# A Level-Depth Correspondence between Verlinde Rings and Subfactors

Jun Yang Harvard University, Cambridge, MA 02138, USA

###### Abstract

We establish a correspondence between the levels of Verlinde rings and the
depths of subfactors. Given the $l$-level Verlinde ring $R_{l}(G)$ of a simple
compact Lie group $G$, the tensor products of fundamental representations give
us the inclusion of a pair of $\text{II}_{1}$ factors $N\subset M$. For the
depth $d$ of $N\subset M$, we first prove $d=l$ for type $A_{n},C_{n}$ and
$B_{2}$. More generally, the depth $d$ is shown to satisfy

$\beta\cdot l\leq d\leq l$ with $\beta\in(0,1)$,

where $\beta$ is uniquely determined by the simple type of $G$. We also show
that the simple $N$-$N$-bimodules contained in $L^{2}(M)$ generate the
Verlinde ring $R_{l}(G)$ as its fusion category.

††✉<EMAIL_ADDRESS>
###### Contents

  1. 1 Introduction
  2. 2 The Verlinde Ring as a Quotient
  3. 3 Tensor Products of Fundamental Representations
  4. 4 Towers of Finite-Dimensional Algebras and Subfactors
  5. 5 The Commutants
  6. 6 The Bimodules and Their Fusion Rule

##  1 Introduction

The Verlinde ring is a fusion category arising from the positive energy
representation of loop groups [24] and also from the representation theory of
quantum groups [1]. There are several distinct approaches to the Verlinde
ring, i.e., from the view of algebraic geometry by G. Faltings [9], operator
algebra by A. Wassermann [27], and twisted K-theory by D. Freed, M. Hopkins
and C. Teleman [11]. It can be shown to be isomorphic to a quotient of the
representation ring $R(G)$ of a compact simple Lie group $G$, or,
equivalently, the representation ring $R(\mathfrak{g})$ of the corresponding
simple complex Lie algebra $\mathfrak{g}$. The kernel of this quotient is
uniquely determined by a positive integer $l$, which is usually called the
level.

Generally speaking, a fusion category usually has a strong correspondence with
subfactors, which denotes the inclusion of a pair of von Neumann algebras of
type $\text{II}_{1}$ with trivial centers (factors). More precisely, we are
always able to construct an inclusion pair of factors $N\subset M$ such that a
certain tensor category within it is isomorphic to a given tensor category
$\mathcal{C}$. T. Hayashi and S. Yamagami [13] first realized all the
$C^{*}$-tensor categories of bimodules over the hyperfinite $\text{II}_{1}$
factor. S. Falguières and S. Vaes [8] showed the representation category of
any compact group arises from the finite index bimodules of some
$\text{II}_{1}$ factor. Then S. Falguières and S. Raum [7] treated the finite
$C^{*}$-tensor category as well. Conversely, starting with a given tensor
category, one can also generate subfactors. S. Sawin [25] first obtained
subfactors from quantum groups with parameters that are not the roots of
unity. For the case of roots of unity, H. Wenzl [28] constructed subfactors
from the tensor product of a module (and its dual) over quantum groups while
F. Xu [29] constructed subfactors through quantum groups and the
$\lambda$-lattices (see S. Popa’s axiomatization [23]).

For subfactors, there is a positive integer called the depth, denoted $d$,
which gives us much information about the inclusion. Given a subfactor
$N\subset M$, one can iterate the basic construction, which plays a central
role in the study of the index $[M:N]$ by V. Jones [16]. We then obtain a
tower of $\text{II}_{1}$ factors:

$N\subset M=M_{1}\subset M_{2}=\langle M_{1},e_{1}\rangle\subset M_{3}=\langle
M_{2},e_{2}\rangle\subset\cdots$,

where $e_{k}$ is the projection $L^{2}(M_{k})\to L^{2}(M_{k-1})$. The depth
$d$ is then defined to be the minimal integer $k$ such that the center of the
commutant $N^{\prime}\cap M_{k}$ has its maximal dimension.

This paper aims to give a subtle construction of subfactors from a Verlinde
ring at level $l$ with the depth equal (or proportional) to $l$. Let
$R_{l}(G)$ be the $l$-level Verlinde ring of a simple simply-connected compact
Lie group $G$.

######  Theorem 1.1.

There is a subfactor $N\subset M$ constructed from the fundamental
representations of $G$ with the depth $d(l)$ which satisfies

$\beta\cdot l\leq d(l)\leq l$ with $\beta\in(0,1)$

for all simple types of $G$. Moreover, $d(l)=l$ for type $A_{n}$, $C_{n}$ and
$B_{2}$.

The motivation for this result originates from joint work [19] of the author
with V. Jones on Motzkin algebras, which can be constructed from
$\operatorname{End}_{G}((V_{0}\oplus V_{1})^{\otimes k})$ with $V_{0},V_{1}$
the trivial and fundamental representation of $G=SU(2)$ or
$U_{q}(\mathfrak{gl}_{2})$ (see also [3]). Actually, their work contains an
equivalent definition based on planar algebra [17]. For any level $l\geq 3$,
they construct a subfactor $N\subset M$ of depth $l$ such that the bimodules
generated from ${}_{N}L^{2}(M)_{N}$ are the $l$-level Verlinde ring of
$SU(2)$, or equivalently, of type $A_{1}$.

In this paper, we generalize this result to simple simply-connected compact
Lie groups or simple complex Lie algebras. We also start with the
$\mathfrak{g}$-module $V_{0}\oplus(\oplus_{i}V(\omega_{i}))$, where $V_{0}$ is
the trivial module and each $V(\omega_{i})$ the is the irreducible
representation with the fundamental weight $\omega_{i}$. It involves the
Littlewood-Richardson problem in studying the decomposition of the tensor
product of the highest weight representations (see [20]). As the trivial
module is included, we obtain an increasing sequence of weights set which will
finally contain all the weights $\lambda$ such that $(\lambda,\theta)\leq l$,
which are the weights in $R_{l}(G)$ ($\theta$ is the highest root). The tower
of the endomorphism algebras gives us a family of factors and also the
commutants of the subfactors. The bimodules are then constructed in a
canonical way from these commutants. We show the bimodules from $N\subset M$
have the same fusion rule as $R_{l}(G)$.

######  Corollary 1.2.

The tensor category generated by the $N$-$N$ bimodules in $L^{2}(M)$ is the
Verlinde ring $R_{l}(G)$.

In Section 2, we have a brief review of the Verlinde rings. In Section 3, we
construct the Verlinde ring from the direct sum of fundamental
representations. In Section 4 and Section 5, we construct the subfactors and
describe the commutants. In Section 6, we construct a family of bimodules and
describe their fusion rule.

Acknowledgements The author is grateful for the comments from D. Bisch, A.
Jaffe, and Y. Kawahigashi. This work was supported in part by the ARO Grant
W911NF-19-1-0302 and the ARO MURI Grant W911NF-20-1-0082.

##  2 The Verlinde Ring as a Quotient

We first have a short review of some facts about complex semisimple Lie
algebra. We mainly refer to [14], for the basic Lie theory and to [2, 21] for
the Verlinde rings.

Let $G$ be a compact, simply-connected, simple Lie group and
$\mathfrak{g}=\mathfrak{g}_{\mathbb{C}}$ be the complexified simple Lie
algebra. Let $\mathfrak{t}$ be the Cartan subalgebra of $\mathfrak{g}$. Denote
the set of integral weights and the set of dominant integral weights by $P$
and $D$ respectively. Let $\Phi$ be the set of roots and
$\Delta=\\{\alpha_{i},\dots,\alpha_{n}\\}$ be the set of simple roots, where
$n=\dim_{\mathbb{C}}\mathfrak{t}$. Let $W=\langle
s_{\alpha_{1}},\dots,s_{\alpha_{n}}\rangle$ be the Weyl group with each
$s_{\alpha_{i}}$ the reflection given by the simple root $\alpha_{i}$.

Let $\theta$ be the highest root and $\rho$ be the half-sum of positive roots.
Let $(\cdot,\cdot)$ be the inner product on
$\mathfrak{t}\cong\mathfrak{t}^{*}$ which is normalized in the sense
$\|\theta\|^{2}=(\theta,\theta)=2$. Let
$\alpha^{\vee}\colon=\frac{2\alpha}{(\alpha,\alpha)}$ be the coroot of
$\alpha\in\Phi$. Define
$\langle\beta,\alpha\rangle=(\beta,\alpha^{\vee})=\frac{2(\beta,\alpha)}{(\alpha,\alpha)}$
for $\alpha,\beta\in\Phi$ (and also defined on $P$). Let
$\omega_{1},\dots,\omega_{n}$ be the fundamental weights, i.e.,
$\langle\omega_{i},\alpha_{j}\rangle=\delta_{i,j}$ for all $1\leq i,j\leq n$.

Define $R(G)$ (or $R(\mathfrak{g})$) to be the representation ring of $G$ (or
$\mathfrak{g}$). It is well-known that $R(G)\cong
R(\mathfrak{g})=\mathbb{Z}[D]$, i.e., the $\mathbb{Z}$-linear span of the
isomorphism classes of highest weight representations indexed by $D$.

Let $V(\lambda)$ be the irreducible representation with the highest weight
$\lambda$, which will also stand for its isomorphism class in
$R(\mathfrak{g})$. For a finite-dimensional $V$ representation of
$\mathfrak{g}$, we let

$\displaystyle\Pi(V)=\text{the set of all weights of }V;$
$\displaystyle\Pi_{h}(V)=\text{the set of all highest weights of the simple
summands of }V.$

For instance, if $V=\oplus_{\lambda\in D}m_{\lambda}\cdot V(\lambda)$ as the
decomposition into irreducible representations, we have
$\Pi_{h}(V)=\\{\lambda\in D|m_{\lambda}\neq 0\\}$. For each $1\leq i\leq n$,
let $V(\omega_{i})$ be the fundamental representation, which is the
irreducible representation with the highest weight $\omega_{i}$.

