On the radical of Cluster tilted algebras (original) (raw)

ON THE RADICAL OF CLUSTER TILTED ALGEBRAS

CLAUDIA CHAIO AND VICTORIA GUAZZELLI

Abstract

We determine the minimal lower bound nn, with n≥1n \geq 1, where the nn-th power of the radical of the module category of a representation-finite cluster tilted algebra vanishes. We give such a bound in terms of the number of vertices of the underline quiver. Consequently, we get the nilpotency index of the radical of the module category for representation-finite self-injective cluster tilted algebras. We also study the non-zero composition of m,m≥2m, m \geq 2, irreducible morphisms between indecomposable modules in representation-finite cluster tilted algebras lying in the (m+1)(m+1)-th power of the radical of their module category.

INTRODUCTION

Let AA be a finite dimensional algebra over an algebraically closed field. The representation theory of an algebra AA deals with the study of the module category of finitely generated AA-modules, mod A\bmod A. A fundamental tool in the study of mod A\bmod A is the Auslander-Reiten theory, based on irreducible morphisms and almost split sequences.

For X,Y∈ mod AX, Y \in \bmod A, we denote by ℜ(X,Y)\Re(X, Y) the set of all morphisms f:X→Yf: X \rightarrow Y such that, for all indecomposable AA-module MM, each pair of morphisms h:M→Xh: M \rightarrow X and h′:Y→Mh^{\prime}: Y \rightarrow M the composition h′fhh^{\prime} f h is not an isomorphism. Inductively, the powers of ℜ(X,Y)\Re(X, Y) are defined.

There is a close connection between irreducible morphisms and the powers of the radical, given by a well-known result proved by R. Bautista which states that if f:X→Yf: X \rightarrow Y is a morphism between indecomposable AA-modules then ff is irreducible if and only if f∈ℜ(X,Y)\ℜ2(X,Y)f \in \Re(X, Y) \backslash \Re^{2}(X, Y), see [2].

In case that ℜn(M,N)=0\Re^{n}(M, N)=0 for some positive integer nn and for all MM and NN in mod A\bmod A, we write this fact by the expression ℜn( mod A)=0\Re^{n}(\bmod A)=0. We recall that an algebra AA is representation-finite (or of finite representation type) if and only if there is a positive integer nn such that ℜn( mod A)\Re^{n}(\bmod A) vanishes, (see [1] p. 183).

In [17], S. Liu defined the notion of degree of an irreducible morphism (see 1.5) which has been a powerful tool to study, between others problems, the one concerning the nilpotency of the radical of a module category of an algebra AA, in case that we deal with finite-dimensional kk-algebras over an algebraically closed field of finite representation type.

If AA is a finite dimensional basic algebra over an algebraically closed field then we know that A≃kQA/IAA \simeq k Q_{A} / I_{A}. In addition, if AA is representation-finite then by [10] we know that all irreducible epimorphisms and all irreducible monomorphisms are of finite left and right degree, respectively. In particular, the irreducible monomorphism ιa:rad⁡(Pa)↪Pa\iota_{a}: \operatorname{rad}\left(P_{a}\right) \hookrightarrow P_{a} where PaP_{a} is the projective module corresponding to the vertex aa in QAQ_{A}, has finite right degree. Dually, the irreducible epimorphism ga:Ia→Ia/soc⁡(Ia)g_{a}: I_{a} \rightarrow I_{a} / \operatorname{soc}\left(I_{a}\right) where IaI_{a} is the injective module corresponding to the vertex aa in QAQ_{A}, has finite left degree. We denote by SaS_{a} the simple AA-module corresponding to the vertex aa in QAQ_{A}.

By [8], we know that for a finite dimensional algebra over an algebraically closed field A≃A \simeq kQA/IAk Q_{A} / I_{A} where AA is representation-finite we can compute the nilpotency index rAr_{A} of ℜ( mod A)\Re(\bmod A) by

[1]

  1. 2000 Mathematics Subject Classification. 16G70, 16G20, 16E10.
    Key words and phrases. irreducible morphisms, radical, projective cover, injective hull. ↩︎

max⁡{ra}a∈(QA)0+1\max \left\{r_{a}\right\}_{a \in\left(Q_{A}\right)_{0}}+1 where rar_{a} is equal to the length of any path of irreducible morphisms between indecomposable modules from the projective PaP_{a} to the injective IaI_{a}, going through the simple SaS_{a}.

