A Note on Mutation Equivalence

Abstract

We focus on the necessary conditions for two totally sign-skew-symmetric matrices B and $B^{\prime }$ to be mutation equivalent, obtaining two specific conditions: the equality of their column greatest common divisor vectors and the equality of $\left|B\right|$ and $|B^{\prime }|$, up to a relabeling of indices, when both matrices are acyclic. As a byproduct, the former condition confirms a conjecture on cluster automorphisms for totally sign-skew-symmetric cluster algebras.

1. Introduction

Cluster algebras, introduced by Fomin and Zelevinsky in the fundamental paper [1] for providing a combinatorial framework to study total positivity and canonical bases associated by Lusztig to semisimple algebraic groups, are thought to be a spectacular advance in mathematics. Many relations between cluster algebras and other branches of mathematics have been discovered, such as periodicities of T-systems and Y-systems, Grassmannians, representation theory of finite dimensional algebras and 2-Calabi–Yau triangulated categories, combinatorics, Poisson geometry, higher Teichmüler spaces, and scattering diagrams.

Roughly speaking, cluster algebras are commutative $\mathbb{Z}$-algebras generated by certain combinatorially defined generators called cluster variables, which are grouped into overlapping clusters. In cluster theory, exchange matrices and their mutations play important roles. Exchange matrices are chosen to be totally sign-skew-symmetric matrices in [1] that include all skew-symmetrizable matrices as examples.

Cluster algebras are determined combinatorially by mutation equivalent classes of the initial exchange matrices. On the one hand, studying mutation equivalent classes can help to determine whether two cluster algebras are the same and further classify cluster algebras. On the other hand, studying mutation equivalent classes can also help to determine whether an exchange matrix is mutation-acyclic or not, since mutation-acyclic cluster algebras have good properties, such as they are finitely generated and they admit unfoldings.

Thus, one key topic is to characterize mutation equivalence, which is rather complex even only for exchange matrices, although the definition of one-step mutation seems quite simple. Up to now, only several particular cases have been figured out.

Related work includes a criterion algorithm classifying mutation equivalent classes of $3\times 3$ skew-symmetrizable matrices [2,3], classification of skew-symmetrizable matrices of finite mutation type [4,5], explicit descriptions of mutation equivalent classes of quivers of type $A,D$ and $\tilde{A}$ [6,7,8], and derived equivalence classification for cluster-tilted algebras of quivers of type $A,D,E$ and $\tilde{A}$ [6,7,9,10]. For more general cases, in [11], Caldero and Keller confirmed a conjecture proposed by Fomin and Zelevinsky in [12] which claimed all mutation equivalent acyclic quivers have the same underlying graph up to an isomorphism.

In this paper, we extend such discussion to totally sign-skew-symmetric matrices to obtain partial results for characterizing their mutation equivalence.

Conclusions and Future Works

The paper is divided into two parts.

In the first part, for each exchange matrix $B_t$, define ${\mathbf{a}}_t=(a_{1;t},\dots ,a_{n;t})$ to be the column greatest common divisor vector of $B_t$, where $a_{k;t}$ equals the positive greatest common divisor of $b_{1k}^t,\dots ,b_{nk}^t$ for any $k\in [1,n]$. Inspired by the work of Seven ([3]), we show the column greatest common divisor vector of any totally sign-skew-symmetric matrix is mutation invariant.

Theorem 1.

Let $\mathcal{A}$ be a totally sign-skew-symmetric cluster algebra with exchange matrices ${\left\{B_t\right\}}_{t\in {\mathbb{T}}_n}$. Then ${\mathbf{a}}_t={\mathbf{a}}_{t^{\prime }}$ for any $t,t^{\prime }\in {\mathbb{T}}_n$.

As an application, this result confirms a conjecture proposed by Chang and Schiffler regarding cluster automorphisms in [13], as introduced by Assem, Schiffler, and Shamchenko in [14]. This conjecture was previously verified in [15] for skew-symmetrizable cluster algebras. More precisely, we have the following result.

Theorem 2.

Let $\mathcal{A}=\mathcal{A}(\mathbf{x},B)$ be a totally sign-skew-symmetric cluster algebra, and $f:\mathcal{A}\to \mathcal{A}$ a $\mathbb{Z}$-algebra homomorphism of $\mathcal{A}$. Then f is a cluster automorphism if and only if f maps a cluster to a cluster, i.e. , there are two clusters $\mathbf{x}$ and $\mathbf{z}$ of $\mathcal{A}$ such that $f\left(\mathbf{x}\right)=\mathbf{z}$.

In the second part, we generalize the characterization of acyclic mutation equivalent skew-symmetric matrices mentioned above to acyclic sign-skew-symmetric matrices via folding. With the unfolding technique developed by Huang and Li (cf. [16]), we study acyclic sign-skew-symmetric matrices using (possibly infinite) strongly locally finite quivers and then check that categorification of a cluster structure associated with some acyclic initial quiver (please refer to [11,17,18] for more details) can be naturally generalized to that of a locally reachable cluster structure associated to some acyclic initial quiver, which thus induces an isomorphism between two mutation equivalent quivers as in the proof in [11]. Before presenting our final main result, we introduce a notation. For an integer matrix $B=\left(b_{ij}\right)$, denote $\left|B\right|:=\left(\right|b_{ij}\left|\right)$ to be the matrix obtained from B by replacing every entry $b_{ij}$ with $|b_{ij}|$.

Theorem 3.

Let B and $B^{\prime }$ be two mutation equivalent acyclic sign-skew symmetric matrices. Then, up to a simultaneous permutation of rows and columns, $\left|B\right|=|B^{\prime }|$ and any seed containing $B^{\prime }$ can be obtained from a seed containing B by a finite mutation sequence at sources and sinks. In particular, the set of acyclic seeds forms a connected subgraph (possibly empty) of the exchange graph.

Theorems 1 and 3 only give necessary conditions to determine whether two exchange matrices are mutation equivalent. It is still an open problem to characterize mutation equivalent classes for all exchange matrices. Our future works will focus on characterizing mutation equivalent classes for some particular exchange matrices.

2. Preliminaries

2.1. Cluster Algebras

Let n be a positive integer. We often denote $[1,n]$ the set $\{1,\dots ,n\}$ and we assume that all matrices in the paper are indecomposable integer matrices.

For any $n\times n$ matrix $B={\left(b_{ij}\right)}_{n\times n}$, we associate a directed (simple) graph ${\mathsf{\Gamma }}_B$ whose vertices are given by $1,2,\dots ,n$ and there is a directed edge $i\to j$ if and only if $b_{ij}>0$.

Definition 1.

Let $B={\left(b_{ij}\right)}_{n\times n}$ be an $n\times n$ matrix. We say

-

B is skew-symmetrizable if there is a diagonal matrix $D=\mathrm{diag}(d_i,i\in [1,n])$ with positive integer diagonal entries such that $DB$ is skew-symmetric, i.e. , $d_i{b}_{ij}=-d_j{b}_{ji}$ for any $i,j\in [1,n]$, and D is called a skew-symmetrizer of B.

-

B is sign-skew-symmetric if either $b_{ij}{b}_{ji}<0$ or $b_{ij}=b_{ji}=0$ for any $i,j\in [1,n]$.

-

B is acyclic if ${\mathsf{\Gamma }}_B$ is acyclic, i.e. , there are no oriented cycles in ${\mathsf{\Gamma }}_B$.

Definition 2.

Let $B={\left(b_{ij}\right)}_{n\times n}$ be a sign-skew-symmetric matrix, and let $j\in [1,n]$. We say the index j is a sourceof B if $b_{ji}\ge 0$ for any $i\in [1,n]$, and we say the index j is a sink of B if $b_{ji}\le 0$ for any $i\in [1,n]$.