Given an integer $l\geq 1$, we define

  * •

the dominant integral weights at level $l$

$D_{l}=\\{\lambda\in D|(\lambda,\theta)\leq l\\}$,

  * •

the affine wall

$H_{\alpha,m}=\\{\lambda\in P|(\lambda,\alpha)=m(l+h^{\vee})\\}$,

for $\alpha\in\Phi$, $m\in\mathbb{Z}$. Let

$H=\cup_{\alpha\in\Phi,m\in\mathbb{Z}}H_{\alpha,m}$.

  * •

the affine Weyl group at level $l$

$W_{l}=$ the group generated by $W$ and the map
$\lambda\mapsto\lambda+(l+h^{\vee})\theta$.

Note the action of $W_{l}$ on $P_{\mathbb{R}}=P\otimes_{\mathbb{Z}}\mathbb{R}$
is defined by $w*\lambda=w(\lambda+\rho)-\rho$ for $w\in W_{l}$ and
$\lambda\in P_{\mathbb{R}}$. We also define the set of minimal-length coset
representatives in $W_{l}/W$ by $W_{l}^{\prime}$.

We define $I_{l}\subset R(\mathfrak{g})$ be the ideal spanned over
$\mathbb{Z}$ by

  1. 1.

$V(\lambda)$ with $\lambda\in D$ and $\lambda+\rho\in H$,

  2. 2.

$V(w^{-1}*\mu)-\epsilon(w)V(\mu)$ with $\mu\in D_{l}$ and $w\in
W_{l}^{\prime}$.

The Verlinde ring at level $l$ of $G$ (or $\mathfrak{g}$) is defined to be the
quotient ring

$R_{l}(G)=R(G)/I_{l}$ (or $R_{l}(\mathfrak{g})=R(\mathfrak{g})/I_{l}$).

We will denote the image of the isomorphism class of $V(\lambda)$ in the
quotient ring by $[V(\lambda)]$. We denote the quotient map by $\pi_{l}$ and
the multiplication (tensor product) in $R_{l}(\mathfrak{g})$ by $\otimes_{l}$.

The following result is well-known (see [2], [21] Chapter 4 and [10] Chapter
2.3).

######  Proposition 2.1.

  1. 1.

$R_{l}(\mathfrak{g})$ has a $\mathbb{Z}$-basis $\\{[V(\lambda)]|\lambda\in
D_{l}\\}$;

  2. 2.

$\pi_{l}(V(\lambda))=[V(\lambda)]$ for $\lambda\in D_{l}$;

  3. 3.

$[V(\lambda)]\otimes_{l}[V(\mu)]=[V(\lambda)\otimes V(\mu)]$ if
$\lambda+\mu\in D_{l}$.

Indeed, these $[V(\lambda)]$ gives the family of positive energy
representations of the loop group $LG=C^{\infty}(S^{1},G)$ at level $l$ (see
[24]). In the following sections, we will treat them as $LG$-modules. We will
use the same notations as above for the weights and representations of $LG$ if
there is no confusion. For instance, $\Pi_{h}([V])$ will denote the highest
weights of the irreducible $LG$-modules in the decomposition of a $LG$-module
$[V]$.

######  Proposition 2.2.

Suppose $\lambda_{1},\cdots,\lambda_{t}\in D$ such that $\sum_{1\leq i\leq
t}\lambda_{i}\in D_{l}$. We have $\Pi_{h}(\otimes_{1\leq i\leq
t}V(\lambda_{i}))\subset D_{l}$.

Proof: Note any weight in $\Pi_{h}(\otimes_{1\leq i\leq t}V(\lambda_{i}))$
must be of the form $\sum_{1\leq i\leq t}\lambda_{i}-\sum_{1\leq j\leq
n}y_{i}\cdot\alpha_{i}$ with each $y_{i}\in\mathbb{Z}_{\geq 0}$. It suffices
to show

$(\left(\sum_{1\leq i\leq t}\lambda_{i}-\sum_{1\leq j\leq
n}y_{i}\cdot\alpha_{i}\right),\theta)\leq l$.

As $\sum_{1\leq i\leq t}\lambda_{i}\in D_{l}$, we have $(\sum_{1\leq i\leq
t}\lambda_{i},\theta)\leq l$. It then suffices to show
$(\alpha_{j},\theta)\geq 0$ for each $1\leq j\leq n$, which follows the fact
that $\theta\in D$.

##  3 Tensor Products of Fundamental Representations

In this section, as $\mathfrak{g}$-modules, we consider how the fundamental
representations generate the irreducible representations of level $l$, which
are the ones with highest weights in $D_{l}=\\{\lambda\in
D|(\lambda,\theta)\leq l\\}$. Then we move to the case of $LG$-modules and the
Verlinde ring.

Consider the $\mathfrak{g}$-module:

$W=V(0)\oplus(\oplus_{1\leq i\leq n}V(\omega_{i}))$,

which is the direct sum of trivial module $V(0)=\mathbb{C}$ and all
fundamental representations $V(\omega_{i})$’s. We have the following
increasing sequence of sets of dominant weights:

$\Pi_{h}(W^{\otimes 0})\subset\Pi_{h}(W^{\otimes 1})\subset\Pi_{h}(W^{\otimes
2})\subset\cdots\subset\Pi_{h}(W^{\otimes k})\subset\Pi_{h}(W^{\otimes
k+1})\subset\cdots$,

where $\Pi_{h}(W^{\otimes 0})=\\{0\\}$ and $\Pi_{h}(W^{\otimes
1})=\\{0,\omega_{1},\cdots,\omega_{n}\\}$ by the definition. Observe $D_{l}$
is a finite set. By Proposition 2.1 and the fact that fundamental
representations generate $R(\mathfrak{g})$, we know there exists some $d(l)$
depending on the simple type of the Lie algebra $\mathfrak{g}$ such that

$d(l)=d_{\mathfrak{g}}(l)=\min\\{k\geq 0|D_{l}\subset\Pi_{h}(W^{\otimes
k})\\}$.

This is equivalent to say:

######  Lemma 3.1.

For each $l\geq 0$, there is an integer $d(l)\geq 0$ such that

$D_{l}\subset\Pi_{h}(W^{\otimes k})$ if and only if $k\geq d(l)$.

Then we pass to $LG$-modules at level $l$, where their highest weights are
always contained in $D_{l}$. We will prove (see Corollary 3.8)

$d(l)=d_{\mathfrak{g}}(l)=\min\\{k\geq 0|D_{l}=\Pi_{h}([W]^{\otimes k})\\}$.

The rest of this section is mainly devoted to the following result:

######  Theorem 3.2.

  1. 1.

For type $A_{n}$, $C_{n}$ or $B_{2}$, $d(l)=l$;

  2. 2.

For type $B_{n}$ ($n\geq 3$), $\lceil\frac{2l}{n}\rceil\leq d(l)\leq l$;

  3. 3.

For type $D_{n}$ ($n\geq 4$), $\lceil\frac{2l}{n-1}\rceil\leq d(l)\leq l$;

  4. 4.

For type $E_{6}$, $E_{7}$ or $E_{8}$, $\lceil\frac{l}{3}\rceil\leq d(l)\leq
l$, $\lceil\frac{l}{5}\rceil\leq d(l)\leq l$, $\lceil\frac{4l}{15}\rceil\leq
d(l)\leq\lfloor\frac{l}{2}\rfloor$ respectively;

  5. 5.

For type $F_{4}$, $\lceil\frac{2l}{5}\rceil\leq d(l)\leq l$;

  6. 6.

For type $G_{2}$, $\lceil\frac{2l}{3}\rceil\leq d(l)\leq l$.

We first consider the map $\varepsilon\colon P\to\mathbb{Z}$ given by

$\varepsilon(\sum_{1\leq i\leq n}x_{i}\omega_{i})=\sum_{1\leq i\leq n}x_{i}$.

For each $k\geq 0$, we define a set of dominant weights

$B_{k}=\\{\lambda=\sum_{1\leq i\leq n}x_{i}\omega_{i}|\sum_{1\leq i\leq
n}x_{i}\leq k,x_{i}\in\mathbb{Z}_{\geq 0}\\}$,

or, equivalently, $B_{k}=D\cap\varepsilon^{-1}([0,k])$.

######  Example 3.3.

For the group $SU(2)$ (type $A_{1}$), $W=V_{0}\oplus V(\omega_{1})$,
$\varepsilon(\Pi_{h}(W^{\otimes k}))=\\{0,1,\cdots,k\\}$ by the Clebsch–Gordan
formula. We can further show $D_{k}=\Pi_{h}(W^{\otimes k})=B_{k}$.

As shown in [12] (see page 351), for $E_{8}$ and its fundamental
representation $\omega_{5}$, $V(\omega_{5})\otimes V(\omega_{5})$ contains
$V(5\omega_{1}+\omega_{7})$. Hence $\varepsilon(5\omega_{1}+\omega_{7})=6$ and
$6\in\varepsilon(\Pi_{h}(V(\omega_{5})\otimes
V(\omega_{5})))\subset\varepsilon({\Pi_{h}(W^{\otimes 2})})$. So
$\Pi_{h}(W^{\otimes k})$ may be strictly larger than $B_{k}$.

######  Proposition 3.4.

For each simple complex Lie algebra $\mathfrak{g}$, we have
$B_{k}\subset\Pi_{h}(W^{\otimes k})$. For $k\geq 1$, we further obtain

  1. 1.

For type $A_{n}$, $C_{n}$ or $B_{2}$, $\Pi_{h}(W^{\otimes k})=B_{k}$;

  2. 2.

For type $B_{n}$ ($n\geq 3$), $B_{k}\subset\Pi_{h}(W^{\otimes k})\subset
B_{\lfloor\frac{nk}{2}\rfloor}$;

  3. 3.

