# Subalgebras of étale algebras in braided fusion categories

Sebastian Burciu  Inst. of Math. “Simion Stoillow" of the Romanian Academy
P.O. Box 1-764, RO-014700, Bucharest, Romania<EMAIL_ADDRESS>
###### Abstract.

In [7, Rem. 3.4] the authors asked the question if any étale subalgebra of an
étale algebra in a braided fusion category is also étale. We give a positive
answer to this question if the braided fusion category $\mathcal{C}$ is
pseudo-unitary and non-degenerate. In the case of a pseudo-unitary fusion
category we also give a new description of the lattice correspondence from [6,
Theorem 4.10]. This new description enables us to describe the two binary
operations on the lattice of fusion subcategories.

This work was supported by a grant of the Ministry of Research, Innovation and
Digitization, CNCS/CCCDI - UEFISCDI, project number PN-III-P4-ID-
PCE-2020-0878, within PNCDI III

##  1\. Introduction

In [7, Rem. 3.4] the authors asked the question if any étale subalgebra of an
étale algebra in a braided fusion category $\mathcal{C}$ is also étale. It
seems that an answer to this question does not appear in the literature at
this moment. We give a positive answer to this question if the braided fusion
category $\mathcal{C}$ is pseudo-unitary and non-degenerate.

For any fusion category $\mathcal{C}$ it was shown in [6, Lemma 3.5] that
${\mathrm{A}}=R(\mathbf{1})$ is a connected étale algebra in
${\mathcal{Z}(\mathcal{C})}$ where $R$ is a right adjoint of the forgetful
functor $F:{\mathcal{Z}(\mathcal{C})}\rightarrow\mathcal{C}$. Moreover, in [6,
Theorem 4.10] it was shown that the lattice of fusion subcategories of
$\mathcal{C}$ is in a reverse one to one correspondence with the lattice of
étale subalgebras of ${\mathrm{A}}$. ${\mathrm{A}}$ is sometimes called the
adjoint algebra of $\mathcal{C}$ in [18].

Recall that given a connected étale subalgebra ${\mathrm{L}}$ of
${\mathrm{A}}$ in [6] it was associated a fusion subcategory
$\beta({\mathrm{L}})$ of $\mathcal{C}$. Under the tensor equivalence
$R:\mathcal{C}\rightarrow{\mathcal{Z}(\mathcal{C})}_{\mathrm{A}}$ the fusion
subcategory $\beta({\mathrm{L}})$ consists of those ${\mathrm{A}}$-modules in
${\mathcal{Z}(\mathcal{C})}$ that are dyslectic as right
${\mathrm{L}}$-modules.

For a pivotal tensor categories, Shimizu has developed in [18] a character
theory for fusion categories, similar to the classical theory of
representations of finite groups and semisimple Hopf algebras. Moreover he
showed that the adjoint algebra ${\mathrm{A}}$ associated to a fusion category
has a natural action on any object of $\mathcal{C}$.

The main goal of this paper is to show that in the case of a pseudo-unitary
fusion category $\mathcal{C}$ all unitary subalgebras of ${\mathrm{A}}$ in
${\mathcal{Z}(\mathcal{C})}$ are connected étale. We also obtain a new
characterization of the fusion subcategories from [6] associated to étale
subalgebras of ${\mathrm{A}}$.

In Section 3, in the case $\mathcal{C}$ is a pseudo-unitary fusion category,
we associate to any unitary subalgebra ${\mathrm{L}}$ of ${\mathrm{A}}$ in
${\mathcal{Z}(\mathcal{C})}$ a fusion subcategory $\mathcal{S}_{\mathrm{L}}$
of $\mathcal{C}$, consisting of those simple objects of $\mathcal{C}$ whose
characters (as defined in [18]) have trivial restriction to ${\mathrm{L}}$. We
also denote by ${{\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}}$ the full abelian
subcategory of $\mathcal{C}$ consisting of those objects that receive a
trivial action from the subalgebra ${\mathrm{L}}$, see Definition 3.4.

It is shown that for any pivotal fusion category $\mathcal{C}$ one has a chain
of inclusions of abelian full subcategories:

(1.1)
$\beta({\mathrm{L}})\subseteq{{\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}}\subseteq\mathcal{S}_{\mathrm{L}}.$

On the other hand, we show in Theorem 3.2 that if $\mathcal{C}$ is a pseudo-
unitary fusion category then
$\mathrm{FPdim}(\mathcal{S}_{\mathrm{L}})=\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}({\mathrm{L}})}$.
Since by [6, Theorem 4.10] we also have
$\mathrm{FPdim}(\beta({\mathrm{L}}))=\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}({\mathrm{L}})}$
this shows that the above chain inclusions of fusion subcategories are in fact
equalities. This chain of equalities implies that
${{\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}}$ is also a fusion subcategory
and that any unitary connected subalgebra of ${\mathrm{A}}$ is also an étale
subalgebra of ${\mathrm{A}}$. These facts can be stated as follows:

######  Theorem 1.1.

Any fusion subcategory of a pseudo-unitary fusion category $\mathcal{C}$ is of
the form ${{\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}}$ for some unitary
subalgebra ${\mathrm{L}}$ of the algebra ${\mathrm{A}}:=R(\mathbf{1})$.
Moreover, every such unitary subalgebra ${\mathrm{L}}$ of ${\mathrm{A}}$ is
étale.

From Theorem 1.1, using some results concerning étale algebras developed in
[6] we deduce that in a non-degenerate braided pseudo-unitary fusion category,
any subalgebra of an étale algebra is also étale. More precisely we prove the
following:

######  Theorem 1.2.

Let $\mathcal{C}$ be a pseudo-unitary non-degenerate braided fusion category
and $A$ an étale algebra of ${\mathcal{B}}$. Then any unitary subalgebra $L$
of $A$ is also étale.

For two subalgebras of the adjoint algebra ${\mathrm{A}}$ we denote by
${\mathrm{L}}{\mathrm{M}}$ the image of ${\mathrm{L}}\otimes{\mathrm{M}}$
under the multiplication
$m:{\mathrm{A}}\otimes{\mathrm{A}}\rightarrow{\mathrm{A}}$. Since
${\mathrm{A}}$ is commutative it follows that
${\mathrm{L}}{\mathrm{M}}={\mathrm{M}}{\mathrm{L}}$ and moreover
${\mathrm{M}}{\mathrm{L}}$ is a subalgebra of ${\mathrm{A}}$. Concerning the
two binary operations on the lattice of fusion subcategories we prove the
following

######  Theorem 1.3.

Let $\mathcal{C}$ be a pseudo-unitary fusion category and
${\mathrm{L}},{\mathrm{M}}$ two unitary subalgebras of the adjoint algebra
${\mathrm{A}}$ of $\mathcal{C}$. With the above notations one has

${\mathcal{S}_{{\mathrm{L}}}}\cap\mathcal{S}_{\mathrm{M}}=\mathcal{S}_{{\mathrm{L}}{\mathrm{M}}},\;{\mathcal{S}_{{\mathrm{L}}}}\vee\mathcal{S}_{\mathrm{M}}=\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}}.$

Shortly, the organization of the paper is as follows. In Section 2 we recall
the basics on fusion categories and also the character theory developed by
Shimizu in [18] for pivotal fusion categories. In Section 3 we construct the
fusion subcategory ${\mathcal{S}_{\mathrm{L}}}$ mentioned above in the pivotal
case. We also define for any fusion category the full abelian subcategory
${\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}$ and prove the inclusion
${\mathcal{S}_{\mathrm{L}}}\supseteq{\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}$
from Equation (1.1). In Section 4 we prove the inclusion
${\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}\supseteq\beta({\mathrm{L}})$ from
Equation (1.1). Also in this section we give the proofs of the above three
theorems. In Subsection 4.6 we also consider as an example the case of fusion
categories coming from representation categories semisimple Hopf algebras.

We work over an algebraically closed field $\Bbbk$ of characteristic zero. All
the Hopf algebras and fusion categories notations follow [9].

##  2\. Preliminaries

In this section we review the basic properties of fusion categories that are
needed through the paper. For the definition and standard theory of monoidal
categories, we refer the reader to [14] and [13]. Given a monoidal category we
define by $\mathcal{O}(\mathcal{C})$ the class of all its objects. Recall that
a left dual object of $X\in\mathcal{O}(\mathcal{C})$ is a triple
$(X^{*},{\operatorname{ev}_{X}},\mathrm{coev}_{X})$ consisting of an object
$X^{*}\in\mathcal{O}(\mathcal{C})$ and morphisms
${\operatorname{ev}_{X}}:X^{*}\otimes X\rightarrow\mathbf{1}$ and
$\mathrm{coev}_{X}:\mathbf{1}\rightarrow X\otimes X^{*}$ such that the
following equalities are satisfied:

(2.1)
$({\operatorname{ev}_{X}}\otimes\operatorname{id}_{{X^{*}}})\circ(\operatorname{id}_{{X^{*}}}\otimes
d)=\operatorname{id}_{{X^{*}}},\;\;(\operatorname{id}_{X}\otimes{\operatorname{ev}_{X}})\circ(\mathrm{coev}_{X}\otimes\operatorname{id}_{X})=\operatorname{id}_{X}.$

Similarly, one can define a right dual of $X$ (which in fact is a left dual of
$X$ in $\mathcal{C}^{rev}$). A monoidal category $\mathcal{C}$ is said to be
rigid if every object of $\mathcal{C}$ has both a left and a right dual
object.

Recall that a finite tensor category  [11] over a field $\Bbbk$ is a rigid
monoidal category $\mathcal{C}$ such that $\mathcal{C}$ is a finite abelian
category, the tensor product
$\otimes:\mathcal{C}\times\mathcal{C}\rightarrow\mathcal{C}$ is $\Bbbk$-linear
in each variable, and $\mathrm{End}_{\mathcal{C}}(\mathbf{1})\simeq\Bbbk$ as
algebras. A fusion category [10] is a semisimple finite tensor category.

For a monoidal category $\mathcal{C}$, the left monoidal center (or the
Drinfeld center) of $\mathcal{C}$ is a category ${\mathcal{Z}(\mathcal{C})}$
defined as follows: An object of ${\mathcal{Z}(\mathcal{C})}$ is a pair
$(V,\sigma_{V})$ consisting of an object $V\in\mathcal{C}$ and a natural
isomorphism

$\sigma_{V,X}:V\otimes X\rightarrow X\otimes V$

for all $X\in\mathcal{O}(\mathcal{C})$, satisfying a part of the hexagon
axiom. A morphism $f:(V,\sigma_{V})\rightarrow(W,\sigma_{W})$ in
${\mathcal{Z}(\mathcal{C})}$ is a morphism in $\mathcal{C}$ such that
$(\operatorname{id}_{X}\otimes
f)\circ\sigma_{V,X}=\sigma_{W,X}\circ(f\otimes\operatorname{id}_{X})$ for all
$X\in\mathcal{C}$. The composition of morphisms is defined in an obvious way.
The category ${\mathcal{Z}(\mathcal{C})}$ is in fact a braided monoidal
category, see, e.g., [13, Chapt. XIII.3] for details.

Let $\mathcal{C}$ be any finite tensor category and $A$ be an algebra in
$\mathcal{C}$. Recall that the algebra $A$ is called connected if
$\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},A)=\Bbbk$. A subalgebra
$B\xrightarrow{\iota}A$ is called unitary if $u_{B}\circ\iota=u_{A}$ where
$u_{B}$ and $u_{A}$ are the units of $B$ and $A$. Recall [6] that an algebra
$A$ of a braided fusion category ${\mathcal{B}}$ is called étale if it is
separable and connected. An algebra $A$ is called separable if its
multiplication $A\otimes A\xrightarrow{m}A$ has a section as $A$-bimodules in
$\mathcal{C}$. This is equivalent by [6, Theorem 3.2] to the fact that the
category of right (or left) $A$-modules ${\mathcal{B}}_{A}$ is a semisimple
category.

Let $A$ be an algebra in a braided fusion category ${\mathcal{B}}$ and $M$ be
a right $A$-module in ${\mathcal{B}}$. Then $M$ is called dyslectic (local) if
$c_{A,M}\circ c_{M,A}\circ\rho_{M}=\rho_{M}$ where $\rho_{M}:M\otimes
A\rightarrow M$ is the module structure of $M$.

Dyslectic modules form a full subcategory of ${\mathcal{B}}_{A}$ which is
usually denoted by ${\mathcal{B}}^{0}_{A}$. This subcategory is closed under
$\otimes_{A}$ and the braiding in ${\mathcal{B}}$ induces a natural braided
structure in ${\mathcal{B}}^{0}_{A}$, see [15, Section 2]. Thus,
${\mathcal{B}}^{0}_{A}$ is a braided fusion category.

###  2.1. The central Hopf comonad of a finite tensor category

Let $\mathcal{C}$ be a fusion category. The forgetful functor
$F:{\mathcal{Z}(\mathcal{C})}\rightarrow\mathcal{C}$ admits a right adjoint
functor $R:\mathcal{C}\rightarrow{\mathcal{Z}(\mathcal{C})}$ and
$Z:=FR:\mathcal{C}\rightarrow\mathcal{C}$ is a Hopf comonad called the central
Hopf comonad associated to $\mathcal{C}$. Moreover, one has that

(2.2) $Z(V)\simeq\int_{X\in\mathcal{C}}X\otimes V\otimes X^{*}.$

see [18, Section 2.6]. We denote by $\pi_{V;X}:Z(V)\rightarrow X\otimes
V\otimes X^{*}$ the universal dinatural transformation associated to the end
$Z(V)$. The Hopf comonad structure of $Z$ can be described in terms of the
dinatural transformation $\pi$. The comultiplication
$\delta:Z\rightarrow{Z}^{2}$ is the unique natural transformation such that

(2.3)
$(\operatorname{id}_{X}\otimes\pi_{V;Y}\otimes\operatorname{id}_{X^{*}})\circ\pi_{Z(V);X}\circ\delta_{V}=\pi_{V;X\otimes
Y}$

The counit of ${\bf{\epsilon}}:Z\rightarrow\operatorname{id}_{\mathcal{C}}$ is
given by ${\bf\epsilon}_{V}:=\pi_{V;1}.$ There is a lax monoidal structure
defined on $Z$ by

$Z_{2}(M,N):Z(M)\otimes Z(N)\rightarrow Z(M\otimes N)$

as the unique map making the following diagram commutative:  ${Z(M)\otimes
Z(N)}$${Z(M\otimes N)}$${(X\otimes M\otimes X^{*})\otimes(X\otimes N\otimes
X^{*})}$${X\otimes M\otimes N\otimes
X^{*}}$$\scriptstyle{Z_{2}{(M,N)}}$$\scriptstyle{{\pi}_{M;\;X}\otimes{\pi}_{N;\;X}}$$\scriptstyle{\pi_{M\otimes
N,\;X}}$$\scriptstyle{\operatorname{id}_{X\otimes
M}\otimes\operatorname{ev}_{X}\otimes\operatorname{id}_{N\otimes X^{*}}}$

which can be written as

(2.4) $\pi_{M\otimes N,\;X}\circ Z_{2}(M,N)=(\operatorname{id}_{X\otimes
M}\otimes\operatorname{ev}_{X}\otimes\operatorname{id}_{N\otimes
X^{*}})\circ({\pi}_{M;\;X}\otimes{\pi}_{N;\;X})$

###  2.2. On the adjoint algebra of a finite tensor category

It is known that ${\mathrm{A}}:=R(\mathbf{1})$ has the structure of central
commutative algebra in $\mathcal{Z}(\mathcal{C})$. The half braiding of
${\mathrm{A}}$, denoted by $c_{A,X}$, is defined by

$A\otimes
X\xrightarrow{\delta_{\mathbf{1}}\otimes\operatorname{id}_{X}}Z(A)\otimes
X\xrightarrow{\pi_{Z(\mathbf{1}),X}\otimes\operatorname{id}_{X}}X\otimes
A\otimes X^{*}\otimes
X\xrightarrow{\operatorname{id}_{X}\otimes\operatorname{id}_{A}\otimes\operatorname{ev}_{X}}X\otimes
A.$

The multiplication $m:{\mathrm{A}}\otimes{\mathrm{A}}\rightarrow{\mathrm{A}}$
and the unit $u_{\mathrm{A}}:\mathbf{1}\rightarrow{\mathrm{A}}$ of the adjoint
algebra ${\mathrm{A}}$ are uniquely determined by by the universal property of
the end $A=Z(\mathbf{1})$ as:

(2.5) $\pi_{\mathbf{1};X}\circ u_{\mathrm{A}}=\mathrm{coev}_{X},$ (2.6)
$\pi_{\mathbf{1};X}\circ
m=(\operatorname{id}_{X}\otimes\operatorname{ev}_{X}\otimes\operatorname{id}_{X^{*}})\circ(\pi_{\mathbf{1};X}\otimes\pi_{\mathbf{1};X}).$

Moreover ${\bf\epsilon}_{\mathbf{1}}:{\mathrm{A}}\rightarrow\mathbf{1}$ is a
morphism of algebras, see [18].