Applying the above mentioned result we give the minimal positive integer mm such that ℜ( mod Γ)\Re(\bmod \Gamma) vanishes, where Γ\Gamma is a cluster tilted algebra of type Δ‾\overline{\Delta}, with Δ\Delta a Dynkin quiver. More precisely, we prove Theorem A and B.

Theorem A. Let C\mathcal{C} be the cluster category of a representation-finite hereditary algebra HH. let Tˉ\bar{T} be an almost complete tilting object in C\mathcal{C} with complements MM and M∗M^{*}. Consider Γ=End⁡C(T)op\Gamma=\operatorname{End}_{\mathcal{C}}(T)^{o p} and Γ′=End⁡C(T′)op\Gamma^{\prime}=\operatorname{End}_{\mathcal{C}}\left(T^{\prime}\right)^{o p} the cluster tilted algebras with T=Tˉ⊕MT=\bar{T} \oplus M and T′=Tˉ⊕M∗T^{\prime}=\bar{T} \oplus M^{*}. Let rΓr_{\Gamma} and rΓ′r_{\Gamma^{\prime}} be the nilpotency indices of ℜ( mod Γ)\Re(\bmod \Gamma) and ℜ( mod Γ′)\Re\left(\bmod \Gamma^{\prime}\right), respectively. Then, rΓ=rΓ′r_{\Gamma}=r_{\Gamma^{\prime}}.

Theorem B. Let Δ\Delta be a Dynkin quiver and let Γ\Gamma be a cluster tilted algebra of type Δ‾\overline{\Delta}. Let rΓr_{\Gamma} be the nilpotency index of ℜ( mod Γ)\Re(\bmod \Gamma). Then the following conditions hold.
(a) If Δ‾=An\overline{\Delta}=A_{n}, then rΓ=nr_{\Gamma}=n for n≥1n \geq 1.
(b) If Δ‾=Dn\overline{\Delta}=D_{n}, then rΓ=2n−3r_{\Gamma}=2 n-3 for n≥4n \geq 4.
© If Δ‾=E6\overline{\Delta}=E_{6}, then rΓ=11r_{\Gamma}=11.
(d) If Δ‾=E7\overline{\Delta}=E_{7}, then rΓ=17r_{\Gamma}=17.
(e) If Δ‾=E8\overline{\Delta}=E_{8}, then rΓ=29r_{\Gamma}=29.

We observe that the nilpotency index of the radical of the module category of a cluster tilted algebra of type Δ‾\overline{\Delta} with Δ\Delta a Dynkin quiver, coincide with the nilpotency index of the radical of the module category of the hereditary algebra kΔk \Delta.

The non-zero composition of nn irreducible morphisms between indecomposable modules could belong to ℜn+1\Re^{n+1}. In the last years, there have been many works done in this direction. The first to give a partial solution to that problem were K. Igusa and G. Todorov in [16], where they proved that if X0→f1X1→⋯→Xn−1→fnXnX_{0} \xrightarrow{f_{1}} X_{1} \rightarrow \cdots \rightarrow X_{n-1} \xrightarrow{f_{n}} X_{n} is a sectional path then fn…f1f_{n} \ldots f_{1} lies in ℜn(X0,Xn)\Re^{n}\left(X_{0}, X_{n}\right) but not in ℜn+1(X0,Xn)\Re^{n+1}\left(X_{0}, X_{n}\right).

In [9], F. U. Coelho, S. Trepode and the first named author characterized when the composition of two irreducible morphisms is non-zero and lies in ℜ3( mod A)\Re^{3}(\bmod A) for AA an artin algebra. In [11], P. Le Meur, S. Trepode and the first named author solved the problem of when the composition of nn irreducible morphisms between indecomposable modules is non-zero and belongs to ℜn+1( mod A)\Re^{n+1}(\bmod A) for finite dimensional kk-algebras over a perfect field kk.

