Derived equivalence classification of $m$-cluster tilted algebras of type $A_{n}$

Cluster categories were introduced in [6] as a representation theoretic framework for cluster algebras of Fomin and Zelevinsky [12]. Independently, in [14] Iyama defines maximal $m$-orthogonal modules for Artin algebras, $m\geqslant 1$, and presents a classification of maximal 1-orthogonal modules for finite type self-injective algebras of tree class $A_{n},B_{n},C_{n}$ and $D_{n}$. Motivated by a generalized version of the combinatorics related to cluster algebras, $m$-cluster categories were defined as a generalization of cluster categories [21], [22]. The cluster categories and their generalizations have been the subject of much recent research.
The $m$-cluster category is defined to be $\mathcal{C}_{m}(H):=\mathcal{D}^{b}(H)/\tau^{-1}[m]$ where $H$ is a finite dimensional hereditary algebra over a field $k$ (which we take to be algebraically closed) whose corresponding quiver has no loops or oriented 2-cycles; $\tau$ is the Auslander-Reiten (AR) translate in the bounded derived category $\mathcal{D}^{b}(H)$; $[1]$ denotes the suspension functor present in the triangulated structure of $\mathcal{D}^{b}(H)$ and $[m]$ denotes its $m$-th power, that is $[1]^{m}=[m]$.
In $m$-cluster categories one can define $m$-cluster tilting objects as follows [21].

It is the endomorphism algebras of these $m$-cluster tilting objects, $\operatorname{End}_{\mathcal{C}_{m}(H)}(T)$, called $m$-cluster tilted algebras, that we study in this paper. We focus on the case where $H$ is an hereditary algebra of Dynkin type $A_{n}$.
Our aim is to generalize a result of Buan and Vatne, [9], on the derived equivalence classification of 1-cluster tilted algebras of type $A_{n}$.
In section 2 we give a local description of $m$-cluster tilted algebras of type $A_{n}$ as quivers with relations using the combinatorics of $m$-cluster complexes [11, section 4] which can be related to $m$-cluster tilting objects. More precisely, it can be proven that the maximal faces, or facets, of the generalized cluster complex correspond bijectively to the $m$-cluster tilting objects in $\mathcal{C}_{m}(kA_{n})$ [21], [18]. Then, the combinatorial model for the generalized cluster complex in type $A_{n}$ can be used to determine the $m$-cluster tilted algebra corresponding to a given $m$-cluster tilting object.
This description of the $m$-cluster tilted algebras based on the combinatorial model for the generalized cluster complex enables us to prove that $m$-cluster tilted algebras of type $A_{n}$ are gentle and that if the quiver of an $m$-cluster tilted algebra of type $A_{n}$ contains cycles they must be of length $m+2$ and must have full relations, that is the composition of any two consecutive arrows in the cycle must be a relation. These $m+2$-cycles play a crucial rôle when considering derived equivalences of $m$-cluster tilted algebras. Details of these statements can be found in section 2.
Our main theorem can be stated as follows.

The statement of theorem 1.2 must allow for the possibility that for $m\geqslant 2$ the $m$-cluster tilted algebras of type $A_{n}$ may be disconnected. That these algebras can be disconnected follows from the combinatorial description in section 2. This is one of the noticeable differences between the $m=1$ and $m\geqslant 2$ cases, since 1-cluster tilted algebras of type $A_{n}$ are connected [9].
After section 2 the paper is structured as follows. In section 3 we provide tilting complexes which prove that there exist derived equivalences between $m$-cluster tilted algebras which preserve $m+2$-cycles with full relations. In this section we also describe how these tilting complexes are related to the combinatorial model of $m$-cluster tilted algebras in type $A_{n}$. Then we use [5, thm 4.1,cor 4.3], which states that the Cartan matrix of a gentle algebra is unimodular equivalent to a diagonal matrix determined by the cycles with full relations which appear in the description of that algebra as a quiver with relations, to identify the necessary conditions for derived equivalence of two connected components of an $m$-cluster tilted algebra of type $A_{n}$. The form of the diagonal matrix is restated in section 3.
Finally, in section 4, we give the algorithm which uses the results of section 3 to mutate any connected component of an $m$-cluster tilted algebra to a particular $m$-cluster tilted algebra, referred to as the normal form (to be defined in section 3), in such a way that a derived equivalence is achieved.