Any object $X\in\mathcal{C}$ is canonically an $A$-module in $\mathcal{C}$,
via the morphism:

(2.7) $\alpha_{X}:A\otimes
X\xrightarrow{\pi_{\mathbf{1};X}\otimes\operatorname{id}_{X}}X\otimes
X^{*}\otimes X\xrightarrow{\operatorname{id}_{X}\otimes ev_{X}}X.$

We denote by $c_{R(M),X}:R(M)\otimes X\rightarrow X\otimes R(M)$ the half
braiding of $R(M)\in{\mathcal{Z}(\mathcal{C})}$. Recall that with the above
braiding ${\mathrm{A}}=R(\mathbf{1})$ becomes a commutative algebra in the
center ${\mathcal{Z}(\mathcal{C})}$. Moreover, by [18, Equation (3.12)] one
has that

(2.8) $\alpha_{X}=(A\otimes X\xrightarrow{c_{{\mathrm{A}},X}}X\otimes
A\xrightarrow{\operatorname{id}_{X}\otimes\epsilon_{\mathbf{1}}}X).$

The vector space
$\mathrm{{CE}(\mathcal{C})}:=\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},A)$ is
called the set of central elements. For $a,b\in\mathrm{{CE}(\mathcal{C})}$ we
set $ab:=m\circ(a\otimes b)$. Then the set $\mathrm{{CE}(\mathcal{C})}$ is a
monoid with respect to this operation. Note that for the algebra unit
$u_{\mathrm{A}}:\mathbf{1}\rightarrow{\mathrm{A}}$ of the algebra
${\mathrm{A}}$ to the unit of the monoid $\mathrm{{CE}(\mathcal{C})}$ is
$F(u_{\mathrm{A}})$.

###  2.3. Internal characters for pivotal strctures

Recall that a pivotal structure $j$ on a tensor category $\mathcal{C}$ is a
tensor isomorphism $j:\operatorname{id}_{\mathcal{C}}\rightarrow(-)^{**}$.
Using the pivotal structure one can construct a right evaluation as follows:

$\widetilde{ev}_{X}:X\otimes X^{*}\xrightarrow{j\otimes id}X^{**}\otimes
X^{*}\xrightarrow{ev_{X^{*}}}\mathbf{1}.$

Then the right partial pivotal trace of $f:A\otimes X\rightarrow B\otimes X$
is defined as follows:

(2.9)
$\mathrm{tr}_{A,B}^{X}:A=A\otimes\mathbf{1}\xrightarrow{\operatorname{id}_{A}\otimes
coev_{X}}A\otimes X\otimes X^{*}\xrightarrow{f\otimes id}B\otimes X\otimes
X^{*}\xrightarrow{\operatorname{id}_{B}\otimes\widetilde{ev}_{X}}B.$

The usual right pivotal trace of an endomorphism $f:X\rightarrow X$ is
obtained as a particular case for $A=B=\mathbf{1}$. In particular, the right
pivotal dimension $\dim^{r}(X)$ of $X$ is defined as the right trace of the
identity of $X$. A pivotal structure a on a tensor category $\mathcal{C}$ is
called spherical if $\dim(V)=\dim(V^{*})$ for any object
$V\in\mathcal{O}(\mathcal{C})$. Given an object $X\in\mathcal{O}(\mathcal{C})$
the internal character $\mathrm{ch}(X)$ is defined as the morphism

$\mathrm{ch}(X):=\mathrm{tr}^{X}_{A,\mathbf{1}}(\alpha_{X}):A\rightarrow\mathbf{1}.$

Therefore using Equation (2.9) one can write that

(2.10) $\mathrm{ch}(X)=A\otimes\mathbf{1}\xrightarrow{\operatorname{id}\otimes
coev_{X}}A\otimes X\otimes X^{*}\xrightarrow{\alpha_{X}\otimes id}X\otimes
X^{*}\xrightarrow{\widetilde{ev}_{X}}\mathbf{1}.$

Then the space
$\mathrm{{CF}}(\mathcal{C}):=\mathrm{{Hom}}_{\mathcal{C}}(A,\mathbf{1})$ is
called the space of class functions of $\mathcal{C}$. For two class functions
$f,g\in\mathrm{{CF}}(\mathcal{C})$ one can define a multiplication by

$f\star g:=f\circ Z(g)\circ\delta_{\mathbf{1}}.$

Here $\delta:Z\rightarrow Z^{2}$ is the comultiplication structure of $Z$
defined in the Equation (2.3). By [18, Theorem 3.10] one has that
$\mathrm{ch}(X\otimes Y)=\mathrm{ch}(X)\mathrm{ch}(Y)$ for any two objects $X$
and $Y$ of $\mathcal{C}$. For a finite tensor category $\mathcal{C}$ the space
of class functions $\mathrm{{CF}}(\mathcal{C})$ is a finite-dimensional
algebra.

###  2.4. The cointegral and integral of a fusion category

Let $\mathcal{C}$ be a fusion category and $A=Z(\mathbf{1})$ be its adjoint
algebra as defined above.

An integral in $\mathcal{C}$ is a morphism $\Lambda:\mathbf{1}\rightarrow A$
in $\mathcal{C}$ such that

$m\circ(\operatorname{id}_{A}\otimes\Lambda)={\bf\epsilon}_{\mathbf{1}}\otimes\Lambda.$

A cointegral in $\mathcal{C}$ is a morphism $\lambda:A\rightarrow\mathbf{1}$
such that

$Z(\lambda)\circ\delta_{\mathbf{1}}=u\otimes\lambda$

where $u:\mathbf{1}\rightarrow A$ is the unit of the algebra $A$. It is well
known that the integral and cointegral of a finite unimodular tensor category
are unique up to a scalar, see [18].

###  2.5. Fourier transform for finite tensor categories

Let $\lambda\in\mathrm{{CF}}(\mathcal{C})$ be a non-zero integral of
$\mathcal{C}$. The Fourier transform for finite tensor categories of
$\mathcal{C}$ associated to $\lambda$ is the linear map

(2.11)
$\mathcal{F}_{\lambda}:\mathrm{{CE}(\mathcal{C})}\rightarrow\mathrm{{CF}}(\mathcal{C})\;\;\text{given
by}\;\;a\mapsto\lambda\leftharpoonup\mathcal{S}(a)$

where
$\mathcal{S}:\mathrm{{CE}(\mathcal{C})}\rightarrow\mathrm{{CE}(\mathcal{C})}$
is the antipodal operator on $\mathrm{{CE}(\mathcal{C})}$, see [18, Definition
3.6]. The Fourier transform is a bijective $\Bbbk$-linear map whose inverse is
given in [18, Equation (5.17)]. Here, the right action denoted by
$\leftharpoonup$ of $\mathrm{{CE}(\mathcal{C})}$ on
$\mathrm{{CF}}(\mathcal{C})$ given by $f\leftharpoonup b=f\circ
m\circ(b\otimes\operatorname{id}_{A})$ for all
$f\in\mathrm{{CF}}(\mathcal{C})$ and $b\in\mathrm{{CE}(\mathcal{C})}$. There
is also a non-degenerate evaluation pairing
$\langle,\;\rangle_{\mathrm{{\mathrm{A}}}}$ given by

(2.12)
$\langle\;,\;\rangle_{\mathrm{{\mathrm{A}}}}:\mathrm{{CF}}(\mathcal{C})\otimes\mathrm{{CE}(\mathcal{C})}\rightarrow\Bbbk,\;\langle\chi,z\rangle\mapsto\chi\circ
z.$

#### Fourier transform for fusion categories

For the rest of this section, suppose that $\mathcal{C}$ is a pivotal fusion
category over an algebraically closed field $\Bbbk$. Furthermore, let
$\mathrm{Irr}(\mathcal{C}):=\\{V_{0},\dots,V_{m}\\}$ be a complete set of
representatives of isomorphism classes of simple objects with
$V_{0}=\mathbf{1}$, the unit object. For $i\in\\{0,\dots,m\\}$, we define
$i^{*}\in\\{0,\dots,m\\}$ by $V_{i}^{*}\simeq V_{i^{*}}$. Then $i\mapsto
i^{*}$ is an involution on $\\{0,\dots,m\\}$. As an object of $\mathcal{C}$,
the adjoint algebra decomposes as

(2.13) $A\simeq\bigoplus_{i=0}^{m}V_{i}\otimes V_{i}^{*}.$

Shimizu has defined in [18] the elements

$E_{i}:\mathbf{1}\xrightarrow{\mathrm{coev}_{V_{i}}}V_{i}\otimes
V_{i}^{*}\hookrightarrow A,\;\;\chi_{i}:A\xrightarrow{\pi_{i}}V_{i}\otimes
V_{i}^{*}\xrightarrow{\widetilde{\mathrm{ev}_{V_{i}}}}\mathbf{1}.$

It is easy to see that $\\{E_{i}\\}_{i=0,\dots,m}$ and
$\\{\chi_{i}\\}_{i=0,\dots,m}$ are bases for $\mathrm{{CE}(\mathcal{C})}$ and
$\mathrm{{CF}}(\mathcal{C})$ respectively, such that

(2.14) $\langle\chi_{i},E_{j}\rangle=d_{i}\delta_{i,j}.$

where $d_{i}:=\dim(V_{i})$. Moreover, $E_{i}E_{j}=\delta_{i,j}$ and
$\mathcal{S}(E_{i})=E_{i^{*}}$, where
$\tilde{S}:\mathrm{{CE}(\mathcal{C})}\rightarrow\mathrm{{CE}(\mathcal{C})}$ is
the antipodal mentioned above. The elements $\chi_{i}$ are called the
irreducible characters of the simple objects $V_{i}$ and
$E_{i}\in\mathrm{{CE}(\mathcal{C})}$ their corresponding primitive
idempotents. Note that $E_{0}$, the idempotent associated to the unit object
$\mathbf{1}$ of $\mathcal{C}$, is the idempotent integral
$\Lambda\in\mathrm{{CE}(\mathcal{C})}$, see [18, Lemma 6.1]. By [18, Equation
(6.8)] one has that the idempotent cointegral of $\mathcal{C}$ can be written
as:

(2.15)
$\lambda_{\mathcal{C}}=\frac{1}{\dim(\mathcal{C})}(\sum_{[V_{i}]\in\mathrm{Irr}(\mathcal{C})}d_{i^{*}}\chi_{i}).$

Since $\Lambda=E_{0}$ it follows by Equation (2.15) that
$\langle\lambda_{\mathcal{C}},\Lambda\rangle=\frac{1}{\dim(\mathcal{C})}$. It
also follows that

(2.16)
$\mathcal{F}_{\lambda}^{-1}(\chi_{i})=\frac{\dim(\mathcal{C})}{d_{i}}E_{i^{*}}$

This equation was proven in [18, Equation (6.10)] under the hypothesis that
the Grothendieck ring of $\mathcal{C}$ is commutative but the proof works word
by word also in the case of an arbitrary pivotal fusion category, with the not
necessarily commutative Grothendieck ring.

For a spherical fusion category $\mathcal{C}$ one has
$\dim(V_{i^{*}})=\dim(V_{i})$ and it follows from [5, Equation (4.7)] that$:$

(2.17)
$\langle\chi,\;\mathcal{F}_{\lambda}^{-1}(\mu)\rangle_{\mathrm{A}}=\dim(\mathcal{C})\tau(\chi\mu).$

for any $\chi,\mu\in\mathrm{{CF}}(\mathcal{C})$.

###  2.6. On the inclusion of class functions of a fusion subcategory

Let ${\mathcal{D}}$ be a fusion subcategory of a given fusion category
$\mathcal{C}$. As in [18, Sect. 4.3], for any object $V$ of $\mathcal{C}$ the
end

$\bar{Z}(V):{\mathcal{D}}^{\mathrm{op}}\times{\mathcal{D}}\rightarrow\mathcal{C},\;\;\bar{Z}(V):=\int_{X\in{\mathcal{D}}}X\otimes
V\otimes X^{*}$

exists and we denote by
$\bar{\pi}_{\mathbf{1}\;X}:{\bar{Z}(\mathbf{1})}\rightarrow X\otimes X^{*}$
the universal dinatural maps defining this end. From the universal property of
${\bar{Z}(\mathbf{1})}$ there is a unique canonical map
$Z(V)\xrightarrow{q_{V}}{\bar{\mathrm{Z}}}(V)$ in $\mathcal{C}$, such that
$\bar{\pi}_{V;\;X}\circ q={\pi}_{V;\;X}$ for any object $X$ of
${\mathcal{D}}$. In [5, Appendix] it is shown that
$q:Z(\mathbf{1})\rightarrow{\bar{Z}(\mathbf{1})}$ induces a map
$q_{\mathbf{1}}^{*}:\mathrm{CF}({\mathcal{D}})\rightarrow\mathrm{{CF}}(\mathcal{C}),\;\chi\mapsto\chi\circ
q_{\mathbf{1}}$ that is a monomorphism of $\Bbbk$-algebras.

###  2.7. The adjunction isomorphisms for
$F:{\mathcal{Z}(\mathcal{C})}\rightarrow\mathcal{C}$

Let $\mathcal{C}$ be any fusion category and
$R:{\mathcal{Z}(\mathcal{C})}\rightarrow\mathcal{C}$ be a right adjoint of the
forgetful functor as in the previous section. It is known that $Z:=FR$ is a
Hopf comonad in $\mathcal{C}$. One has that the category of $Z$-comodules in
$\mathcal{C}$ is equivalent to the center ${\mathcal{Z}(\mathcal{C})}$, see
[8] and [2]. Moreover, in this case, the comodule forgetful functor
$\mathcal{C}^{Z}\rightarrow\mathcal{C}$ identifies to
$F:{\mathcal{Z}(\mathcal{C})}\rightarrow\mathcal{C}$. This shows that the
adjunction isomorphisms are given by

(2.18)
$\psi_{V,\mathbf{1}}:\mathrm{{Hom}}_{\mathcal{C}}(F(V),\mathbf{1})\rightarrow\mathrm{{Hom}}_{{\mathcal{Z}(\mathcal{C})}}(V,{\mathrm{A}}),\;f\mapsto
R(f)\circ\eta_{V},$

and its inverse is given by

(2.19)
$\psi_{V,\mathbf{1}}^{-1}:\mathrm{{Hom}}_{{\mathcal{Z}(\mathcal{C})}}(V,{\mathrm{A}})\rightarrow\mathrm{{Hom}}_{\mathcal{C}}(F(V),\mathbf{1}),\;g\mapsto{\bf\epsilon}_{\mathbf{1}}\circ
F(g).$

for any object $V\in{\mathcal{Z}(\mathcal{C})}$. Here $\eta_{V}:V\rightarrow
R(F(V))$ is the unit of the adjunction which coincides to the $Z$-comodule
structure of $V\in{\mathcal{Z}(\mathcal{C})}$.

###  2.8. Frobenius-Perron dimensions and pseudo-unitarity

For a simple object $X$ of a fusion category $\mathcal{C}$ the Frobenius-
Perron eigenvalue of the left multiplication by $[X]$ on the Grothendieck ring
$K_{0}(\mathcal{C})$ is denoted by $\mathrm{FPdim}(X)$ and it is called the
Frobenius-Perron dimension of $X$. Recall that $\mathrm{FPdim}(X)$ is a
positive real algebraic number and the Frobenius-Perron dimension extends
linearly to an algebra morphism
$\mathrm{FPdim}:K_{0}(\mathcal{C})\otimes_{\mathbb{Z}}\mathbb{C}\rightarrow\mathbb{C}$.
The Frobenius-Perron dimension $\mathrm{FPdim}(\mathcal{C})$ of $\mathcal{C}$
is defined as

$\mathrm{FPdim}(\mathcal{C}):=\sum_{X\in\mathrm{Irr}(\mathcal{C})}\mathrm{FPdim}(X)^{2}.$

A fusion category $\mathcal{C}$ is called pseudo-unitary if
$\mathrm{FPdim}(\mathcal{C})=\dim(\mathcal{C})$. In this case, $\mathcal{C}$
admits a unique (canonical) spherical structure with respect to which the
categorical dimensions of simple objects are all positive, see [10,
Proposition 8.23]. With respect to this spherical structure, the categorical
dimension of any object coincides with its Frobenius-Perron dimension, i.e.
$\mathrm{FPdim}(X)=\dim(X)$ for any object $X\in\mathcal{O}(\mathcal{C})$.
Moreover, every full fusion subcategory of $\mathcal{C}$ is pseudo-unitary.

##  3\. On the subcategory ${\mathcal{S}_{\mathrm{L}}}$ via character theory

Let as above $\mathcal{C}$ be a pivotal fusion category and
$F:{\mathcal{Z}(\mathcal{C})}\rightarrow\mathcal{C}$ be the forgetful functor
with right adjoint $R:\mathcal{C}\rightarrow{\mathcal{Z}(\mathcal{C})}$. Let
also ${\mathrm{A}}:=R(\mathbf{1})$ be its adjoint algebra and
$\iota:{\mathrm{L}}\hookrightarrow{\mathrm{A}}$ be a unitary subalgebra of
${\mathrm{A}}$. We also denote by $L$ and $A$ the images of $L$ and $A$ under
the forgetful functor $F$. Since ${\mathrm{A}}$ is connected it follows that
${\mathrm{L}}$ is also connected. Note also that ${\mathrm{L}}$ is a
commutative algebra in ${\mathcal{Z}(\mathcal{C})}$ as a subalgebra of
${\mathrm{A}}$. Since ${\mathcal{Z}(\mathcal{C})}$ is a fusion category there
is also a projection ${\mathrm{A}}\xrightarrow{\pi}{\mathrm{L}}$ in
${\mathcal{Z}(\mathcal{C})}$ of ${\mathrm{A}}$ onto the object ${\mathrm{L}}$
such that $\pi\circ\iota=\operatorname{id}_{\mathrm{L}}$. Since $\mathcal{C}$
is a fusion category, the left adjoint of $F$ is also isomorphic to $R$ (see
[17]), and therefore
$\mathrm{{Hom}}_{\mathcal{Z}(\mathcal{C})}(\mathbf{1},{\mathrm{A}})=\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},\mathbf{1})\cong\Bbbk$
. Thus the unit $u_{\mathrm{L}}$ of the subalgebra ${\mathrm{L}}$ can be
written as $\pi\circ u_{\mathrm{A}}$, where
${\mathrm{A}}\xrightarrow{\pi}{\mathrm{L}}$ is the projection in
${\mathcal{Z}(\mathcal{C})}$ of ${\mathrm{A}}$ into ${\mathrm{L}}$.

