discussion,the biunitary connection Φ we construct here is unique up to gauge equivalence. C (Φ) and End † ( M , F, G ) We have already seen the method to construct a Markov lattice from C (Φ) above or from M in § M is an indecomposable semisimple C ∗ A − B bimodulecategory. In this section, by using the similar technique as in § efinition 5.5.1. Suppose M is an indecomposable semisimple C ∗ T LJ ( d ) − T LJ ( d ) bi-module category, where X = 1 + ⊗ X ⊗ − , Y = 1 + ⊗ Y ⊗ − are the generators of T LJ ( d )and T LJ ( d ) respectively. Define F = X (cid:3) − , F = X (cid:3) − , G = − (cid:1) Y , G = − (cid:1) Y , which areendofunctors on M . Note that ( F, F ) and (
G, G ) are adjoint pairs, with unit ev F , ev G inducedby ev X , ev Y and counit coev F , coev G induced by coev X , coev Y .Define End † ( M , F, G ) to be the full subcategory of End † ( M ) Cauchy tensor generated by F, F , G, G , so it is a rigid C ∗ tensor category.We warn the reader that End † ( M , F, G ) will only be multitensor (dim(End(id M )) < ∞ )when M is finitely semisimple. Definition 5.5.2 (Biunitary connection in End † ( M , F, G )) . Note that the bimodule associator α X, − ,Y : ( X (cid:3) − ) (cid:1) Y → X (cid:3) ( − (cid:1) Y ) is a unitary, which induces a natural isomorphismΦ F,G : F ⊗ G → G ⊗ F , where F ⊗ G := G ◦ F . Then Φ G,F : G ⊗ F → F ⊗ G is equal to the90 ◦ rotation Φ rF,G defined as follows:Φ rF,G := (id F ⊗ id G ⊗ ev F ) ◦ (id F ⊗ Φ F,G ⊗ id F ) ◦ (coev F ⊗ id G ⊗ id F ) . It is easy to show that Φ
F,G is vertical and horizontal unitary and so is Φ
G,F .Similar to § † ( M , F, G ) and 2-category C (Φ)are unitarily equivalent. Construction 5.5.3.
We construct C (Φ) from End † ( M , F, G ) functorially.(a) Let V be a set of representatives of all simple objects P ∈ M such that P = 1 + (cid:3) P (cid:1) + ; V be the set of representatives of all simple objects Q ∈ M such that Q = 1 − (cid:3) Q (cid:1) + ; V be the set of representatives of all simple objects R ∈ M such that R = 1 − (cid:3) R (cid:1) − ; V be the set of representatives of all simple objects S ∈ M such that S = 1 + (cid:3) S (cid:1) − .Then the objects are the sets V i,j , i, j = 0 , V = V (cid:116) V (cid:116) V (cid:116) V .(b) 1-morphism: The 1-morphism of C (Φ) is the object of End † ( M , F, G ). The way to constructthe corresponding V × V -bigraded Hilbert space from an endofunctor is the same as inConstruction 4.7.4. The same for the dual 1-morphism and tensor structure/composition.(c) 2-morphism: The 2-morphism of C (Φ) is the morphism of End † ( M , F, G ).(d) 1-morphism generator: Define K := H J + ⊗ H F K = H J + ⊗ H F K := H J − ⊗ H F K = H J − ⊗ H F L := H I + ⊗ H G L = H I + ⊗ H G L := H I − ⊗ H G L = H I − ⊗ H G (e) ev and coev. The same as in Construction 4.7.4(h).(f) Biunitary connection: Φ : K ⊗ L → L ⊗ K is defined as Φ F,G : F ⊗ G → G ⊗ F . Thecheck that Φ is vertical and horizontal unitary is left to the reader. Construction 5.5.4.