For type $D_{n}$ ($n\geq 4$), $B_{k}\subset\Pi_{h}(W^{\otimes k})\subset
B_{\lfloor\frac{(n-1)k}{2}\rfloor}$;

  4. 4.

For type $E_{6}$, $E_{7}$ or $E_{8}$, $B_{k}\subset\Pi_{h}(W^{\otimes
k})\subset B_{3k}$, $B_{5k}$ or $B_{\lfloor\frac{15k}{2}\rfloor}$
respectively;

  5. 5.

For type $F_{4}$, $B_{k}\subset\Pi_{h}(W^{\otimes k})\subset
B_{\lfloor\frac{5k}{2}\rfloor}$;

  6. 6.

For type $G_{2}$, $B_{k}\subset\Pi_{h}(W^{\otimes k})\subset
B_{\lfloor\frac{3k}{2}\rfloor}$.

Proof:

Let us first prove $B_{k}\subset\Pi_{h}(W^{\otimes k})$ by induction. It is
straightforward to check $B_{1}\subset\Pi_{h}(W)$. We take
$\lambda=\sum_{1\leq i\leq n}x_{i}\omega_{i}\in\Pi_{h}(W^{\otimes k})$ such
that $\sum_{1\leq i\leq n}x_{i}=k$. Consider the tensor product
$V(\lambda)\otimes V(\omega_{j})$. It has a simple summand with the highest
weight $\omega_{j}+\sum_{1\leq i\leq n}x_{i}\omega_{i}$, which is in
$B_{k+1}$. This shows that $\Pi_{h}(W^{\otimes k+1})$ contains all the weights
of the form $\sum_{1\leq i\leq n}x_{i}\omega_{i}$ with $\sum_{1\leq i\leq
n}x_{i}=k+1$.

Meanwhile, $W^{\otimes k}$ is a proper subspace of $W^{\otimes k+1}$ as $W$
contains the trivial representation, which is to say
$B_{k}\subset\Pi_{h}(W^{\otimes k})\subset\Pi_{h}(W^{\otimes k+1})$. Hence
$B_{k+1}\subset\Pi_{h}(W^{\otimes k+1})$.

Now we describe an upper bound of $\Pi_{h}(W^{\otimes k})$. Observe
$\Pi_{h}(W^{\otimes k})$ consists the elements of the form

$\mu=\sum_{1\leq i\leq n}x_{i}\omega_{i}-\sum_{1\leq i\leq n}y_{i}\alpha_{i}$,
with $\sum x_{i}\leq k$ and $x_{i},y_{i}\geq 0$,

which subjects to the conditions $\langle\mu,\alpha_{j}\rangle\geq 0$ for all
$1\leq j\leq n$. This is equivalent to the linear inequalities

$\vec{y}\cdot A\leq\vec{x}$,

where $\vec{x}=(x_{1},\cdots,x_{n})$, $\vec{y}=(y_{1},\cdots,y_{n})$ and
$A=[\langle\alpha_{i},\alpha_{j}\rangle]_{n\times n}$ is the Cartan matrix of
$\mathfrak{g}$. Hence we have

  1. 1.

$\sum_{1\leq j\leq n}y_{j}\langle\alpha_{j},\alpha_{i}\rangle\leq x_{i}$ for
$1\leq i\leq n$ (by $\mu\in D$);

  2. 2.

$\varepsilon(\sum_{1\leq i\leq n}y_{i}\alpha_{i})\leq\sum_{i}x_{i}\leq k$ (by
$\mu\in D$),

  3. 3.

$y_{i}\geq 0$ for $1\leq i\leq n$ (by $\sum_{1\leq i\leq n}x_{i}\omega_{i}$
are highest).

Please note $\sum_{1\leq j\leq n}\langle\alpha_{i},\alpha_{j}\rangle\geq 0$
for each $i$ in type $A_{n}$, $B_{2}$ or $C_{n}$,. Hence each
$-y_{i}\alpha_{i}$ contributes $-y_{i}\sum_{1\leq j\leq
j}\langle\alpha_{i},\alpha_{j}\rangle$ to $\varepsilon(\mu)$, which is non-
positive. We have $\varepsilon(\mu)\leq\sum_{1\leq i\leq n}x_{i}$ and
$\Pi_{h}(W^{\otimes k})=B_{k}$.

For the remaining types, we apply the simplex method [22] and induction on the
rank $n$ to get the maximal values of $\varepsilon(\mu)$. We leave it to the
reader to check the linear inequalities.

Assume $\theta=\sum_{1\leq i\leq n}c_{i}\cdot\alpha_{i}$ as a
$\mathbb{Z}$-linear combination of simple roots.

######  Lemma 3.5.

If $\min_{1\leq i\leq n}\\{\frac{c_{i}\|\alpha_{i}\|^{2}}{2}\\}=c$, we have
$D_{l}\subset B_{k}$ if and only if $k\geq\lfloor\frac{l}{c}\rfloor$.

Proof: Without loss of generality, we assume $c=1$. Let $\lambda=\sum_{1\leq
i\leq n}n_{i}\omega_{i}$ and observe

$\displaystyle(\lambda,\theta)$ $\displaystyle=\sum_{1\leq i\leq
n}c_{i}(\lambda,\alpha_{i})=\sum_{1\leq i\leq
n}c_{i}\cdot\frac{2(\lambda,\alpha_{i})}{(\alpha_{i},\alpha_{i})}\frac{(\alpha_{i},\alpha_{i})}{2}$
$\displaystyle=\sum_{1\leq i\leq
n}c_{i}\cdot\frac{(\alpha_{i},\alpha_{i})}{2}\langle\lambda,\alpha_{i}\rangle=\sum_{1\leq
i\leq n}\frac{c_{i}\|\alpha_{i}\|^{2}}{2}\cdot x_{i}.$

Hence $D_{l}=\\{\lambda=\sum_{1\leq i\leq n}x_{i}\omega_{i}|\sum_{1\leq i\leq
n}\frac{c_{i}\|\alpha_{i}\|^{2}}{2}\cdot x_{i}\leq l\\}$.

The inclusion $D_{l}\subset B_{l}$ is straightforward by Lemma 3.5 as all
$\frac{c_{i}\|\alpha_{i}\|^{2}}{2}\geq 1$. As $B_{k}\subset B_{k+1}$, it
suffices to show $D_{l}\not\subset B_{l-1}$. Suppose
$\frac{c_{j}\|\alpha_{j}\|^{2}}{2}=1$ for some $j$. Then $l\cdot\omega_{j}\in
D_{l}$ but $l\cdot\omega_{j}\notin B_{l-1}$.

For $c\geq 1$, $D_{l}=B_{\lfloor\frac{l}{c}\rfloor}$ is clear. Suppose
$\frac{c_{j}\|\alpha_{j}\|^{2}}{2}=c$ for some $j$. We have
$\lfloor\frac{l}{c}\rfloor\cdot\omega_{j}\in D_{l}$ but
$\lfloor\frac{l}{c}\rfloor\cdot\omega_{j}\notin
B_{\lfloor\frac{l}{c}\rfloor-1}$.

We now consider the simple types of $\mathfrak{g}$ for the construction above.
We refer to [5] Chapter VI.4 for notations and more details.

######  Proposition 3.6.

Assume the highest root $\theta=\sum_{1\leq i\leq n}c_{i}\cdot\alpha_{i}$.
Then we have

$\min_{1\leq i\leq
n}\\{\frac{c_{i}\|\alpha_{i}\|^{2}}{2}\\}=\begin{cases}1,&\text{if
}\mathfrak{g}\text{ is of type
}A_{n},B_{n},C_{n},D_{n},E_{6},E_{7},F_{4}\text{ or }G_{2}\\\ 2,&\text{if
}\mathfrak{g}\text{ is of type }E_{8}.\end{cases}$

Proof:

  1. 1.

Type $A_{n}$: We have all $c_{i}=1$ as $\theta=\sum_{1\leq i\leq
n}\alpha_{i}=\varepsilon_{1}-\varepsilon_{n+1}$. The Euclidean inner product
is normalized in the sense of $\|\theta\|^{2}=2$. Hence
$\frac{c_{i}\|\alpha_{i}\|^{2}}{2}=1$ for each $i$.

  2. 2.

Type $B_{n}$: We have all $c_{1}=1$ and $c_{i}=2$ for $2\leq i\leq n$ as
$\theta=\alpha_{1}+2\alpha_{2}+\cdots+2\alpha_{n}=\varepsilon_{1}+\varepsilon_{2}$.
The Euclidean inner product is normalized in the sense of $\|\theta\|^{2}=2$.
Hence $\min_{1\leq i\leq
n}\\{\frac{c_{i}\|\alpha_{i}\|^{2}}{2}\\}=\frac{c_{1}\|\alpha_{1}\|^{2}}{2}=1$.

  3. 3.

Type $C_{n}$: We have all $c_{n}=1$ and $c_{i}=2$ for $1\leq i\leq n-1$ as
$\theta=2\alpha_{1}+\cdots+2\alpha_{n-1}+\alpha_{n}=2\varepsilon_{1}$. The
normalized inner product is one-half of the Euclidean one. So
$\|\alpha_{i}\|^{2}=1$ for $1\leq i\leq n-1$ and $\|\alpha_{n}\|^{2}=2$. Hence
$\frac{c_{i}\|\alpha_{i}\|^{2}}{2}=1$ for each $i$.

  4. 4.

Type $D_{n}$: We have all $c_{1}=c_{n-1}=c_{n}=1$ and $c_{i}=2$ for $2\leq
i\leq n-2$ as
$\theta=\alpha_{1}+2\alpha_{2}+\cdots+2\alpha_{n-2}+\alpha_{n-1}+\alpha_{n}=\varepsilon_{1}+\varepsilon_{2}$.
The Euclidean inner product is normalized in the sense of $\|\theta\|^{2}=2$
and we have $\|\alpha_{i}\|^{2}=2$ for each $i$. Hence
$\frac{c_{i}\|\alpha_{i}\|^{2}}{2}=1$ when $i=1,n-1,n$ are the minimal values
.