Define also a character space of ${\mathrm{L}}$ by
${\mathrm{CF}}(L):=\mathrm{{Hom}}_{\mathcal{C}}(L,\mathbf{1})$. Then there are
well-defined restriction maps:

(3.1)
${\mathrm{Res}}:=\mathrm{{Hom}}_{\mathcal{C}}(F(\iota_{L})\text{,}\;-):\mathrm{{CF}}(\mathcal{C})\rightarrow{\mathrm{CF}}(L),$
(3.2)
$\mathrm{Res}^{A}_{L}:=\mathrm{{Hom}}_{{\mathcal{Z}(\mathcal{C})}}(\iota_{L}\text{,}\;-):\mathrm{{Hom}}_{\mathcal{Z}(\mathcal{C})}({\mathrm{A}},{\mathrm{A}})\rightarrow\mathrm{{Hom}}_{\mathcal{Z}(\mathcal{C})}({\mathrm{L}},{\mathrm{A}}).$

Since $Z=FR$ is a comonad it follows that the adjunction isomorphisms $F\dashv
R$ are given in this case by

(3.3)
$\psi_{{\mathrm{L}},\mathbf{1}}:\mathrm{{Hom}}_{\mathcal{C}}(L,\mathbf{1})\xrightarrow{}\mathrm{{Hom}}_{{\mathcal{Z}(\mathcal{C})}}({\mathrm{L}},{\mathrm{A}}),\;\;\mu\mapsto
R(\mu)\circ\eta_{\mathrm{L}}.$

where $\eta_{\mathrm{L}}$ is the unit of the adjunction
$L\xrightarrow{\eta_{\mathrm{L}}}R(F({\mathrm{L}}))$.

Moreover, its inverse is given by

(3.4)
$\psi_{{\mathrm{L}},\mathbf{1}}^{-1}:\mathrm{{Hom}}_{{\mathcal{Z}(\mathcal{C})}}({\mathrm{L}},{\mathrm{A}})\rightarrow\mathrm{{Hom}}_{\mathcal{C}}(L,\mathbf{1}),\;g\mapsto{\bf\epsilon}_{\mathbf{1}}\circ
g.$

Naturality in the first variable of the adjunction isomorphisms
$\psi_{(-),\;\mathbf{1}}$ of $F\dashv R$ implies that the following diagram is
commutative:

(3.5)
${{\mathrm{CF}}(L)=\mathrm{{Hom}}_{\mathcal{C}}(L,\mathbf{1})}$${\mathrm{{Hom}}_{\mathcal{Z}(\mathcal{C})}({\mathrm{L}},{\mathrm{A}})}$${\mathrm{{CF}}(\mathcal{C})=\mathrm{{Hom}}_{\mathcal{C}}(A,\mathbf{1})}$${\mathrm{{Hom}}_{\mathcal{Z}(\mathcal{C})}({\mathrm{A}},{\mathrm{A}}).}$$\scriptstyle{\psi_{{\mathrm{L}},\mathbf{1}}}$$\scriptstyle{{\mathrm{Res}}}$$\scriptstyle{\psi_{{\mathrm{A}},\mathbf{1}}}$$\scriptstyle{\mathrm{Res}^{A}_{L}}$

Note that $\mathrm{End}_{\mathcal{Z}(\mathcal{C})}({\mathrm{A}})$ is an
algebra with $f\star g:=f\circ g$. Moreover,
$\mathrm{{Hom}}_{\mathcal{Z}(\mathcal{C})}({\mathrm{L}},{\mathrm{A}})$ is a
left $\mathrm{End}_{\mathcal{Z}(\mathcal{C})}({\mathrm{A}})$-module via
$f.\mu=f\circ\mu$ for any
$f\in\mathrm{End}_{\mathcal{Z}(\mathcal{C})}({\mathrm{A}})$ and
$\mu\in\mathrm{{Hom}}_{\mathcal{Z}(\mathcal{C})}({\mathrm{L}},{\mathrm{A}})$.
Then the commutativity of diagram (3.5) also gives that ${\mathrm{CF}}(L)$ is
a left $\mathrm{{CF}}(\mathcal{C})$-module via

(3.6)
$\chi.\mu=\psi_{{\mathrm{A}},\mathbf{1}}^{-1}(\psi_{{\mathrm{L}},\mathbf{1}}(\chi)\circ\psi_{{\mathrm{L}},\mathbf{1}}(\mu)),\;\text{for
any}\;\chi\in\mathrm{{CF}}(\mathcal{C}),\mu\in{\mathrm{CF}}(L).$

On the other hand, since the restriction map
${\mathrm{Res}^{A}_{L}}:\mathrm{{Hom}}_{\mathcal{Z}(\mathcal{C})}({\mathrm{A}},{\mathrm{A}})\rightarrow\mathrm{{Hom}}_{\mathcal{Z}(\mathcal{C})}({\mathrm{L}},{\mathrm{A}})$
is an $\mathrm{End}_{\mathcal{Z}(\mathcal{C})}({\mathrm{A}})$-module
homomorphism it follows that ${\mathrm{Res}}$ is also a
$\mathrm{{CF}}(\mathcal{C})$-module homomorphism. This can be written as:

(3.7)
$\chi.{\mathrm{Res}}(\chi^{\prime})={\mathrm{Res}}(\chi\star\chi^{\prime}),$

for any two $\chi,\chi^{\prime}\in\mathrm{{CF}}(\mathcal{C})$. In particular,
for $\chi^{\prime}={\bf\epsilon}_{\mathbf{1}}$ the unit of
$\mathrm{{CF}}(\mathcal{C})$ it follows that for any
$\chi\in\mathrm{{CF}}(\mathcal{C})$ one has

(3.8) $\chi.{\mathrm{Res}}({\bf\epsilon}_{\mathbf{1}})={\mathrm{Res}}(\chi).$

Define also the central subspace of $\mathrm{{CE}(\mathcal{C})}$ associated to
${\mathrm{L}}$ as the vector space
${\mathrm{CE}}(L):=\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},L)$. Denote by

(3.9)
$\iota^{e}:{\mathrm{CE}}(L)=\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},L)\hookrightarrow\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},A)=\mathrm{{CE}(\mathcal{C})}$

the canonical inclusion given by $\iota^{e}(z)=F(\iota_{\mathrm{L}})\circ z$
for any $z\in{\mathrm{CE}}(L)$.

For $z,z^{\prime}\in{\mathrm{CE}}(L)$, one can set
$zz^{\prime}:=m_{L}\circ(z\otimes z^{\prime})$. Then as in the case of
$\mathrm{{CE}(\mathcal{C})}$ it is easy to see that ${\mathrm{CE}}(L)$ is a
monoid with respect to this operation. Moreover, it is easy to see that
$\iota^{e}$ is a unitary algebra embedding. Since $\mathrm{{CE}(\mathcal{C})}$
is a semisimple commutative algebra over an algebraically closed field, see
[18], it follows that ${\mathrm{CE}}(L)$ is also a semisimple commutative
algebra.

Let $\\{\ell_{s}\\}_{s\in\\{0,\dots,r\\}}$ be the (central) primitive
idempotents of ${\mathrm{CE}}(L)$. One can write
$\iota^{e}_{L}(\ell_{s})=\sum_{i\in{\mathcal{B}}_{s}}E_{i}$ for the
decomposition of the primitive idempotents of ${\mathrm{CE}}(L)$ inside
$\mathrm{{CE}(\mathcal{C})}$. In this way we get a partition
$\\{0,\dots,m\\}={\mathcal{B}}_{0}\sqcup\dots\sqcup{\mathcal{B}}_{r}.$ Through
the rest of the paper, by abuse of notations we also denote the corresponding
partition of the irreducible characters,
$\mathrm{Irr}(\mathcal{C})={\mathcal{B}}_{0}\sqcup\dots\sqcup{\mathcal{B}}_{r}$
with the same symbols ${\mathcal{B}}_{s}$, with $0\leq s\leq r$. Without loss
of generality we may suppose that the irreducible character
$\chi_{0}={\bf\epsilon}_{\mathbf{1}}$ of the unit object of $\mathcal{C}$
satisfies $\chi_{0}\in{\mathcal{B}}_{0}$ i.e.

(3.10) $\iota^{e}(\ell_{0})=\Lambda+\sum_{\\{i\in{\mathcal{B}}_{0},\;i\neq
0\\}}E_{i}$

where $\Lambda$ is the idempotent integral of $\mathcal{C}$.

######  Remark 3.11.

Note that the map $\pi:{\mathrm{A}}\rightarrow{\mathrm{L}}$ induces also two
maps at the level of characters and centers. One is a canonical embedding of
vector spaces
$\pi^{c}_{{}_{\uparrow}}:{\mathrm{CF}}(L)\hookrightarrow\mathrm{{CF}}(\mathcal{C}),\;\mu\mapsto\mu\circ
F(\pi).$ The other one is a surjective linear map
$\pi^{e}_{{}_{\downarrow}}:\mathrm{{CE}(\mathcal{C})}\rightarrow{\mathrm{CE}}(L),\;z\mapsto\pi\circ
z$. Clearly one has
${\mathrm{Res}}\circ\pi^{c}_{{}_{\uparrow}}=\operatorname{id}_{{\mathrm{CF}}(L)}$
and
$\pi^{e}_{{}_{\downarrow}}\circ\iota^{e}=\operatorname{id}_{{\mathrm{CE}}(L)}$.

There is also a nondegenerate canonical paring given by

(3.12)
$\langle\;,\;\rangle_{\mathrm{L}}:{\mathrm{CF}}(L)\otimes{\mathrm{CE}}(L)\rightarrow\Bbbk,\;\langle\alpha,z\rangle\mapsto\alpha\circ
z.$

Recall also the non-degenerate evaluation pairing
$\langle,\;\rangle_{\mathrm{{\mathrm{A}}}}$ given by

(3.13)
$\langle\;,\;\rangle_{\mathrm{{\mathrm{A}}}}:\mathrm{{CF}}(\mathcal{C})\otimes\mathrm{{CE}(\mathcal{C})}\rightarrow\Bbbk,\;\langle\chi,z\rangle\mapsto\chi\circ
z.$

######  Lemma 3.14.

One has the following compatibility properties between pairings:

(3.15)
$\langle{\mathrm{Res}}(\chi),\;\bar{z}\rangle_{\mathrm{L}}=\langle\chi,\;{\iota^{e}}(\bar{z})\rangle_{\mathrm{{\mathrm{A}}}}.$

(3.16)
$\langle\mu,\pi^{e}_{{}_{\downarrow}}(z)\rangle_{\mathrm{L}}=\langle\pi^{c}_{{}_{\uparrow}}(\mu),z\rangle_{\mathrm{A}}$

for any $\bar{z}\in{\mathrm{CE}}(L)$, $z\in\mathrm{{CE}(\mathcal{C})}$,
$\chi\in\mathrm{{CF}}(\mathcal{C})$ and any $\mu\in{\mathrm{CF}}(L)$

###### Proof.

Straightforward. ∎

######  Proposition 3.17.

For any two irreducible characters
$\chi_{i},\chi_{j}\in\mathrm{Irr}(\mathcal{C})$ one has that
$\chi_{i},\chi_{j}\in{\mathcal{B}}_{s}$ for some $0\leq s\leq r$ if and only
if

$\frac{{\mathrm{Res}}(\chi_{i})}{d_{i}}=\frac{{\mathrm{Res}}(\chi_{j})}{d_{j}}.$

###### Proof.

Let ${\eta_{s}}\in{\mathrm{CF}}(L)$ be the dual basis of $\ell_{s}$ with
respect to the non-degenerate pairing $\langle,\;\rangle_{\mathrm{L}}$. Thus
one has $\langle\eta_{t},\;\ell_{s}\rangle_{\mathrm{L}}=\delta_{s,t}$, for any
$0\leq s,t\leq r$. We will show that if $\chi_{i}\in{\mathcal{B}}_{s}$ then

(3.18) $\frac{{\mathrm{Res}}(\chi_{i})}{d_{i}}={\eta_{s}}.$

This would finish the proof of the proposition.

Suppose now that $\chi_{i}\in{\mathcal{B}}_{s}$. Then by Equation (3.15) one
has
$\langle{\mathrm{Res}}(\chi_{i}),\ell_{s}\rangle_{\mathrm{L}}=\langle\chi_{i},\;{\iota^{e}}(\ell_{s})\rangle_{\mathrm{{\mathrm{A}}}}=\langle\chi_{i},\;\sum_{j\in{\mathcal{B}}_{s}}E_{j}\rangle_{\mathrm{{\mathrm{A}}}}=d_{i}.$
On the other hand,
$\langle{\mathrm{Res}}(\chi_{i}),\ell_{t}\rangle_{\mathrm{L}}=\langle\chi_{i},\;{\iota^{e}}(\ell_{t})\rangle_{\mathrm{{\mathrm{A}}}}=0,\text{if}\;t\neq
s.$ Therefore $\frac{1}{d_{i}}{\mathrm{Res}}(\chi_{i})={\eta_{s}}$ for any
irreducible character $\chi_{i}\in{\mathcal{B}}_{s}$. ∎

Recall that for the character $\chi_{0}$ of the unit object $\mathbf{1}$ of
$\mathcal{C}$ we assumed that $\chi_{0}\in{\mathcal{B}}_{0}$ . We denote by
${\underline{\epsilon}_{L}}:={\mathrm{Res}}(\chi_{0})$ its restriction to $L$.
Note that the above proposition implies ${\underline{\epsilon}_{L}}=\eta_{0}$.
Moreover the proof shows that $\chi_{i}\in{\mathcal{B}}_{0}$ if and only if

(3.19) ${\mathrm{Res}}(\chi_{i})=d_{i}{\underline{\epsilon}_{L}}.$

######  Remark 3.20.

We have chosen to add the underline bar in the notation
${\underline{\epsilon}_{L}}:L\rightarrow\mathbf{1}$ of the restriction
${\mathrm{Res}}({\bf\epsilon}_{\mathbf{1}})$ in order to distinguished it from
the counit ${\bf\epsilon}_{L}:Z(L)\rightarrow L$ of the comonad $Z$ evaluated
at $L$.

###  3.1. On the right $L$-module maps in ${\mathrm{CF}}(L)$

Since ${\bf\epsilon}_{\mathbf{1}}:A\rightarrow\mathbf{1}$ is an algebra
morphism in $\mathcal{C}$ it follows that $\mathbf{1}$ can be considered a
right $A$-module in $\mathcal{C}$ via the morphism
${\bf\epsilon}_{\mathbf{1}}$. As explained in [18, Subsection 5.2] the
integral $\Lambda:\mathbf{1}\rightarrow A$ is the unique (up to scalar)
morphism of right $A$-modules in $\mathcal{C}$.

The next Lemma is an analogue of [18, Lemma 6.1] and its proof follows the
same lines.

######  Lemma 3.21.

Let $\mathcal{C}$ be a fusion category and ${\mathrm{L}}$ be a unitary
subalgebra of ${\mathrm{A}}$ in ${\mathcal{Z}(\mathcal{C})}$.

  1. (1)

An arrow $a:\mathbf{1}\rightarrow L$ in $\mathcal{C}$ is a morphism of right
$L$-modules in $\mathcal{C}$ if and only if for any $z\in{\mathrm{CE}}(L)$ one
has:

(3.22) $za=\langle{\underline{\epsilon}_{L}},z\rangle_{\mathrm{L}}a.$

  2. (2)

$\ell_{0}$ is a morphism of right $L$-modules in $\mathcal{C}$.

###### Proof.

The proof of the first item is straightforward, see also [18, Lemma 6.1].

Since ${\mathrm{CE}}(L)$ is a product of copies of the field $\Bbbk$ an
element $\phi\in{\mathrm{CE}}(L)$ satisfying Equation (3.22) is unique up to a
scalar. Moreover, note that the idempotent $\ell_{0}$ satisfies this equation.
Indeed, since ${\mathrm{CE}}(L)$ is a semisimple algebra there is
$\mu\in\widehat{{\mathrm{CE}}(L)}$ such that $z\ell_{0}=\mu(z)\ell_{0}$ for
all $z\in{\mathrm{CE}}(L)$. Since
$\langle{\underline{\epsilon}_{L}},\ell_{0}\rangle_{\mathrm{L}}=\langle{\bf\epsilon}_{\mathbf{1}},\iota^{e}(\ell_{0})\rangle_{\mathrm{A}}=1$
it follows that

$\displaystyle\mu(z)$ $\displaystyle=$
$\displaystyle\langle{\underline{\epsilon}_{L}},\mu(z)\ell_{0}\rangle_{\mathrm{L}}=\langle{\underline{\epsilon}_{L}},z\ell_{0}\rangle_{\mathrm{A}}=\langle{\bf\epsilon}_{\mathbf{1}},\iota^{e}(z\ell_{0})\rangle_{\mathrm{A}}$
$\displaystyle=$
$\displaystyle\langle{\bf\epsilon}_{\mathbf{1}},\iota^{e}(z)\rangle_{\mathrm{A}}\langle{\bf\epsilon}_{\mathbf{1}},\iota^{e}(\ell_{0})\rangle_{\mathrm{A}}=\langle{\bf\epsilon}_{\mathbf{1}},\iota^{e}(z)\rangle_{\mathrm{A}}$
$\displaystyle=$
$\displaystyle\langle{\underline{\epsilon}_{L}},z\rangle_{\mathrm{L}}.$

∎

### On the full subcategory ${\mathcal{S}_{{\mathrm{L}}}}$

We denote by ${\mathcal{S}_{{\mathrm{L}}}}$ the full abelian subcategory of
$\mathcal{C}$ generated by the simple objects $V_{i}$ of $\mathcal{C}$ whose
irreducible characters $\chi_{i}$ satisfy $\chi_{i}\in{\mathcal{B}}_{0}$.