For any matrix $B={\left(b_{ij}\right)}_{n\times n}$, we define the mutation$B^{\prime }={\left(b_{ij}^{\prime }\right)}_{n\times n}$ of the matrix B at k, denoted by ${\mu }_k\left(B\right)$, which is given by

$$b_{ij}^{\prime }=\left\{\begin{array}{cc}-b_{ij},\hfill & \mathrm{if} i=k \mathrm{or} j=k;\hfill \\ b_{ij}+\mathrm{sgn}\left(b_{ik}\right){\left[b_{ik}{b}_{kj}\right]}_{+},\hfill & \mathrm{otherwise},\hfill \end{array}\right.$$

where ${\left[a\right]}_{+}=\max \{a,0\}$. A sign-skew-symmetric matrix B is said to be totally sign-skew-symmetric if the matrix obtained from B by applying an arbitrary finite sequence of mutations is sign-skew-symmetric. It is easy to check that skew-symmetric matrices and skew-symmetrizable matrices are totally sign-skew-symmetric. In [1], Fomin and Zelevinsky also found many examples of totally sign-skew-symmetric and non-skew-symmetrizable matrices as illustrated in the following example.

Example 1.

Let $a,b$, and c be positive integers such that $abc\ge 3$. Then the matrix $B(a,b,c)$ is totally sign-skew-symmetric and is not skew-symmetrizable, where $B(a,b,c)$ is given by

$$\left(\begin{array}{ccc}0& 2a& -2ab\\ -bc& 0& 2b\\ c& -ac& 0\end{array}\right).$$

Let $\mathcal{F}=\mathbb{Q}(u_1,\dots ,u_n)$ be the field of rational functions in n variables $u_1,\dots ,u_n$.

Definition 3

(Labeled seeds and unlabeled seeds). A labeled seed is a pair $\Sigma =(\mathbf{x},B)$ in which $\mathbf{x}=(x_1,\dots ,x_n)$ is an n-tuple of free generators of $\mathcal{F}$, and B is an $n\times n$ totally sign-skew-symmetric matrix. Here, we call $\mathbf{x}$ a cluster, entries in $\mathbf{x}$ are called cluster variables , and B an exchange matrix.

Two labeled seeds determine the same (unlabeled) seed if there is a permutation transforming one to the other.

Definition 4

(Acyclic seeds). A seed $(\mathbf{x},B)$ is said to be acyclic if ${\mathsf{\Gamma }}_B$ is acyclic.

The mutation of seeds, introduced by Fomin and Zelevinsky in [1], transforms a seed into another seed.

Definition 5

(Mutation of seeds). For $k\in [1,n]$, define another pair $({\mathbf{x}}^{\prime },B^{\prime })={\mu }_k(\mathbf{x},B)$ which is called the mutation of $(\mathbf{x},B)$ at k and obtained by the following rules:

(1) 

${\mathbf{x}}^{\prime }=(x_1^{\prime },\dots ,x_n^{\prime })$ is given by

$$x_k^{\prime }={\displaystyle \frac{{\displaystyle \prod _{i=1}^n}{x}_i^{{\left[b_{ik}\right]}_{+}}+{\displaystyle \prod _{i=1}^n}{x}_i^{{[-b_{ik}]}_{+}}}{x_k}}$$

and $x_i^{\prime }=x_i$ for $i\ne k$;

(2) 

$B^{\prime }={\mu }_k\left(B\right)$.

A direct check shows that the mutation of seeds is an involution, i.e., ${\mu }_k{\mu }_k(\mathbf{x},B)=(\mathbf{x},B)$ for any k and any seed. Let ${\mathbb{T}}_n$ be the n-regular tree whose edges are labeled by the numbers $1,\dots ,n$ so that the n edges emanating from each vertex receive different labels.

Definition 6.

A cluster pattern is an assignment of a labeled seed $({\mathbf{x}}_t,B_t)$ to every vertex $t\in {\mathbb{T}}_n$ such that the seeds assigned to the endpoints of any edge $t{\displaystyle \frac{k}{}}{t}^{\prime }$ are obtained from each other by the mutation in direction k. We often write the seed $({\mathbf{x}}_t,B_t)$ as

$${\mathbf{x}}_t=(x_{1,t},\dots ,x_{n,t}),\hspace{1em}{B}_t={\left(b_{ij,t}\right)}_{n\times n}.$$

Now we are ready to define cluster algebras.

Definition 7.

The cluster algebra$\mathcal{A}$ is defined to be the $\mathbb{Z}$-subalgebra of $\mathcal{F}$ generated by all cluster variables appeared in the clusters on ${\mathbb{T}}_n$. We also say $\mathcal{A}$ has rank n.

Note that the rank of a cluster algebra is not necessarily equal to the rank of its exchange matrix. The exchange graph $E\left(\mathcal{A}\right)$ of a cluster algebra $\mathcal{A}$ is the n-regular graph whose vertices are seeds and two (unlabeled) seeds are connected by a single edge if and only if they are related by a single mutation.

Fomin and Zelevinsky proved the Laurent phenomenon for totally sign-skew-symmetric cluster algebras in [1] which claims that every cluster variable can be expressed as a Laurent polynomial in the initial cluster variables and they also conjectured that the coefficients are positive. The positivity conjecture was proved by Lee and Schiffler for skew-symmetric cluster algebras in [19], by Gross, Hacking, Keel, and Kontsevich for the skew-symmetrizable cluster algebras in [20], and by Li and Pan for the totally sign-skew-symmetric cluster algebras in [21].

Theorem 4

(Laurent phenomenon and positivity [1,19,20,21]). Let $\mathcal{A}=\mathcal{A}(\mathbf{x},B)$ be the cluster algebra. Then, for any $i\in [1,n]$ and $t\in {\mathbb{T}}_n$, we have that

$$x_{i,t}\in {\mathbb{Z}}_{\ge 0}[x_{1;t_0}^{\pm 1},x_{2;t_0}^{\pm 1},\dots ,x_{n;t_0}^{\pm 1}],$$

and hence

$$\mathcal{A}\subseteq \mathbb{Z}[x_{1;t_0}^{\pm 1},x_{2;t_0}^{\pm 1},\cdots ,x_{n;t_0}^{\pm 1}].$$

2.2. Cluster Automorphisms

For totally sign-skew-symmetric cluster algebras, we can naturally define cluster automorphisms as follows.

Definition 8

([14]). Let $\mathcal{A}=\mathcal{A}(\mathbf{x},B)$ be a totally sign-skew-symmetric cluster algebra and $f:\mathcal{A}\to \mathcal{A}$ be an automorphism of $\mathbb{Z}$-algebras. The map f is called a cluster automorphism of $\mathcal{A}$ if there exists another seed $(\mathbf{z},B^{\prime })$ of $\mathcal{A}$ such that

(1) 

$f\left(\mathbf{x}\right)=\mathbf{z}$ and f maps frozen variables to frozen variables;

(2) 

$f\left({\mu }_x\left(\mathbf{x}\right)\right)={\mu }_{f\left(x\right)}\left(\mathbf{z}\right)$ for any exchangeable cluster variable $x\in \mathbf{x}$.

In [14], Assem, Schiffler, and Shamchenko proved that cluster automorphisms have some equivalent characterizations.

Proposition 1

([14]). Let f be a $\mathbb{Z}$-algebra automorphism of $\mathcal{A}$. Then, the following conditions are equivalent:

(i) 

f is a cluster automorphism of $\mathcal{A}$;

(ii) 

f satisfies (1)(2) in Definition 8 for every seed;

(iii) 

f maps each cluster to a cluster;

(iv) 

There exists a seed $({\mathbf{x}}^{\prime },B^{\prime })$ such that $f\left({\mathbf{x}}^{\prime }\right)$ is a cluster, and $B\left(f\left({\mathbf{x}}^{\prime }\right)\right)=B^{\prime }$ or $-B^{\prime }$.