For the convenience to the reader, we also provide the construction from C (Φ) to End † ( M , F, G ):(a) Object: The object are the 1-morphisms in C (Φ). In particular, the generator F = K ⊕ K , F = K ⊕ K , G = L ⊕ L and G = L ⊕ L ; the unit I + = 1 + (cid:3) − = C | V (cid:116) V | , I − = 1 − (cid:3) − = C | V (cid:116) V | , J + = − (cid:1) + = C | V (cid:116) V | and J − = − (cid:1) − = C | V (cid:116) V | .(b) Morphism: The morphisms are the 2-morphisms in C (Φ).(c) The associator: Note that F ⊗ G = ( K ⊕ K ) ⊗ ( L ⊕ L ) = K ⊗ L and G ⊗ F =( L ⊕ L ) ⊗ ( K ⊕ K ) = L ⊗ K , the associator Φ F,G : F ⊗ G → G ⊗ F is defined as thebiunitary connection Φ : K ⊗ L → L ⊗ K . All the 8 cases of associators are defined asΦ g , where g ∈ (cid:104) r, †(cid:105) . 75 heorem 5.5.5. There is a bijective correspondence between equivalence classes of the follow-ing:
Indecomposable semisimple C ∗ T LJ ( d ) − T LJ ( d ) bimodulecategories M ∼ = W ∗ C (Γ , ω ; Φ) of BigHilb ,where Γ is a balanced ( d , d )-fair squarepartite graph with edge-weighting ω and Φa biunitary connection Equivalence on the left hand side is unitary equivalence; equivalence on the right hand sideis isomorphism on the edge-weighted square-partite graph and gauge equivalence on biunitaryconnection.Proof.
We can prove this correspondence for the version with base point by using the Markovlattice. According to Construction 5.5.3, the correspondence holds without fixing the base point.As for the equivalence, combining Remark 5.2.3, Definition 5.2.9 and the last paragraph inConstruction 5.4.8, the isomorphism on the edge-weighted graph (Γ , ω ) and gauge equivalence onΦ corresponds to the unitary equivalence on C (Φ), which corresponds to the unitary equivalenceon T LJ ( d ) − T LJ ( d ) bimodule category M based on Construction 5.4.8 and Remark 4.7.7. In this chapter, we finally discuss the tracial/pivotal case for (bi)module categories. As anapplication, we prove the module embedding theorem for (infinite depth) graph planar algebra.
Definition 6.1.1. [Sc13] Let C be a rigid C* (multi)tensor category with the canonical sphericalunitary dual functor. We call M a semisimple pivotal C* C− module category, if there exists apivotal trace tr M compatible with the spherical structure on C , i.e.,tr M m (cid:1) c ( f ) = tr M m ((id m (cid:1) coev † c ) ◦ ( f (cid:1) id c ) ◦ (id m (cid:1) coev c )) , for all f ∈ End( m (cid:1) c ), where m ∈ M , c ∈ C . Remark 6.1.2. If f ∈ End( c ) , c ∈ C and m ∈ M ,tr M m (cid:1) c (id m (cid:1) f ) = tr M m ((id m (cid:1) coev † c ) ◦ ((id m (cid:1) f ) (cid:1) id c ) ◦ (id m (cid:1) coev c ))= tr M m (id m (cid:1) (coev † c ◦ ( f (cid:1) id c ◦ coev c )))= tr M m (id m (cid:1) tr A c ( f ))= tr M m (id m ) · tr A c ( f ) . Here we call tr M m (id m ) the dimension of object m . Remark 6.1.3. [Sc13, § C is fusion and M is indecomposable, then the pivotal tracetr M is unique up to scalar. Definition 6.1.4 (Tracial Markov tower) . We call M a tracial Markov tower if M a Markovtower equipped with a unital trace tr on (cid:83) n ≥ M n and the conditional expectation E n aretrace-preserving, i.e., tr ◦ E n = tron M n , n ≥
0. 76 efinition 6.1.5.
We call M a tracial standard A − module, where A is a standard λ -lattice, iftr M | A = tr A and M is a standard A − module, see Definition 2.1.1.Let A be a standard λ -lattice. If we start with a tracial standard A -module M , combiningthe construction in § A − module category. Furthermore, from this pivotal planar A − module category, wecan construct an indecomposable semisimple pivotal C* A− module category with a choice ofsimple base object. The following is the theorem. Theorem 6.1.6.