######  Theorem 3.1.

Let $\mathcal{C}$ be a pseudo-unitary fusion category and ${\mathrm{L}}$ be a
unitary subalgebra of the adjoint algebra ${\mathrm{A}}$ of $\mathcal{C}$.
With the above notations one has that ${\mathcal{S}_{\mathrm{L}}}$ is a fusion
subcategory of $\mathcal{C}$.

###### Proof.

Suppose that $M,M^{\prime}\in\mathrm{Irr}({\mathcal{S}_{\mathrm{L}}})$ are two
simple objects with characters
$\chi_{M}=\chi,\;\chi_{M^{\prime}}=\chi^{\prime}$. Then
${\mathrm{Res}}(\chi)=d(\chi){\underline{\epsilon}_{L}}$ and
${\mathrm{Res}}(\chi^{\prime})=d(\chi){\underline{\epsilon}_{L}}$. It follows
that

${\mathrm{Res}}(\chi\star\chi^{\prime})=\chi.{\mathrm{Res}}(\chi^{\prime})=d(\chi^{\prime})\chi.{\underline{\epsilon}_{L}}\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{chdeps}\mkern-1.5mu)}}{{=}}d(\chi){\mathrm{Res}}(\chi^{\prime})=d(\chi)d(\chi^{\prime}){\underline{\epsilon}_{L}}$

On the other hand suppose that
$\chi\star\chi^{\prime}=\sum_{\chi_{u}\in\mathrm{Irr}(\mathcal{C})}N^{u}_{M,M^{\prime}}\chi_{u}$
with some $N^{u}_{M,M^{\prime}}\geq 0$. It follows that

$\displaystyle{\mathrm{Res}}(\chi\star\chi^{\prime})$ $\displaystyle=$
$\displaystyle\sum_{\chi_{u}\in\mathrm{Irr}(\mathcal{C})}N^{u}_{M,M^{\prime}}{\mathrm{Res}}(\chi_{u})=\sum_{s=0}^{r}\sum_{u\in\mathcal{B}_{s}}N^{u}_{M,M^{\prime}}d_{u}{\mathrm{Res}}(\frac{\chi_{u}}{d_{u}})$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{resd}\mkern-1.5mu)}}{{=}}$
$\displaystyle\sum_{s=0}^{r}\big{(}\sum_{u\in\mathcal{B}_{s}}N^{u}_{M,M^{\prime}}d_{u}\big{)}{\eta_{s}}$

where $\\{\eta_{s}\\}$ is the dual basis of $\ell_{s}$ with respect to the
pairing $\langle,\;\rangle_{\mathrm{L}}$. Since $\eta_{s}$ are linearly
independent it follows that for any $s\neq 0$ one has

$\sum_{u\in\mathcal{B}_{s}}N^{u}_{M,M^{\prime}}d_{u}=0$

Since $\mathcal{C}$ is pseudo-unitary one has $d_{u}>0$ and therefore in this
case $N^{u}_{M,M^{\prime}}=0$ for any $u\in\mathcal{B}_{s}$. Thus all the
simple constituents of $M\otimes M^{\prime}$ are in full abelian subcategory
${\mathcal{S}_{\mathrm{L}}}$. ∎

### On the $\mathrm{FPdim}({\mathcal{S}_{{\mathrm{L}}}})$.

In this subsection we prove the following:

######  Theorem 3.2.

Let $\mathcal{C}$ be a pseudo-unitary fusion category and ${\mathrm{L}}$ be a
unitary subalgebra of the adjoint algebra ${\mathrm{A}}$ of $\mathcal{C}$.
With the above notations one has that

$\mathrm{FPdim}({\mathcal{S}_{\mathrm{L}}})=\frac{\dim(\mathcal{C})}{\dim({\mathrm{L}})}.$

First we need to fix several notations. Let $\mathcal{C}$ be a fusion category
and $F:{\mathcal{Z}(\mathcal{C})}\rightarrow\mathcal{C}$ be the forgetful
functor. As above, let $R$ be a right adjoint of $F$. We may suppose that
${\mathrm{A}}=\bigoplus_{j=0}^{r}{\mathcal{C}^{(j)}}$ is the decomposition of
${\mathrm{A}}$ in homogenous components in ${\mathcal{Z}(\mathcal{C})}$.
Define $\mathcal{J}:=\\{0,\dots,r\\}$ the set of indices of homogenous
components of ${\mathrm{A}}$.

We may write each homogenous component as
${\mathcal{C}^{(j)}}=\bigoplus_{s=1}^{m_{j}}{\mathcal{C}^{(j)}_{s}}$ where
${\mathcal{C}^{(j)}_{s}}$ are the simple sub-objects of ${\mathrm{A}}$
entering in the homogenous component ${\mathcal{C}^{(j)}}$. Therefore as an
object of ${\mathcal{Z}(\mathcal{C})}$ one has a decomposition in simple
objects

(3.23)
${\mathrm{A}}=\bigoplus_{j\in\mathcal{J}}\bigoplus_{s\in\mathcal{M}_{j}}{\mathcal{C}^{(j)}_{s}}$

where ${\mathcal{C}^{(j)}_{s}}$ and $\mathcal{C}^{(j)}_{t}$ are isomorphic
simple ${\mathcal{Z}(\mathcal{C})}$-submodules of ${\mathrm{A}}$ and
$\mathcal{M}_{j}:=\\{1,\dots,m_{j}\\}$. One has
$\mathrm{End}_{\mathcal{Z}(\mathcal{C})}({\mathcal{C}^{(j)}})\simeq
M_{m_{j}}(\Bbbk)$ as algebras and therefore
$\mathrm{End}_{\mathcal{Z}(\mathcal{C})}({\mathrm{A}})\simeq\bigoplus_{j\in\mathcal{J}}M_{m_{j}}(\Bbbk).$
By the natural isomorphism ${\psi_{{\mathrm{A}},\mathbf{1}}}$ from Equation
(3.4) one also has that
$\mathrm{{CF}}(\mathcal{C})\simeq\bigoplus_{j\in\mathcal{J}}M_{m_{j}}(\Bbbk)$
as a semisimple algebra.

Without loss of generality we may also suppose that
$\mathcal{C}^{(0)}=\mathbf{1}_{\mathcal{Z}(\mathcal{C})}$ is the simple
trivial object of ${\mathrm{A}}$. Since
$\mathrm{{Hom}}_{\mathcal{Z}(\mathcal{C})}({\mathrm{A}},\mathbf{1})\simeq\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},\mathbf{1})=\Bbbk$
it follows that $m_{0}=1$.

Define
$\widetilde{F^{j}_{st}}\in\mathrm{End}_{\mathcal{Z}(\mathcal{C})}({\mathrm{A}})$
as the unique endomorphisms of ${\mathrm{A}}$ that send (as identity)
$\mathcal{C}^{(j)}_{t}$ into ${\mathcal{C}^{(j)}_{s}}$ and that are zero on
the other conjugacy classes $\mathcal{C}^{(j)}_{t^{\prime}}$ with
$t^{\prime}\neq t$. Then these endomorphisms form a linear basis
$\mathrm{End}_{\mathcal{Z}(\mathcal{C})}({\mathrm{A}})$ such as

$\widetilde{F^{j}_{st}}\circ\widetilde{F^{j^{\prime}}_{s^{\prime},t^{\prime}}}=\delta_{j,j^{\prime}}\delta_{s^{\prime},t}\widetilde{F^{j}_{st^{\prime}}}$

Denote
$F^{j}_{st}:=\psi_{{\mathrm{A}},\mathbf{1}}^{-1}(\widetilde{F^{j}_{st}})\in\mathrm{{CF}}(\mathcal{C})$.
It follows that $F^{j}_{st}$ are the standard matrix entries in the matrix-
block $M_{m_{j}}(\Bbbk)$. Thus

(3.24)
$F^{j}_{st}F^{j^{\prime}}_{s^{\prime}t^{\prime}}=\delta_{j,j^{\prime}}\delta_{s^{\prime},t}F^{j}_{st^{\prime}},$

and the corresponding primitive central idempotent corresponding to the unit
of the block is $F^{j}=\sum_{s=0}^{m_{j}}F^{j}_{ss}$. Note that the
decompositions of ${\mathrm{A}}$ from Equation (3.23) are in bijection with
the matrix bases $\\{F^{j}_{st}\\}$ of $\mathrm{{CF}}(\mathcal{C})$.

It is well known that the block $\Bbbk=M_{m_{0}}(\Bbbk)$ from the above
decomposition corresponds to the central primitive idempotent $F^{0}=\lambda$,
the cointegral of $\mathcal{C}$.

By [18, Lemma 6.2] the Grothendieck ring $\mathrm{Gr}_{\Bbbk}(\mathcal{C})$ is
a symmetric Frobenius algebra with non-degenerate trace
$\tau:\mathrm{Gr}_{\Bbbk}(\mathcal{C})\rightarrow\mathbb{C}$ given by
$[X]\mapsto\dim_{\Bbbk}\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},X)$. For a
pivotal fusion category one has $\tau([X])=\langle\chi(X),\Lambda\rangle$ for
any object $X\in\mathcal{O}(\mathcal{C})$, where $\Lambda$ is an idempotent
cointegral associated to $\mathcal{C}$.

Then the corresponding associative non-degenerate bilinear form on
$\mathrm{{CF}}(\mathcal{C})$ is given by
$\beta_{\tau}(\chi,\mu):=\langle\chi\mu,\Lambda\rangle$. A pair of dual bases
for $\beta_{\tau}$ is given by $\\{\chi_{i},\chi_{i^{*}}\\}$.

Suppose that on the matrix-block decomposition of $\mathrm{{CF}}(\mathcal{C})$
one has

(3.25)
$\beta_{\tau}(\chi,\;\mu)=\sum_{j}\frac{1}{n_{j}}\mathrm{tr}_{j}(\chi\mu).$

where $\mathrm{tr}_{j}$ is the usual trace on the mtrix algebra
$M_{m_{j}}(\Bbbk)$. Since $\\{F^{j}_{st},\;n_{j}F^{j}_{ts}\\}$ is another pair
of dual bases for $\beta_{\tau}(-,\;-)$ one can write:

(3.26) $\sum_{j=0}^{m}\;\sum_{s,t\in\mathcal{M}_{j}}n_{j}F^{j}_{st}\otimes
F^{j}_{ts}=\sum_{i=0}^{m}\chi_{i}\otimes\chi_{i^{*}}.$

######  Lemma 3.27.

For a pivotal fusion category, with the above notations one has

(3.28) $n_{j}=\frac{\dim(\mathcal{C})}{\dim(\mathcal{C}^{(j)}_{s})}.$

###### Proof.

By [18, Proposition 5.17] one has that for any
$f\in\mathrm{{CF}}(\mathcal{C})$

$\mathrm{tr}(\psi_{{\mathrm{A}},\mathbf{1}}^{-1}(f))=\langle
f,\Lambda\rangle\langle\lambda,u\rangle$

where $(\Lambda,\lambda)$ is any pair of a integral and cointegral of
$\mathcal{C}$ such that $\langle\lambda,\Lambda\rangle=1$. One can chose such
a pair with $\langle\lambda,u\rangle=\dim(\mathcal{C})$ and $\Lambda$ an
idempotent integral, i.e ${\bf\epsilon}_{\mathbf{1}}(\Lambda)=1$.

In particular, for $f=F^{j}$ one has that
$\psi_{{\mathrm{A}},\mathbf{1}}^{-1}(F^{j})$ is the projection on the
homogenous component $\mathcal{C}^{(j)}$. Therefore the above trace formula
can be written as

$m_{j}\dim(\mathcal{C}^{(j)}_{s})=\mathrm{tr}(\psi_{{\mathrm{A}},\mathbf{1}}^{-1}(F^{j}))=\langle
F^{j},\Lambda\rangle\dim(\mathcal{C}).$

On the other hand, from Equation (3.25) one has

(3.29) $\tau(F^{j}_{ss})=\langle
F^{j}_{ss},\Lambda\rangle=\frac{1}{n_{j}}\mathrm{tr}_{j}(F^{j}_{ss})=\frac{1}{n_{j}}$

Thus $\langle
F^{j},\Lambda\rangle=\tau(F^{j})=\sum_{s=0}^{m_{j}}\tau(F^{j}_{ss})=\frac{m_{j}}{n_{j}}$
and the result follows. ∎

######  Definition 3.3.

Denote by
${\mathrm{C}}^{j}_{st}:=\mathcal{F}_{\lambda}^{-1}(F^{j}_{st})\in\mathrm{{CE}(\mathcal{C})}$
and call this element the conjugacy class sum corresponding to $F^{j}_{st}$.
Here we use the cointegral $\lambda$ with $\langle\lambda,u\rangle=1$.

######  Lemma 3.30.

Let $\mathcal{C}$ be a spherical fusion category. With the above notations one
has that:

(3.31) $\langle
F^{j}_{st},\;{\mathrm{C}}^{i}_{uv}\rangle_{\mathrm{A}}=\delta_{i,j}\delta_{v,s}\delta_{u,t}{\dim(\mathcal{C}^{(j)}_{s})}.$

###### Proof.

In particular, for $\chi=F^{j}_{st}$ and $\mu=F^{j}_{ts}$ in Equation (2.17)
one has that

$\displaystyle\langle F^{j}_{st},\;{\mathrm{C}}^{i}_{uv}\rangle_{\mathrm{A}}$
$\displaystyle=$ $\displaystyle\langle
F^{j}_{st},\;\mathcal{F}_{\lambda}^{-1}(F^{i}_{uv})\rangle_{\mathrm{A}}\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{comp1}\mkern-1.5mu)}}{{=}}\dim(\mathcal{C})\tau(F^{j}_{st}F^{i}_{uv})$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{mcfa}\mkern-1.5mu)}}{{=}}$
$\displaystyle\delta_{i,j}\delta_{u,t}\dim(\mathcal{C})\tau(F^{i}_{sv})\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{tau}\mkern-1.5mu)}}{{=}}\delta_{i,j}\delta_{v,s}\delta_{u,t}\frac{\dim(\mathcal{C})}{n_{j}}.$

Then use the formula (3.29). ∎

######  Lemma 3.32.

Let $\mathcal{C}$ be a spherical non-degenerate fusion category. With the
above notations one has

(3.33)
$\langle{\bf\epsilon}_{\mathbf{1}},{\mathrm{C}}^{j}_{st}\rangle_{\mathrm{A}}=\delta_{s,t}\dim(\mathcal{C}^{(j)}_{s})$

###### Proof.

Note that ${\bf\epsilon}_{\mathbf{1}}=\chi_{0}=\sum_{j}F^{j}$ is the unit of
$\mathrm{{CF}}(\mathcal{C})$. Equation (2.17) for
$\chi={\bf\epsilon}_{\mathbf{1}}$ and $\mu=F^{j}_{ts}$ gives:

$\langle{\bf\epsilon}_{\mathbf{1}},{\mathrm{C}}^{j}_{st}\rangle_{\mathrm{A}}\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{comp1}\mkern-1.5mu)}}{{=}}\dim(\mathcal{C})\tau(F^{j}_{st})\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{nj}\mkern-1.5mu)}}{{=}}\delta_{s,t}\frac{\dim(\mathcal{C})}{n_{j}}\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{njclass}\mkern-1.5mu)}}{{=}}\delta_{s,t}\dim(\mathcal{C}^{(j)}_{s}).$

∎

######  Lemma 3.34.

With the above notations one has

  1. (1)

$F^{j}_{st}\in\mathrm{{Hom}}_{\mathcal{C}}(\mathcal{C}^{(j)}_{t},\mathbf{1})$

  2. (2)

$\mathrm{C}^{j}_{st}\in\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},\mathcal{C}^{(j)}_{s})$.

  3. (3)

On the homogenous components the Fourier transform sends
$\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},F(\mathcal{C}^{(j)})$ into
$\mathrm{{Hom}}_{\mathcal{C}}(F(\mathcal{C}^{(j)}),\mathbf{1})$.

###### Proof.

By the adjunction isomorphism $\psi^{-1}_{{\mathrm{A}},\mathbf{1}}$ from
Equation (2.19) one has
$F^{j}_{st}={\bf\epsilon}_{\mathbf{1}}\circ\widetilde{F^{j}_{st}}$ which gives
the proof for the first item.

On the other hand Equation (2.17) shows that also
$\\{F^{j}_{st},\frac{n_{j}}{\dim(\mathcal{C})}{\mathrm{C}}^{j}_{ts}\\}$ are
also dual bases for $\langle,\;\rangle_{\mathrm{A}}$. Since for $s\neq t$ the
pairing $\langle,\;\rangle_{\mathrm{A}}$ is zero on
$\mathrm{{Hom}}_{\mathcal{C}}(\mathcal{C}^{(j)}_{t},\mathbf{1})\times\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},F({\mathcal{C}^{(j)}_{s}}))$
it follows that
$\mathrm{C}^{j}_{ts}\in\mathrm{{Hom}}_{\mathcal{C}}(\mathcal{C}^{(j)}_{t},\mathbf{1})$.
∎

Recall the embedding
$q^{*}:\mathrm{CF}({\mathcal{D}})\rightarrow\mathrm{{CF}}(\mathcal{C})$ from
Subsection 2.6. By abuse of notations we denote by the same symbol
$\lambda_{\mathcal{D}}$, the image of
$\lambda_{\mathcal{D}}\in\mathrm{CF}({\mathcal{D}})$ under $q^{*}$.