As a consequence of the results of this work, we obtain when the composition of nn irreducible morphisms between indecomposable AA-modules belongs to the n+1n+1 power of the radical of their module category, for a representation-finite cluster tilted algebra AA. More precisely, we prove Theorem C.

Theorem C. Let Γ\Gamma be a representation-finite cluster tilted algebra. Consider the irreducible morphisms hi:Xi→Xi+1h_{i}: X_{i} \rightarrow X_{i+1}, with Xi∈ind⁡ΓX_{i} \in \operatorname{ind} \Gamma for 1≤i≤m1 \leq i \leq m. Then hm…h1∈ℜm+1(X1,Xm+1)h_{m} \ldots h_{1} \in \Re^{m+1}\left(X_{1}, X_{m+1}\right) if and only if hm…h1=0h_{m} \ldots h_{1}=0.

The authors thankfully acknowledge partial support from CONICET and Universidad Nacional de Mar del Plata, Argentina. The authors also thanks Ana Garcia Elsener for useful conversations. The first author is a CONICET researcher.

1. Preliminaries

Throughout this work, by an algebra we mean a finite dimensional basic kk-algebra over an algebraically closed field, kk.

1.1. Notions on quivers and algebras

A quiver QQ is given by a set of vertices Q0Q_{0} and a set of arrows Q1Q_{1}, together with two maps s,e:Q1→Q0s, e: Q_{1} \rightarrow Q_{0}. Given an arrow α∈Q1\alpha \in Q_{1}, we write s(α)s(\alpha) the starting vertex of α\alpha and e(α)e(\alpha) the ending vertex of α\alpha. We denote by Qˉ\bar{Q} the underlying graph of QQ. For each algebra AA there is a quiver QAQ_{A}, called the ordinary quiver of AA, such that AA is the quotient of the path algebra kQAk Q_{A} by an admissible ideal.

Let AA be an algebra. We denote by mod A\bmod A the category of finitely generated left AA-modules and by ind AA the full subcategory of mod A\bmod A which consists of one representative of each isomorphism class of indecomposable AA-modules.

We say that AA is a representation-finite algebra if there is only a finite number of isomorphisms classes of indecomposable A-modules.

We denote by ΓA\Gamma_{A} the Auslander-Reiten quiver of mod A\bmod A, and by τ\tau the Auslander-Reiten translation DTr with inverse TrD denoted by τ−1\tau^{-1}.

1.2. On the radical of a module category

A morphism f:X→Yf: X \rightarrow Y, with X,Y∈ mod AX, Y \in \bmod A, is called irreducible provided it does not split and whenever f=ghf=g h, then either hh is a split monomorphism or gg is a split epimorphism.

If X,Y∈ mod AX, Y \in \bmod A, the ideal ℜ(X,Y)\Re(X, Y) of Hom⁡(X,Y)\operatorname{Hom}(X, Y) is the set of all the morphisms f:X→Yf: X \rightarrow Y such that, for each M∈ind⁡AM \in \operatorname{ind} A, each h:M→Xh: M \rightarrow X and each h′:Y→Mh^{\prime}: Y \rightarrow M the composition h′fhh^{\prime} f h is not an isomorphism. For n≥2n \geq 2, the powers of ℜ(X,Y)\Re(X, Y) are inductively defined. By ℜ∞(X,Y)\Re^{\infty}(X, Y) we denote the intersection of all powers ℜi(X,Y)\Re^{i}(X, Y) of ℜ(X,Y)\Re(X, Y), with i≥1i \geq 1.

By [2], it is known that for X,Y∈ind⁡AX, Y \in \operatorname{ind} A, a morphism f:X→Yf: X \rightarrow Y is irreducible if and only if f∈ℜ(X,Y)\ℜ2(X,Y)f \in \Re(X, Y) \backslash \Re^{2}(X, Y).