Now let us consider the exceptional types. Note the $E_{6},E_{7}$ type can be
embedded into $E_{8}$ as subsystems.

  1. 5.

Type $E_{8}$:
$\theta=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+6\alpha_{4}+5\alpha_{5}+4\alpha_{6}+3\alpha_{7}+2\alpha_{8}=\varepsilon_{1}+\varepsilon_{8}$.
We have all $\|\alpha_{i}\|^{2}=2$. Hence
$\frac{c_{i}\|\alpha_{i}\|^{2}}{2}=2$ are minimal with value when $i=1,8$.

  2. 6.

Type $E_{7}$:
$\theta=2\alpha_{1}+2\alpha_{2}+3\alpha_{3}+4\alpha_{4}+3\alpha_{5}+2\alpha_{6}+\alpha_{7}=\varepsilon_{8}-\varepsilon_{7}$.
We have all $\|\alpha_{i}\|^{2}=2$. Hence
$\frac{c_{i}\|\alpha_{i}\|^{2}}{2}=1$ are minimal with value when $i=7$.

  3. 7.

Type $E_{6}$:
$\theta=\alpha_{1}+2\alpha_{2}+2\alpha_{3}+3\alpha_{4}+2\alpha_{5}+\alpha_{6}=1/2(\varepsilon_{1}+\cdots+\varepsilon_{5}-\varepsilon_{6}-\varepsilon_{7}+\varepsilon_{8})$.
We have all $\|\alpha_{i}\|^{2}=2$. Hence
$\frac{c_{i}\|\alpha_{i}\|^{2}}{2}=1$ are minimal with value when $i=1,6$.

  4. 8.

Type $F_{4}$:
$\theta=2\alpha_{1}+3\alpha_{2}+4\alpha_{3}+2\alpha_{4}=\varepsilon_{1}+\varepsilon_{2}$.
We have $\|\alpha_{1}\|^{2}=\|\alpha_{2}\|^{2}=2$ and
$\|\alpha_{3}\|^{2}=\|\alpha_{4}\|^{2}=1$. Hence
$\frac{c_{i}\|\alpha_{i}\|^{2}}{2}=1$ is minimal with value when $i=4$.

  5. 9.

Type $G_{2}$:
$\theta=3\alpha_{1}+2\alpha_{2}=-\varepsilon_{1}-\varepsilon_{2}+2\varepsilon_{3}$.
The normalized inner product is $1/3$ of the Euclidean one. We have
$\|\alpha_{1}\|^{2}=2/3$ and $\|\alpha_{2}\|^{2}=2$. Hence
$\frac{c_{i}\|\alpha_{i}\|^{2}}{2}=1$ is minimal with value when $i=1$.

######  Proposition 3.7.

For any simple complex Lie algebra $\mathfrak{g}$, $d(l)\leq l$.

In particular, if $\mathfrak{g}$ is of type $E_{8}$, $d(l)\leq\lfloor
l/2\rfloor$.

Proof: By Proposition 3.6 and Lemma 3.5, we know that $c=1$ for all the types
except $E_{8}$ (for which $c=2$). It then follows Lemma 3.4.

Now we pass to the Verlinde ring $R_{l}(\mathfrak{g})$ or, equivalently, the
category of $LG$-modules.

######  Corollary 3.8.

$d(l)=\min\\{k\geq 0|D_{l}=\Pi_{h}([W]^{\otimes k})\\}$.

Proof: By Proposition 2.2, we know $\Pi_{h}(W^{k})\subset D_{l}$ when
$B_{k}\subset D_{l}$. Then, by 2 of Proposition 2.1, we conclude

$\operatorname{Hom}_{\mathfrak{g}}(V_{\mu},W^{\otimes
k})=\operatorname{Hom}_{LG}([V_{\mu}],[W]^{\otimes k})$,

for all $\mu\in D_{l}$. This shows $W^{\otimes k}$ contains $V(\mu)$ if and
only if $[W]^{\otimes k}$ contains $[V(\mu)]$ for each $\mu\in D_{l}$.

Proof: [Proof of Theorem 3.2] The upper bound is given in Proposition 3.7.

For type $A_{n}$, $C_{n}$ or $B_{2}$, it follows the fact
$B_{k}=\Pi_{h}(W^{\otimes k})$ (see 1 of Proposition 3.4), Lemma 3.5 and
Proposition 3.6.

For the remaining types except $E_{8}$, let us assume $d(l)\geq s$. Let
$\beta$ be the value given in Proposition 3.4, i.e, $\beta=\frac{n}{2}$ for
$B_{n}$,$\frac{n-1}{2}$ for $C_{n}$, $3,5$ for $E_{6},E_{7}$, $\frac{5}{2}$
for $F_{4}$ or $\frac{3}{2}$ for $G_{2}$. We have

$\displaystyle d(l)>k$ $\displaystyle\Leftrightarrow\text{There exists
}\exists\lambda\in D_{l},\lambda\notin\Pi_{h}(W^{\otimes k})$
$\displaystyle\Leftarrow\exists\lambda\in D_{l},\lambda\notin B_{\lfloor\beta
k\rfloor}$ $\displaystyle\Leftarrow\lfloor\beta k\rfloor\leq l-1$
$\displaystyle\Leftrightarrow\beta k<l\Leftrightarrow k<\beta^{-1}l.$

Here we apply Proposition 3.4 and Lemma 3.5 in the second and third lines
respectively. Then we conclude $d(l)\geq\beta^{-1}l$ or, equivalently
$d(l)\geq\lceil\beta^{-1}l\rceil$.

The inequality of $d(l)$ for $E_{8}$ follows similarly by Corollary 3.5 and
Proposition 3.4.

##  4 Towers of Finite-Dimensional Algebras and Subfactors

From this section, we fix the positive integer $l$and suppose $|D_{l}|=m$. Let
$V_{1},\dots,V_{m}$ denote the simple $LG$-modules in the Verlinde ring
$R_{l}(\mathfrak{G})$.

For each $k\geq 0$, define a finite-dimensional $C^{*}$-algebra

$A_{k}=\operatorname{End}(W^{\otimes k})=\operatorname{Hom}(W^{\otimes
k},W^{\otimes k})$,

where we let $A_{0}=\mathbb{C}$. Observe
$\operatorname{Hom}(V_{i},V_{j})=\mathbb{C}\delta_{i,j}$ and $\dim Z(A_{k})$
is the number of isomorphism classes of simple modules contained in
$W^{\otimes k}$. By Proposition 3.6, we know $\dim Z(A_{k})=m$ when $k\geq l$
for all the types except $E_{8}$, or, $k\geq\lfloor l/2\rfloor$ for type
$E_{8}$.

The left inclusion $i_{k}\colon A_{k}\to A_{k+1}$ There is a natural inclusion
$i_{k}:A_{k}\hookrightarrow A_{k+1}$ defined as
$i_{k}(f)=f\otimes\text{id}_{W}$. We denote the inclusion matrix of the pair
$A_{k}\subset A_{k+1}$ by $T(k)=[t(k)_{i,j}]\in M_{m\times m}(\mathbb{Z})$,
which is given by

$t(k)_{i,j}=\dim_{\mathbb{C}}\operatorname{Hom}(V_{i}\otimes W,V_{j})$.

######  Lemma 4.1.

For $k\geq d(l)$, the inclusion matrices are identical, i.e. $T_{k}=T$ for
$k\geq d(l)$. Moreover, $T$ is symmetric and irreducible.

Proof: We first claim $W$ is self-dual. It is well-known that the dual of a
simple $\mathfrak{g}$-module $V(\lambda)$ is given by $V(-w_{0}(\lambda))$,
where $w_{0}$ is the longest element in the Weyl group $W$.

Note $w_{0}$ sends the positive Weyl chamber to the negative one. Observe
$-w_{0}^{-1}(\alpha_{i})$ is still a simple root and $-w_{0}^{-1}$ acts as a
permutation of $\Delta$, say $-w_{0}^{-1}(\alpha_{i})=\alpha_{\sigma(i)}$ for
some $\sigma\in S_{n}$. We have

$\displaystyle\langle-w_{0}^{-1}(\lambda_{i}),\alpha_{j}\rangle$
$\displaystyle=\frac{2(-w_{0}^{-1}(\lambda_{i}),\alpha_{j})}{(\alpha_{j},\alpha_{j})}=\frac{2(-\lambda_{i},w_{0}^{-1}(\alpha_{j}))}{(w_{0}^{-1}(\alpha_{j}),w_{0}^{-1}(\alpha_{j}))}$
$\displaystyle=\frac{\lambda_{i},\alpha_{\sigma(j)})}{(\alpha_{\sigma(j)},\alpha_{\sigma(j)})}=\delta_{i,{\sigma(j)}}.$

Hence $-w_{0}^{-1}(\lambda_{i})=\lambda_{\sigma(i)}$ and $W$ is self-dual.

Thus we obtain

$\displaystyle t(k)_{i,j}$ $\displaystyle=\dim\operatorname{Hom}(V_{i}\otimes
W,V_{j})=\dim\operatorname{Hom}(V_{i},V_{j})\otimes W^{*})$
$\displaystyle=\dim\operatorname{Hom}(V_{i},V_{j}\otimes
W)=\dim\operatorname{Hom}(V_{j}\otimes W,V_{i})=t(k)_{j,i},$

which is independent with $k$ by Proposition 3.7 once $k\geq d(l)$. Hence
$T(k)=T(k)^{t}=T$ for some $T$ if $k\geq d(l)$.