Indeed, if a cluster automorphism maps a seed to another seed with the same exchange matrix, then it maps every seed to another seed with the same exchange matrix.

Corollary 1

([14]). Let $f:\mathcal{A}\to \mathcal{A}$ be a cluster automorphism of $\mathcal{A}$. Fix a seed $({\mathbf{x}}^{\prime },B^{\prime })$ satisfying $B\left(f\left({\mathbf{x}}^{\prime }\right)\right)=B^{\prime }$ or $-B^{\prime }$. Then,

(i) 

If $B\left(f\left({\mathbf{x}}^{\prime }\right)\right)=B^{\prime }$, then for any seed $({\mathbf{x}}^{″},B^{″})$ of $\mathcal{A}$, we have $B\left(f\left({\mathbf{x}}^{″}\right)\right)=B^{″}$.

(ii) 

If $B\left(f\left({\mathbf{x}}^{\prime }\right)\right)=-B^{\prime }$, then for any seed $({\mathbf{x}}^{″},B^{″})$ of $\mathcal{A}$, we have $B\left(f\left({\mathbf{x}}^{″}\right)\right)=-B^{″}$.

2.3. Representation Categories and Derived Categories

A quiver $Q=(Q_0,Q_1)$, consisting of the vertex set $Q_0$ and the arrow set $Q_1$, is a directed graph whose directed edges are called arrows. For basic concepts and notations on quivers and their representations, we refer to [22]. In this paper, we denote by Q a (possibly infinite) strongly locally finite quiver without infinite paths, that is, there are finitely many arrows incident to x and finitely many paths from x to y for any vertices $x,y\in Q_0$. In particular, Q is acyclic.

Let ${\mathrm{rep}}^b\left(Q\right)$ be the category of all finite dimensional representations of Q and ${\mathcal{D}}^b\left({\mathrm{rep}}^b\left(Q\right)\right)$ be the bounded derived category of ${\mathrm{rep}}^b\left(Q\right)$. Denote by $\mathrm{supp} M$ the support of the representation M, see [22] for more details.

Liu and Paguette depicted the Auslander–Reiten quivers ${\mathsf{\Gamma }}_{{\mathrm{rep}}^b\left(Q\right)}$ and ${\mathsf{\Gamma }}_{{\mathcal{D}}^b\left({\mathrm{rep}}^b\left(Q\right)\right)}$ of ${\mathrm{rep}}^b\left(Q\right)$ and ${\mathcal{D}}^b\left({\mathrm{rep}}^b\left(Q\right)\right)$, respectively, in [23]. Here are some we need later.

A connected component $\mathsf{\Gamma }$ of an Auslander–Rieten quiver is called standard if the full subcategory generated by objects lying in $\mathsf{\Gamma }$ is equivalent to the mesh category of $\mathsf{\Gamma }$.

Theorem 5

([23]). Let Q be a connected strongly locally finite quiver without infinite paths.

1. 

The preprojective component ${\mathcal{P}}_Q$ of ${\mathsf{\Gamma }}_{{\mathrm{rep}}^b\left(Q\right)}$ embeds in $\mathbb{N}{Q}^{op}$ with a left-most section $P_Q$ whose vertices are parameterized by the isomorphism classes of indecomposable projective representations. In particular, ${\mathcal{P}}_Q\cong \mathbb{N}{Q}^{op}$ if Q is moreover not of finite Dynkin type.

2. 

The preinjective component ${\mathcal{I}}_Q$ of ${\mathsf{\Gamma }}_{{\mathrm{rep}}^b\left(Q\right)}$ embeds in ${\mathbb{N}}^{-}{Q}^{op}$ with a right-most section $I_Q$ whose vertices are parameterized by the isomorphism classes of indecomposable injective representations. In particular, ${\mathcal{I}}_Q\cong {\mathbb{N}}^{-}{Q}^{op}$ if Q is moreover not of finite Dynkin type.

3. 

A regular component of ${\mathsf{\Gamma }}_{{\mathrm{rep}}^b\left(Q\right)}$ is isomorphic to $\mathbb{Z}{\mathcal{A}}_{\infty }$ if it exists and Q is not of finite Euclidean type.

4. 

${\mathsf{\Gamma }}_{{\mathrm{rep}}^b\left(Q\right)}$ does not admit regular components when the underlying graph of Q is ${\mathcal{A}}_{\infty }$.

5. 

The preprojective component ${\mathcal{P}}_Q$ and the preinjective components ${\mathcal{I}}_Q$ of ${\mathsf{\Gamma }}_{{\mathrm{rep}}^b\left(Q\right)}$ are both standard.

Theorem 6

([23]). Let Q be a connected strongly locally finite quiver without infinite paths. Then any connected component of ${\mathsf{\Gamma }}_{{\mathcal{D}}^b\left({\mathrm{rep}}^b\left(Q\right)\right)}$ is either the shift of a regular component of ${\mathsf{\Gamma }}_{{\mathrm{rep}}^b\left(Q\right)}$ or the shift of the connecting component $C_Q$, where $C_Q$ is obtained by connecting the components ${\mathcal{P}}_Q$ and ${\mathcal{I}}_Q[-1]$ in ${\mathsf{\Gamma }}_{{\mathrm{rep}}^b\left(Q\right)}$ via arrows $I_x[-1]\to P_y$ for each arrow $x\to y$ in Q. Moreover, each shift of $C_Q$ is standard.

2.4. Cluster Categories

Let k be a field and let $\mathcal{C}$ be a triangulated Hom-finite k-linear category with split idempotents and shift functor $\left[1\right]$. Let $D={\mathrm{Hom}}_k(-,k)$. The triangulated category $\mathcal{C}$ is called 2-Calabi–Yau if for any objects $X,Y\in \mathcal{C}$, there is a natural isomorphism

$${\mathrm{Hom}}_{\mathcal{C}}(X,Y)\cong D{\mathrm{Hom}}_{\mathcal{C}}(Y,X\left[2\right]),$$

functorial in both X and Y.

Definition 9.

Let $\mathcal{C}$ be a 2-Calabi–Yau triangulated category.

1. 

An object X of $\mathcal{C}$ is called rigid if ${\mathrm{Hom}}_{\mathcal{C}}(X,X\left[1\right])=0$.

2. 

A full subcategory of $\mathcal{C}$ is called strictly additive if it is closed under isomorphisms, finite direct sums, and taking direct summands.

3. 

A strictly additive subcategory $\mathcal{T}$ of $\mathcal{C}$ is called rigid if any object $X\in \mathcal{C}$ is rigid.

4. 

A strictly additive subcategory $\mathcal{T}$ of $\mathcal{C}$ is called cluster-tilting if $\mathcal{T}$ is functorially finite and for any object $X\in \mathcal{C}$, ${\mathrm{Hom}}_{\mathcal{C}}(\mathcal{T},X\left[1\right])=0$ if and only if $X\in \mathcal{T}$.

By definition, a cluster-tilting subcategory is maximal rigid.

For a strictly additive subcategory $\mathcal{T}$ of $\mathcal{C}$, it is a Krull–Schmidt additive category. We denote by $Q_{\mathcal{T}}$ the Auslander–Reiten quiver of $\mathcal{T}$. For any indecomposable object $M\in \mathcal{T}$, denote by ${\mathcal{T}}_M$ the full additive subcategory of $\mathcal{T}$ generated by the indecomposable objects not isomorphic to M. Note that ${\mathcal{T}}_M$ is also strictly additive in $\mathcal{C}$.

