Slightly trivial extensions of a fusion category

TL;DR

This paper introduces the concept of slightly trivial extensions of fusion categories, exploring their properties and providing examples from rank 3 fusion categories, thus advancing understanding of fusion category extensions.

Contribution

It defines and studies slightly trivial extensions of fusion categories, offering initial insights and concrete examples from rank 3 categories.

Findings

Defined slightly trivial extensions of fusion categories.

Provided two explicit examples from rank 3 categories.

Laid groundwork for further exploration of fusion category extensions.

Abstract

We introduce and study the notion of slightly trivial extensions of a fusion category which can be viewed as the first level of complexity of extensions. We also provide two examples of slightly trivial extensions which arise from rank $3$ fusion categories.

Equations

$$\begin{array}{r}{X}_i\otimes X_j=\underset{t=0}{\overset{n}{⨁}}{N}_{ij}^t{X}_t.\end{array}$$

$$\begin{array}{rl}({\delta }_g\otimes X_i)\otimes ({\delta }_h\otimes X_j)& =({\delta }_g\otimes {\delta }_h)\otimes (X_i\otimes X_j)\\ & ={\delta }_{gh}\otimes (X_i\otimes X_j)\\ & ={\delta }_{gh}\otimes \underset{t=0}{\overset{n}{⨁}}{N}_{ij}^t{X}_t\\ & =\underset{t=0}{\overset{n}{⨁}}{N}_{ij}^t{\delta }_{gh}\otimes X_t.\end{array}$$

$$\mathrm{FPdim}({\oplus }_{i=2}^n{N}^i{X}_i^g)=\mathrm{FPdim}(X\otimes X_1^g)-\mathrm{FPdim}(X_1^g)<\mathrm{FPdim}(X_1^g).$$

$$\mathrm{FPdim}(X\otimes X_1^g)-\mathrm{FPdim}(X_k^g)=\mathrm{FPdim}({\oplus }_{i>k}^n{N}^i{X}_i^g)<\mathrm{FPdim}(X_k^g).$$

$$\begin{array}{r}\alpha \otimes \alpha =\text{1}\oplus \beta ,\beta \otimes \beta =\text{1}\oplus \alpha \oplus \beta ,\alpha \otimes \beta =\beta \otimes \alpha =\alpha \oplus \beta .\end{array}$$

$$\delta \otimes \delta \cong \text{1},\delta \otimes X\cong X\otimes \delta \cong X,X\otimes X\cong \text{1}\oplus \delta .$$

Full text

1. Introduction

Let $G$ be a finite group and $e$ be the identity of $G$. A fusion category ${\mathcal{C}}$ is $G$-graded if there is a decomposition ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ of ${\mathcal{C}}$ into a direct sum of full abelian subcategories, such that $({\mathcal{C}}_{g})^{*}={\mathcal{C}}_{g^{-1}}$ and ${\mathcal{C}}_{g}\otimes{\mathcal{C}}_{h}\subseteq{\mathcal{C}}_{gh}$ for all $g,h\in G$.

The grading ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ is called faithful if ${\mathcal{C}}_{g}\neq 0$ for all $g\in G$. A fusion category ${\mathcal{C}}$ is called a $G$-extension of ${\mathcal{D}}$ if there is a faithful grading ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$, such that the trivial component ${\mathcal{C}}_{e}$ is equivalent to ${\mathcal{D}}$. If $G$ is trivial then ${\mathcal{C}}$ is equivalent to ${\mathcal{D}}$. In this case, we call ${\mathcal{C}}$ is a trivial extension of ${\mathcal{D}}$.

In this paper, we introduce the new notion of slightly trivial extensions of a fusion category, which can be viewed as the first level of complexity of extensions, except the trivial extension. Let ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ be a $G$-extension of a fusion category ${\mathcal{D}}$. We obtain a necessary and sufficient condition for ${\mathcal{C}}$ being a slightly trivial extension of ${\mathcal{D}}$. In particular, if the largest pointed fusion subcategory ${\mathcal{D}}_{pt}$ is trivial and ${\mathcal{C}}$ is a slightly trivial extension of ${\mathcal{D}}$, then ${\mathcal{C}}={\mathcal{C}}_{pt}\bullet{\mathcal{D}}$ admits an exact factorization. When ${\mathcal{C}}$ is a slightly trivial extension of ${\mathcal{D}}$ and the Grothendieck ring of ${\mathcal{C}}$ is commutative, we also obtain the fusion rules of ${\mathcal{C}}$ in terms of that of ${\mathcal{D}}$. In the last section of this paper, we provide two examples of slightly trivial extensions which arise from rank $3$ fusion categories.