######  Proposition 3.35.

Let $\mathcal{C}$ be a non-degenerate spherical fusion category and
${\mathcal{D}}\subseteq\mathcal{C}$ a fusion subcategory of $\mathcal{C}$.
Suppose that

$\lambda_{\mathcal{D}}=\sum_{(j,s,t)\in\mathcal{L}_{\mathcal{D}}}\beta^{j}_{st}F^{j}_{st}$

where all $\beta^{j}_{st}\in\Bbbk$ are non-zero scalars and
$\mathcal{L}_{\mathcal{D}}\subseteq\sqcup_{j\in\mathcal{J}}\\{j\\}\times\mathcal{M}_{j}\times\mathcal{M}_{j}$
is a subset of indices. Then

(3.36)
$\frac{\dim(\mathcal{C})}{\dim({\mathcal{D}})}=\sum_{(j,s,s)\in\mathcal{L}_{{\mathcal{D}}}}\beta^{j}_{ss}\dim(\mathcal{C}^{(j)}_{s}).$

###### Proof.

We show that both terms of Equation (3.36) equal
$\langle{\bf\epsilon}_{\mathbf{1}},\ell_{\mathcal{D}}\rangle_{\mathrm{A}}$
where
$\ell_{{\mathcal{D}}}:=\mathcal{F}_{\lambda}^{-1}(\lambda_{\mathcal{D}})$.
Here the cointegral $\lambda$ is chosen such that
$\langle\lambda,u\rangle_{\mathrm{A}}=1$, where $u=F(\eta_{\mathrm{A}})$ is
the unit of $\mathrm{{CE}(\mathcal{C})}$. Applying
$\mathcal{F}_{\lambda}^{-1}$ to the above formula for $\lambda_{\mathcal{D}}$
it follows that the element $\ell_{{\mathcal{D}}}$ has the formula

(3.37)
$\ell_{{\mathcal{D}}}=\sum_{(j,s,t)\in\mathcal{L}_{\mathcal{D}}}\beta^{j}_{st}C^{j}_{st}.$

Equation (3.37) gives that

$\langle{\bf\epsilon}_{\mathbf{1}},\ell_{\mathcal{D}}\rangle_{\mathrm{A}}=\sum_{(j,s,t)\in\mathcal{L}_{{\mathcal{D}}}}\beta^{j}_{st}\langle{\bf\epsilon}_{\mathbf{1}},\;{\mathrm{C}}^{j}_{st}\rangle_{\mathrm{A}}\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{epsucj}\mkern-1.5mu)}}{{=}}\sum_{(j,s,s)\in\mathcal{L}_{{\mathcal{D}}}}\beta^{j}_{ss}\dim(\mathcal{C}^{(j)}_{s}).$

On the other hand from Equation (2.15) one has

$\displaystyle\ell_{\mathcal{D}}$ $\displaystyle=$
$\displaystyle\mathcal{F}^{-1}_{\lambda}(\lambda_{\mathcal{D}})=\frac{1}{\dim({\mathcal{D}})}\big{(}\sum_{\chi_{i}\in\mathrm{Irr}({\mathcal{D}})}d_{i^{*}}\mathcal{F}^{-1}_{\lambda}(\chi_{i})\big{)}$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{610}\mkern-1.5mu)}}{{=}}$
$\displaystyle\frac{1}{\dim({\mathcal{D}})}\big{(}\sum_{\chi_{i}\in\mathrm{Irr}({\mathcal{D}})}d_{i^{*}}\frac{\dim(\mathcal{C})}{d_{i}}E_{i^{*}}\big{)}.$

Since $\mathcal{C}$ is spherical it follows that

(3.38)
$\ell_{\mathcal{D}}=\frac{\dim(\mathcal{C})}{\dim({\mathcal{D}})}\big{(}\sum_{\chi_{i}\in\mathrm{Irr}({\mathcal{D}})}E_{i}\big{)}.$

By applying ${\bf\epsilon}_{\mathbf{1}}$ to Equation (3.38) it follows that
$\langle{\bf\epsilon}_{\mathbf{1}},\ell_{\mathcal{D}}\rangle_{\mathrm{A}}=\frac{\dim(\mathcal{C})}{\dim({\mathcal{D}})}$.
Indeed, note that $\langle\epsilon_{\mathbf{1}},\;E_{j}\rangle=\delta_{j,\;0}$
since we assumed that $\chi_{0}={\bf\epsilon}_{\mathbf{1}}$. ∎

######  Remark 3.39.

Note that the above result holds for any decomposition of ${\mathrm{A}}$ from
Equation (3.23), thus basically for any matrix basis $F^{j}_{st}$ of
$\mathrm{{CF}}(\mathcal{C})$.

######  Remark 3.40.

In [5] it was studied the central element
$\ell_{\mathcal{D}}:=\mathcal{F}^{-1}_{\lambda}(\lambda_{\mathcal{D}})\in\mathrm{{CE}(\mathcal{C})}$
in the case of a fusion category with commutative Grothendieck ring.

To simplify the notations, through the rest of the paper, we denote
$C_{st}:=C_{st}^{j}$ for the class sum $C_{st}^{j}$ whenever it is specified
that $s,t\in\mathcal{M}_{j}$, case in which the index $j$ is implicitly
understood. The same notation $F_{st}:=F^{j}_{st}$ will be used in the same
circumstances.

Note that the central subspace of
$\mathrm{{CE}(\mathcal{C})}:=\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},A)$
associated to ${\mathrm{L}}$ can be described as
$\mathrm{{CE}(\mathcal{C})}:=\bigoplus_{j\in\mathcal{J}}\bigoplus_{s\in\mathcal{M}_{j}}\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},F({\mathcal{C}^{(j)}_{s}})).$
Therefore, a linear basis for $\mathrm{{CE}(\mathcal{C})}$ is given by
${\mathrm{C}}^{j}_{st}$ with $j\in\mathcal{J}$ and $s,t\in\mathcal{M}_{j}$.

###  3.2. On unitary subalgebras of ${\mathrm{A}}$

Let ${\mathrm{L}}$ be a subalgebra of ${\mathrm{A}}$ in
${\mathcal{Z}(\mathcal{C})}$. Since ${\mathrm{A}}$ is a semisimple object of
${\mathcal{Z}(\mathcal{C})}$ we may choose the simple direct summands
$\mathcal{C}^{(j)}_{s}$ from Equation (3.23) such that

(3.41)
${\mathrm{L}}=\bigoplus_{j\in\mathcal{J}_{L}}\bigoplus_{s\in\mathcal{L}_{j}}{\mathcal{C}^{(j)}_{s}}$

for some subset $\mathcal{J}_{L}\subseteq\mathcal{J}$ and a subset
$\mathcal{L}_{j}\subseteq\mathcal{M}_{j}:=\\{1,\dots,m_{j}\\}$. Denote by
$\widetilde{J}_{\mathrm{L}}:=\sqcup_{j\in\mathcal{J}_{L}}\\{j\\}\times\mathcal{L}_{j}$.

By the naturality of the adjunction isomorphisms
$\psi_{{\mathcal{C}^{(j)}_{s}},\mathbf{1}}$ it follows that via the inclusion
$\iota^{e}:{\mathrm{CE}}(L)\hookrightarrow\mathrm{{CE}(\mathcal{C})}$ one has

(3.42)
${\mathrm{CE}}(L):=\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},L)=\bigoplus_{j\in\mathcal{J}_{L}}\bigoplus_{s\in\mathcal{L}_{j}}\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},F({\mathcal{C}^{(j)}_{s}})).$

Therefore we may denote by the same symbol
${\mathrm{C}}^{j}_{st}\in\mathrm{{Hom}}_{\mathcal{C}}(\mathbf{1},F({\mathcal{C}^{(j)}_{s}}))$
the elements of the linear basis of $\mathrm{CE}(\mathcal{C})$ that belong to
${\mathrm{CE}}(L)\subseteq\mathrm{{CE}(\mathcal{C})}$ if
$s\in\mathcal{L}_{j}$. In other words,
$\iota^{e}({\mathrm{C}}^{j}_{st})={\mathrm{C}}^{j}_{st}$ if
$j\in\mathcal{J}_{\mathrm{L}}$ and $s\in\mathcal{L}_{j}$.

Thus, a linear basis for ${\mathrm{CE}}(L)$ is given by
${\mathrm{C}}^{j}_{st}:=\mathcal{F}_{\lambda}^{-1}(F^{j}_{st})$ with
$j\in\mathcal{J}_{L}$, $s\in\mathcal{L}_{j}$ and $t\in\mathcal{M}_{j}$.

Note also that
$\mathrm{{CF}}(\mathcal{C})=\bigoplus_{j\in\mathcal{J},s\in\mathcal{M}_{j}}\mathrm{{Hom}}_{\mathcal{C}}(F({\mathcal{C}^{(j)}_{s}}),\mathbf{1})$.
Recall that there is also a projection
$\pi:{\mathrm{A}}\rightarrow{\mathrm{L}}$ in ${\mathcal{Z}(\mathcal{C})}$
since this is a semisimple category. Then, as above, the induced linear
morphism
$\pi^{c}_{{}_{\uparrow}}:=F(\pi)^{*}:\mathrm{{Hom}}_{\mathcal{C}}(F({\mathrm{L}}),\mathbf{1})\rightarrow\mathrm{{Hom}}_{\mathcal{C}}(F({\mathrm{A}}),\mathbf{1})$
gives an embedding
${\mathrm{CF}}(L)\hookrightarrow\mathrm{{CF}}(\mathcal{C})$. Via this
embedding one has
${\mathrm{CF}}(L)=\bigoplus_{(j,s)\in\widetilde{\mathcal{J}}_{L}}\mathrm{{Hom}}_{\mathcal{C}}(F({\mathcal{C}^{(j)}_{s}}),\mathbf{1})$
and we may denote by the same symbol $F^{j}_{st}$ the same elements
corresponding to ${\mathrm{CF}}(L)$ when
${(j,t)\in\widetilde{\mathcal{J}}_{L}}$.

The following three lemmas are easy to verify.

######  Lemma 3.43.

A basis for ${\mathrm{CF}}(L)$ is given by $F^{j}_{st}$ with
$(j,t)\in\widetilde{\mathcal{J}}_{L}$ and $s\in\mathcal{M}_{j}$.

######  Lemma 3.44.

With the above identifications one has

(3.45)
${\mathrm{Res}}(F^{j}_{st})=F^{j}_{st},\;\text{if}\;(j,t)\in\widetilde{\mathcal{J}}_{L},\;\text{for
any,}\;0\leq s\leq m_{j}$

and

(3.46)
${\mathrm{Res}}(F^{j}_{st})=0,\;\text{if}\;(j,t)\notin\widetilde{\mathcal{J}}_{L},\;\;\text{for
any,}\;0\leq s\leq m_{j}.$

######  Lemma 3.47.

For any $j\in\mathcal{J}_{L}$ one has
$\pi^{e}_{{}_{\downarrow}}(C^{j}_{st})=C^{j}_{st}$ if
$j\in\mathcal{J}_{\mathrm{L}}$, $s\in\mathcal{L}_{j}$ and,
$\pi^{e}_{{}_{\downarrow}}(C^{j}_{st})=0$ otherwise.

For any irreducible character $\chi_{i}$ write

(3.48)
$\chi_{i}=\sum_{j=0}^{m}\;\sum_{s,t\in\mathcal{M}_{j}}{\alpha(i)^{j}_{st}}F^{j}_{st}$

for some scalars ${\alpha(i)^{j}_{st}}\in\Bbbk$. Applying
$\mathcal{F}_{\lambda}^{-1}$ to this Equation and using Equation (2.16) one
has that

(3.49)
$\frac{\dim(\mathcal{C})}{{d_{i}}}E_{i^{*}}=\sum_{j=0}^{m}\;\sum_{s,t\in\mathcal{M}_{j}}\alpha(i)^{j}_{st}{\mathrm{C}}^{j}_{st}$

which gives that

(3.50)
$E_{i^{*}}=\frac{1}{\dim(\mathcal{C})}\sum_{j=0}^{m}\;\sum_{s,t\in\mathcal{M}_{j}}d_{i}\alpha(i)^{j}_{st}{\mathrm{C}}^{j}_{st}.$

Denote by

(3.51) $\beta^{j}_{st}:=\sum_{i\in{\mathcal{B}}_{0}}d_{i}\alpha(i)^{j}_{st}.$

######  Lemma 3.52.

Let $\mathcal{C}$ be a fusion category and ${\mathrm{L}}$ be a unitary
subalgebra of ${\mathrm{A}}$. With the above notations one has

(3.53) $\beta^{j}_{st}=0,\;\text{for
any}\;(j,s)\notin\widetilde{\mathcal{J}_{L}},\;\text{and any}\;0\leq t\leq
m_{j}.$

###### Proof.

Recall the linear basis $\ell_{k}$ of ${\mathrm{CE}}(L)$, with $0\leq k\leq r$
. Applying Equation (3.49) one obtains that for any $0\leq k\leq r$ one has

$\iota^{e}(\ell_{k})=\sum_{\chi_{i}\in\mathrm{Irr}({\mathcal{B}}_{k})}E_{i}\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{einc}\mkern-1.5mu)}}{{=}}\frac{1}{{\dim(\mathcal{C})}}\sum_{j=0}^{m}\;\sum_{s,t\in\mathcal{M}_{j}}\bigg{(}\sum_{\chi_{i}\in\mathrm{Irr}({\mathcal{B}}_{k})}{d_{i^{*}}\alpha(i^{*})^{j}_{st}}\bigg{)}{\mathrm{C}}^{j}_{st}.$

Since
$\\{{\mathrm{C}}^{j}_{st}\\}_{{j\in\mathcal{J}_{L}},s\in\mathcal{L}_{j},t\in\mathcal{M}_{j}}$
is a basis of ${\mathrm{CE}}(L)$ it follows that

(3.54)
$\sum_{\chi_{i}\in\mathrm{Irr}({\mathcal{B}}_{k})}{d_{i^{*}}\alpha(i^{*})^{j}_{st}}=0$

if $j\notin\mathcal{J}_{L}$ or if $j\in\mathcal{J}_{L}$ but
$s\notin\mathcal{L}_{j}$. Note that for $k=0$ this can also be written as
$\beta^{j}_{st}=0,\;\text{for
any}\;(j,s)\notin\widetilde{\mathcal{J}_{L}},\;\text{and any}\;0\leq t\leq
m_{j}$ since ${\mathcal{S}_{{\mathrm{L}}}}={\mathcal{B}}_{0}$ is a fusion
subcategory. ∎

######  Lemma 3.55.

With the above notations one has

(3.56)
${\underline{\epsilon}_{L}}=\sum_{(j,t)\in\widetilde{\mathcal{J}}_{L}}F^{j}_{tt}.$

For any $\chi_{i}\in{\mathcal{B}}_{0}$ one has

(3.57) $\alpha(i)^{j}_{st}=\delta_{s,t}d_{i},\text{for
any}\;(j,t)\in\widetilde{\mathcal{J}}_{L},\;\text{and any},\;1\leq s\leq
m_{j}.$

###### Proof.

Since ${\bf\epsilon}_{\mathbf{1}}=\sum_{j=0}^{m}F^{j}$ is the unit of
$\mathrm{{CF}}(\mathcal{C})$ it follows by Lemma 3.44 that
${\underline{\epsilon}_{L}}:={\mathrm{Res}}({\bf\epsilon}_{\mathbf{1}})=\sum_{(j,t)\in\widetilde{\mathcal{J}}_{L}}F^{j}_{tt}$.
If $\chi_{i}\in{\mathcal{B}}_{0}$ then by Equation (3.19) one has
${\mathrm{Res}}(\chi_{i})=d_{i}{\underline{\epsilon}_{L}}=d_{i}\big{(}\sum_{(j,t)\in\widetilde{\mathcal{J}}_{L}}F^{j}_{tt}\big{)}$.
On the other hand

$\displaystyle{\mathrm{Res}}(\chi_{i})$ $\displaystyle=$
$\displaystyle\sum_{j\in\mathcal{J}}\sum_{s\in\mathcal{M}_{j}}\sum_{t\in\mathcal{M}_{j}}{\alpha(i)^{j}_{st}}{\mathrm{Res}}(F^{j}_{st})$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{resal}\mkern-1.5mu)}}{{=}}$
$\displaystyle\sum_{j\in\mathcal{J}_{L}}\sum_{t\in\mathcal{L}_{j}}\sum_{s\in\mathcal{M}_{j}}{\alpha(i)^{j}_{st}}F^{j}_{st}.$

Comparing the two expressions for ${\mathrm{Res}}(\chi_{i})$ it follows that
for $(j,t)\in\widetilde{\mathcal{J}}_{L}$ one has
$\alpha(i)^{j}_{st}=d_{i}\delta_{s,t}$. ∎

######  Proposition 3.58.

Suppose that $\mathcal{C}$ is a spherical fusion category and ${\mathrm{L}}$
be a unitary subalgebra of ${\mathrm{A}}$. With the above notations, for the
idempotent integral of ${\mathcal{S}_{{\mathrm{L}}}}$ one has that

(3.59)
$\lambda_{\mathcal{S}_{\mathrm{L}}}=\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j}}F^{j}_{ss}+\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j},t\notin\mathcal{L}_{j}}\beta^{j}_{st}F^{j}_{st}$

for some scalars $\beta^{j}_{st}\in\Bbbk$.