Definition 10 

([24]). Let $\mathcal{C}$ be a 2-Calabi–Yau triangulated category. A non-empty collection χ of strictly additive subcategories of $\mathcal{C}$ is called a weak cluster structure if for any $\mathcal{T}\in \chi$ and any indecomposable object $M\in \mathcal{T}$, the following holds.

1. 

Up to isomorphisms, there is a unique indecomposable object $M^{*}\ncong M\in \mathcal{C}$ such that the strictly additive subcategory ${\mu }_M\left(\mathcal{T}\right)$ of $\mathcal{C}$ generated by ${\mathcal{T}}_M$ and $M^{*}$ belongs to χ.

2. 

There exist two exact triangles in $\mathcal{C}$,

$$M\stackrel{f}{\to }B\stackrel{g}{\to }{M}^{*}\to M\left[1\right] \mathrm{and} M^{*}\stackrel{h}{\to }{B}^{\prime }\stackrel{l}{\to }M\to M^{*}\left[1\right],$$

where f and h are minimal left ${\mathcal{T}}_M$-approximations while g and l are minimal right ${\mathcal{T}}_M$-approximations in $\mathcal{C}$.

A weak cluster structure χ is called a cluster structure if moreover

3. 

$Q_{\mathcal{T}}$ does not contain loops or 2-cycles and $Q_{{\mu }_M\left(\mathcal{T}\right)}={\mu }_M\left(Q_{\mathcal{T}}\right)$, where the quiver mutation is in the sense of Fomin–Zelevinsky mutation in [1].

In particular, we have the following cluster categories which turn out to be 2-Calabi–Yau triangulated. Denote by ${\mathcal{C}}_Q$ the orbit category

$$D^b\left({\mathrm{rep}}^b\left(Q\right)\right)/F,$$

where $F={\tau }_D^{-1}\circ \left[1\right]$. More precisely, the objects of ${\mathcal{C}}_Q$ are the same as those of $D^b\left({\mathrm{rep}}^b\left(Q\right)\right)$ and

$${\mathrm{Hom}}_{{\mathcal{C}}_Q}(X,Y)=\underset{i\in \mathbb{Z}}{⨁}{\mathrm{Hom}}_{D^b\left({\mathrm{rep}}^b\left(Q\right)\right)}(X,F^{i}Y)$$

for any $X,Y\in {\mathcal{C}}_Q$. The composition of morphisms is given by

$${\left(g_i\right)}_{i\in \mathbb{Z}}\circ {\left(f_i\right)}_{i\in \mathbb{Z}}={(\sum _{p+q=i}{F}^p\left(g_q\right)\circ f_p)}_{i\in \mathbb{Z}}.$$

There is a canonical projection (triangle) functor for ${\mathcal{C}}_Q$ as an orbit category

$$\pi :D^b\left({\mathrm{rep}}^b\left(Q\right)\right)\to {\mathcal{C}}_Q$$

sending a morphism $f:X\to Y$ to ${\left(f_i\right)}_{i\in \mathbb{Z}}:X\to Y$, where $f_i=f$ if $i=0$ and $f_i=0$ otherwise.

Theorem 7

([25]). Let Q be a strongly locally finite quiver without infinite paths. Then ${\mathcal{C}}_Q$ is a Hom-finite Krull–Schmidt 2-Calabi–Yau triangulated k-category.

The following well-known lemma can be checked by definition.

Lemma 1.

Let Q be a strongly locally finite quiver without infinite paths and $M,N\in {\mathrm{rep}}^b\left(Q\right)$.

$${\mathrm{Hom}}_{{\mathcal{C}}_Q(M,N)}\cong {\mathrm{Hom}}_{D^b\left({\mathrm{rep}}^b\left(Q\right)\right)}(M,N)\oplus D{\mathrm{Hom}}_{D^b\left({\mathrm{rep}}^b\left(Q\right)\right)}(N,{\tau }_D^{2}M).$$

Theorem 8

([11,17,18]). Let Q be a finite acyclic quiver. The collection of cluster-tilting subcategories of ${\mathcal{C}}_Q$ is a cluster structure.

Proposition 2

([23]). Let Q be a strongly locally finite quiver without infinite paths. The strictly additive subcategory of ${\mathcal{C}}_Q$ generated by $\{P_x\left[1\right]\mid x\in Q_0\}$ is a cluster-tilting subcategory.

Theorem 9

([26,27]). Let Q be a finite acyclic quiver and $\mathcal{T}$ be a cluster-tilting subcategory of ${\mathcal{C}}_Q$ such that $Q_{\mathcal{T}}$ is acyclic. There is an equivalence

$${\mathcal{C}}_Q/\left(\mathcal{T}\left[1\right]\right)\stackrel{\sim }{\to }{\mathrm{rep}}^b\left(Q_{\mathcal{T}}^{op}\right)$$

induced by ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(T,-)$, taking T to $k{Q}_{\mathcal{T}}^{op}$, where $T=\oplus T_i$ with $T_i$ being the representative of the isomorphism class of indecomposable objects in $\mathcal{T}$.

Proof. 

Let $\mathsf{\Gamma }={\mathrm{End}}_{{\mathcal{C}}_{\mathrm{Q}}}{\left(\mathrm{T}\right)}^{\mathrm{op}}$. By [Theorem 2.2, [27], the functor ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(T,-)$ induces an equivalence: ${\mathcal{C}}_Q/\left(\mathcal{T}\left[1\right]\right)\stackrel{\sim }{\to }\mathrm{mod} \mathsf{\Gamma }$ taking T to $k{Q}_{\mathcal{T}}^{op}$. By [II. Theorem 3.7, [22], $\mathsf{\Gamma }\cong k{Q}_{\mathcal{T}}^{op}/I$ for some admissible ideal I. Since $Q_{\mathcal{T}}$ is acyclic, $\mathsf{\Gamma }$ has finite global dimension. Since $\mathsf{\Gamma }$ is also a cluster-tilted algebra, $\mathsf{\Gamma }$ is hereditary, thanks to [Corollary 2.1, [26]. Thus, $\mathsf{\Gamma }\cong k{Q}_{\mathcal{T}}^{op}$ and $\mathrm{mod}\mathsf{\Gamma }\stackrel{\sim }{\to }{\mathrm{rep}}^b\left(Q_{\mathcal{T}}^{op}\right)$. This completes the proof. □

3. Proof of Theorems 1 and 2

Let us firstly prove Theorem 1 which gives a necessary condition for $BD$ mutation equivalent to B for an exchange matrix B and a diagonal matrix D.

Proof of Theorem 1.

With an induction on the length of the path from t to $t_0$, it suffices to prove ${\mathbf{a}}_t={\mathbf{a}}_{t^{\prime }}$ when t and $t^{\prime }$ are connected by an edge labeled k.

Since $b_k^{t^{\prime }}=-b_k^t$, $a_{k;t^{\prime }}=a_{k;t}$. Assume $j\ne k$. Then for any $i\in [1,n]$,

$$b_{ij}^{t^{\prime }}=b_{ij}^t+sign\left(b_{ik}^t\right){\left[b_{ik}^t{b}_{kj}^t\right]}_{+},$$

so $a_{j;t}|b_{ij}^{t^{\prime }}$ as $a_{j;t}|b_{ij}^t$ and $a_{j;t}|b_{kj}^t$. Hence $a_{j;t}|a_{j;t^{\prime }}$. Dually, since the mutation is an involution, we have $a_{j;t^{\prime }}|a_{j;t}$ as well. Therefore, $a_{j;t}=a_{j;t^{\prime }}$.

This completes the proof. □

The following result follows easily from Proposition 1.

Corollary 2.

Let $\mathcal{A}$ be a totally sign-skew-symmetric cluster algebra with an exchange matrix B and $D=diag(d_1,\cdots ,d_n)$ be a diagonal matrix. If $BD$ is mutation equivalent to B, then $D=\pm I$, where I is the identity matrix.