For the irreducibility of $T$, it suffices to show the associated graph is
strongly connected. This is equivalent to $\sum_{s=1}^{S}T^{s}$ is positive
for sufficiently large $S$. Suppose $T^{s}=[t_{i,j}^{(s)}]$ and fix a pair of
indices $(i,j)$. There exist positive integers $a,b$ such that $W^{\otimes
a},W^{\otimes b}$ have the summands $V_{i}$, $V_{j}$ respectively. Let $s=a+b$
and we obtain

$\displaystyle t^{(s)}_{i,j}$
$\displaystyle=\dim\operatorname{Hom}(V_{i}\otimes W^{\otimes
a+b},V_{j})=\dim\operatorname{Hom}(V_{i}\otimes W^{\otimes b},V_{j}\otimes
W^{\otimes a})$ $\displaystyle\geq\dim\operatorname{Hom}(V_{i}\otimes
V_{j},V_{j}\otimes V_{i})\geq 1.$

Hence the associated graph is strongly connected.

######  Proposition 4.2.

The algebra $\cup_{k\geq 0}A_{k}$ admits a unique tracial state.

Proof: By the Perron-Frobenius theorem, the inclusion matrix $T=T_{k}$ ($k\geq
d+1$) admits an eigenvalue $\beta\in\mathbb{R}_{+}$ such that $|\beta|$ is
strictly greater than the others. Its eigenvector $V_{\beta}$ has all its
components positive. As $T$ is irreducible by Lemma 4.1, one can show the
space of tracial states is a singleton and hence contains a factor trace (see
[26] Chapter XIX, Lemma 3.9).

This trace will yield the hyperfinite $\text{II}_{1}$ factor as its completion
in the GNS construction. We denote this hyperfinite $\text{II}_{1}$ factor by
$M$ and the trace by $\operatorname{tr}$.

The conditional expectation $E_{k+1}$ For each $A_{k}$, we consider its
completion with respect to $tr$, which is Hilbert space and will be denoted as
$L^{2}(A_{k},\operatorname{tr})$. Let
$e_{k+1}:L^{2}(A_{k},\operatorname{tr})\to L^{2}(A_{k-1},\operatorname{tr})$
be the orthogonal projection, which is comes from the embedding $i_{k-1}$. The
projection $e_{k+1}$ will certainly induce a map $E_{k+1}:A_{k}\to A_{k-1}$
called the conditional expectation. Consider the action of $A_{k}$ and
$e_{k+1}$ on $L^{2}(A_{k})$. They generate a von Neumann
$(A_{k}\cup\\{e_{k+1}\\})^{\prime\prime}$, denoted $\langle
A_{k},e_{k+1}\rangle$. This is the basic construction of finite-dimensional
$C^{*}$-algebras.

######  Lemma 4.3.

We have $\langle A_{k},e_{k+1}\rangle\subset A_{k+1}$. If $k\geq d+1$,
$\langle A_{k},e_{k+1}\rangle=A_{k+1}$.

Proof: Note the inclusion matrix $T^{\langle
A_{k},e_{k+1}\rangle}_{A_{k}}=(T^{A_{k}}_{A_{k-1}})^{\rm t}=T_{k-1}^{\rm t}$
for $A_{k}\subset\langle A_{k},e_{k+1}\rangle$. It suffices to show
$T_{k}-T_{k-1}^{\rm t}$ is positive in general and $T^{\langle
A_{k},e_{k+1}\rangle}_{A_{k}}=T_{k-1}^{\rm t}=T_{k-1}$ if $k\geq d+1$.

Note that $W$ contains the trivial representation $V_{0}$ as a summand. Hence
the number of any irreducible object $V_{i}$ at depth $k$ is no greater than
that at depth $k+1$. So $t(k+1)_{i,j}\geq t(k)_{j,i}$ or equivalently
$T_{k+1}-T_{k}^{\rm t}$ is positive, which implies $A_{k+1}$ always contains
the algebra $\langle A_{k},e_{k}\rangle$.

Observe $T_{k+1}=T_{k}={T_{k}}^{\rm t}$ is symmetric when $k\geq d+1$. By [16]
Lemma 4.4.1, $A_{k+1}$ is the basic construction of the pair
$A_{k-1}\subset^{e_{k}}A_{k}$.

The right inclusion $i_{k,j+k}\colon A_{k}\to A_{j+k}$ There is another
natural inclusion $i_{k,j+k}\colon A_{k}\subset A_{j+k}$ defined by

$i_{k,j+k}(f)=\text{id}_{W^{\otimes j}}\otimes f$

for $j\geq 0$, which is in $\operatorname{End}(W^{\otimes(j+k)})=A_{j+k}$ for
$f\in\operatorname{End}(W^{\otimes k})=A_{k}$. Thus it induces an inclusion
$i_{j}^{R}:\cup_{k\geq 0}{A_{k}}\subset\cup_{k\geq 0}A_{j+k}$ (here $R$
denotes the inclusion on the right side). Indeed, this inclusion is a
composite of $i_{k,k+1},i_{k+1,k+2},\dots,i_{k+j-1,k+j}$ and can be shown to
be trace-preserving.

Now let us consider the inclusion $i_{j}^{R}$ which maps the triple
$A_{k-1}\subset A_{k}\subset\langle A_{k},e_{k+1}\rangle$ to $A_{j+k-1}\subset
A_{j+k}\subset A_{j+k+1}$.

######  Corollary 4.4.

Within $B(L^{2}(A_{j+k}))$, we have $i_{j}^{R}(e_{k+1})=e_{j+k+1}$.

Proof: Consider the restriction on the subspace
$L^{2}(i_{j}^{R}(A_{k}))\subset L^{2}(A_{j+k})$, we have $i_{j}^{R}(e_{k+1})$
is the orthogonal projection from $L^{2}(i_{j}^{R}(A_{k}))$ to
$L^{2}(i_{j}^{R}(A_{k-1}))$.

Moreover, for $x\in i_{j}^{R}(A_{k-1})$, we have $[i_{j}^{R}(e_{k+1}),x]=0$.
It is clear that $i_{j}^{R}(e_{k+1})$ commutes with the elements in
$A_{j+k-1}$. This implies $i_{j}^{R}(e_{k+1})$ acts as the same as $e_{j+k}$
on $L^{2}(A_{j+k})$, which is the unique projection.

##  5 The Commutants

Consider the complex algebra $\cup_{k\geq 0}A_{j+k}$. By Proposition 4.2,
$\cup_{k\geq 0}A_{j+k}$ also admits a factor trace. The GNS construction gives
us a hyperfinite $\text{II}_{1}$ factor, denoted $M_{j}$. Moreover, as
$A_{k}\subset A_{j+k}$ for each $k\geq 0$, $M$ is a subfactor of $M_{j}$. Thus
we get an increasing tower of factors

$M=M_{0}\subset M_{1}\subset M_{2}\subset M_{3}\subset\dots$.

The commutants $M^{\prime}\cap M_{j}$ will be discussed with commuting
squares. We refer to [15] for some basic facts about commuting squares and
their properties. Now we consider the following diagram

$\displaystyle A_{j+k}~{}~{}$ $\displaystyle\subset^{i_{j+k}}{}{}{}{}$
$\displaystyle A_{j+k+1}$ $\displaystyle\cup^{i_{k,j+k}}~{}~{}$
$\displaystyle\cup^{i_{k+1,j+k+1}}~{}~{}$ $\displaystyle A_{k}~{}~{}$
$\displaystyle\subset^{i_{k}}{}{}{}{}$ $\displaystyle A_{k+1}$

with $j\geq 0$. Please note the horizontal embeddings are the left inclusions
while the vertical ones are the right inclusions.

######  Lemma 5.1.

We have $E_{j+k+2}(i_{k+1,j+k+1}(A_{k+1}))=A_{k}$. Hence the diagram above is
a commuting square.

Proof: The inclusion $E_{j+k+2}(i_{k+1,j+k+1}(A_{k+1}))\subset
i_{k,j+k}(A_{k})$ is straightfoward.

Note as $W=V_{0}\oplus W_{0}$ with $W_{0}=\oplus_{i}V(\omega_{i})$, we have
$W^{\otimes(k-j)}=(W^{\otimes(k-j)}\otimes
V_{0})\oplus(W^{\otimes(k-j)}\otimes W_{0})$. For
$i_{k,j+k}(g)=\text{id}_{W^{\otimes j}}\otimes g\in i_{k,j+k}(A_{k})$ with
$g\in A_{k}$, we define an element $\bar{g}\in A_{k+1}$ by

$\bar{g}=\begin{bmatrix}g&0\\\
0&0\end{bmatrix}\in\operatorname{End}{((W^{\otimes k}\otimes
V_{0})\oplus(W^{\otimes k}\otimes W_{0}))}$

with respect to the decomposition of $W^{\otimes k}$ above. Then we have
$E_{j+k+2}(i_{k+1,j+k+1}(\bar{g}))=E_{j+k+2}(\text{id}_{W^{\otimes
j}}\otimes\bar{g})=i_{k,j+k}(g)$.

######  Lemma 5.2.

If $k\geq d$, the commuting square is symmetric.

Proof: By [15] Corollary 5.4.4, it suffices to show the inclusion matrices
have the following relation:

$(T_{A_{k}}^{A_{k+1}})^{\rm
t}T_{A_{k}}^{A_{j+k}}=T_{A_{k+1}}^{A_{k+j+1}}(T_{A_{j+k}}^{A_{j+k+1}})^{\rm
t}$.

By Lemma 4.1, if $k\geq d$, we have $\dim\mathcal{Z}({A_{k}})=m$. So all these
inclusion matrices are $T=T_{k}$ that we obtained in the proof of Lemma 4.1,
which is a symmetric one in $\text{Mat}_{m}(\mathbb{Z})$.