###### Proof.

Recall that
$\dim({\mathcal{S}_{\mathrm{L}}})=\sum_{i\in{\mathrm{Irr}({\mathcal{S}_{{\mathrm{L}}}})}}d_{i^{*}}d_{i}$.
Then since $\mathcal{C}$ is spherical one has that

$\displaystyle\lambda_{\mathcal{S}_{\mathrm{L}}}$ $\displaystyle=$
$\displaystyle\frac{1}{\dim(\mathcal{S}_{\mathrm{L}})}\big{(}\sum_{\chi_{i}\in{\mathrm{Irr}({\mathcal{S}_{{\mathrm{L}}}})}}d_{i^{*}}\chi_{i}\big{)}$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{chi}\mkern-1.5mu)}}{{=}}$
$\displaystyle\frac{1}{\dim(\mathcal{S}_{\mathrm{L}})}\bigg{(}\sum_{\chi_{i}\in{\mathrm{Irr}({\mathcal{S}_{{\mathrm{L}}}})}}d_{i^{*}}\big{(}\sum_{j=0}^{m}\sum_{s,t\in\mathcal{M}_{j}}\alpha(i)^{j}_{st}F^{j}_{st}\big{)}\bigg{)}$
$\displaystyle=$
$\displaystyle\frac{1}{\dim(\mathcal{S}_{\mathrm{L}})}\sum_{j=0}^{m}\sum_{s,t\in\mathcal{M}_{j}}\big{(}\sum_{\chi_{i}\in{\mathrm{Irr}({\mathcal{S}_{{\mathrm{L}}}})}}d_{i}^{*}\alpha(i)^{j}_{st}\big{)}F^{j}_{st}$
$\displaystyle=$
$\displaystyle\frac{1}{\dim(\mathcal{S}_{\mathrm{L}})}\sum_{j=0}^{m}\sum_{s,t\in\mathcal{M}_{j}}\beta^{j}_{st}F^{j}_{st}$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{idt}\mkern-1.5mu)}}{{=}}$
$\displaystyle\frac{1}{\dim(\mathcal{S}_{\mathrm{L}})}\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j}}\sum_{t\in\mathcal{M}_{j}}\beta^{j}_{st}F^{j}_{st}$
$\displaystyle=$
$\displaystyle\frac{1}{\dim(\mathcal{S}_{\mathrm{L}})}\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j}}\sum_{t\in\mathcal{L}_{j}}\beta^{j}_{st}F^{j}_{st}$
$\displaystyle+$
$\displaystyle\frac{1}{\dim(\mathcal{S}_{\mathrm{L}})}\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j}}\sum_{t\in\mathcal{M}_{j}\setminus\mathcal{L}_{j}}\beta^{j}_{st}F^{j}_{st}$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{lastid}\mkern-1.5mu)}}{{=}}$
$\displaystyle\frac{1}{\dim(\mathcal{S}_{\mathrm{L}})}\bigg{(}\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j}}\dim(\mathcal{S}_{\mathrm{L}})F^{j}_{ss}\bigg{)}$
$\displaystyle+$
$\displaystyle\frac{1}{\dim(\mathcal{S}_{\mathrm{L}})}\sum_{j\in\mathcal{J}_{L}}\sum_{t\in\mathcal{L}_{j}}\sum_{s\in\mathcal{M}_{j}\setminus\mathcal{L}_{j}}\beta^{j}_{st}F^{j}_{st}$
$\displaystyle=$
$\displaystyle\sum_{j\in\mathcal{J}_{L}}\sum_{t\in\mathcal{L}_{j}}F^{j}_{ss}+\frac{1}{\dim(\mathcal{S}_{\mathrm{L}})}\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j},t\notin\mathcal{L}_{j}}\beta^{j}_{st}F^{j}_{st}$

where the scalars
$\beta^{j}_{st}:=\big{(}\sum_{\chi_{i}\in{\mathrm{Irr}({\mathcal{S}_{{\mathrm{L}}}})}}d_{i}^{*}\alpha(i)^{j}_{st}\big{)}$
are defined as in Equation (3.51). ∎

### Proof of Theorem 3.2

###### Proof.

Note that Proposition 3.58 shows that the set of indices
$\mathcal{L}_{{\mathcal{S}_{{\mathrm{L}}}}}$ of ${\mathcal{S}_{{\mathrm{L}}}}$
coincides to
$\sqcup_{j\in\mathcal{J}_{L}}\\{j\\}\times\mathcal{M}_{j}\times\mathcal{L}_{j}$.
Then by Equation (3.36) one has that

$\frac{\dim(\mathcal{C})}{\dim({\mathcal{S}_{{\mathrm{L}}}})}=\sum_{(j,s)\in\widetilde{\mathcal{J}}_{L}}\dim(\mathcal{C}^{(j)}_{s})=\dim({\mathrm{L}}).$

which finishes the proof of Theorem 3.2 since $\mathcal{C}$ is a pseudo-
unitary fusion category.∎

######  Corollary 3.60.

Let $\mathcal{C}$ be a pseudo-unitary fusion category and ${\mathrm{M}}$,
${\mathrm{L}}$ be two unitary subalgebras of ${\mathrm{A}}$. Then
$\mathcal{S}_{\mathrm{M}}={\mathcal{S}_{{\mathrm{L}}}}$ if and only if
${\mathrm{M}}={\mathrm{L}}$

###### Proof.

Note that since ${\mathrm{A}}$ is a semisimple object one can choose the the
simple direct summands ${\mathcal{C}^{(j)}_{s}}$ such that Equation (3.61) is
satisfied for both ${\mathrm{M}}$ and ${\mathrm{L}}$, i.e.

(3.61)
${\mathrm{L}}=\bigoplus_{j\in\mathcal{J}_{L}}\bigoplus_{s\in\mathcal{L}^{1}_{j}}{\mathcal{C}^{(j)}_{s}},\;{\mathrm{M}}=\bigoplus_{j\in\mathcal{J}_{\mathrm{M}}}\bigoplus_{s\in\mathcal{L}^{2}_{j}}{\mathcal{C}^{(j)}_{s}}.$

for some subsets
$\mathcal{J}_{\mathrm{L}},\mathcal{J}_{\mathrm{M}}\subseteq\mathcal{J}$ and
$\mathcal{L}^{1}_{j},\mathcal{L}^{2}_{j}\subseteq\mathcal{M}_{j}$. One has
${\mathcal{S}_{{\mathrm{L}}}}=\mathcal{S}_{\mathrm{M}}$ if and only if
$\lambda_{\mathcal{S}_{{\mathrm{L}}}}=\lambda_{\mathcal{S}_{\mathrm{M}}}$.
Then the result follows from Equation (3.59). ∎

###  3.3. On the abelian full subcategories
${\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}$

Let $M$ be any object of $\mathcal{C}$. We denote by
${\alpha_{M}\big{|}_{{}_{L}}}$ the restriction to the subalgebra $L$ of the
action ${{\alpha_{M}}}$ of $A$ on $M$ defined in Equation (2.7). We say that
an object $M$ of $\mathcal{C}$ receives a trivial $L$-action if

(3.62)
${\alpha_{M}\big{|}_{{}_{L}}}={\underline{\epsilon}_{L}}\otimes\operatorname{id}_{M}.$

######  Definition 3.4.

Let $\mathcal{C}$ be any finite tensor category and ${\mathrm{L}}$ be a
unitary subalgebra of ${\mathrm{A}}$. We define
${{\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}}$ as the full abelian subcategory
of $\mathcal{C}$ whose objects $M$ receive a trivial action ${{\alpha_{M}}}$.

Next proposition holds for any finite tensor category, not necessarily fusion.

######  Proposition 3.63.

Let $\mathcal{C}$ be a pivotal finite tensor category and ${\mathrm{L}}$ be a
unitary subalgebra of ${\mathrm{A}}$. Then as abelian subcategories of
$\mathcal{C}$ one has

${{\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}}\subseteq\mathcal{S}_{\mathrm{L}}.$

###### Proof.

Note that by definition of the dimension one has

$d(M)\operatorname{id}_{\mathbf{1}}=(\mathbf{1}\xrightarrow{\mathrm{coev}_{M}}M\otimes
M^{*}\xrightarrow{\widetilde{ev_{M}}}\mathbf{1})$

where $\widetilde{ev_{M}}=(M\otimes M^{*}\xrightarrow{j_{M}}M^{**}\otimes
M^{*}\xrightarrow{ev_{M^{*}}}\mathbf{1})$ is defined using the pivotal
structure $j_{M}$. This implies

(3.64) $d(M){\underline{\epsilon}_{L}}=(L\simeq
L\otimes\mathbf{1}\xrightarrow{{\underline{\epsilon}_{L}}\otimes\operatorname{id}_{\mathbf{1}}}\mathbf{1}\xrightarrow{\mathrm{coev}_{M}}M\otimes
M^{*}\xrightarrow{\widetilde{ev_{M}}}\mathbf{1}).$

Suppose that $M\in\mathcal{O}(\mathcal{C})$ satisfies Equation (3.62). We will
show that $M\in\mathcal{O}({\mathcal{S}_{{\mathrm{L}}}})$. Since for any two
morphisms in $\mathcal{C}$ one has $f\otimes g=(f\otimes 1)\circ(1\otimes g)$,
it follows that

(3.65) $(\operatorname{id}_{A}\otimes\operatorname{coev}_{M})\circ
F(\iota_{L})=(F(\iota_{L})\otimes\operatorname{id}_{M}\otimes\operatorname{id}_{M^{*}})\circ(\operatorname{id}_{L}\otimes\operatorname{coev}_{M}).$

Then the restriction ${\mathrm{Res}}(\chi(M))$ can be written as:

$\displaystyle\hskip-19.91684pt{\mathrm{Res}}(\chi(M))$ $\displaystyle=$
$\displaystyle(L\xrightarrow{{F(\iota_{\mathrm{L}})}}A\xrightarrow{\chi(M)}\mathbf{1})=$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{chx}\mkern-1.5mu)}}{{=}}$
$\displaystyle(L\xrightarrow{{F(\iota_{\mathrm{L}})}}A\simeq
A\otimes\mathbf{1}\xrightarrow{1_{A}\otimes\mathrm{coev}_{M}}A\otimes M\otimes
M^{*}\rightarrow$
$\displaystyle\xrightarrow{{{\alpha_{M}}}\otimes\operatorname{id}_{M^{*}}}$
$\displaystyle M\otimes
M^{*}\xrightarrow{\widetilde{\operatorname{ev}_{M}}}\mathbf{1})=$
$\displaystyle\stackrel{{\scriptstyle\scriptscriptstyle(\mkern-1.5mu\ref{int1}\mkern-1.5mu)}}{{=}}$
$\displaystyle(L\simeq
L\otimes\mathbf{1}\xrightarrow{\operatorname{id}_{L}\otimes\mathrm{coev}_{M}}L\otimes
M\otimes
M^{*}\xrightarrow{{\alpha_{M}\big{|}_{{}_{L}}}\otimes\operatorname{id}_{M^{*}}}M\otimes
M^{*}\xrightarrow{\widetilde{\operatorname{ev}_{M}}}\mathbf{1}).$

Replacing from Equation (3.62) ${\alpha_{M}\big{|}_{{}_{L}}}$ by
${\underline{\epsilon}_{L}}\otimes\operatorname{id}_{M}$ one obtains that
${\mathrm{Res}}(\chi(M))$ can be written as

$(L\simeq
L\otimes\mathbf{1}\xrightarrow{\operatorname{id}_{L}\otimes\mathrm{coev}_{M}}L\otimes
M\otimes
M^{*}\xrightarrow{{\underline{\epsilon}_{L}}\otimes\operatorname{id}_{M}\otimes\operatorname{id}_{M^{*}}}M\otimes
M^{*}\xrightarrow{\tilde{\operatorname{ev}_{M}}}\mathbf{1}).$

Using Equation (3.64) this shows that
${\mathrm{Res}}(\chi(M))=d(M){\underline{\epsilon}_{L}}$, i.e
$M\in\mathcal{O}(\mathcal{S}_{\mathrm{L}})$. ∎

##  4\. Proof of Theorem 1.1

Let $\mathcal{C}$ be a finite tensor category and
$F:{\mathcal{Z}(\mathcal{C})}\rightarrow\mathcal{C}$ be the forgetful functor.
As above we denote by $R$ a right adjoint of $F$. The monoidal structure of
the adjoint functor $R:\mathcal{C}\rightarrow{\mathcal{Z}(\mathcal{C})}$ gives
two actions of the algebra ${\mathrm{A}}=R(\mathbf{1})$ on the object
$R(M)\in{\mathcal{Z}(\mathcal{C})}$. The left action is defined by
$\rho^{l}_{R(M)}=R_{2}(M,\mathbf{1}):R(M)\otimes R(\mathbf{1})\rightarrow
R(M\otimes\mathbf{1})\simeq R(M)$ and similarly the right action is defined by
$\rho^{r}_{R(M)}=R_{2}(\mathbf{1},M):R(\mathbf{1})\otimes R(M)\rightarrow
R(\mathbf{1}\otimes M)\simeq R(M).$ By [16, Lemma 6.4] for any finite tensor
category one has

(4.1) $\rho^{r}_{R(M)}=\rho^{l}_{R(M)}\circ c_{R(M),{\mathrm{A}}}$

where $c_{R(M),-}$ is the braiding of $R(M)\in{\mathcal{Z}(\mathcal{C})}$. The
author also has warned that in general
$\rho^{l}_{R(M)}\neq\rho^{r}_{R(M)}\circ c_{{\mathrm{A}},R(M)}$.

######  Lemma 4.2.

Let $\mathcal{C}$ be a finite tensor category. For any object $M$ of
$\mathcal{C}$ with the above notations one has

(4.3)
$\epsilon_{M}\circ\rho^{l}_{R(M)}={\bf\epsilon}_{\mathbf{1}}\otimes\epsilon_{M},\;\;\epsilon_{M}\circ\rho^{r}_{R(M)}={\bf\epsilon}_{\mathbf{1}}\otimes\epsilon_{M}$

###### Proof.

It follows from the definition of the actions
$\rho^{l}_{R(M)},\rho^{r}_{R(M)}$ and Equation (2.4). ∎

By [6, Lemma 3.5] ${\mathrm{A}}$ is an étale algebra in
${\mathcal{Z}(\mathcal{C})}$ and the right adjoint functor $R$ induces a
tensor equivalence
$\mathcal{C}\xrightarrow{R}{\mathcal{Z}(\mathcal{C})}_{{\mathrm{A}}}$. Recall
also that [6, Theorem 4.10] gives an (inclusion reversing) bijection between
the lattice of étale subalgebras of ${\mathrm{A}}$ and that of fusion
subcategories of $\mathcal{C}$. This bijection can be described as follows:

Given an étale subalgebra ${\mathrm{L}}$ of the adjoint algebra ${\mathrm{A}}$
one considers the full subcategory $\beta({\mathrm{L}})$ of $\mathcal{C}$ as
the category generated by those right ${\mathrm{A}}$-modules in
${\mathcal{Z}(\mathcal{C})}$ that are dyslectic as right
${\mathrm{L}}$-modules with respect to ${\mathrm{L}}$. Its inverse associates
to a fusion subcategory ${\mathcal{D}}$ of $\mathcal{C}$ the subalgebra
${\mathrm{L}}_{\mathcal{D}}=J_{\mathcal{D}}(\mathbf{1})$ where
$J_{\mathcal{D}}$ is the right adjoint of the forgetful functor
$F_{\mathcal{D}}:{\mathcal{Z}(\mathcal{C})}\rightarrow\mathcal{Z}_{\mathcal{C}}({\mathcal{D}}).$
Recall that $\mathcal{Z}_{\mathcal{C}}({\mathcal{D}})$ is the relative center
of ${\mathcal{D}}$ as a fusion subcategory of $\mathcal{C}$, see [6]. By the
proof of [6, Theorem 4.10] one has

(4.4)
$\mathrm{FPdim}(\beta({\mathrm{L}}))=\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}({\mathrm{L}})}$

for any connected étale subalgebra ${\mathrm{L}}$ of ${\mathrm{A}}$. Using
Equation (4.1), since ${\mathrm{L}}\in{\mathcal{Z}(\mathcal{C})}$ is a sub-
object of ${\mathrm{A}}$ we note that for any $M\in\mathcal{O}(\mathcal{C})$,
the dyslecticity of $R(M)$ as a right ${\mathrm{L}}$-module is equivalent to

(4.5)
$\rho^{l}_{R(M)}\big{|}_{{}_{\mathrm{L}}}=\rho^{r}_{R(M)}\big{|}_{{}_{\mathrm{L}}}\circ
c_{{\mathrm{L}},R(M)}$

For any left object $M$ of $\mathcal{C}$ we denote by
${\alpha_{M}\big{|}_{{}_{L}}}$ the restriction of the action
${{\alpha_{M}}}:A\otimes M\rightarrow M$ of $A$ from Equation (2.7) to $L$.
Recall that by Equation (2.8) one has
${{\alpha_{M}}}=(\operatorname{id}_{M}\otimes{\bf\epsilon}_{\mathbf{1}})\circ
c_{{\mathrm{A}},M}$. Since ${\mathrm{L}}$ is a subalgebra of ${\mathrm{A}}$ in
${\mathcal{Z}(\mathcal{C})}$ this formula can be restricted to ${\mathrm{L}}$
as:

(4.6)
${{\alpha_{M}\big{|}_{{}_{L}}}}=(\operatorname{id}_{M}\otimes{\underline{\epsilon}_{L}})\circ
c_{{\mathrm{L}},M},$

where as above ${\underline{\epsilon}_{L}}$ denotes the restriction of
${\bf\epsilon}_{\mathbf{1}}:A\rightarrow\mathbf{1}$ to $L$.