Proof. 

By Proposition 1, the column greatest common divisor vector of B equals that of $B^{\prime }$, hence $d_i=\pm 1$ for each $i\in [1,n]$.

Notice that if $b_{ij}\ne 0$, then $sgn\left(b_{ij}\right)=-sgn\left(b_{ji}\right)$. Since $BD$ is also sign-skew-symmetric, then $sgn\left(b_{ij}{d}_j\right)=-sgn\left(b_{ji}{d}_i\right)$. Hence, in this case $sgn\left(d_i\right)=sgn\left(d_j\right)$, and we therefore have that $d_i=d_j$. Since B is indecomposable, for any $i,j\in [1,n]$ such that $i\ne j$, there are indices $i=i_1,i_2,\dots ,i_k=j$ such that $b_{i_1{i}_2}\dots b_{i_{k-1}{i}_k}\ne 0$. Therefore $sgn\left(d_i\right)=sgn\left(d_{i_1}\right)=\cdots =sgn\left(d_{i_k}\right)=sgn\left(d_j\right).$ Thus, $D=\pm I$. □

Let us prove Theorem 2.

Proof of Theorem 2.

The proof is similar to that of [Theorem 3.6, [15]. For the convenience of the reader, we repeat it here.

Let $\mathcal{A}$ be a totally sign-skew-symmetric cluster algebra, and $f:\mathcal{A}\to \mathcal{A}$ be a $\mathbb{Z}$-algebra homomorphism. Assume that there exists another seed $(\mathbf{z},B^{\prime })$ of $\mathcal{A}$ such that $f\left(\mathbf{x}\right)=\mathbf{z}$, then $f\left({\mu }_x\left(\mathbf{x}\right)\right)={\mu }_{f\left(x\right)}\left(\mathbf{z}\right)$ for any $x\in \mathbf{x}$. Without loss of generality, we assume that $f\left(x_i\right)=z_i$ for $i\in [1,n]$. For any $k\in [1,n]$, denote by $x_k^{\prime }$ and $z_k^{\prime }$ the new obtained cluster variables in ${\mu }_k(\mathbf{x},B)$ and ${\mu }_k(\mathbf{z},B^{\prime })$, respectively.

Then we have

$$f\left(x_k^{\prime }\right)=f({\displaystyle \frac{{\displaystyle \prod _{i=1}^n}{x}_i^{{\left[b_{ik}\right]}_{+}}+{\displaystyle \prod _{i=1}^n}{x}_i^{{[-b_{ik}]}_{+}}}{x_k}})={\displaystyle \frac{{\displaystyle \prod _{i=1}^n}{z}_i^{{\left[b_{ik}\right]}_{+}}+{\displaystyle \prod _{i=1}^n}{z}_i^{{[-b_{ik}]}_{+}}}{z_k}}={\displaystyle \frac{{\displaystyle \prod _{i=1}^n}{z}_i^{{\left[b_{ik}\right]}_{+}}+{\displaystyle \prod _{i=1}^n}{z}_i^{{[-b_{ik}]}_{+}}}{{\displaystyle \prod _{i=1}^n}{z}_i^{{\left[b_{ik}^{\prime }\right]}_{+}}+{\displaystyle \prod _{i=1}^n}{z}_i^{{[-b_{ik}^{\prime }]}_{+}}}}{z}_k^{\prime }$$

Due to the Laurent phenomenon, $f\left(x_k^{\prime }\right)\in \mathcal{A}\subseteq \mathbb{Z}[z_1^{\pm 1},\cdots ,{\left(z_k^{\prime }\right)}^{\pm 1},\cdots ,z_n^{\pm 1}]$, so

$${\displaystyle \frac{{\displaystyle \prod _{i=1}^n}{z}_i^{{\left[b_{ik}\right]}_{+}}+{\displaystyle \prod _{i=1}^n}{z}_i^{{[-b_{ik}]}_{+}}}{{\displaystyle \prod _{i=1}^n}{z}_i^{{\left[b_{ik}^{\prime }\right]}_{+}}+{\displaystyle \prod _{i=1}^n}{z}_i^{{[-b_{ik}^{\prime }]}_{+}}}}\in \mathbb{Z}[z_1^{\pm 1},\cdots ,z_n^{\pm 1}].$$

Since both ${\displaystyle \prod _{i=1}^n}{z}_i^{{\left[b_{ik}\right]}_{+}}+{\displaystyle \prod _{i=1}^n}{z}_i^{{[-b_{ik}]}_{+}}$ and ${\displaystyle \prod _{i=1}^n}{z}_i^{{\left[b_{ik}^{\prime }\right]}_{+}}+{\displaystyle \prod _{i=1}^n}{z}_i^{{[-b_{ik}^{\prime }]}_{+}}$ cannot be divided by any variable in $\mathbf{z}$, we can see that

$${\displaystyle \frac{{\displaystyle \prod _{i=1}^m}{z}_i^{{\left[b_{ik}\right]}_{+}}+{\displaystyle \prod _{i=1}^m}{z}_i^{{[-b_{ik}]}_{+}}}{{\displaystyle \prod _{i=1}^m}{z}_i^{{\left[b_{ik}^{\prime }\right]}_{+}}+{\displaystyle \prod _{i=1}^m}{z}_i^{{[-b_{ik}^{\prime }]}_{+}}}}\in \mathbb{Z}[z_1,\cdots ,z_m],$$

which is only possible when ${\mathbf{b}}_k=d_k{\mathbf{b}}_k^{\prime }$ for some odd integer $d_k$ by [Lemma 3.1, [28], where ${\mathbf{b}}_k$ and ${\mathbf{b}}_k^{\prime }$ is the k-th column of B and $B^{\prime }$, respectively. Hence there is an $n\times n$ integer matrix $D=diag(d_1,\cdots ,d_n)$ such that $B=B^{\prime }D$.

By Corollary 2, we know that $D=I$ or $D=-I$, thus $B^{\prime }=B$ or $B^{\prime }=-B$. By Proposition 1, f is a cluster automorphism. □

4. Proof of Theorem 3

Let us briefly recall some basic concepts and some results of unfolding theory mainly referred to [16]. Let Q be a strongly locally finite quiver with a group action G such that the G-orbits are finite. For a vertex i of Q, a G-loop at i is an arrow $i\to g·i$ for some $g\in G$, and a G-2-cycle at i is a pair of arrows $i\to j$ and $j\to h·i$ for some $j\notin \left[i\right]:=\{g·i|g\in G\}$ and some $h\in G$. We say Q is G-foldable if there are no G-loops or G-2-cycles at each vertex i of Q.

Let Q be G-foldable. For each vertex i, in [16], the orbit mutation${\mu }_{\left[i\right]}$ is defined as a sequence of mutations at each vertex in $\left[i\right]$ which commutes with each other; see [Definition 2.1, [16] for more details. We say Q is globally G-foldable if any $Q^{\prime }$ obtained from Q by a finite sequence of orbit mutations is G-foldable.

Let Q be G-foldable. One can also obtain a sign-skew-symmetric matrix $B\left(Q^G\right)=\left(b_{\left[i\right]\left[j\right]}\right)$, where $b_{\left[i\right]\left[j\right]}:={\sum }_{i^{\prime }\in \left[i\right]}(a_{i^{\prime }j}-a_{j{i}^{\prime }})$ and $a_{i^{\prime }j}$ is the number of arrows from $i^{\prime }$ to j in Q. For a sign-skew-symmetric matrix B, we say $(Q,G)$ is an unfolding of B if Q is globally G-admissible, and $B=B\left(Q^G\right)$. It was proved in [16] that every acyclic sign-skew-symmetric matrix B admits an unfolding $(Q,G)$ such that Q is strongly locally finite with no infinite paths called the universal covering of B.