We recall the next proposition fundamental for our results.
Proposition 1.3. [1, V, Proposition 7.4] Let MM and NN be indecomposable modules in mod A\bmod A and let ff be a morphism in ℜn(M,N)\Re^{n}(M, N), with n≥2n \geq 2. Then, the following conditions hold.
(i) There exist a natural number ss, indecomposables AA-modules X1,…,XsX_{1}, \ldots, X_{s}, morphisms fi∈f_{i} \in ℜ(M,Xi)\Re\left(M, X_{i}\right) and morphisms gi:Xi→Ng_{i}: X_{i} \rightarrow N, with each gig_{i} a sum of compositions of n−1n-1 irreducible morphisms between indecomposable modules such that f=∑i=1sgifif=\sum_{i=1}^{s} g_{i} f_{i}.
(ii) If f∈ℜn(M,N)\ℜn+1(M,N)f \in \Re^{n}(M, N) \backslash \Re^{n+1}(M, N), then at leats one of the fif_{i} in (i)(i) is irreducible.

It is well known by a result of M. Auslander that an algebra AA is representation-finite if and only if ℜ∞( mod A)=0\Re^{\infty}(\bmod A)=0. That is, there is a positive integer nn such that ℜn(X,Y)=0\Re^{n}(X, Y)=0 for all X,YX, Y AA-modules. The minimal positive integer mm such that ℜm( mod A)=0\Re^{m}(\bmod A)=0 is called the nilpotency index of ℜ( mod A)\Re(\bmod A). We denote such an index by rAr_{A}.

1.4. Basic definitions of paths

A path in mod A\bmod A is a sequence M0→f1M1→f2M2→…→Mn−1→fnMnM_{0} \xrightarrow{f_{1}} M_{1} \xrightarrow{f_{2}} M_{2} \rightarrow \ldots \rightarrow M_{n-1} \xrightarrow{f_{n}} M_{n} of non-zero nonisomorphisms f1,…,fnf_{1}, \ldots, f_{n} between indecomposable AA-modules with n≥1n \geq 1. In case that f1,…,fnf_{1}, \ldots, f_{n} are irreducible morphisms, we say that the path is in ΓA\Gamma_{A} or equivalently that it is a path in ΓA\Gamma_{A}. The length of a path in ΓA\Gamma_{A} is defined as the number of irreducible morphisms (not necessarily different) involved in the path.

Let us recall that paths in ΓA\Gamma_{A} having the same starting vertex and the same ending vertex are called parallel paths.

Let Γ\Gamma be a component of ΓA\Gamma_{A}. We say that Γ\Gamma is a component with length if parallel paths in Γ\Gamma have the same length. Otherwise, it is called a component without length, see [12].

By a directed component we mean a component Γ\Gamma that there is no sequence M0→f1M1→f2M2→M_{0} \xrightarrow{f_{1}} M_{1} \xrightarrow{f_{2}} M_{2} \rightarrow …→Mn−1→fnMn\ldots \rightarrow M_{n-1} \xrightarrow{f_{n}} M_{n} of non-zero non-isomorphisms f1,…,fnf_{1}, \ldots, f_{n} between indecomposable AA-modules with M0=MnM_{0}=M_{n}.

Given a directed component Γ\Gamma of ΓA\Gamma_{A}, its orbit graph O(Γ)O(\Gamma) is a graph defined as follows: the points of O(Γ)O(\Gamma) are the τ\tau-orbits O(M)O(M) of the indecomposable modules M in Γ\Gamma. There is an edge between O(M)O(M) and O(N)O(N) in O(Γ)O(\Gamma) if there are positive integer n,mn, m and either an irreducible morphism from τmM\tau^{m} M to τnN\tau^{n} N or from τnN\tau^{n} N to τmM\tau^{m} M in mod A\bmod A.

Note that if the orbit graph O(Γ)O(\Gamma) is of tree-type, then Γ\Gamma is a simply connected translation quiver, and by [3] we know that Γ\Gamma is a component with length.

1.5. On the nilpotency index of the radical of a module category

We say that the depth of a morphism f:M→Nf: M \rightarrow N in mod A\bmod A is infinite if f∈ℜ∞(M,N)f \in \Re^{\infty}(M, N); otherwise, the depth of ff is the integer n≥0n \geq 0 for which f∈ℜn(M,N)f \in \Re^{n}(M, N) but f∉ℜn+1(M,N)f \notin \Re^{n+1}(M, N). We denote the depth of ff by dp⁡(f)\operatorname{dp}(f).