We begin with a brief summary of the combinatorial background from [11, section 4],[14, section 4], [19].
The $m$-co-ordinate system, on the translation quiver $\mathbb{Z}A_{n}$ is defined as shown in the diagram below.

We now recall the definition of the mesh category associated to the translation quiver $\mathbb{Z}A_{n}$. The objects of this category are the vertices of $\mathbb{Z}A_{n}$ and morphisms are the arrows of $\mathbb{Z}A_{n}$, subject to the mesh relations. For each arrow $\alpha:x\rightarrow y$ denote by $\sigma(\alpha)$ the unique arrow $\sigma(\alpha):\tau(y)\rightarrow x$. Notice that the uniqueness of $\sigma(\alpha)$ follows from the bijection between the arrows starting at $x$ and the arrows ending at $x$. The mesh relations are given by,

for each vertex $y\in(\mathbb{Z}A_{n})_{0}$.
Next for a vertex $x=[rm,sm+1]\in\mathbb{Z}A_{n}$, where $r\in\mathbb{Z}$ and $r+1\leqslant s\leqslant r+n$, $H^{-}(x)$ and $H^{+}(x)$ as the vertices of $\mathbb{Z}A_{n}$ enclosed by and on the boundary of the regions specified in the following figure.

Define the following automorphisms on $\mathbb{Z}A_{n}$:

Further, $\tau_{i}:=\tau\omega^{i-1}$.

Results by D. Happel show that for a quiver $Q$ of Dynkin type $\Delta$ the AR-quiver of the bounded derived category $\mathcal{D}^{b}(kQ)$ is isomorphic as a stable translation quiver to $\mathbb{Z}\Delta$ [13, I,5.6]. From this result it follows that $\omega$ as defined above in type $A_{n}$ is the action of the inverse of the suspension $[1]$ in $\mathcal{D}^{b}(kA_{n})$ expressed in terms of the $m$-co-ordinate system. Also the automorphism $\tau$ on $\mathbb{Z}A_{n}$ is the action of the AR-translate in $\mathcal{D}^{b}(kA_{n})$ expressed in terms of the $m$-co-ordinate system. It is also proven in [13, I,5.6] that for $Q$ a quiver of Dynkin type the mesh category $k(\mathbb{Z}\Delta)$ is equivalent to $\operatorname{ind}\mathcal{D}^{b}(kQ)$, where $Q$ is of type $\Delta$. Here $\operatorname{ind}\mathcal{D}^{b}(kQ)$ denotes the subcategory of $\mathcal{D}^{b}(kQ)$ whose objects are the indecomposable objects of $\mathcal{D}^{b}(kQ)$ and whose morphisms are the irreducible morphisms between indecomposable objects in $\mathcal{D}^{b}(kQ)$.
We can construct the quotient translation quiver $\mathbb{Z}A_{n}/\langle\tau_{m+1}\rangle$ by identifying the vertices of $\mathbb{Z}A_{n}$ with their $\tau_{m+1}$-shifts.

Now we present some polygonal combinatorics which appears in [11] in connection with the generalized cluster complexes of type $A_{n}$. We denote by $P$ a (regular convex) $N$-gon where $N\in\mathbb{N}$.

We label the vertices of $P$ from 0 to $N-1$ in an anti-clockwise direction, then we denote diagonals in $P$ by $d(i,j)$ where $d(i,j)$ meets the $P$ at the vertices labeled $i$ and $j$.

The following lemma is easy to prove.

Lemma 2.6 implies that an $(n+1)m+2$-gon, $P$, can be divided into $m+2$-gons by non-crossing $m$-allowable diagonals.

We will need the next proposition which appears in [18] in the proof of proposition 2.14.

The next result appears in [6, sec. 1] (for $m=1$, but which holds for any value of $m\geqslant 1$) relates the mesh category associated to the quotient translation quiver $\mathbb{Z}A_{n}/\langle\tau_{m+1}\rangle$ to the full subcategory category $\operatorname{ind}(\mathcal{C}_{m}(kA_{n}))$ of $\mathcal{C}_{m}$.