Definition 11.

For a (possibly infinite) mutation sequence $\mu =\cdots {\mu }_{i_2}{\mu }_{i_1}$ and a full subquiver Γ of Q, denote ${}_{\mu }\mathsf{\Gamma }={\underleftarrow{\lim }}_{\left(i\right)}\mathsf{\Gamma }$, ${\mathsf{\Gamma }}^{\mu }=\underleftarrow{\lim } {\mathsf{\Gamma }}^{\left(i\right)}$ and ${\mathsf{\Gamma }}_{\mu }=\underleftarrow{\lim } {\mathsf{\Gamma }}_{\left(i\right)}$, where ${}_{\left(0\right)}\mathsf{\Gamma }={\mathsf{\Gamma }}^{\left(0\right)}={\mathsf{\Gamma }}_{\left(0\right)}=\mathsf{\Gamma }$, ${}_{(j+1)}\mathsf{\Gamma }\mathsf{\Gamma }$ is the full subquiver of Q with

$${}_{(j+1)}{\mathsf{\Gamma }}_0=\left\{\begin{array}{cc}{}_{\left(j\right)}{\mathsf{\Gamma }}_0\cup \left\{i_{j+1}\right\},\hfill & \mathrm{if} i_{j+1} \mathit{is} \mathit{connected} \mathit{to}{}_{\left(j\right)}\mathsf{\Gamma } \mathit{in} {\mu }_{i_j}\cdots {\mu }_{i_1}\left(Q\right);\hfill \\ {}_{\left(j\right)}{\mathsf{\Gamma }}_0,\hfill & \mathit{otherwise}.\hfill \end{array}\right.$$

${\mathsf{\Gamma }}^{(j+1)}$ is the full subquiver of Q with

$${\mathsf{\Gamma }}_0^{(j+1)}={\mathsf{\Gamma }}_0^{\left(j\right)}\cup \{x\in Q_0\mid x \mathit{is} \mathit{connected} \mathit{to} i_{j+1} \mathrm{in} {\mu }_{i_j}\cdots {\mu }_{i_1}\left(Q\right)\},$$

${\mathsf{\Gamma }}_{(j+1)}$ is the full subquiver of Q with

$${\left({\mathsf{\Gamma }}_{(j+1)}\right)}_0={\left(({\mu }_{i_{j+1}}\cdots {\mu }_{i_1}\left(Q\right)){|}_{{\mathsf{\Gamma }}_{\left(j\right)}}\right)}_0^{{\mu }_{i_1}\cdots {\mu }_{i_{j+1}}}$$

and ${\mu }_{i_{j+1}}\cdots {\mu }_{i_1}\left(Q\right){\left)\right|}_{{\mathsf{\Gamma }}_{\left(j\right)}}$ denotes the full subquiver of ${\mu }_{i_{j+1}}\cdots {\mu }_{i_1}\left(Q\right))$ with vertex set ${\left({\mathsf{\Gamma }}_{\left(j\right)}\right)}_0$.

Definition 12.

A mutation sequence μ is called locally admissible to Q if μ contains finitely many ${\mu }_i$ for each $i\in Q_0$ and both ${}_{\mu }\mathsf{\Gamma }$ and ${\mathsf{\Gamma }}_{\mu }$ are finite for any finite full subquiver Γ of Q.

According to the definition of seed mutation, for any mutation sequence $\mu$ locally admissible to Q, any finite full subquiver $\mathsf{\Gamma }$ of Q and any full subquiver ${\mathsf{\Gamma }}^{\prime }$ of Q containing ${\mathsf{\Gamma }}_{\mu }$, ${\left(\mu {|}_{{\mathsf{\Gamma }}^{\prime }}\left({\Sigma }_{t_0}{|}_{{\mathsf{\Gamma }}^{\prime }}\right)\right)|}_{\mathsf{\Gamma }}={\left(\mu {|}_{{\mathsf{\Gamma }}_{\mu }}\left({\Sigma }_{t_0}{|}_{{\mathsf{\Gamma }}_{\mu }}\right)\right)|}_{\mathsf{\Gamma }}$, where ${\Sigma |}_{\mathsf{\Gamma }}$ represents the seed induced from $\Sigma$ with those parameterized by ${\mathsf{\Gamma }}_0$ and ${\mu |}_{{\mathsf{\Gamma }}^{\prime }}$ denotes the mutation sequence obtained from $\mu$ by restricting to mutations in ${\mathsf{\Gamma }}^{\prime }$.

In this case, denote by $\mu \left({\Sigma }_{t_0}\right)$ the uniquely determined seed satisfying

$${\mu \left({\Sigma }_{t_0}\right)|}_{\mathsf{\Gamma }}=({\mu |}_{{\mathsf{\Gamma }}_{\mu }}{\left({\Sigma }_{t_0}{|}_{{\mathsf{\Gamma }}_{\mu }}\right))|}_{\mathsf{\Gamma }}$$

for any full subquiver $\mathsf{\Gamma }$ of Q.

Finite orbit mutation sequences appear among the most common examples of locally admissible mutation sequences.

Lemma 2.

Let Q be a globally G-foldable quiver. Then any finite sequence of orbit mutations is locally admissible to Q.

Proof. 

Denote by ${\mu }^{\left(r\right)}={\mu }_{\left[i_r\right]}\cdots {\mu }_{\left[i_1\right]}$ the finite orbit mutation sequence. Take an induction on r.

It is trivial when $r=0$. Assume ${\mu }^{(r-1)}={\mu }_{\left[i_{r-1}\right]}\cdots {\mu }_{\left[i_1\right]}$ is locally admissible to Q. So for any full subquiver $\mathsf{\Gamma }$ of Q, ${\mu }^{(r-1)}$ contains finitely many ${\mu }_i$ for each $i\in Q_0$ and both ${}_{{\mu }^{(r-1)}}\mathsf{\Gamma }$ and ${\mathsf{\Gamma }}_{{\mu }^{(r-1)}}$ are finite full subquivers of Q. Hence ${\mu }^{\left(r\right)}$ contains finitely many ${\mu }_i$ for each $i\in Q_0$. Moreover, as ${\mu }_x$ commutes with ${\mu }_y$ when $x,y\in \left[i_r\right]$,

$${{}_{{\mu }^{\left(r\right)}}\mathsf{\Gamma }}_0={{}_{{\mu }^{(r-1)}}\mathsf{\Gamma }}_0\cup {}_{\left(r\right)}I$$

and

$${\left({\mathsf{\Gamma }}_{{\mu }^{\left(r\right)}}\right)}_0={\left({\mathsf{\Gamma }}_{{\mu }^{(r-1)}}\right)}_0\cup {\displaystyle \bigcup _{j=1}^r}{I}^{\left(j\right)},$$

where

$${}_{\left(r\right)}I=\{x\in \left[i_r\right]\mid x \mathrm{is} \mathrm{connected} \mathrm{to}{}_{{\mu }^{(r-1)}}\mathsf{\Gamma } \mathrm{in} {\mu }^{(r-1)}\left(Q\right)\},$$

$$I^{\left(r\right)}=\{x\in Q_0\mid x \mathrm{is} \mathrm{connected} \mathrm{to} {\left({\mathsf{\Gamma }}_{{\mu }^{(r-1)}}\right)}_0\cap \left[i_r\right] \mathrm{in} {\mu }^{(r-1)}\left(Q\right)\}$$

and

$$I^{\left(j\right)}=\{x\in Q_0\mid x \mathrm{is} \mathrm{connected} \mathrm{to} ({\left({\mathsf{\Gamma }}_{{\mu }^{(r-1)}}\right)}_0\cup {\displaystyle \bigcup _{s=j+1}^r}{I}^{\left(s\right)})\cap \left[i_j\right] \mathrm{in} {\mu }^{(j-1)}\left(Q\right)\}.$$

Since ${}_{{\mu }^{(r-1)}}\mathsf{\Gamma }$ and ${\mathsf{\Gamma }}_{{\mu }^{(r-1)}}$ are finite by inductive assumption and Q is globally G-foldable, it can be verified by induction that ${}_{\left(r\right)}I$ and $I^{\left(j\right)}$ are finite for any $j\in [1,r]$. Hence ${}_{{\mu }^{\left(r\right)}}\mathsf{\Gamma }$ and ${\mathsf{\Gamma }}_{{\mu }^{\left(r\right)}}$ are finite. So ${\mu }^{\left(r\right)}$ is locally admissible to Q. □

For any full subquiver $\mathsf{\Gamma }$ of Q, denote by ${\mathbf{P}}_{\mathsf{\Gamma }}$ the strictly additive subcategory generated by $\{P_x\mid x\in {\mathsf{\Gamma }}_0\}$. For convenience, we will omit the subscript when it equals Q.