Now we consider the following towers of $C^{*}$-algebras:

$\displaystyle A_{j+d}$ $\displaystyle\subset$
$\displaystyle~{}~{}~{}~{}A_{j+d+1}$ $\displaystyle\subset$
$\displaystyle~{}~{}~{}~{}A_{j+d+2}$ $\displaystyle\subset$
$\displaystyle\cdots$ $\displaystyle\cup$ $\displaystyle~{}~{}~{}~{}~{}\cup$
$\displaystyle~{}~{}~{}~{}~{}\cup$ $\displaystyle A_{d}$
$\displaystyle\subset$ $\displaystyle~{}~{}~{}~{}A_{d+1}$
$\displaystyle\subset$ $\displaystyle~{}~{}~{}~{}A_{d+2}$
$\displaystyle\subset$ $\displaystyle\cdots$

We have $A_{j+k+1}=\langle A_{j+k},e_{j+k+1}\rangle$. As shown before, the
unions of these two rows give a pair of $\text{II}_{1}$ factors
$M=M_{0}\subset M_{j}$.

######  Proposition 5.3.

With the definition of $A_{k},M,M_{j}$ above, we have

$M^{\prime}\cap M_{j}\cong A_{j}$

for all $j\geq 0$.

Proof: We have already checked that the first one of the commuting squares
above is symmetric and the two rows are the towers obtained from basic
constructions. By Lemma 4.3, for $k\geq 1$, we know
$i_{d+k+1,j+d+k+1}(A_{d+k+1})$ is equal to $\langle
i_{d+k,j+d+k}(A_{d+k}),e_{j+d+k+1}\rangle$ by $i(e_{d+k+1})=e_{j+d+k+1}$ in
Corollary 4.4. By the Ocneanu Compactness theorem (see [15] Theorem 5.7.1), we
have

$M^{\prime}\cap M_{j}=(i_{d+1,j+d+1}(A_{d+1}))^{\prime}\cap A_{j+d}$.

It suffices to show the right-hand side is just $A_{j}$.

As shown in Lemma 4.3, we have $A_{k+1}$ always contains all $e_{i}$ with
$2\leq i\leq k+1$. Within the embedding $i_{k+1,j+k+1}:A_{k+1}\to A_{j+k+1}$,
it can be shown that the projections $\\{e_{i}\\}_{2\leq i\leq k+1}$ are
mapped to $\\{e_{j+i}\\}_{2\leq i\leq k+1}$ respectively. So
$(i_{d+1,j+d+1}(A_{d+1}))^{\prime}\cap
A_{j+l}\subset\\{e_{j+1},\dots,e_{j+d+1}\\}^{\prime}\cap A_{j+l}$. Then, by
[16] Proposition 4.1.4, we get $\\{e_{j+1},\dots,e_{j+d+1}\\}^{\prime}\cap
A_{j+d}=A_{j}$.

Moreover, the inclusion $A_{j}\subset(i_{d+1,j+d+1}(A_{d+1}))^{\prime}\cap
A_{j+d}$ is straightforward. Hence we have $M^{\prime}\cap M_{j}=A_{j}$.

The right conditional expectation $E^{\prime}_{j+k+2}\colon A_{j+k+1}\to
A_{j+k}$ There is another conditional expectation $E^{\prime}_{j+k+2}\colon
A_{j+k+1}\to A_{j+k}$ while identifying $A_{j+k}$ as a subalgebra by the
inclusion $i_{j+k,j+k+1}\colon f\mapsto\text{id}_{W}\otimes f$ for $f\in
A_{j+k}$. (Please note the differences between these $E^{\prime}_{k}$ and
$E_{k}$’s, where $E_{k}$ comes from the left inclusion $i_{k}\colon f\mapsto
f\otimes\text{id}_{W}$, see Section 4). These $E^{\prime}_{j+k+2}$’s induce a
map $E^{\prime}_{j+1}:\cup_{k\geq 0}A_{j+k+1}\to\cup_{k\geq 0}A_{j+k}$ and
further yield a conditional expectation

$E^{\prime}_{j+2}:M_{j+1}\to M_{j}$.

Let $\xi_{j}$ be the canonical cyclic trace vector in $L^{2}(M_{j})$. By
identifying $M_{j+1}$ with the algebra of left action operator on
$L^{2}(M_{j+1})$, $E^{\prime}_{j+2}$ extends to a projection
$e^{\prime}_{j+2}$ via $e_{j+2}(x\xi_{j})=E^{\prime}_{j+2}(x)\xi_{j}$.

######  Corollary 5.4.

We have $M_{j+1}=\langle M_{j},e^{\prime}_{j+1}\rangle$ for $j\geq 1$.

Proof: It follows the fact that $A_{j+k+1}$ is the algebra obtained from the
basic construction of the pair $A_{j+k-1}\subset A_{j+k}$ with the conditional
expectation $e^{\prime}_{j+k}$ if $j+k-1\geq d$.

We will denote these $e_{j}^{{}^{\prime}}$’s by $e_{j}$ in the discussion of
the infinite-dimensional algebras (factors). We may obtain a tower of
hyperfinite factors from the basic constructions:

$M=M_{0}\subset M_{1}\subset^{e_{2}}M_{2}\subset^{e_{3}}M_{3}\subset\dots$.

Please note our indices of $e_{k}$ start from $k=2$, which makes $e_{k}\in
A_{k}$ and $e_{k}\notin A_{k-1}$.

##  6 The Bimodules and Their Fusion Rule

We first have a review of bimodules over $\text{II}_{1}$ factors. One may
refer to [4, 6] for more details.

Let $A$ and $B$ be $\text{II}_{1}$ factors. An $A\text{-}B$ bimodule
${}_{A}{H}_{B}$ is a pair of commuting normal (unital) representations
$\pi_{L},\pi_{R}$ of $A$ and $B^{\text{op}}$ respectively on the Hilbert space
$H$. Here $B^{\text{op}}$ is the opposite algebra of $B$, i.e $b_{1}\cdot
b_{2}=b_{2}b_{1}$, which is also a $\text{II}_{1}$ factor. Note that
${}_{A}{H}_{B}$ is a left $A$-module and right $B$-module with the action
denoted as $\pi_{L}(a)\pi_{R}(b)\xi=a\cdot\xi\cdot b$ with $a\in A,b\in
B,\xi\in H$. We say ${}_{A}{H}_{B}$ is bifinite if the left dimension
$\dim_{A}^{L}{H}<\infty$ and right dimension $\dim_{B}^{R}{H}<\infty$.

######  Definition 6.1.

Let $H,K$ be two $A\text{-}B$ bimodules. We say $H,K$ are equivalent if we
have a unitary $u:H\to K$ such that $u(a\cdot\xi\cdot b)=a\cdot u(\xi)\cdot b$
for all $a\in A,b\in B,\xi\in H$ and denoted by
${}_{A}{H}_{B}\cong{}_{A}{K}_{B}$. Moreover, we denote by

$\operatorname{Hom}_{A,B}(H,K)=\\{T\in B(H,K)|T(a\cdot\xi\cdot b)=a\cdot
T(\xi)\cdot b\text{~{}for all~{}}a\in A,b\in B,\xi\in H\\}$

the space of $A\text{-}B$ intertwiners from $H$ to $K$. Let
$\operatorname{Hom}_{A,B}(H)=\operatorname{Hom}_{A,B}(H,H)$ And we call an
$A\text{-}B$ bimodule $H$ irreducible if
$\operatorname{Hom}_{A,B}(H)=\mathbb{C}$.

Note that $\operatorname{Hom}_{A,B}(H)\subset B(H)$ is a von Neumann algebra.
For a $A$-module $H$, $v\in H$ is called $A$-bounded if we have a positive
constant $c_{v}$ such that

$\|xv\|\leq c_{v}\|x\|_{2}$ for all $x\in A$,

where $\|x\|_{2}=\operatorname{tr}(x^{*}x)^{1/2}$. We write $H^{\text{bdd}}$
for the set of all $A$-bounded vectors in $H$. It can be shown to be a dense
subspace of $H$ and also invariant under the action of $A$ and $A^{\prime}$
which leads to the following result (see [6] and [18]). A proof is also
provided below for completeness.

######  Lemma 6.1.

Assume ${}_{A}{H}_{B}$ is bifinite, then

  1. 1.

A vector $v\in H$ is $A$-bounded if and only if it is $B$-bounded;

  2. 2.

$\operatorname{Hom}_{A,B}(H)$ is a finite dimensional von Neumann algebra.

Proof: We only prove 2. For the proof of $1$, see [18]. Note that
$\operatorname{Hom}_{A,B}(H)=A^{\prime}\cap(B^{\text{op}})^{\prime}\cap B(H)$
is centainly a von Neumann algebra. If ${}_{A}{H}_{B}$ is bifinite, we have
$A\subset(B^{\text{op}})^{\prime}\cap B(H)$ by the commuting action. This
imlies an inclusion of $\text{II}_{1}$ factors where

$[(B^{\text{op}})^{\prime}\cap
B(H):A]=\frac{\dim_{B^{\text{op}}}(H)}{\dim_{A}(H)}=\frac{1}{\dim_{A}(H)\dim_{B}(H)}<\infty$.

Hence $\operatorname{Hom}_{A,B}(H)=A^{\prime}\cap(B^{\text{op}})^{\prime}\cap
B(H)$ is a relative commutant of a pair of factors with finite index. So, by
[16], it is finite-dimensional.

######  Corollary 6.2.

If ${}_{A}{H}_{B}$ is bifinite and $p$ is a projection in
$\operatorname{Hom}_{A,B}(H)$, then $Hp$ is an irreducible $A\text{-}B$
bimodule if and only if $p$ is minimal.

Proof: If $p$ is minimal,
$\operatorname{Hom}_{A,B}(Hp)=p\operatorname{Hom}_{A,B}(H)=\mathbb{C}p\cong\mathbb{C}$.
Otherwise, assume $p=p_{1}+p_{2}$ is a decomposition into two subprojections,
then $Hp=Hp_{1}\oplus Hp_{2}$, which is a direct sum of $A\text{-}B$
bimodules.