There is a bijective correspondence between equivalence classes of the follow-ing:
Tracial Markov towers M =( M i ) i ≥ with dim( M ) = 1 asstandard right modules over astandard λ -lattice A ∼ = Pairs ( M , Z ) with M an indecomposablesemisimple pivotal C* right A− module cat-egory together with a choice of simple baseobject Z = Z (cid:1) + A Equivalence on the left hand side is trace-preserving ∗ -isomorphism on the tracial Markov toweras standard A − module; equivalence on the right hand side is the pivotal unitary A− moduleequivalence on their Cauchy completions which maps simple base object to simple base object. Let us look at the balanced d -fair bipartite graph (Λ , ω ) from the tracial Markov tower M .Since the evaluation and coevaluation are compatible with the trace, the edge-weighting comesfrom a vertex-weighting, see [JP19, Prop. 6.8]. To be precise, Definition 6.1.7 (Vertex weighting) . Let Λ be a bipartite graph. Let ν : V (Λ) → (0 , ∞ )be a weighting on the vertices of Λ which satisfies the Frobenius-Perron condition: for each P ∈ V (Λ), (cid:88) { Q ∈ V (Λ): P,Q adjacent } ν ( Q ) = d · ν ( P ) . In the sum on the left hand side, ν ( Q ) has number of edges between P → Q copies.From an undirected bipartite graph, one can obtain a directed graph with involution [HP17,Def. 2.20]. Then for e : P → Q , define w ( e ) := ν ( P ) ν ( Q ) . The d -fairness and balance condition inDefinition 4.1.2 follows automatically. Remark 6.1.8.
Suppose M is an indecomposable semisimple C ∗ pivotal A− module categorywith fusion/principal graph Λ whose vertices are simple objects of M . We can define the vertexweighting for simple object P as ν ( P ) := Tr P (id P ). Remark 6.1.9.
Note that M being a pivotal A− module is equivalent to the dagger ten-sor functor A →
End † ( M ) being pivotal [GMPPS18, Thm. 3.70], so that its essential imageEnd † ( M , F ) has a unitary pivotal structure from the pivotal structure in A , where F = − (cid:1) X isthe generator. We also denote the corresponding 2-subcategory of BigHilb as C ( K, φ ) or C (Λ , ν ). Jones’ planar algebra, as a form of standard invariant, is a method to construct and classifyfinite index type II subfactors. The module embedding theorem has been known to VaughanJones since he first defined the graph planar algebra [Jo00]. The proof for finite depth caseappears in [JP10, CHPS18, GMPPS18]. Many nontrivial examples of subfactors have been77onstructed inspired by this theorem, including the Extended Haagerup subfactor and its rela-tives [BPMS12, GMPPS18].In our setting, the bipartite graph Λ can be infinite depth. We refer the reader to [Bu10]for the definition of the infinite depth bipartite graph planar algebra. Theorem 6.2.1.
The planar algebra constructed from
End † ( M , F ) with generator F mentionedin Remark 6.1.9 is isomorphic to the graph planar algebra of bipartite graph Λ , where M is anindecomposable semisimple pivotal C ∗ A− module category, A is a 2-shaded rigid C ∗ multitensorcategory with generator X = 1 + ⊗ X ⊗ − , Λ is the (possibly infinite) fusion graph for M withrespect to the generator X , where the vertex weighting ν on Λ comes from the trace Tr M as inRemark 6.1.8.Proof. Here we provide the sketch of the proof. From the unitary pivotal dagger functor
A →
End † ( M ), we obtain a rigid C ∗ tensor category End † ( M , F ) with pivotal structure withgenerator F = − (cid:1) X in the sense of § § § † ( M , F ), we can construct the 2-category C (Λ , ν )discussed in Remark 6.1.9 with its generating Hilb -enriched graph Λ, which is equivalent in-formation. Similar to [GMPPS18, § C (Λ , ν ) with generator Λ is ∗ -isomorphic to the graph planar algebra G • (in the sense of Burstein [Bu10]) of the fusiongraph Λ with vertex weighting ν , which corresponds to F in the sense of Remark 6.1.8.Note that there is a well-know correspondence between [Gh11, DGG14, Pe18]: (cid:26) Subfactor planaralgebras P • (cid:27) ∼ = Pairs ( A , X ) with A a 2-shaded rigid C ∗ multitensorcategory with a generator X , i.e., 1 A = 1 + ⊕ − , 1 + , − are simple and X = 1 + ⊗ X ⊗ − Finally, the pivotal dagger tensor functor
A →
End † ( M , F ) gives a planar algebra em-bedding from the subfactor planar algebra A • to the graph planar algebra G • of its principalgraph.If we choose M = A + = 1 + ⊗ A ⊗ + to be the A− module category, we obtain the moduleembedding theorem: Corollary 6.2.2.