Next, we recall the definition of degree of an irreducible morphism given by S. Liu in [17].
Let f:X→Yf: X \rightarrow Y be an irreducible morphism in mod A\bmod A, with XX or YY indecomposable. The left degree dl(f)d_{l}(f) of ff is infinite, if for each integer n≥1n \geq 1, each module Z∈ind⁡AZ \in \operatorname{ind} A and each morphism g:Z→Xg: Z \rightarrow X with dp⁡(g)=n\operatorname{dp}(g)=n we have that fg∉ℜn+2(Z,Y)f g \notin \Re^{n+2}(Z, Y). Otherwise, the left degree of ff is the least natural mm such that there is an AA-module ZZ and a morphism g:Z→Xg: Z \rightarrow X with dp⁡(g)=m\operatorname{dp}(g)=m such that fg∈ℜm+2(Z,Y)f g \in \Re^{m+2}(Z, Y).

The right degree dr(f)d_{r}(f) of an irreducible morphism ff is dually defined.
In order to compute the nilpotency index of the radical of any module category we shall strongly used [8, Theorem A]. For the convenience of the reader, we state below such a result.

Let A=kQA/IAA=k Q_{A} / I_{A} be a representation-finite algebra. Let a∈(QA)0a \in\left(Q_{A}\right)_{0} and Pa,IaP_{a}, I_{a} and SaS_{a} be the projective, injective and simple AA-modules, respectively, corresponding to the vertex aa.

For each a∈(QA)0a \in\left(Q_{A}\right)_{0}, let nan_{a} be the number defined as follows:

Dually, for each a∈(QA)0a \in\left(Q_{A}\right)_{0}, let mam_{a} be the number defined as follows:

We write ra=ma+nar_{a}=m_{a}+n_{a}.
Theorem 1.6. [8, Theorem A] Let A≃kQA/IAA \simeq k Q_{A} / I_{A} be a finite dimensional algebra over an algebraically closed field and assume that AA is a representation-finite algebra. Then the nilpotency index rAr_{A} of ℜ( mod A)\Re(\bmod A) is rA=max⁡{ra}a∈(QA)0+1r_{A}=\max \left\{r_{a}\right\}_{a \in\left(Q_{A}\right)_{0}}+1.

Following [8, Remark 1], rar_{a} is equal to the length of any path of irreducible morphisms between indecomposable modules from the projective PaP_{a} to the injective IaI_{a}, going through the simple SaS_{a}.

Finally, we recall a result of [8] that shall be useful in this work.
Lemma 1.7. [8, Lemma 2.4] Let A≅kQ/IA \cong k Q / I be a representation-finite algebra. Given a∈Q0a \in Q_{0}, consider rar_{a} the number defined as above. Then, the following conditions hold.

(a) Every non-zero morphism f:Pa→Iaf: P_{a} \rightarrow I_{a} that factors through the simple AA-module SaS_{a} is such that dp⁡(f)=ra\operatorname{dp}(f)=r_{a}.
(b) Every non-zero morphism f:Pa→Iaf: P_{a} \rightarrow I_{a} which does not factor through the simple AA - module SaS_{a} is such that dp⁡(f)=k\operatorname{dp}(f)=k, with 0≤k<ra0 \leq k<r_{a}.

1.8. The cluster category

Let HH be a hereditary algebra. We denote by D=Db( mod H)\mathcal{D}=\mathcal{D}^{b}(\bmod H) the bounded derived category of mod H\bmod H. The cluster category, C\mathcal{C}, is defined as the quotient D/F\mathcal{D} / F, where FF is the composition τD−1[1]\tau_{\mathcal{D}}^{-1}[1] of the suspension functor and the Auslander-Reiten translation in D\mathcal{D}. The objects of C\mathcal{C} are the FF-orbits of the objects in D\mathcal{D}, and the morphisms in C\mathcal{C} are defined as

Hom⁡C(X~,Y~)=∐i∈ZHom⁡D(FiX,Y)\operatorname{Hom}_{\mathcal{C}}(\widetilde{X}, \widetilde{Y})=\coprod_{i \in \mathbb{Z}} \operatorname{Hom}_{\mathcal{D}}\left(F^{i} X, Y\right)

where X,YX, Y are objects in D\mathcal{D} and X~,Y~\widetilde{X}, \widetilde{Y} are the corresponding objects in C\mathcal{C}. By [4, Proposition 1.5], the summands of (1) are almost all zero.