Now let $A,B,C$ be $\text{II}_{1}$ factors. Given an $A\text{-}B$ bimodule
${}_{A}{H}_{B}$ and a $B\text{-}C$ bimodule ${}_{B}{K}_{C}$, we define the
$A\text{-}C$ bimodule of their tensor as [6], which is given by the completion
of the algebraic tensor product
${}_{A}{H}^{\text{bdd}}_{B}\otimes{}_{B}{K}^{\text{bdd}}_{C}$ of bounded
subspace with respect to the inner product defined by

$\langle v_{1}\otimes u_{1},v_{2}\otimes v_{2}\rangle=\langle v_{1}\langle
u_{1},u_{2}\rangle_{B},v_{2}\rangle$

Here $\langle u_{1},u_{2}\rangle_{B}\in B$ is uniquely determined by

$\operatorname{tr}(x\langle u_{1},u_{2}\rangle_{B})=\langle
xu_{1},u_{2}\rangle_{B}$ for all $x\in B$.

It is easy to check the following properties [6]:

  1. 1.

$\langle\lambda u_{1}+\mu u_{2},u_{3}\rangle_{B}=\lambda\langle
u_{1},u_{3}\rangle_{B}+\mu\langle u_{2},u_{3}\rangle_{B}$,

  2. 2.

$\langle u_{1},u_{2}\rangle_{B}=\langle u_{2},u_{1}\rangle_{B}^{*}$,

  3. 3.

$\langle xu_{1},u_{2}\rangle_{B}=x\langle u_{1},u_{2}\rangle_{B}$ and $\langle
u_{1},xu_{2}\rangle_{B}=\langle u_{1},u_{2}\rangle_{B}x^{*}$.

One may refer to [4] for general descriptions of bimodules.

Consider the tower of $\text{II}_{1}$ factors

$M=M_{0}\subset^{e_{1}}M_{1}\subset^{e_{2}}M_{2}\subset\cdots$

with $e_{k}\in M_{k}$ by iterating the basic constructions $M_{k-1}\subset
M_{k}\subset^{e_{k+1}}M_{k+1}=\langle
M_{k},e_{k+1}\rangle=(M_{k}\cup\\{e_{k+1}\\})^{\prime\prime}\subset
B(L^{2}(M_{k}))$. Observe the $M\text{-}M$ bimodule $L^{2}(M_{j})$ with the
action induced from the two sided action of $A_{k}$ on $A_{j+k}$ is given by

$a\cdot\xi\cdot b=i_{k,j+k}(a)\xi(i_{k,j+k}(b^{*}))$

with $a,b\in A_{k},\xi\in A_{k,j+k}$. We define a projection

$g_{k}=D^{k(k-1)}(e_{k+1}e_{k}\dots e_{2})(e_{k+2}e_{k+1}\dots
e_{3})\cdots(e_{2k}e_{2k-1}\cdots e_{k+1})$,

where $D=\sqrt{[M_{1}:M]}$. We have $M\subset M_{k}\subset^{g_{k}}M_{2k}$ is
the basic construction [4]. We can also define the actions $\pi_{k}$ of
$M_{k},M_{2k}$ on $L^{2}(M_{k})$ as following:

  1. 1.

$\pi_{k}(x)(\hat{z})=\widehat{xz}$, for all $\hat{z}\in\widehat{M_{k}}\subset
L^{2}(M_{k})$,

  2. 2.

$\pi_{k}(xg_{k}y)(\hat{z})=x\widehat{E^{M_{k}}_{N}(yz)}$ for all $xg_{k}y\in
M_{2k}$ and $x,y,z\in M_{k}$.

######  Proposition 6.3 ([4]).

Let $p,q\in M^{\prime}\cap M_{2k}$ be two equivalent projections and
$M_{2k}\subset^{e_{2k+1}}\subset M_{2k+1}\subset^{e_{2k+2}}M_{2k+2}$. Then we
have

$\displaystyle\pi_{k}(p)L^{2}(M_{k})$
$\displaystyle\cong\pi_{k}(q)L^{2}(M_{k}),\text{~{}and}$
$\displaystyle\pi_{k}(p)L^{2}(M_{k})$
$\displaystyle\cong\pi_{k+1}(pe_{2k+2})L^{2}(M_{k+1})$

as $M\text{-}M$ bimodules.

Let $J_{k}:L^{2}(M_{k})\to L^{2}(M_{k})$ be the modular conjugation defined by
$J_{k}(\hat{x})=\hat{x^{*}}$. Then we have $J^{2}_{k}=\text{id}$ and
$J_{k}\pi_{k}(M)^{\prime}J_{k}=\pi_{k}(M_{2k})$.

Now we will construct the shifts between the higher commutants. Let
$\gamma_{k}:M^{\prime}\cap M_{2k}\to M^{\prime}\cap M_{2k}$ be the surjective
linear $*$-antiisomorphism defined by
$\pi_{k}(\gamma_{k}(x))=J_{k}\pi_{k}(x)^{*}J_{k}$. Then we get a trace
preserving, surjective $*$-isomorphism $sh_{2k}$ given by

$sh_{2k}=\gamma_{2j+2k}\gamma_{2j}:M^{\prime}\cap M_{2j}\to
M_{2k}^{\prime}\cap M_{2j+2k}$.

Then we obtain the following proposition, which generalizes [4] Theorem 4.6.c.

######  Theorem 6.4.

Let $p\in M^{\prime}\cap M_{2j}$, $q\in M^{\prime}\cap M_{2k}$ be projections
and $\operatorname{sh}_{2j}:M^{\prime}\cap M_{2k}\to M_{2j}^{\prime}\cap
M_{2j+2k}$ be the shift as above. Then,

$\pi_{j}(p)L^{2}(M_{j})\otimes\pi_{k}(q)L^{2}(M_{k})\cong\pi_{j+k}(p\operatorname{sh}_{2j}(q))L^{2}(M_{j+k})$

as $M\text{-}M$ bimodules. And $p\operatorname{sh}_{2j}(q)\in M^{\prime}\cap
M_{2j+2k}$ is a projection with trace
$\operatorname{tr}_{M_{2j+2k}}(p\operatorname{sh}_{2j}(q))=\operatorname{tr}_{M_{2j}}(p)\operatorname{tr}_{M_{2k}}(q)$.

Proof: Observe that $p$ and $sh_{2j}(q)$ are commuting projections in
$M^{\prime}\cap M_{2j+2k}$, so $psh_{2j}(q)$ is also a projection with the
trace as stated above.

Without loss of generality, we assume $j\geq k$. We have $qe_{2k+2}\dots
e_{2j}\in M^{\prime}\cap M_{2j}$. By [4] Theorem 4.6 c). we obtain

$\pi_{j}(p)L^{2}(M_{j})\otimes\pi_{j}(qe_{2k+2}\dots
e_{2j})L^{2}(M_{k})\cong\pi_{2j}(p\operatorname{sh}_{2j}(qe_{2k+2}\dots
e_{2j}))L^{2}(M_{j+k})$.

And by Proposition 6.3, we have $\pi_{j}(qe_{2k+2}\dots
e_{2j})L^{2}(M_{j})\cong\pi_{k}(q)L^{2}(M_{k})$.

We can show that $sh_{2j}(e_{i})=e_{2j+i}$. Note that $q$ commutes with all
$e_{2k+2},\dots,e_{2_{j}}$, we have $p\operatorname{sh}_{2j}(qe_{2k+2}\dots
e_{2j})=p\operatorname{sh}_{2j}(q)e_{2j+2k+2}\dots e_{4j}$. Then by
Proposition 6.3 again, we obtain
$\pi_{2j}(p\operatorname{sh}_{2j}(qe_{2k+2}\dots
e_{2j}))L^{2}(M_{j+k})\cong\pi_{j+k}(p\operatorname{sh}_{2j}(q))L^{2}(M_{j+k})$,
which completes the proof.

The construction of bimodules Let $M_{j}$’s be the $\text{II}_{1}$ factors
that are constructed in Section 4 and Section 5. Let us consider the Jones
tower of $\text{II}_{1}$ factors

$M=M_{0}\subset M_{1}\subset^{e_{2}}M_{2}\subset\cdots$

By Proposition 5.3 and [4] Proposition 3.2, we have
$\operatorname{Hom}_{M\text{-}M}({}_{M}{L^{2}(M_{j})}_{n})=M^{\prime}\cap
M_{2j}\cong A_{2j}$.

Recall each $V_{i}$ must be in a summand of $W^{\otimes k}$ when $k\geq d(l)$
by Lemma 3.1. For each simple object $V_{i}$ in $R_{l}(\mathfrak{g})$, we
define

$k_{i}=\min\\{k\geq 0|\operatorname{Hom}(V_{i},W^{\otimes k})\neq 0\\}$,

which is the minimal integer $k$ such that $W^{\otimes k}$ contains $V_{i}$.
Note if $V_{i}$ is fundamental, $k_{i}=1$.

Define a map $\phi:\\{1,\dots,m\\}\to\mathbb{Z}_{2}$ by $\phi(i)=k_{i}\mod 2$.
It should be mentioned that $\phi(1)=0$ as $V_{1}=W^{\otimes 0}$ and
$\phi(i)=1$ if $V_{i}$ is fundamental as $W$ is the direct sum of fundamental
ones. We are now able to construct the simple bimodules as follows. For any
central projection $p\in A_{k}$, we let $z(p)$ denote the projection in
$\mathcal{Z}(A_{k})$ which is equivalent to $p$ in $A_{k}$. As all these
$A_{k}$’s are multi-matrix algebras, $z(p)$ would be a sum of diagonal
matrices with only $0$ and $1$ on the diagonals.