2. Preliminaries

2.1. Frobenius-Perron dimensions

Let ${\mathcal{C}}$ be a fusion category. We shall denote by $\operatorname{Irr}({\mathcal{C}})$ the set of isomorphism classes of simple objects in ${\mathcal{C}}$. The set $\operatorname{Irr}({\mathcal{C}})$ is a basis of the Grothendieck ring of ${\mathcal{C}}$. The cardinal number of $\operatorname{Irr}({\mathcal{C}})$ is called the rank of ${\mathcal{C}}$ and is denoted by $\operatorname{rk}({\mathcal{C}})$.

Let ${\mathcal{C}}$ be a fusion category and let $X$ be a simple object of ${\mathcal{C}}$. The Frobenius-Perron dimension $\operatorname{FPdim}(X)$ of $X$ is defined as the Frobenius-Perron eigenvalue of the matrix of left multiplication by the class of $X$ in the basis $\operatorname{Irr}({\mathcal{C}})$. The Frobenius-Perron dimension of ${\mathcal{C}}$ is $\operatorname{FPdim}({\mathcal{C}})=\oplus_{X\in\operatorname{Irr}({\mathcal{C}})}(\operatorname{FPdim}X)^{2}$. The following lemma is due to Müger, see [2, Remark 8.4].

Lemma 2.1**.**

Let $X$ be a simple object of a fusion category. Then $\operatorname{FPdim}(X)\geq 1$. In particular, if $\operatorname{FPdim}(X)<2$ then $\operatorname{FPdim}(X)=2\cos(\frac{\pi}{n})$ for some integer $n\geq 3$.

A simple object $X$ of a fusion category is invertible if $X\otimes X^{*}=\textbf{1}$, where $X^{*}$ is the dual of $X$ and 1 is the trivial simple object of ${\mathcal{C}}$. It is obvious that a simple object $X$ is invertible if and only if $\operatorname{FPdim}(X)=1$. A fusion category is pointed if all of its simple objects are invertible. Let ${\mathcal{C}}$ be a fusion category. We shall denote by ${\mathcal{C}}_{pt}$ the largest pointed fusion subcategory of ${\mathcal{C}}$, and by $G({\mathcal{C}})$ the group of isomorphism classes of invertible objects of ${\mathcal{C}}$. It is clear that $G({\mathcal{C}})$ generates ${\mathcal{C}}_{pt}$ as a fusion subcategory.

The class of pointed fusion categories has been classified (see e. g. [5]). Let $G$ be a finite group, $\omega$ be a cohomology class in $H^{3}(G,k^{*})$, and $\operatorname{Vec}_{G}^{\omega}$ be the category of finite dimensional vector spaces graded by $G$ with associativity determined by $\omega$. It is shown that a pointed fusion category is equivalent to some $\operatorname{Vec}_{G}^{\omega}$.

2.2. Exact factorizations of fusion categories

Let ${\mathcal{C}}$ be a fusion category, and let ${\mathcal{A}},{\mathcal{B}}$ be fusion subcategories of ${\mathcal{C}}$. Let ${\mathcal{A}}{\mathcal{B}}$ be the full abelian (not necessarily tensor) subcategory of ${\mathcal{C}}$ spanned by direct summands in $X\otimes Y$, where $X\in{\mathcal{A}}$ and $Y\in{\mathcal{B}}$. We say that ${\mathcal{C}}$ factorizes into a product of ${\mathcal{A}}$ and ${\mathcal{B}}$ if ${\mathcal{C}}={\mathcal{A}}{\mathcal{B}}$. A factorization ${\mathcal{C}}={\mathcal{A}}{\mathcal{B}}$ of ${\mathcal{C}}$ is called exact if $A\cap{\mathcal{B}}=\operatorname{Vec}$, and is denoted by ${\mathcal{C}}={\mathcal{A}}\bullet{\mathcal{B}}$, see [3].