######  Proposition 4.7.

Let ${\mathrm{L}}$ be a unitary subalgebra of ${\mathrm{A}}$ in
${\mathcal{Z}(\mathcal{C})}$. With the above notations if $M$ is an object of
$\mathcal{C}$ such that $R(M)$ is dyslectic with respect to ${\mathrm{L}}$
then the following diagram is commutative: ${{\mathrm{L}}\otimes
Z(M)}$${Z(M)}$${M}$$\scriptstyle{\alpha_{Z(M)}\big{|}_{{}_{\mathrm{L}}}}$$\scriptstyle{{\underline{\epsilon}_{L}}\otimes\epsilon_{M}}$$\scriptstyle{\epsilon_{M}}$

which can be written as:

(4.8)
$\epsilon_{M}\circ\alpha_{Z(M)}\big{|}_{{}_{\mathrm{L}}}={\underline{\epsilon}_{L}}\otimes\epsilon_{M}.$

###### Proof.

Assume that $R(M)$ is a dyslectic right ${\mathrm{L}}$-module in
${\mathcal{Z}(\mathcal{C})}$. From Equation (4.5) it follows that

(4.9)
$\epsilon_{M}\circ\rho^{l}_{R(M)}\big{|}_{{}_{\mathrm{L}}}=\epsilon_{M}\circ\rho^{r}_{{Z(M)}}\circ\rho^{r}_{R(M)}\big{|}_{{}_{\mathrm{L}}}\circ
c_{{\mathrm{L}},R(M)}.$

By Equation (4.3) the above equality can be written as

(4.10)
${\underline{\epsilon}_{L}}\otimes\epsilon_{M}=({\underline{\epsilon}_{L}}\otimes\epsilon_{M})\circ
c_{{\mathrm{L}},R(M)}.$

On the other hand by Equation (4.6) one has

$(\epsilon_{M}\otimes{\underline{\epsilon}_{L}})\circ
c_{{\mathrm{L}},R(M)}=(\epsilon_{M}\otimes\operatorname{id}_{\mathbf{1}})\circ[(\operatorname{id}_{R(M)}\otimes{\underline{\epsilon}_{L}})\circ
c_{{\mathrm{L}},Z(M)}]=(\epsilon_{M}\otimes\operatorname{id}_{\mathbf{1}})\circ{\alpha_{Z(M)}\big{|}_{{}_{\mathrm{L}}}}.$

Thus Equation (4.10) can be written as
${\underline{\epsilon}_{L}}\otimes\epsilon_{M}=\epsilon_{M}\circ\alpha_{Z(M)}\big{|}_{{}_{\mathrm{L}}}$.
∎

###  4.1. Proof of the inclusion
$\beta({\mathrm{L}})\subseteq{{\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}}$

For any unitary subalgebra ${\mathrm{L}}$ of ${\mathrm{A}}$ we denote by
$\beta({\mathrm{L}})$ the full subcategory of $\mathcal{C}$ which by the
tensor equivalence
$R:\mathcal{C}\rightarrow{\mathcal{Z}(\mathcal{C})}_{\mathrm{A}}$ correspond
to the right ${\mathrm{A}}$-modules that are dyslectic with respect to
${\mathrm{L}}$. Note that as mentioned above, if ${\mathrm{L}}$ is an étale
subalgebra then $\beta({\mathrm{L}})$ is a fusion subcategory of $\mathcal{C}$
by [6, Theorem 4.10].

######  Proposition 4.11.

Let $\mathcal{C}$ be a fusion category and ${\mathrm{L}}$ be a unitary
subalgebra of $A$. With the above notations one has
$\beta({\mathrm{L}})\subseteq{{\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}}$.

###### Proof.

Suppose that $M$ is an object of $\beta({\mathrm{L}})$, i.e a dyslectic right
${\mathrm{L}}$-module. By the previous Proposition the action of
${\mathrm{L}}$ on $Z(M)$ satisfies
$\epsilon_{M}\circ\alpha_{Z(M)}\big{|}_{{}_{\mathrm{L}}}={\underline{\epsilon}_{L}}\otimes\epsilon_{M}$.

Since $\epsilon_{M}:Z(M)\rightarrow M$ is a morphism in $\mathcal{C}$ it
follows that it is also an $A$-linear morphism, see [18], and by restriction
it is also $L$-linear. Therefore we have

(4.12)
$\epsilon_{M}\circ\alpha_{Z(M)}\big{|}_{{}_{\mathrm{L}}}={\alpha_{M}\big{|}_{{}_{\mathrm{L}}}}\circ(\operatorname{id}_{\mathrm{L}}\otimes\epsilon_{M})$

Then Equation 4.8 can be written as

(4.13) $({\mathrm{L}}\otimes
Z(M)\xrightarrow{{\underline{\epsilon}_{L}}\otimes\epsilon_{M}}M)=({\mathrm{L}}\otimes
Z(M)\xrightarrow{\operatorname{id}_{\mathrm{L}}\otimes\epsilon_{M}}{\mathrm{L}}\otimes
M\xrightarrow{{{\alpha_{M}}}}M).$

Then [1, Proposition 5.1] applied to $\mathcal{C}^{\mathrm{op}}$ implies that
the counit $\epsilon_{M}$ is an epimorphism. Since $\mathcal{C}$ is semisimple
it follows that $\epsilon_{M}:Z(M)\rightarrow M$ has a section
$r_{M}:M\rightarrow Z(M)$ in $\mathcal{C}$, i.e. a morphism satisfying
$\epsilon_{M}\circ r_{M}=\operatorname{id}_{M}$. Composing Equation (4.13)
with $(\operatorname{id}_{\mathrm{L}}\otimes r_{M})$ one obtains that

$\displaystyle({\mathrm{L}}\otimes
M\xrightarrow{\operatorname{id}_{\mathrm{L}}\otimes r_{M}}{\mathrm{L}}\otimes
Z(M)\xrightarrow{{\underline{\epsilon}_{L}}\otimes\epsilon_{M}}M)=$
$\displaystyle=$ $\displaystyle({\mathrm{L}}\otimes
M\xrightarrow{\operatorname{id}_{\mathrm{L}}\otimes r_{M}}{\mathrm{L}}\otimes
Z(M)\xrightarrow{\operatorname{id}_{\mathrm{L}}\otimes\epsilon_{M}}{\mathrm{L}}\otimes
M\xrightarrow{\alpha_{M}}M)$

which can be written as

$({\mathrm{L}}\otimes
M\xrightarrow{{\underline{\epsilon}_{L}}\otimes\operatorname{id}_{M}}M)=({\mathrm{L}}\otimes
M\xrightarrow{{{\alpha_{M}}}}M).$

This shows that $M\in{{\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}}$. ∎

###  4.2. Proof of Theorem 1.1

###### Proof.

First, we show that
$\beta({\mathrm{M}})=\mathcal{C}_{\mathrm{M}}^{\mathrm{triv}}=\mathcal{S}_{{\mathrm{M}}}$
for any étale subalgebra ${\mathrm{M}}$ of ${\mathrm{A}}$. Indeed, Proposition
3.63 and Proposition 4.11 imply that
$\beta({\mathrm{M}})\subseteq\mathcal{C}^{\mathrm{triv}}_{\mathrm{M}}\subseteq\mathcal{S}_{{\mathrm{M}}}$.
Moreover by Theorem 3.2 the two fusion subcategories $\beta({{\mathrm{M}}})$
and $\mathcal{S}_{{\mathrm{M}}}$ have the same Frobenius-Perron dimensions,
namely $\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}({\mathrm{M}})}$.

On the other hand for any unitary subalgebra ${\mathrm{L}}$ of ${\mathrm{A}}$
we know that $\mathcal{S}_{\mathrm{L}}$ is a fusion subcategory of
$\mathcal{C}$ with dimension
$\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}({\mathrm{L}})}$. The
bijective correspondence $\beta$ from [6] gives that
${\mathcal{S}_{{\mathrm{L}}}}={\beta(M)}$ for some étale subalgebra
${\mathrm{L}}$ of ${\mathrm{A}}$. The first part of the proof implies that
${\mathcal{S}_{{\mathrm{L}}}}=\mathcal{S}_{\mathrm{M}}$ and Corollary 3.60
implies that ${\mathrm{M}}={\mathrm{L}}$. ∎

###  4.3. Proof of Theorem 1.2

Let $\mathcal{C}$ be a braided fusion category
$F:\mathcal{C}\rightarrow{\mathcal{D}}$ a tensor functor to an arbitrary
fusion category ${\mathcal{D}}$. Recall that a central structure for $F$ is a
braided functor
$\mathcal{C}\xrightarrow{F^{\prime}}\mathcal{Z}({\mathcal{D}})$ such that the
composition
$\mathcal{C}\xrightarrow{F^{\prime}}\mathcal{Z}({\mathcal{D}})\xrightarrow{\mathrm{Forg}}{\mathcal{D}}$
coincides to $F$. A functor $F$ admitting such a structure is called central.
It follows by [6, Lemma 3.5] that $R(\mathbf{1})$ is an étale algebra in
$\mathcal{C}$ where $R$ is a right adjoint $R$ of the central functor $F$.

###### Proof.

By [6, Lemma 3.29] one has that the free functor
$F_{A}:\mathcal{C}\rightarrow\mathcal{C}_{A}$ is a central functor and
moreover, $A=R_{A}(\mathbf{1})$ where $R_{A}$ is the right adjoint of $F_{A}$.
Since $\mathcal{C}$ is non-degenerate, there is an equivalence of categories
${\mathcal{Z}(\mathcal{C})}\simeq\mathcal{C}\boxtimes\mathcal{C}^{\operatorname{rev}}$.
By [6, Corollary 3.30] there is also an equivalence of categories
$\mathcal{Z}(\mathcal{C}_{A})\simeq\mathcal{C}\boxtimes(\mathcal{C}^{0}_{A})^{\operatorname{rev}}.$
Moreover, under this equivalence, the free functor $F_{A}$ identifies as

$(F_{A}:\mathcal{C}\rightarrow\mathcal{C}_{A})=(\mathcal{C}=\mathcal{C}\boxtimes\mathrm{Vec}\xrightarrow{\iota}\mathcal{C}\boxtimes(\mathcal{C}^{0}_{A})^{\operatorname{rev}}\simeq\mathcal{Z}(\mathcal{C}_{A})\xrightarrow{\mathrm{Forg}}\mathcal{C}_{A}).$

which shows that $R_{A}=\iota^{\mathrm{ra}}\circ R$ where $R$ is the right
adjoint functor of
$\mathrm{Forg}:\mathcal{Z}(\mathcal{C}_{A})\rightarrow\mathcal{C}_{A}$ and
$\iota^{\mathrm{ra}}$ is the right adjoint of the inclusion
$\iota:\mathcal{C}\hookrightarrow\mathcal{C}\boxtimes(\mathcal{C}^{0}_{A})^{\operatorname{rev}}$.
Then $A=R_{A}(\mathbf{1})=\iota^{\mathrm{ra}}(R(\mathbf{1}))$ is a subalgebra
of $R(\mathbf{1})$ as being the largest sub-object of
$\mathcal{Z}(\mathcal{C}_{A})$ that belongs to $\mathcal{C}$. Then any unitary
subalgebra of $A$ in $\mathcal{C}$ is also a subalgebra of $R(\mathbf{1})$ in
$\mathcal{Z}(\mathcal{C}_{A})$ and therefore étale by Theorem 1.1. ∎

######  Remark 4.14.

It is interesting to investigate if the result of Theorem 1.2 can be deduced
by directly investigating the semisimplicity of the corresponding module
category $\mathcal{C}_{M}$. However, note that our proof gives a new
perspective on the bijective correspondence from [6]. It shows that at least
in the pseudo-unitary case, analogue to the group representations situation,
fusion subcategories can be obtained via trivial actions or corresponding
kernels of objects. Another natural question at this stage is if the above
result holds without the pseudo-unitarity assumption on the category or if one
can remove the non-degeneracy assumption in the above statement.

###  4.4. A diagonal formula for $\lambda_{\mathcal{D}}$

In this subsection we assume that $\mathcal{C}$ is pseudo-unitary. For a
unitary subalgebra ${\mathrm{L}}$ of ${\mathrm{A}}$ in
${\mathcal{Z}(\mathcal{C})}$ we let
${\mathcal{D}}={\mathcal{S}_{{\mathrm{L}}}}$. Clearly, the algebra inclusion
$\iota:{\mathrm{L}}\hookrightarrow{\mathrm{A}}$ is an inclusion in the module
category ${\mathcal{Z}(\mathcal{C})}_{\mathrm{L}}$. Since ${\mathrm{L}}$ is an
étale algebra it follows by [6, Theorem 3.2] that
${\mathcal{Z}(\mathcal{C})}\;_{\mathrm{L}}$ is a semisimple category. The
semisimplicity of this category implies that $\iota$ admits a retract
$\pi:{\mathrm{L}}\rightarrow{\mathrm{A}}$ in the same category. For the rest
of this section we assume that $\pi$ is such a retract.

It is easy to check that
$\mathcal{F}_{\lambda}(\Lambda)=\frac{1}{\dim(\mathcal{C})}{\bf\epsilon}_{\mathbf{1}}.$
Since
${\bf\epsilon}_{\mathbf{1}}=\sum_{j\in\mathcal{J}}\sum_{s\in\mathcal{M}_{j}}F^{j}_{ss}$
follows that

(4.15)
$\dim(\mathcal{C})\Lambda=\sum_{j\in\mathcal{J},s\in\mathcal{M}_{j}}C^{j}_{ss}.$

Applying $\pi^{e}_{{}_{\downarrow}}$ to the Equation (4.15) it follows that

(4.16)
$\pi^{e}_{{}_{\downarrow}}(\Lambda)=\frac{1}{\dim(\mathcal{C})}\big{(}\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j}}C^{j}_{ss}\big{)}$

######  Lemma 4.17.

With the above chosen map $\pi$ one has

(4.18)
$\pi^{e}_{{}_{\downarrow}}(\Lambda)=\frac{1}{\dim({\mathcal{D}})}\ell_{0}.$

###### Proof.

Since by construction
$\pi^{e}_{{}_{\downarrow}}(\Lambda):=\mathbf{1}\xrightarrow{\Lambda}A\xrightarrow{\pi}L$
is a morphism of right $L$-modules in $\mathcal{C}$ it follows by Lemma 3.21
that there is $\alpha\in\Bbbk$ such that
$\pi^{e}_{{}_{\downarrow}}(\Lambda)=\alpha\ell_{0}.$ Equation (4.16) implies
that
$\dim(\mathcal{C})\alpha\ell_{0}=\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j}}C_{{s}{s}}$
Applying ${\underline{\epsilon}_{L}}$ to this equation one has
$\dim(\mathcal{C})\alpha=\dim(L)$ which implies that
$\alpha=\frac{\dim(L)}{\dim(\mathcal{C})}=\frac{1}{\dim({\mathcal{D}})}$. ∎

######  Remark 4.19.

Equation (4.16) implies now that

(4.20)
$\ell_{0}={\dim({\mathcal{D}})}\pi^{e}_{{}_{\downarrow}}(\Lambda)=\frac{\dim({\mathcal{D}})}{\dim(\mathcal{C})}(\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j}}C_{{s}{s}})$

######  Proposition 4.21.

With the above notations it follows that

(4.22)
$\lambda_{{\mathcal{D}}}=\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j}}F^{j}_{ss}.$

###### Proof.

Since
${\ell_{{\mathcal{D}}}}:={\mathcal{F}_{\lambda}^{-1}}(\lambda_{\mathcal{D}})$
one has by Equation (3.59) that

(4.23)
${\ell_{{\mathcal{D}}}}=\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j}}C^{j}_{ss}+\frac{1}{\dim({\mathcal{S}_{{\mathrm{L}}}})}\big{(}\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j},t\notin\mathcal{L}_{j}}\beta^{j}_{st}C^{j}_{st}\big{)}.$

Using Lemma 3.47 it follows that

(4.24)
$\pi^{e}_{{}_{\downarrow}}({\ell_{{\mathcal{D}}}})=\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j}}C^{j}_{ss}+\frac{1}{\dim({\mathcal{S}_{{\mathrm{L}}}})}\big{(}\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}_{j},t\notin\mathcal{L}_{j}}\beta^{j}_{st}C^{j}_{st}\big{)}.$

On the other hand
$\ell_{0}=\pi^{e}_{{}_{\downarrow}}(\iota^{e}(\ell_{0}))=\pi^{e}_{{}_{\downarrow}}(\sum_{i\in{\mathcal{B}}_{0}}E_{i})=\frac{\dim({\mathcal{D}})}{\dim(\mathcal{C})}\pi^{e}_{{}_{\downarrow}}({\ell_{\mathcal{D}}}).$
Comparing Equations (4.20) and (4.24) it follows that $\beta^{j}_{st}=0$ for
$s\neq t$. ∎

###  4.5. On the lattice of fusion subcategories

In this subsection we prove Theorem 1.3. It follows from the next three
results presented below. For two subalgebras of the adjoint algebra
${\mathrm{A}}$ we denote by ${\mathrm{L}}{\mathrm{M}}$ the image of
${\mathrm{L}}\otimes{\mathrm{M}}$ under the multiplication
$m:{\mathrm{A}}\otimes{\mathrm{A}}\rightarrow{\mathrm{A}}$. Since
${\mathrm{A}}$ is commutative it follows that
${\mathrm{L}}{\mathrm{M}}={\mathrm{M}}{\mathrm{L}}$ and moreover
${\mathrm{M}}{\mathrm{L}}$ is a subalgebra of ${\mathrm{A}}$. Note that
${\mathrm{L}}{\mathrm{M}}$ is the smallest unitary subalgebra of
${\mathrm{A}}$ containing both ${\mathrm{L}}$ and ${\mathrm{M}}$.