  * •

If $\phi(i)=0$, i.e. $k_{i}$ is even, say $k_{i}=2r_{i}$. We take a minimal
projection $g_{i}$ in $A_{2r_{i}}=M^{\prime}\cap M_{2r_{i}}$ such that
$z(g_{i})$ is the projection from $W^{\otimes 2r_{i}}$ on $V_{i}$. We let
$H_{i}=\pi_{r_{i}}(g_{i})L^{2}(M_{r_{i}})$.

  * •

If $\phi(i)=1$, i.e. $k_{i}$ is odd, say $k_{i}=2{r_{i}}-1$. We take a minimal
projection $g_{i}^{\prime}\in A_{2{r_{i}}-1}=M^{\prime}\cap M_{2{r_{i}}-1}$
such that $z(g_{i})$ is also the projection from $W^{\otimes 2r_{i}-1}$ on
$V_{i}$. Define $g_{i}=g_{i}^{\prime}\otimes\text{id}_{V_{0}}\in
A_{{k_{i}}+1}=M^{\prime}\cap M_{2r_{i}}$ and let
$H_{i}=\pi_{r_{i}}(g_{i})L^{2}(M_{r_{i}})$.

By Proposition 6.3, these bimodules only depend on the equivalence class of
the projections but not the particular choice of the minimal projection
$g_{i}$. In particular, we let $H_{1}$ denote the standard bimodule
$L^{2}(M)$, which corresponds to the unique nontrivial projection in
$\operatorname{End}(W^{\otimes 0})\cong\mathbb{C}$.

The construction of the fusion category $\text{Bimod}(M,M_{1})$ Define a
category

$\text{Bimod}(M,M_{1})=\\{$the equivalence classes of $M$-$M$ bimodules in
$\cup_{j}L^{2}(M_{j})\\}$,

where $M_{j}$ is obtained from the basic construction of $M_{j-2}\subset
M_{j-1}$ for each $j\geq 2$. It is well-known to be the tensor category
generated by the equivalence class $\pi_{j}(p)L^{2}(M_{j})$ with a minimal
projection $p\in M^{\prime}\cap M_{2j}$ for $j\geq 0$. It can also be shown
$\text{Bimod}(M,M_{1})$ is generated by the fundamental ones:
$H_{i}=\pi_{1}(p_{i})L^{2}(M_{1})$ with the projection $p_{i}\colon
W=V(0)\oplus(\oplus_{1\leq k\leq n}V(\omega_{k}))\to V(\omega_{i})$.

######  Lemma 6.5.

$\text{Bimod}(M,M_{1})$ is a fusion category with simple objects $H_{i}$’s
defined above.

Proof: By Theorem 6.4, they are closed under tensor products. Since the
inclusion $M\subset M_{1}$ of $\text{II}_{1}$ factors is of the finite depth
$d(l)$, there are finitely many simple objects. These objects are in one-to-
one correspondence with the (equivalence classes of) minimal projections in
the higher commutants $M^{\prime}\cap M_{2i}$ [4], which give us the bimodule
$H_{i}$’s.

The rest of this section is mainly devoted to proving the following theorem.

######  Theorem 6.6.

As a fusion category, $\text{Bimod}(M,M_{1})\cong R_{l}(G)$ .

The proof is based on several statements below.

######  Lemma 6.7.

Take any $f\in A_{2k}=M\cap M_{2k}$, we have
$\operatorname{sh}_{2j}(f)=i_{2j,2j+2k}(f)$.

Proof: Observe that $i_{2j,2j+2k}(A_{2k})\subset A_{2j+2k}$ and it commutes
with $A_{2j}$, we have $i_{2j,2j+2k}(A_{2k})\subset M_{2j}^{\prime}\cap
M_{2j+2k}$ which can be further shown to be a surjective, trace preserving,
$*$-isomorphism. Then the proof reduces to the construction of the isomorphism
$sh_{2j}$.

A functor $\Psi\colon\text{Bimod}(M,M_{1})\to R_{l}(G)$ is defined as follows:

$\Psi(\pi_{j}(p)L^{2}(M_{j}))=p(W^{\otimes 2j})$.

where $p\in M\cap M_{2j}$ for some $j$.

######  Lemma 6.8.

We have $\Psi(H_{i})=V_{i}$ for all $1\leq i\leq n$.

Proof: If $\phi(i)=0$, it is straightforward by the construction of $H_{i}$’s
above. If $\phi(i)=1$, we have
$\Psi(H_{i})=V_{i}\otimes\text{id}_{V_{0}}(W)=V_{i}\otimes V_{0}=V_{i}$ by
Lemma 6.7.

######  Lemma 6.9.

$\Psi(\pi_{j}(p)L^{2}(M_{j}))$ depends only on the isomophism class of $p$.
Hence $\Psi$ is well-defined.

Proof: Let $p$ be (equivalent to) a minimal projection in the $i$-th simple
summand of $A_{2j}$. Assume there is another projection $p^{\prime}\in M\cap
M_{2j^{\prime}}$ which is equivalent to $p$. Then it is also equivalent to a
minimal projection in the $i$-th simple summand of $A_{2j^{\prime}}$. Assume
$j\geq j^{\prime}$, then $p$ is equivalent to
$p^{\prime}e_{2j^{\prime}+2}\dots e_{2j}$ in $M^{\prime}\cap M_{2j}$. We have
$\pi_{j}(p)L^{2}(M_{j})\cong\pi_{j^{\prime}}(p^{\prime})L^{2}(M_{j}^{\prime})$
and both of them are mapped to $V_{i}$ under the functor $\Psi$.

Now it is clear that $\Psi^{-1}(V_{k})$ is the equivalence class of the
minimal projections in the $k$-th simple summand.

######  Proposition 6.10.

The functor $\Psi$ preserves tensor products.

Proof: Take any two projections $p\in M_{2j},q\in M_{2k}$. By Theorem 6.4, we
have
$\pi_{j}(p)L^{2}(M_{j})\otimes\pi_{k}(q)L^{2}(M_{k})\cong\pi_{j+k}(p\operatorname{sh}_{2j}(q))L^{2}(M_{j+k})$.
On the other hand, $p\operatorname{sh}_{2j}(q)=p\cdot
i_{2j,2j+2k}(q)=p\cdot(\text{id}_{W^{2j}}\otimes q)$. Hence
$p\operatorname{sh}_{2j}(q)(W^{\otimes
2j+2k})=p\cdot(\text{id}_{W^{2j}}\otimes q)(W^{\otimes 2j+2k})=p(W^{\otimes
2j})\otimes q(W^{\otimes 2k})$, which completes the proof.

######  Lemma 6.11.

The functor $\Psi$ preserves direct sums.

Proof: Now we take two irreducible bimodules $H_{j},H_{k}$ so that
$\Psi(H_{j})=V_{j},\Psi(H_{k})=V_{k}$. Let us consider $H_{j}\oplus H_{k}$
which is
$\pi_{r_{j}}(g_{j})L^{2}(M_{r_{j}})\oplus\pi_{r_{k}}(g_{k})L^{2}(M_{r_{k}})$.
Assume $j\geq k$, by Proposition 6.3, we have this is also the bimodule
$\pi_{r_{j}}(g_{j}\oplus g_{k}e_{2k+2}\dots e_{2j})L^{2}(M_{r_{j}})$ which is
a direct sum of two bimodules. For the first one,
$\Psi(\pi_{r_{j}}(g_{j})L^{2}(M_{r_{j}})=V_{j}$ is clear. And by Proposition
6.3 again, we have $\Psi(\pi_{r_{j}}(g_{k}e_{2k+2}\dots
e_{2r_{j}}))=\Psi(\pi_{r_{k}}(g_{k})L^{2}(M_{r_{k}}))=\Psi(H_{k})=V_{k}$.
Hence $\Psi(H_{j}\oplus H_{k})=\Psi(H_{j})\oplus\Psi(H_{k})$.

Proof: [Proof of Theorem 6.6] Take any two irreducible representations
$V_{i},V_{j}$ of $LG$. Assume we have the following decomposition of their
tensor product:

$V_{i}\otimes V_{j}=\oplus_{k=0}^{m}m_{i,j}^{k}\cdot V_{k}$,
$m_{i,j}^{k}\in\mathbb{Z}_{\geq 0}$.

We want to show that $H_{i}\otimes H_{j}$ has the same decomposition into the
irreducible $M$-$M$ bimodules $H_{k}$’s.

By Theorem 6.4, note $g_{i}\in M^{\prime}\cap M_{2r_{i}}$ and $g_{j}\in
M^{\prime}\cap M_{2r_{j}}$, we obtain

$\pi_{r_{i}}(g_{i})L^{2}(M_{r_{i}})\otimes\pi_{r_{j}}(g_{j})L^{2}(M_{r_{j}})\cong\pi_{r_{i}+r_{j}}(g_{i}\operatorname{sh}_{2r_{i}}(g_{j}))L^{2}(M_{r_{i}+r_{j}})$,

where $g_{i}\operatorname{sh}_{2r_{i}}(g_{j})$ is a projection $\in
M^{\prime}\cap M_{2r_{i}+2r_{j}}=A_{2r_{i}+2r_{j}}$ and $g_{i}$ commutes with
the minimal projection $sh_{2r_{i}}(g_{j})\in M_{2r_{i}}^{\prime}\cap
M_{2r_{i}+2r_{j}}$. We then have
$\Psi(\pi_{r_{i}+r_{j}}(g_{i}\operatorname{sh}_{2r_{i}}(g_{j}))L^{2}(M_{r_{i}+r_{j}})=V_{i}\otimes
V_{j}$ by the fact that

$z(g_{i})z(\operatorname{sh}_{2r_{i}}(g_{j}))(W^{\otimes
2r_{i}+2r_{j}})=z(g_{i})(W^{\otimes 2r_{i}})\otimes z(g_{j})(W^{\otimes
2r_{j}})$.

By taking $\Phi^{-1}$, we obtain $H_{i}\otimes
H_{j}=\oplus_{k=0}^{m}m_{i,j}^{k}\cdot H_{k}$.

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