By [3, Theorem 3.8], ${\mathcal{C}}={\mathcal{A}}\bullet{\mathcal{B}}$ is an exact factorization if and only every simple object of ${\mathcal{C}}$ can be uniquely expressed in the form $X\otimes Y$, where $X\in\operatorname{Irr}({\mathcal{A}})$ and $\operatorname{Irr}({\mathcal{B}})$.

3. Slightly trivial extensions

Let ${\mathcal{D}}$ be a fusion category with $\operatorname{Irr}({\mathcal{D}})=\{\textbf{1}=X_{0},X_{1},\cdots,X_{n}\}$. Let ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ be an extension of ${\mathcal{D}}$. We say that the component ${\mathcal{C}}_{g}$ is similar to ${\mathcal{D}}$ if there exists an invertible simple object $\delta_{g}\in{\mathcal{C}}_{g}$ such that $\operatorname{Irr}({\mathcal{C}}_{g})=\{\delta_{g}=\delta_{g}\otimes\textbf{1},\delta_{g}\otimes X_{1},\cdots,\delta_{g}\otimes X_{n}\}$. If all components are similar to ${\mathcal{D}}$ then we say that ${\mathcal{C}}$ is a slightly trivial extension of ${\mathcal{D}}$.

Proposition 3.1**.**

Let ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ be an extension of a fusion category ${\mathcal{D}}$. Then the component ${\mathcal{C}}_{g}$ is similar to ${\mathcal{D}}$ if and only if ${\mathcal{C}}_{g}$ contains an invertible simple object. In particular, ${\mathcal{C}}$ is a slightly trivial extension of ${\mathcal{D}}$ if and only if every component contains an invertible simple object.

Proof.

Assume that ${\mathcal{C}}_{g}$ contains an invertible simple object $\delta_{g}$. Let $\operatorname{Irr}({\mathcal{D}})=\{\textbf{1},X_{1},\cdots,X_{n}\}$. Then $\delta_{g},\delta_{g}\otimes X_{1},\cdots,\delta_{g}\otimes X_{n}$ are nonisomorphic simple objects in ${\mathcal{C}}_{g}$. Since $\operatorname{FPdim}(\delta_{g})=\operatorname{FPdim}(\textbf{1})$, $\operatorname{FPdim}(\delta_{g}\otimes X_{i})=\operatorname{FPdim}(X_{i})$ and $\operatorname{FPdim}({\mathcal{D}})=\operatorname{FPdim}({\mathcal{C}}_{g})$, we conclude that $\operatorname{Irr}({\mathcal{C}}_{g})=\{\delta_{g},\delta_{g}\otimes X_{1},\cdots,\delta_{g}\otimes X_{n}\}$. Hence ${\mathcal{C}}_{g}$ is similar to ${\mathcal{D}}$.

The other direction is obvious, by the definition of a slightly trivial extension. ∎

Let $1=d_{0}<d_{1}<\cdots<d_{s}$ be positive real numbers, and let $n_{0},n_{1},\cdots,n_{s}$ be positive integers. A fusion category is said of type $(d_{0},n_{0};d_{1},n_{1};\cdots;d_{s},n_{s})$ if $n_{i}$ is the number of the non-isomorphic simple objects of Frobenius-Perron dimension $d_{i}$, for all $0\leq i\leq s$.

Remark 3.2*.*

Let ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ be a slightly trivial extension of a fusion category ${\mathcal{D}}$.

(1) Every component ${\mathcal{C}}_{g}$ contains an invertible simple object $\delta_{g}$. Set $\operatorname{Irr}({\mathcal{D}})=\{\textbf{1},X_{1},\cdots,X_{n}\}$ and $\Delta_{G}=\{\delta_{g}\in{\mathcal{C}}_{g}|g\in G\}$, where $\delta_{e}=\textbf{1}$. The proof of Proposition 3.1 shows that $\operatorname{Irr}({\mathcal{C}})=\{\delta_{g},\delta_{g}\otimes X_{1},\cdots,\delta_{g}\otimes X_{n}|\delta_{g}\in\Delta_{G}\}$.