######  Proposition 4.25.

Let $\mathcal{C}$ be a pseudo unitary fusion category. With the above
notations one has

${\mathcal{S}_{{\mathrm{L}}}}\cap\mathcal{S}_{\mathrm{M}}=\mathcal{S}_{{\mathrm{L}}{\mathrm{M}}}.$

###### Proof.

Corollary 3.60 implies that
$\mathcal{S}_{{\mathrm{L}}{\mathrm{M}}}\subseteq\mathcal{S}_{{\mathrm{L}}}\cap\mathcal{S}_{{\mathrm{M}}}$
since ${\mathrm{L}},{\mathrm{M}}\subseteq{\mathrm{A}}$. On the other hand, if
$V\in\mathcal{S}_{{\mathrm{L}}}\cap\mathcal{S}_{{\mathrm{M}}}$, then
${\mathrm{L}}{\mathrm{M}}$ acts trivially on $V$, by the associativity of the
action $\alpha_{V}$ of $A$ on $V$. This shows that
$\mathcal{S}_{{\mathrm{L}}{\mathrm{M}}}\supseteq\mathcal{S}_{{\mathrm{L}}}\cap\mathcal{S}_{{\mathrm{M}}}$.
∎

######  Lemma 4.26.

Let $\mathcal{C}$ be a pseudo-unitary fusion category and
${\mathrm{L}},{\mathrm{M}}$ be two unitary subalgebras of ${\mathrm{A}}$.
Then:

(4.27)
$\mathrm{FPdim}({\mathcal{S}_{{\mathrm{L}}}}\vee\mathcal{S}_{\mathrm{M}})\geq\mathrm{FPdim}(\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}})$

###### Proof.

Suppose
${{\mathcal{S}_{{\mathrm{L}}}}\vee\mathcal{S}_{\mathrm{M}}}=\mathcal{S}_{\mathrm{P}}$
for some subalgebra ${\mathrm{P}}\subseteq{\mathrm{A}}$. One has
${\mathcal{S}_{{\mathrm{L}}}}\subseteq\mathcal{S}_{\mathrm{P}}$ which implies
that ${\mathrm{P}}\subseteq{\mathrm{L}}$ since $\beta$ is a reversing order
bijection. Similarly one deduces that ${\mathrm{P}}\subseteq{\mathrm{M}}$, and
therefore $P\subseteq{\mathrm{L}}\cap{\mathrm{M}}$.

Since ${\mathrm{A}}$ is a semisimple object, similarly to Corollary 3.60 one
can choose a decomposition of ${\mathrm{A}}$ from equation (3.23) such that

${\mathrm{L}}=\bigoplus_{j\in\mathcal{J}_{\mathrm{L}}}\bigoplus_{s\in\mathcal{L}^{1}_{j}}\mathcal{C}^{(j)}_{s},\;{\mathrm{M}}=\bigoplus_{j\in\mathcal{J}_{\mathrm{M}}}\bigoplus_{s\in\mathcal{L}^{2}_{j}}\mathcal{C}^{(j)}_{s},\;{\mathrm{P}}=\bigoplus_{j\in\mathcal{J}_{\mathrm{P}}}\bigoplus_{s\in\mathcal{L}^{3}_{j}}\mathcal{C}^{(j)}_{s}.$

for some subsets
$\mathcal{L}^{i}_{j}\subseteq\mathcal{M}_{j}=\\{0,1,\dots,m_{j}\\}$ for
$i=1,2,3$. Note also that

${\mathrm{L}}\cap{\mathrm{M}}=\bigoplus_{j\in\mathcal{J}_{\mathrm{L}}\cap\mathcal{J}_{\mathrm{M}}}\bigoplus_{s\in\mathcal{L}^{1}_{j}\cap\mathcal{L}^{2}_{j}}\mathcal{C}^{(j)}_{s}.$

Since ${\mathrm{P}}\subseteq{\mathrm{L}}\cap{\mathrm{M}}$ it follows that
$\mathcal{J}_{\mathrm{P}}\subseteq\mathcal{J}_{\mathrm{L}}\cap\mathcal{J}_{\mathrm{M}}$
and $\mathcal{L}_{j}^{3}\subseteq\mathcal{L}_{j}^{1}\cap\mathcal{L}_{j}^{2}$
for any $j\in\mathcal{J}_{\mathrm{P}}$.

By Equation (4.22) one has

(4.28)
$\lambda_{{\mathcal{S}_{{\mathrm{L}}}}}=\sum_{j\in\mathcal{J}_{L}}\sum_{s\in\mathcal{L}^{1}_{j}}F^{j}_{ss},\;\lambda_{\mathcal{S}_{\mathrm{M}}}=\sum_{j\in\mathcal{J}_{\mathrm{M}}}\sum_{s\in\mathcal{L}^{2}_{j}}F^{j}_{ss},\;\lambda_{\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}}}=\sum_{j\in\mathcal{J}_{\mathrm{L}}\cap\mathcal{J}_{\mathrm{M}}}\sum_{s\in\mathcal{L}^{1}_{j}\cap\mathcal{L}^{2}_{j}}F^{j}_{ss}.$

Also

$\lambda_{\mathcal{S}_{{\mathrm{P}}}}=\sum_{j\in\mathcal{J}_{\mathrm{P}}}\sum_{s\in\mathcal{L}^{3}_{j}\cap\mathcal{L}^{2}_{j}}F^{j}_{ss}.$

By Equation (3.29) and Equation (3.28) one has

$\tau(F^{j}_{ss})=\frac{1}{n_{j}}=\frac{\dim(\mathcal{C}^{(j)}_{s})}{\dim(\mathcal{C})}>0$

since $\mathcal{C}$ is pseudo-unitary. It follows that

$\tau(\lambda_{\mathcal{S}_{\mathrm{P}}})=\sum_{j\in\mathcal{J}_{\mathrm{P}}}\sum_{s\in\mathcal{L}^{3}_{j}}\tau(F^{j}_{ss})=\sum_{j\in\mathcal{J}_{\mathrm{P}}}\frac{|\mathcal{L}^{3}_{j}|}{n_{j}}.$

Similarly one deduces that

$\tau(\lambda_{\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}}})=\sum_{j\in\mathcal{J}_{\mathrm{L}}\cap\mathcal{J}_{\mathrm{M}}}\frac{|\mathcal{L}^{1}_{j}\cap\mathcal{L}^{2}_{j}|}{n_{j}}.$

Since
$\mathcal{J}_{\mathrm{P}}\subseteq\mathcal{J}_{\mathrm{L}}\cap\mathcal{J}_{\mathrm{M}}$
and for any $j\in\mathcal{J}_{\mathrm{P}}$ one has
$\mathcal{L}_{j}^{3}\subseteq\mathcal{L}_{j}^{1}\cap\mathcal{L}_{j}^{2}$ it
follows that

(4.29)
$\tau(\lambda_{\mathcal{S}_{\mathrm{P}}})\leq\tau(\lambda_{\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}}}).$

On the other hand, since
$\lambda_{\mathcal{S}_{\mathrm{P}}}=\frac{1}{\dim({\mathcal{S}_{\mathrm{P}}})}\big{(}\sum_{\chi_{i}\in\mathrm{Irr}(\mathcal{S}_{\mathrm{P}})}d_{i}\chi_{i}\big{)}$
we have that
$\tau(\lambda_{\mathcal{S}_{\mathrm{P}}})=\frac{1}{\dim({\mathcal{S}_{\mathrm{P}}})}$
since $\tau(\chi_{i})=0$ if $\chi_{i}\neq{\bf\epsilon}_{\mathbf{1}}$.
Similarly
$\tau(\lambda_{\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}}})=\frac{1}{\dim({\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}}})}$.
Then Equation (4.29) can be written as

$\frac{1}{\dim({\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}}})}\geq\frac{1}{\dim(\mathcal{S}_{\mathrm{P}})}.$

which proves the lemma. ∎

######  Proposition 4.30.

Let $\mathcal{C}$ be a pseudo unitary fusion category. With the above
notations one has

${\mathcal{S}_{{\mathrm{L}}}}\vee\mathcal{S}_{\mathrm{M}}=\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}}$

###### Proof.

Since
${\mathcal{S}_{{\mathrm{L}}}}\subseteq\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}}$
and
$\mathcal{S}_{\mathrm{M}}\subseteq\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}}$
it follows that
${\mathcal{S}_{{\mathrm{L}}}}\vee\mathcal{S}_{\mathrm{M}}\subseteq\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}}$.
The equality follows from the Lemma above. ∎

######  Proposition 4.31.

With the above notations one has

$\mathrm{FPdim}(LM)\leq\frac{\mathrm{FPdim}({\mathrm{L}})\mathrm{FPdim}({\mathrm{M}})}{\mathrm{FPdim}({\mathrm{L}}\cap{\mathrm{M}})}.$

Moreover one has equality if and only if
${\mathcal{S}_{{\mathrm{L}}}}\mathcal{S}_{\mathrm{M}}=\mathcal{S}_{\mathrm{M}}{\mathcal{S}_{{\mathrm{L}}}}$.

###### Proof.

From [12, Lemma 3.2] one has

$\mathrm{FPdim}({\mathcal{S}_{{\mathrm{L}}}}\mathcal{S}_{\mathrm{M}})=\frac{\mathrm{FPdim}({\mathcal{S}_{{\mathrm{L}}}})\mathrm{FPdim}(\mathcal{S}_{\mathrm{M}})}{\mathrm{FPdim}({\mathcal{S}_{{\mathrm{L}}}}\cap\mathcal{S}_{\mathrm{M}})}$

which gives that

$\mathrm{FPdim}({\mathcal{S}_{{\mathrm{L}}}}\mathcal{S}_{\mathrm{M}})=\frac{\mathrm{FPdim}(\mathcal{C})\mathrm{FPdim}({\mathrm{L}}{\mathrm{M}})}{\mathrm{FPdim}({\mathrm{L}})\mathrm{FPdim}({\mathrm{M}})}$

Since
${\mathcal{S}_{{\mathrm{L}}}}\mathcal{S}_{\mathrm{M}}\subseteq{\mathcal{S}_{{\mathrm{L}}}}\vee\mathcal{S}_{\mathrm{M}}$,
one has
$\mathrm{FPdim}({\mathcal{S}_{{\mathrm{L}}}}\mathcal{S}_{\mathrm{M}})\leq\mathrm{FPdim}({\mathcal{S}_{{\mathrm{L}}}}\vee\mathcal{S}_{\mathrm{M}})$.
Note that

$\mathrm{FPdim}({\mathcal{S}_{{\mathrm{L}}}}\vee\mathcal{S}_{\mathrm{M}})=\mathrm{FPdim}(\mathcal{S}_{{\mathrm{L}}\cap{\mathrm{M}}})=\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}({\mathrm{L}}\cap{\mathrm{M}})}.$

Thus

$\frac{\mathrm{FPdim}(\mathcal{C})}{\mathrm{FPdim}({\mathrm{L}}\cap{\mathrm{M}})}\geq\frac{\mathrm{FPdim}(\mathcal{C})\mathrm{FPdim}({\mathrm{L}}{\mathrm{M}})}{\mathrm{FPdim}({\mathrm{L}})\mathrm{FPdim}({\mathrm{M}})}$

which gives the above inequality. One has equality above if and only if
${\mathcal{S}_{{\mathrm{L}}}}\mathcal{S}_{\mathrm{M}}={\mathcal{S}_{{\mathrm{L}}}}\vee\mathcal{S}_{\mathrm{M}}$
which is equivalent to
${\mathcal{S}_{{\mathrm{L}}}}\mathcal{S}_{\mathrm{M}}=\mathcal{S}_{\mathrm{M}}{\mathcal{S}_{{\mathrm{L}}}}$.
∎

######  Corollary 4.32.

If $\mathcal{C}$ is a pseudo-unitary fusion category with a commutative
grothendieck ring then: One has that

$\mathrm{FPdim}({\mathrm{L}}{\mathrm{M}})=\frac{\mathrm{FPdim}({\mathrm{L}})\mathrm{FPdim}({\mathrm{M}})}{\mathrm{FPdim}({\mathrm{L}}\cap{\mathrm{M}})}$

for any tow uniray subalgebras ${\mathrm{L}}$ and ${\mathrm{M}}$ of
${\mathrm{A}}$.

###### Proof.

It is like Maschke’s theorem: ∎

###  4.6. Examples from semisimple Hopf algebras

Let $H$ be a semisimple Hopf algebra and let $\mathcal{C}=\mathrm{Rep}(H)$ be
the fusion category of its finite dimensional representations over $\Bbbk$. As
in [18, Section 3.7] we identify the left monoidal center of $\mathcal{C}$
with the category ${\mathcal{Z}(\mathcal{C})}\simeq^{H}_{H}\mathcal{YD}$ of
left-left Yetter-Drinfeld modules of $H$. The objects of the former category
are pairs $(V,\delta)$ where $V$ is a left $H$-modules and
$\delta:V\rightarrow H\otimes V$ defines a left $H$-comodules structure on $V$
such that

$(hv){{}_{-1}}\otimes(hv){{}_{0}}=h_{1}v{{}_{-1}}S(h_{3})\otimes
h_{2}v{{}_{0}}$

for any $h\in H$ and any $v\in V$. A morphism in the category
${}^{H}_{H}\mathcal{YD}$ is a linear map which is simultaneously a morphism of
left $H$-modules and of left $H$-comodules. The left half-braiding of
$V\in\;^{H}_{H}\mathcal{YD}$ can be written as

$c_{V,M}:V\otimes M\rightarrow M\otimes V,\;v\otimes m\mapsto v_{-1}m\otimes
v_{0}$

for any object $M\in\mathrm{Rep}(H)$. The forgetful functor

$F:^{H}_{H}\mathcal{YD}\rightarrow H-\text{mod}$

admits a right adjoint

$R:H-\text{mod}\rightarrow^{H}_{H}\mathcal{YD}$

which can be written as $R(M)=H\otimes M$ with

$h.(l\otimes m)=h_{1}lS(h_{3})\otimes h_{2}m,\;\delta(h\otimes
m)=h_{1}\otimes(h_{2}\otimes m).$

Thus, with the above notations, the Hopf comonad $Z:=FR$ can be written as
$Z(M)=F(R(M))=H\otimes M$ with the left $H$-module structure defined above.
Then the left half-braiding of $R(M)$ becomes:

$c_{{Z(M)},X}:{Z(M)}\otimes X\rightarrow X\otimes{Z(M)},\;(h\otimes m)\otimes
x\mapsto h_{1}x\otimes(h_{2}\otimes m)$

for any $X\in\mathrm{Rep}(H)$. In particular one has that
${\mathrm{A}}:=R(\mathbf{1})$ is the pair $(H_{\operatorname{ad}},\Delta)$
where $H_{\operatorname{ad}}$ coincides to $H$ as vector spaces, and the
structure as left $H$-module on $H_{\operatorname{ad}}$ is given by the left
adjoint action $h.a=h_{1}aS(h_{2})$. Moreover the left comodule structure on
$H_{\operatorname{ad}}$ is given by the comultiplication
$\Delta:H_{\operatorname{ad}}\rightarrow H\otimes H_{\operatorname{ad}}$. The
multiplication and the unit of $R(\mathbf{1})$ are given the usual
multiplication and unit of $H$. The universal dinatural projections from
Subsection 2.1 are given by:

$\pi_{\mathbf{1};\;M}:H_{\operatorname{ad}}\rightarrow M\otimes
M^{*},\;\;h\mapsto\sum_{i}hm_{i}\otimes m_{i}^{*},$

where $m_{i}$ is a $\Bbbk$-linear basis on $M$ and $\\{m_{i}^{*}\\}$ is the
corresponding dual basis in $M^{*}$. In this case the natural action of
$A=H_{\operatorname{ad}}$ from Equation (2.7) can be written as
$\alpha_{M}:A\otimes M\rightarrow M,\alpha_{M}(a\otimes m)=a.m$, for each
object $M\in\mathrm{Rep}(H)$. Moreover, by considering the standard pivotal
structure on $\mathrm{Rep}(H)$, it follows that
$\mathrm{{CF}}(\mathcal{C})=C(H)$, the character ring of $H$, and
$\mathrm{{CE}(\mathcal{C})}=Z(H)$, the center of $H$.

In this case, unitary subalgebras of
${\mathrm{A}}\in{\mathcal{Z}(\mathcal{C})}$ are in bijection with the left
normal coideal subalgebras of $H$. Recall that ${\mathrm{L}}\subseteq H$ is a
left normal coideal subalgebra of $H$ if it is a subalgebra of $H$ with
$\Delta({\mathrm{L}})\subseteq H\otimes{\mathrm{L}}$ and ${\mathrm{L}}$ is
closed under the left adjoint action.

Moreover, for a semisimple Hopf algebra $H$ it follows that
$\beta({\mathrm{L}})={\mathcal{S}_{{\mathrm{L}}}}={\mathcal{C}_{\mathrm{L}}^{\mathrm{triv}}}=\mathrm{Rep}(H//{\mathrm{L}})$
is the fusion subcategory of $\mathrm{Rep}(H)$ consisting of those $H$-modules
receiving trivial action from ${\mathrm{L}}$. The above equalities in this
case were previously shown in [4].

The character space ${\mathrm{CF}}(L):=\mathrm{{Hom}}_{H}(L,\mathbf{1})$
defined in Section 3 it can be described by
$\mu\in{\mathrm{CF}}(L)\iff\mu(h_{1}lS(h_{2}))=\epsilon(h)\mu(l)$, for all
$h\in H$ and $l\in L$. Note that there is an inclusion of

${\mathrm{CF}}(L):=\mathrm{{Hom}}_{H}(L,\mathbf{1})\xhookrightarrow{\nu}\mathrm{{Hom}}_{L}(L,\mathbf{1})=C(L),$

into the character space (trace space) $C(L)$ of left $L$-modules.

Denote by ${\mathrm{Res}}^{A}_{L}:H-\mathrm{mod}\rightarrow L-\mathrm{mod}$
the restriction functor of $H$-modules to the subalgebra $L$. This functor
induces a map
${({\mathrm{Res}}^{A}_{L})}_{*}:\mathrm{{CF}}(\mathcal{C})\rightarrow C(L)$ at
the level of characters. It is easy to see that the image of this map is
${\mathrm{CF}}(L)\xhookrightarrow{\nu}C(L)$.

Note also that the central space ${\mathrm{CE}}(L)$ becomes in this case
${\mathrm{CE}}(L)=\mathrm{{Hom}}_{H}(\mathbf{1},L)=L\cap Z(H)$ and the
inclusion $\iota^{e}:{\mathrm{CE}}(L)\rightarrow\mathrm{{CE}(\mathcal{C})}$ is
the usual space inclusion $L\cap Z(H)\hookrightarrow Z(H)$. Note also that in
this case $\ell_{\mathcal{D}}$ is a multiple of the integral $\Lambda_{L}$
defined in [3].

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