From now on let $P$ denote a regular convex polygon with $(n+1)m+2$ vertices. Any division of $P$ by $m$-allowable diagonals will be assumed to be maximal and all $m$-cluster-tilted algebras will be of type $A_{n}$. We will denote by $\mathcal{C}_{m}$ the $m$-cluster category $\mathcal{D}^{b}(kA_{n})/\tau^{-1}[m]$. Given a division, $\mathcal{T}$, of a regular convex $(n+1)m+2$-gon into $m+2$-gons by non-crossing $m$-allowable diagonals we describe a finite quiver, $Q_{\mathcal{T}}$. We also describe a set of relations, $\mathcal{I}_{\mathcal{T}}$, in the path algebra, $kQ_{\mathcal{T}}$, and show the algebra $kQ_{\mathcal{T}}/\mathcal{I}_{\mathcal{T}}$ is isomorphic to an $m$-cluster-tilted algebra. We will show that all $m$-cluster-tilted algebras can be realized in this way.

Let $\mathcal{T}$ denote the set of $m$-allowable diagonals in a division of $P$. The vertices of the quiver $Q_{\mathcal{T}}$ are in one-to-one correspondence with the elements of $\mathcal{T}$. For any two vertices $i$ and $j$ in $Q_{\mathcal{T}}$, there is an arrow $i\rightarrow j$ if and only if:

1. (1)

   the corresponding $m$-allowable diagonals, $d_{i}$ and $d_{j}$ share a vertex in the $(n+1)m+2$-gon.

2. (2)

   $d_{i}$ and $d_{j}$ are edges of the same $m+2$-gon in the division.

3. (3)

   $d_{j}$ follows $d_{i}$ in a clockwise direction.

In [21] the following statement is proven.

The result also follows from [18, thm 1.01].
For a given division $\mathcal{T}$ of $P$ let $T$ denote the corresponding $m$-cluster-tilting object under the bijection of proposition 2.13 and let $\operatorname{End}_{\mathcal{C}_{m}}(T)\cong kQ/I$ denote the $m$-cluster-tilted algebra.

Next we define $\mathcal{I}_{\mathcal{T}}$. Given consecutive arrows $i\stackrel{{\scriptstyle\alpha}}{{\rightarrow}}j\stackrel{{\scriptstyle\beta}}{{\rightarrow}}k$ in $Q_{\mathcal{T}}$ we define the path to be zero if $d_{i}$, $d_{j}$ and $d_{k}$ are in the same $m+2$-gon in the division of $P$. Let $\mathcal{I}_{\mathcal{T}}$ denote the ideal in the path algebra $kQ_{\mathcal{T}}$ generated by such relations.

The following proposition was known for the $m=1$ case [10], here we state the result for $m\geqslant 1$.

We have now proven that $kQ_{\mathcal{T}}$/$\mathcal{I}_{\mathcal{T}}$ is isomorphic to the $m$-cluster-tilted algebra corresponding to $T$. Since every $m$-cluster-tilting object can be described by a division of $P$ [21] every $m$-cluster-tilted algebra can be realized in this way.

The remainder of this paper will demonstrate theorem 1.2.
We begin with the following observation.

Suppose that we have some division of $P$. Notice that each $m$-allowable diagonal, $d$, is a common edge of two $m+2$-gons and that for a given $d_{0}$ (as in figure the next example) there are $m$-allowable diagonals $d_{1},d_{2},\ldots,d_{m}$ of $P$ which we view as anti-clockwise “rotations” of $d_{0}$ inside the two $m+2$-gons of which it is an edge. This is a generalization of the well know concept of quadrilateral exchange in triangulations of polygons, indeed quadrilateral exchange is the $m=1$ case.
It is clear that for any given $m$-allowable diagonal, $d_{0}\in\mathcal{T}$, there exist $m$ distinct $m$-allowable diagonals $d_{1},d_{2},\ldots,d_{m}$ which could replace $d_{0}$ and provide a distinct division of $P$.

For the most part we will be interested in applying $\mu_{m}$ or $\mu_{m}^{-1}$ which can be thought of as “rotations”of an $m$-allowable diagonal in the anti-clockwise and clockwise directions respectively. Given a division of $P$ we can apply the elementary polygonal moves to the $m$-allowable diagonals in the division to create distinct divisions. As in section 2 an $m$-cluster-tilted algebra can be associated with the original division and with the new division and we say that we have mutated the algebra corresponding to the original division to the algebra corresponding to the new division.