(2) The proof of Proposition 3.1 also shows that ${\mathcal{C}}_{g}$ and ${\mathcal{C}}_{h}$ have the same type, for all $g,h\in G$.

Corollary 3.3**.**

Let ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ be a slightly trivial extension of a fusion category ${\mathcal{D}}$. Assume that ${\mathcal{D}}_{pt}$ is trivial. Then ${\mathcal{C}}={\mathcal{C}}_{pt}\bullet{\mathcal{D}}$ is an exact factorization of ${\mathcal{C}}_{pt}$ and ${\mathcal{D}}$.

Proof.

Since ${\mathcal{D}}_{pt}$ is trivial, the proof of Theorem 3.1 shows that every component ${\mathcal{C}}_{g}$ exactly contains only one invertible simple object. Let $\delta_{g}$ be the invertible simple object in ${\mathcal{C}}_{g}$. Then $\Delta_{G}=\{\delta_{g}|g\in G\}=G({\mathcal{C}})$. Hence every simple object of ${\mathcal{C}}$ can be expressed in the form $X\otimes Y$, where $X\in{\mathcal{C}}_{pt}$ and $Y\in{\mathcal{D}}$ are simple objects, by Remark 3.2. The result then follows from [3, Theorem 3.8]. ∎

Let ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ be an extension of a fusion category ${\mathcal{D}}$. If $\delta_{g}\in{\mathcal{C}}_{g},\delta_{h}\in{\mathcal{C}}_{h}$ are invertible simple objects then $\delta_{g}\otimes\delta_{h}\in{\mathcal{C}}_{gh}$ is also an invertible simple object. We shall write $\delta_{gh}=\delta_{g}\otimes\delta_{h}$ in the following corollary.

Corollary 3.4**.**

Let ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ be a slightly trivial extension of a fusion category ${\mathcal{D}}$. Suppose that the Grothendieck ring of ${\mathcal{C}}$ is commutative. Then the fusion rules of ${\mathcal{C}}$ are determined by that of ${\mathcal{D}}$.

Proof.

Let $\operatorname{Irr}({\mathcal{D}})=\{\textbf{1}=X_{0},X_{1},\cdots,X_{n}\}$. We assume the fusion rules of ${\mathcal{D}}$ are determined by the following equation:

[TABLE]

Let $\delta_{g}\in{\mathcal{C}}_{g},\delta_{h}\in{\mathcal{C}}_{h}$ are invertible simple objects. Then $\delta_{g},\delta_{g}\otimes X_{1},\cdots,\delta_{g}\otimes X_{n}$ and $\delta_{h},\delta_{h}\otimes X_{1},\cdots,\delta_{h}\otimes X_{n}$ are all nonisomorphic simple objects in ${\mathcal{C}}_{g}$ and ${\mathcal{C}}_{h}$, respectively. Then

[TABLE]

∎

4. Examples of slightly trivial extensions

In this section, we shall need the following lemma.

Lemma 4.1**.**

[2, Proposition 8.20]** Let ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ be a $G$-extension of a fusion category ${\mathcal{D}}$. Then $\operatorname{FPdim}({\mathcal{C}}_{g})=\operatorname{FPdim}({\mathcal{C}}_{h})$ for all $g,h\in G$ and $\operatorname{FPdim}({\mathcal{C}})=|G|\operatorname{FPdim}({\mathcal{D}})$ .

For a fusion category ${\mathcal{C}}$, we shall denote by $\mathrm{cd}({\mathcal{C}})$ the set of Frobenius-Perron dimensions of simple objects of ${\mathcal{C}}$.

Lemma 4.2**.**

Let ${\mathcal{D}}$ be a fusion category and ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ be an extension of ${\mathcal{D}}$. Assume that ${\mathcal{D}}$ contains a simple object $X$ such that $1<\operatorname{FPdim}(X)<2$. Then, for every $g\in G$, component ${\mathcal{C}}_{g}$ has one of the following properties.

(1)  ${\mathcal{C}}_{g}$ contains an invertible simple object.