Definition 13.

Let Q be a strongly locally finite quiver without infinite paths. A strictly additive subcategory $\mathcal{T}$ of ${\mathcal{C}}_Q$ is called locally reachable if there is a set of indecomposable objects $\{M_i\mid i\in I\}$ with either $I=[1,m]$ for some positive integer m or $I={\mathbb{Z}}_{>0}$, such that $M_i\in {\mu }_{M_{i-1}}\cdots {\mu }_{M_1}\left(\mathbf{P}\left[1\right]\right)$ for any $i\in I$, the induced (possibly infinite) mutation sequence μ is locally admissible to Q and $\mathcal{T}=\mu \left(\mathbf{P}\right[1\left]\right)$, that is, ${\mathcal{T}|}_{\mathsf{\Gamma }}=\mu {{|}_{\Delta }\left({\mathbf{P}}_{\Delta }\left[1\right]\right)|}_{\mathsf{\Gamma }}$, where ${\mathcal{T}|}_{\mathsf{\Gamma }}$ denotes the additive subcategory generated by indecomposable rigid objects corresponding to vertices parameterized by ${\mathsf{\Gamma }}_0$, $\mathsf{\Delta }$ is the full subquiver of Q with ${\Delta }_0={\cup }_{x\in {\left({\mathsf{\Gamma }}_{\mu }\right)}_0}\mathrm{supp}{P}_x$, and ${\mathbf{P}}_{\Delta }\left[1\right]$ is viewed as a cluster-tilting subcategory in ${\mathcal{C}}_{\Delta }$.

A locally reachable (weak) cluster structure is a (weak) cluster structure containing only locally reachable cluster subcategories.

In the following, we will not distinguish mutations for seeds or cluster-tilting subcategories as there is no risk of confusion.

The following result proves a criterion for locally reachable subcategories to form a cluster structure.

Theorem 10

([24]). Let Q be a strongly locally finite quiver without infinite paths. Locally reachable subcategories of ${\mathcal{C}}_Q$ are all cluster-tilting and determine a locally reachable weak cluster structure on $\mathcal{C}$. They form a locally reachable cluster structure on $\mathcal{C}$ if and only if their associated quivers do not contain loops or 2-cycles.

Lemma 3.

Let Q be a strongly locally finite quiver without infinite paths and $\mathcal{T}$ be a locally reachable cluster-tilting subcategory. For any $M\in {\mathcal{C}}_Q$, up to isomorphisms, ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(X,M)={\mathrm{Hom}}_{{\mathcal{C}}_Q}(M,X)=0$ for all but finitely many indecomposable objects $X\in \mathcal{T}$.

Proof. 

It is trivial when Q is finite. Assume Q is infinite, $\mathcal{T}=\mu \left(\mathbf{P}\right[1\left]\right)$ with $\mu$ a locally admissible mutation sequence to Q and $M\in {\mathrm{rep}}^b\left(Q\right)\bigsqcup \mathbf{P}\left[1\right]$ is indecomposable.

When $\mu =\varnothing$, $X\cong P_x\left[1\right]$ for some $x\in Q_0$. If $M\cong P_y\left[1\right]\in \mathbf{P}\left[1\right]$, by Lemma 1, then

$${\mathrm{Hom}}_{{\mathcal{C}}_Q}(X,M)\cong {\mathrm{Hom}}_{D^b\left({\mathrm{rep}}^b\left(Q\right)\right)}(P_x,P_y)\oplus D{\mathrm{Hom}}_{D^b\left({\mathrm{rep}}^b\left(Q\right)\right)}(P_y,{\tau }^2{P}_x).$$

Hence, following from Theorem 6, ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(X,M)\ne 0$ if and only if the string from y to x is a path. The set consisting of such x is finite as Q is strongly locally finite and does not admit infinite paths. If $M\in {\mathrm{rep}}^b\left(Q\right)$, then

$$\begin{array}{cc}{\mathrm{Hom}}_{{\mathcal{C}}_Q}(X,M)\hfill & \cong {\mathrm{Hom}}_{{\mathcal{C}}_Q}(I_x[-1],M)\hfill \\ \hfill & \cong D{\mathrm{Hom}}_{{\mathcal{C}}_Q}(M,\tau I_x)\hfill \\ \hfill & \cong D{\mathrm{Hom}}_{D^b\left({\mathrm{rep}}^b\left(Q\right)\right)}(M,\tau I_x)\oplus {\mathrm{Hom}}_{D^b\left({\mathrm{rep}}^b\left(Q\right)\right)}(I_x,\tau M).\hfill \end{array}$$

So ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(X,M)\ne 0$ if and only if M is not injective and $x\in \mathrm{supp}{\tau }^{-1}M$. In summary, up to isomorphisms, ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(X,M)=0$ for all but finitely many indecomposable objects $X\cong P_x\left[1\right]$ for some $x\in Q_0$. Dually it holds for ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(M,X)$. So, we can find a finite full subquiver $\mathsf{\Gamma }$ of Q satisfying ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(P_x\left[1\right],M)={\mathrm{Hom}}_{{\mathcal{C}}_Q}(M,P_x\left[1\right])=0$ if $x\notin {\mathsf{\Gamma }}_0$.

For any locally reachable cluster-tilting subcategory ${\mathcal{T}}^{\prime }$ of ${\mathcal{C}}_Q$ and any indecomposable object $N\in {\mathcal{T}}^{\prime }$, there exist two exact triangles in ${\mathcal{C}}_Q$:

$$N\to B\to N^{*}\to N\left[1\right]\hspace{1em}\mathrm{and}\hspace{1em}{N}^{*}\to B^{\prime }\to N\to N^{*}\left[1\right]$$

such that $B,B^{\prime }\in {\mathcal{T}}_N^{\prime }$ and ${\mu }_N\left({\mathcal{T}}^{\prime }\right)$ is generated by ${\mathcal{T}}_N^{\prime }$ and $N^{*}$. Since ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(X,Y)\cong {\mathrm{Hom}}_{{\mathcal{C}}_Q}(\tau X,\tau Y)$ for any $X,Y\in {\mathcal{C}}_Q$ and $X\to Y\to Z\to X\left[1\right]$ is an exact triangle in ${\mathcal{C}}_Q$ if and only if $\tau X\to \tau Y\to \tau Z\to \tau X\left[1\right]$ is also, without loss of generality, we may assume ${\tau }^{i}X\in {\mathrm{rep}}^b\left(Q\right)$ when $i\in [-2,2]$ and $X\in \{N,N^{*},B,B^{\prime },M\}$. Then, there exist two exact sequences in ${\mathrm{rep}}^b\left(Q\right)$:

$$0\to N\to B\to N^{*}\to 0\hspace{1em}\mathrm{and}\hspace{1em}0\to N^{*}\to B^{\prime }\to N\to 0.$$

By Lemma 1, ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(N^{*},M)={\mathrm{Hom}}_{{\mathcal{C}}_Q}(M,N^{*})=0$ if ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(L,M)={\mathrm{Hom}}_{{\mathcal{C}}_Q}(M,L)=0$ when L equals B or $B^{\prime }$.