We now wish to show that using the elementary polygonal moves we can induce algebra mutations such that the original and new $m$-cluster-tilted algebras are derived equivalent. The following theorem, due to Bessenrodt and Holm [5] plays a crucial role in determining when this is possible.

Recall that the Cartan matrix for a finite dimensional algebra $\Lambda\cong kQ/I$ is given by $C_{\Lambda}=[c_{i,j}]_{1\leqslant i,j,\leqslant|Q_{0}|}$ where $c_{i,j}=dim_{k}\operatorname{Hom}_{\Lambda}(P_{i},P_{j})=dim_{k}\operatorname{Hom}_{\Lambda}(e_{i}\Lambda,e_{j}\Lambda)=dim_{k}\,\,e_{j}\Lambda e_{i}$. Here $e_{i}$ is a principle idempotent and $e_{i}\Lambda\cong P_{i}$ are the indecomposable projectives.
With the above conventions the columns of the Cartan matrix are the dimension vectors of the indecomposable projective right $\Lambda$ modules. For an $m$-cluster tilted algebra of type $A_{n}$ we have the following basic properties of the Cartan matrix which follow from the combinatorial descriptions in section 2.

• •

  The entries of $C_{\Lambda}$ are either 0 or 1.

• •

  There are at most $n$ entries which are 1 in any given row or column. In the case where there are $n$ entries which are 1 in some column then the $m$-cluster tilted algebra in question is isomorphic to the path algebra of the linear orientation of the Dynkin diagram $A_{n}$.

• •

  For a vertex $i$ in a cycle there are at least two entries which are 1 in the $i$-th row.

It is well known that the unimodular equivalence class of the Cartan matrix of a finite dimensional algebra is a derived invariant. Therefore since the $m$-cluster-tilted algebras of type $A_{n}$ are gentle we have, by theorem 3.7, that any two $m$-cluster-tilted algebras which are derived equivalent have the same number of $m+2$-cycles. Hence in order to mutate an $m$-cluster-tilted algebra and induce a derived equivalence we must ensure that we preserve the number of $m+2$-cycles.

It is our aim to show that each connected component of an $m$-cluster-tilted algebra with a given number, $r\in\mathbb{N}_{0}$ of $m+2$-cycles is derived equivalent to the following normal form having the same number of $m+2$-cycles.

We remark that there are other possible choices for the normal form in which the number of vertices of types A and B would be different.

To prove our claim that all connected components of $m$-cluster-tilted algebras are derived equivalent to the quiver with relations specified by the normal form we employ mutations induced by the elementary polygonal moves which result in derived equivalent original and mutated algebras.
Shortly we give a list containing local descriptions of some algebra mutations induced by the elementary polygonal moves which preserve the number of $m+2$-cycles occurring in the quiver of an $m$-cluster-tilted algebra. We will use these extensively in section 4. Before presenting the list some explanation of the information contained in the figures below is required.
Denote by $\Lambda$ the original $m$-cluster-tilted algebra and by $\Lambda^{\prime}$ the mutated algebra. Each figure contains four diagrams. Two of these are parts of quivers with relations, the first of which is some local situation in $\Lambda$. The circled vertex in this first diagram denotes the vertex of the quiver at which the algebra mutation corresponding to the elementary move takes place. The second of the quiver diagrams shows the resultant local situation in $\Lambda^{\prime}$. The curved dotted line appearing in both quiver diagrams indicates an $m+2$-cycle.
The third and fourth diagrams are the local polygonal configurations corresponding to $\Lambda$ and $\Lambda^{\prime}$ respectively. The dashed line in the fourth diagram illustrates the new position of the $m$-allowable diagonal corresponding to the mutation vertex after the application of an elementary polygonal move.
The edges which appear in bold typeface correspond to boundary edges of $P$. It should be noted that one edge in bold typeface may correspond to as many as $m+1$ edges of $P$. Each figure is general in the sense that it is independent of the values of $m$ and $n$, though of course we assume $n$ is large enough to allow the configurations we show to exist.
In some figures we may make further assumptions on the quiver of $\Lambda$, if this is the case then these assumptions will be stated in a short remark.
The $(D)$ which sometimes appears below the numbering denotes a mutation which has arisen from $\mu_{m}^{-1}$ at the mutation vertex.

There is no relation involving an arrow in $Q_{2}$ and the arrow going into the mutation vertex (and not in the cycle) which is shown.