(2)  There exists $Y\in\operatorname{Irr}({\mathcal{C}}_{g})$ such that $X\otimes Y$ is a simple object, where $\operatorname{FPdim}(Y)$ is the smallest one in $cd({\mathcal{C}})$.

Proof.

Assume that part (1) does not hold true and prove (2). In this case, $\operatorname{FPdim}(Y)\geq\sqrt{2}$ for every simple object $Y$ in ${\mathcal{C}}_{g}$, by Lemma 2.1. Assume that $\operatorname{Irr}({\mathcal{C}}_{g})=\{X_{1}^{g},\cdots,X_{n}^{g}\}$. We may reorder them such that $\operatorname{FPdim}(X_{1}^{g})\leq\cdots\leq\operatorname{FPdim}(X_{n}^{g})$. Since $X\otimes X_{1}^{g}\in{\mathcal{C}}_{g}$, we may set $X\otimes X_{1}^{g}=\oplus_{i=1}^{n}N^{i}X_{i}^{g}$. Since $\operatorname{FPdim}(X_{1}^{g})<\operatorname{FPdim}(X\otimes X_{1}^{g})<2\operatorname{FPdim}(X_{1}^{g})$, we have $N^{1}=1$ or [math]. If $N^{1}=1$ then

[TABLE]

This is impossible since $\operatorname{FPdim}(X_{i}^{g})\geq\operatorname{FPdim}(X_{1}^{g})$ for all $i\geq 2$. Therefore, we have $N^{1}=0$ and $X\otimes X_{1}^{g}=\oplus_{i=2}^{n}N^{i}X_{i}^{g}$.

Since $\operatorname{FPdim}(X_{i}^{g})\leq\operatorname{FPdim}(X\otimes X_{1}^{g})<2\operatorname{FPdim}(X_{i}^{g})$, for all $i\geq 2$, we have $N^{i}=1$ or [math]. Let $k$ be the smallest number such that $N^{k}=1$. Then

[TABLE]

By the minimality of $k$, we get $N^{i}=0$ for all $i>k$. Hence we have $X\otimes X_{1}^{g}=X_{k}^{g}$ is simple. ∎

Let ${\mathcal{D}}$ be a rank $3$ fusion category with $\operatorname{Irr}({\mathcal{D}})=\{\textbf{1},\alpha,\beta\}$. The fusion rules of ${\mathcal{D}}$ are determined by

[TABLE]

It is not hard to check that $\operatorname{FPdim}(\alpha)=2\cos(\frac{\pi}{7})$ and $\operatorname{FPdim}(\beta)=4\cos(\frac{\pi}{7})^{2}-1$. It is proved in [6] that ${\mathcal{D}}$ is a modular category.

Theorem 4.3**.**

Let ${\mathcal{D}}$ be a fusion category with the fusion rules (4.1) and let ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ be an extension of ${\mathcal{D}}$. Then

(1)  ${\mathcal{C}}$ is a slightly trivial extension of ${\mathcal{D}}$.

(2)  ${\mathcal{C}}={\mathcal{C}}_{pt}\bullet{\mathcal{D}}$ is an exact factorization of ${\mathcal{C}}_{pt}$ and ${\mathcal{D}}$.

Proof.

(1)  By Theorem 3.1, it suffices to prove that every component contains one invertible simple object.

Suppose on the contrary that there exists a component ${\mathcal{C}}_{g}$ such that every simple object in ${\mathcal{C}}_{g}$ is not invertible. Let $X\in{\mathcal{C}}_{g}$ be a simple object such that $\operatorname{FPdim}(X)$ is smallest in $cd({\mathcal{C}}_{g})$. Then Lemma 4.2 shows that $\alpha\otimes X$ is a simple object in ${\mathcal{C}}_{g}$. That is, ${\mathcal{C}}_{g}$ contains at least $2$ simple objects. By Lemma 2.1, $\operatorname{FPdim}(X)\geq\sqrt{2}$ and hence $\operatorname{FPdim}(\alpha\otimes X)\geq 2\sqrt{2}\cos(\frac{\pi}{7})$.

If $\operatorname{rk}({\mathcal{C}}_{g})\geq 3$ then $\operatorname{FPdim}({\mathcal{C}}_{g})\geq 4+8\cos(\frac{\pi}{7})^{2}>\operatorname{FPdim}({\mathcal{D}})$. This is impossible by Lemma 4.1. Hence ${\mathcal{C}}_{g}$ only contains $2$ simple objects.