Therefore, if an indecomposable object $X\in \mathcal{T}$ admitting either ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(X,M)\ne 0$ or ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(M,X)\ne 0$, then ${X\in \mu |}_{\Delta }{\left({\mathbf{P}}_{\Delta }\left[1\right]\right)|}_{{}_{\mu }\mathsf{\Gamma }}={\mathcal{T}|}_{{}_{\mu }\mathsf{\Gamma }}$, where $\Delta ={\cup }_{x\in {\left({{(}_{\mu }\mathsf{\Gamma })}_{\mu }\right)}_0}\mathrm{supp}{P}_x$. So the set consisting of the isomorphism classes of such X is finite since Q is a strongly locally finite quiver without infinite paths and $\mu$ is locally admissible to Q. □

Lemma 4.

Let Q be a strongly locally finite quiver without infinite paths. Locally reachable subcategories of ${\mathcal{C}}_Q$ form a locally reachable cluster structure.

Proof. 

By Theorem 10, it suffices to check $Q_{\mathcal{T}}$ does not contain loops or 2-cycles for any locally reachable subcategory $\mathcal{T}$ of ${\mathcal{C}}_Q$. Assume on the contrary, there is a 2-cycle $\left[T_i\right]\to \left[T_j\right]\to \left[T_i\right]$ in $Q_{\mathcal{T}}$ for $\mathcal{T}=\mu \left(\mathbf{P}\right[1\left]\right)$ with $\mu$ a locally admissible mutation sequence to Q, where $\left[T_i\right]$ and $\left[T_j\right]$ are not necessarily distinguished.

Following from Lemma 3, there is a finite full subquiver $\mathsf{\Gamma }$ of Q such that ${X\in \mathcal{T}|}_{\mathsf{\Gamma }}$ if $X\in \mathcal{T}$ is an indecomposable object with ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(X,T_i)\ne 0$ or ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(T_i,X)\ne 0$. Since ${\mathcal{T}|}_{\mathsf{\Gamma }}=\mu {{|}_{\Delta }\left({\mathbf{P}}_{\Delta }\left[1\right]\right)|}_{\mathsf{\Gamma }}$, where $\Delta ={\cup }_{x\in {\left({\mathsf{\Gamma }}_{\mu }\right)}_0}\mathrm{supp} P_x$, there should be a 2-cycle $\left[T_i\right]\to \left[T_j\right]\to \left[T_i\right]$ in $Q_{{\mu |}_{\Delta }\left({\mathbf{P}}_{\Delta }\left[1\right]\right)}$, which contradicts Theorem 8. □

The following result extends Theorem 9 to strongly locally finite quivers without infinite paths.

Theorem 11.

Let Q be a strongly locally finite quiver without infinite paths. There is an equivalence ${\mathcal{C}}_Q/\left(\mathcal{T}\left[1\right]\right)\stackrel{\sim }{\to }{\mathrm{rep}}^b\left(Q_{\mathcal{T}}^{op}\right)$ taking $\mathcal{T}$ to $k{Q}_{\mathcal{T}}^{op}$.

Proof. 

For any $M\in {\mathcal{C}}_Q$, due to Lemma 3, there is a finite full subquiver $\mathsf{\Gamma }$ of Q satisfying ${\mathrm{Hom}}_{{\mathcal{C}}_Q}(X,M)=0$ if X is indecomposable and ${X\notin \mathcal{T}|}_{\mathsf{\Gamma }}$. Since ${\mathcal{T}|}_{\mathsf{\Gamma }}=\left(\mu {|}_{{\mathsf{\Gamma }}_{\mu }}\left({\mathbf{P}}_{{\mathsf{\Gamma }}_{\mu }}\left[1\right]\right)\right){|}_{\mathsf{\Gamma }}$, then

$$\underset{T_i}{⨁}{\mathrm{Hom}}_{{\mathcal{C}}_Q}(T_i,M)\cong {\mathrm{Hom}}_{{\mathcal{C}}_{{\mathsf{\Gamma }}_{\mu }}}(\underset{T_i^{\prime }}{⨁}{T}_i^{\prime },M),$$

where $T_i$ runs over representatives of the isomorphism classes of indecomposable objects in $\mathcal{T}$ while $T_i^{\prime }$ runs over representatives of the isomorphism classes of indecomposable objects in $\left(\mu {|}_{{\mathsf{\Gamma }}_{\mu }}\left({\mathbf{P}}_{{\mathsf{\Gamma }}_{\mu }}\left[1\right]\right)\right){|}_{\mathsf{\Gamma }}$. Then, following from Theorem 9, ${⨁}_{T_i}{\mathrm{Hom}}_{{\mathcal{C}}_Q}(T_i,-): {\mathcal{C}}_Q/\left(\mathcal{T}\left[1\right]\right)\to {\mathrm{rep}}^b\left(Q_{\mathcal{T}}^{op}\right)$ is an equivalence taking $\mathcal{T}$ to $k{Q}_{\mathcal{T}}^{op}$. □

Let us prove Theorem 3.

Proof of Theorem 3. 

Let Q and $Q^{\prime }$ be the universal coverings of B and $B^{\prime }$, respectively. Since B and $B^{\prime }$ are mutation equivalent, there is a finite mutation sequence $\mu$ such that $B^{\prime }=\mu \left(B\right)$, which lifts to a finite orbit mutation sequence $\tilde{\mu }$ from Q to $Q^{\prime }$.

Following from Lemma 2, $\tilde{\mu }$ is locally admissible. Hence $\tilde{\mu }{\left(Q\right)}^{op}$ equals the quiver of the locally reachable cluster-tilting subcategory $\mathcal{T}=\mu \left({\oplus }_{x\in Q_0}{P}_x\left[1\right]\right)$ in ${\mathcal{C}}_Q$ due to Theorem 10. By Theorem 11, there is an equivalence ${\mathcal{C}}_Q/\left(\mathcal{T}\left[1\right]\right)\to {\mathrm{rep}}^b\left(Q_{\mathcal{T}}^{op}\right)$ taking $\mathcal{T}$ to $k{Q}_{\mathcal{T}}^{op}$, which induces an isomorphism between Auslander–Reiten quivers of ${\mathcal{C}}_Q$ and ${\mathcal{C}}_{Q_{\mathcal{T}}^{op}}$.

By Theorems 5 and 6, the connecting component is the only connected component of a cluster category ${\mathcal{C}}_{\mathrm{R}}$ isomorphic to $\mathbb{Z}\mathrm{R}$ for any strongly locally finite quiver R without infinite paths other than quivers of finite Dynkin type. So the above isomorphism maps the connecting component of ${\mathcal{C}}_Q$ to that of ${\mathcal{C}}_{Q_{\mathcal{T}}^{op}}$, which induces that the subquiver with vertices in $\mathcal{T}$ is contained in the connecting component $C_Q$ and the underlying graph of $Q_{\mathcal{T}}^{op}={\left(Q^{\prime }\right)}^{op}$ equals that of $Q^{op}$. Hence, up to an isomorphism, Q and $Q^{\prime }$ can be obtained from each other via a locally admissible orbit mutation sequence at sources and sinks, which leads to a finite mutation sequence from B to $B^{\prime }$ at sources and sinks up to a simultaneous permutation of rows and columns.

This finishes the proof. □