Every subfactor planar algebra P • embeds into the graph planar algebra ofits principal graph. Definition 6.3.1.
Let C , D be rigid C ∗ (multi)tensor categories with canonical unitary dualfunctors respectively. We call M a semisimple pivotal C ∗ C − D bimodule category, if thereexists a pivotal trace tr M compatible with the spherical structures in C and D , i.e.,tr M a (cid:3) m ( f ) = tr M m ((ev † a (cid:3) id m ) ◦ (id a (cid:3) f ) ◦ (ev a (cid:3) id m ))tr M m (cid:1) b ( f ) = tr M m ((id m (cid:1) coev † b ) ◦ ( f (cid:1) id b ) ◦ (id m (cid:1) coev b )) , for f ∈ End( a (cid:3) m (cid:1) b ), where m ∈ M , a ∈ C , b ∈ D . Definition 6.3.2. (Tracial Markov lattice) We call M a tracial Markov lattice if M is a Markovlattice equipped with a unital trace tr on (cid:83) i,j ≥ M i,j and the conditional expectation E M,li,j , E
M,ri,j are trace-preserving, i.e., tr ◦ E M,li,j = tr , tr ◦ E M,ri,j = tron M i,j , i, j ≥
0. 78 efinition 6.3.3.
We call M a tracial standard A − B bimodule, where A, B are standard λ -lattices, if tr M | A = tr A , tr M | B = tr B and M is a standard A − B bimodule, see Definition3.2.1.Similar to Theorem 6.1.6, we have the following theorem: Theorem 6.3.4.
There is a bijective correspondence between equivalence classes of the follow-ing:
Tracial Markov lattice M =( M i,j ) i,j ≥ with dim( M , ) = 1as a standard A − B bimoduleover standard λ -lattices A, B ∼ = Pairs ( M , Z ) with M an indecomposablesemisimple C ∗ pivotal A − B bimodule cat-egory together with a choice of simple baseobject Z = 1 + A (cid:3) Z (cid:1) + B Equivalence on the left hand side is the trace-preserving ∗ -isomorphism on the tracial Markovlattice as standard A − B bimodule; equivalence on the right hand side is the pivotal unitary A − B bimodule equivalence between their Cauchy completions which maps the simple base objectto simple base object.
Let us look at the balanced ( d , d )-fair square-partite graph (Λ , ω ) from the tracial Markovlattice M . Similar to the tracial Markov tower case, the edge-weighting comes from the vertex-weighting. To be precise,For P ∈ V (cid:116) V , (cid:88) { e : P → Q : Q ∈ V (cid:116) V } ν ( Q ) = d · ν ( P )For P ∈ V (cid:116) V , (cid:88) { e : P → Q : Q ∈ V (cid:116) V } ν ( Q ) = d · ν ( P ) . Remark 6.3.5.
As for the biunitary connection, the computation does not change at all. Infact, the biunitary connection is independent of the pivotal structure, see Proposition 5.2.11(2)and § References [AV15] Arano, Yuki; Vaes, Stefaan. C ∗ -tensor categories and subfactors for totally disconnected groups. Operator algebras and applications—the Abel Symposium 2015, 1–43, Abel Symp., 12, Springer,2017.
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