Let $\operatorname{Irr}({\mathcal{C}}_{g})=\{X,Y\}$ with $\operatorname{FPdim}(X)\leq\operatorname{FPdim}(Y)$. By Lemma 4.2, $Y=\alpha\otimes X$. By [4, Lemma 4.10], the decompositions of $\alpha\otimes\alpha$ and $X\otimes X^{*}$ can not have common summands. This implies that $X\otimes X^{*}=\textbf{1}\oplus\alpha$ and hence $\operatorname{FPdim}(X)=\sqrt{1+2\cos(\frac{\pi}{7})}$. So we have $\operatorname{FPdim}({\mathcal{C}}_{g})=\operatorname{FPdim}(X)^{2}+\operatorname{FPdim}(Y)^{2}=1+2\cos(\frac{\pi}{7})+4\cos(\frac{\pi}{7})^{2}+8\cos(\frac{\pi}{7})^{3}$, which is not equal to $\operatorname{FPdim}({\mathcal{D}})$. It is also impossible by Lemma 4.1. This proves part (1).

(2)  Part (2) follows from Part (1) and Corollary 3.3, since ${\mathcal{D}}_{pt}$ is trivial. ∎

An Ising category $\mathcal{I}$ is a fusion category which is not pointed and has Frobenius-Perron dimension $4$. Let $\mathcal{I}$ be an Ising category then $\operatorname{Irr}(\mathcal{I})$ consists of three simple objects: 1, $\delta$ and $X$, where 1 is the trivial simple object, $\delta$ is an invertible object and $X$ is a non-invertible object. They obey the following fusion rules:

[TABLE]

It is easy to check that $\operatorname{FPdim}(\delta)=1$, $\operatorname{FPdim}(X)=\sqrt{2}$ and $X$ is self-dual. It is known that any Ising category $\mathcal{I}$ is a modular category. See [1, Appendix B] for more details on Ising categories.

Theorem 4.4**.**

Let ${\mathcal{C}}=\oplus_{g\in G}{\mathcal{C}}_{g}$ be an extension of an Ising category $\mathcal{I}$. Then ${\mathcal{C}}$ is a slightly trivial extension of $\mathcal{I}$.

Proof.

As before, we set $\operatorname{Irr}(\mathcal{I})=\{\textbf{1},\delta,X\}$. By Proposition 3.1, it suffices to prove that every component ${\mathcal{C}}_{g}$ contains at least one invertible object.

Suppose on the contrary that every simple object in ${\mathcal{C}}_{g}$ is non-invertible. Let $X_{g}$ be a simple object in ${\mathcal{C}}_{g}$ such that $\operatorname{FPdim}(X_{g})$ is smallest in $cd({\mathcal{C}}_{g})$. Then $X\otimes X_{g}$ is simple by Lemma 4.2. By Lemma 2.1, $\operatorname{FPdim}(X_{g})\geq\sqrt{2}$. Hence $\operatorname{FPdim}({\mathcal{C}}_{g})\geq\operatorname{FPdim}(X_{g})^{2}+\operatorname{FPdim}(X\otimes X_{g})^{2}\geq 6$. Since $\operatorname{FPdim}({\mathcal{C}}_{g})=\operatorname{FPdim}(\mathcal{I})$, we conclude that $X_{g},X\otimes X_{g}$ are exactly the only two simple objects in ${\mathcal{C}}_{g}$ and $\operatorname{FPdim}(X_{g})=\sqrt{2}$. Since $X_{g}\otimes X_{g}^{*}\in\mathcal{I}$, the only possible decomposition of $X_{g}\otimes X_{g}^{*}$ is $\textbf{1}\oplus\delta$, which coincides with $X\otimes X^{*}$. This is impossible by [4, Lemma 4.10]. This proves that every component ${\mathcal{C}}_{g}$ contains at least one invertible object.

∎

Acknowledgements

The author is partially supported by the startup foundation for introducing talent of NUIST (Grant No. 2018R039) and the Natural Science Foundation of China (Grant No. 11201231)