In the two figures immediately above we assume that there is no relation involving an arrow in $Q_{3}$ and the arrow into, or our of, the mutation vertex.

Here we assume there is no relation between an arrow from $Q_{2}$ and the arrow going into the mutation vertex (which is not in the cycle) and that there is no relation between the arrow going out of the mutation vertex (which is not in the cycle) and any arrow in $Q_{3}$.

There is no relation involving an arrow in $Q_{2}$ and the arrow going out of the mutation vertex (and not in the cycle) which is shown.

In the two figures immediately above we assume that there is no relation involving an arrow in $Q_{3}$ and the arrow into, or our of, the mutation vertex.

Here we assume there is no relation between an arrow from $Q_{2}$ and the arrow going into the mutation vertex (which is not in the cycle) and that there is no relation between the arrow going out of the mutation vertex (which is not in the cycle) and any arrow in $Q_{3}$.

There are two groups of diagrams. The first group are those numbered (i) to (xiii). The second consists of the remaining figures (xiv) to (xxvi). The two groups will be used at different stages in the algorithm described in section 4 which reduces any connected component of an $m$-cluster-tilted algebra of type $A_{n}$ to our chosen normal form in definition 3.9.

In order to demonstrate that in each of the local algebra mutations above the original and mutated algebras are derived equivalent we now provide explicit tilting complexes which achieve this. There are two such complexes, one corresponding to each of the elementary polygonal moves in definition 3.4.
Before stating the next theorem we must define the notations used. Let $\Lambda\cong kQ_{\mathcal{T}}/\mathcal{I}_{\mathcal{T}}$ be an $m$-cluster-tilted algebra and let $P_{i}$ denote the finitely generated projective right $\Lambda$-module, $e_{i}\Lambda$ where $e_{i}$ is the primitive idempotent corresponding to $i\in(Q_{\mathcal{T}})_{0}$ (the set of vertices of $Q_{\mathcal{T}}$).
Suppose we mutate $\Lambda$ to $\Lambda^{\prime}$ via an elementary polygonal move, then we distinguish certain vertices in $({Q_{\mathcal{T}}})_{0}$. We denote by $mut$ the mutation vertex. If there exist arrows in $Q_{\mathcal{T}}$ whose target is $mut$ then we label the source $in_{t}$ and the arrow $i_{t}$, where $1\leqslant t\leqslant 2$ (since $\Lambda$ is gentle there can be at most two ingoing and two outgoing arrows). We denote by $out_{u}$, $1\leqslant u\leqslant 2$, the targets of any arrows out of $mut$ and label such arrows $o_{u}$. Since the $m$-cluster-tilted algebras in this case are gentle we may adopt the convention that should the paths $o_{2}i_{1}$ and $o_{1}i_{2}$ exist they are zero.
Further, should there exist an arrow $\gamma$ such that $i_{t}\gamma=0$ then we label the source of $\gamma$ by $pre_{t}$. Similarly, if there exists an arrow $\delta$ such that $\delta o_{u}=0$ then we label the target of $\delta$ by $post_{u}$.
Next, $P(\Lambda)$ denotes the full subcategory of finitely generated projective right $\Lambda$-modules; $K^{b}(P(\Lambda))$ denotes the homotopy category of bounded complexes of finitely generated projective right $\Lambda$-modules; and $\mathcal{D}^{b}(\Lambda)$ is the bounded derived category of finitely generated $\Lambda$-modules. Our last notational convention is that given an arrow $\alpha:i\rightarrow j$ between any two vertices $i,j\in(Q_{\mathcal{T}})_{0}$, the map $P_{i}\rightarrow P_{j}$ between the projective indecomposable $\Lambda$-modules induced by left multiplication by $\alpha$ will also be denoted $\alpha$.

Part of the proof of theorem 3.12 (and part of the proof of theorem 3.15) involves determining the existence of maps between certain indecomposable summands of a tilting complex. In order to do this we use Happel’s alternating sum formula [13, chapter III, 1.3, 1.4].

The next two results prove that when the elementary polygonal move $\mu_{m}$ defined in definition 3.4 is applied in the figures labeled (i)-(v),(viii)-(xiii),(ixx) and (xxi) above there is a derived equivalence between the initial algebra $\Lambda$ and the new algebra $\Lambda^{\prime}$.