We recall some basic and useful properties of C\mathcal{C}.
(i) C\mathcal{C} is a Krull-Schmidt category.
(ii) C\mathcal{C} is a triangulated category, whose suspension functor over C\mathcal{C} is denoted by [1].
(iii) C\mathcal{C} has Auslander-Reiten triangles, which are induced by the Auslander-Reiten triangles of D\mathcal{D}. We also denoted the Auslander-Reiten translation of C\mathcal{C} by τ\tau.
Remark 1.9. We deduce by (iii) that the irreducible morphisms in C\mathcal{C} are induced by the irreducible morphisms in D\mathcal{D}. Moreover, the non-zero paths of irreducible morphisms between indecomposable objects in C\mathcal{C} are induced by non-zero paths of irreducible morphisms between indecomposable objects in D\mathcal{D}, and both paths have the same length.

We denote by S\mathcal{S} the set ind⁡( mod H∨H[1])\operatorname{ind}(\bmod H \vee H[1]) consisting of the indecomposable HH-modules together with the objects P[1]P[1], where PP is an indecomposable projective HH-module. We can see the set S\mathcal{S} as the fundamental domain of C\mathcal{C} for the action of FF in D\mathcal{D}, containing exactly one representative object from each FF-orbit in ind D\mathcal{D}.

It is known that given XX and YY objects in S\mathcal{S}, then Hom⁡D(FiX,Y)=0\operatorname{Hom}_{\mathcal{D}}\left(F^{i} X, Y\right)=0 for all i≠−1,0i \neq-1,0. Moreover, if HH is a representation-finite algebra, then at least one, Hom⁡D(F−1X,Y)\operatorname{Hom}_{\mathcal{D}}\left(F^{-1} X, Y\right) or Hom⁡D(X,Y)\operatorname{Hom}_{\mathcal{D}}(X, Y) vanishes.

1.10. On tilting objects

An object TT in C\mathcal{C} is said to be a tilting object if Ext⁡C1(T,T)=0\operatorname{Ext}_{\mathcal{C}}^{1}(T, T)=0 and TT is maximal with that property, that is, if Ext⁡C1(T⊕X,T⊕X)=0\operatorname{Ext}_{\mathcal{C}}^{1}(T \oplus X, T \oplus X)=0 then X∈add⁡TX \in \operatorname{add} T.

We say that an object Tˉ\bar{T} in C\mathcal{C} is an almost complete tilting object if Ext⁡C1(Tˉ,Tˉ)=0\operatorname{Ext}_{\mathcal{C}}^{1}(\bar{T}, \bar{T})=0 and there is an indecomposable object XX, which is called complement of Tˉ\bar{T}, such that Tˉ⊕X\bar{T} \oplus X is a tilting object. It is known that an almost complete tilting object Tˉ\bar{T} in C\mathcal{C} has exactly two non-isomorphic complements. We denote them by MM and M∗M^{*}.

The algebra End⁡C(T)op\operatorname{End}_{\mathcal{C}}(T)^{o p}, where TT is a tilting object in C\mathcal{C}, is called a cluster tilted algebra of type Qˉ\bar{Q}, where QQ is the quiver whose path algebra is the hereditary algebra HH, that is, H=kQH=k Q.

We denote by Γ\Gamma the cluster tilted algebra End⁡C(T)op\operatorname{End}_{\mathcal{C}}(T)^{o p}, and by Γ′\Gamma^{\prime} the cluster tilted algebra End⁡C(T′)op\operatorname{End}_{\mathcal{C}}\left(T^{\prime}\right)^{o p}, with T=Tˉ⊕MT=\bar{T} \oplus M and T′=Tˉ⊕M∗T^{\prime}=\bar{T} \oplus M^{*} where Tˉ\bar{T} is an almost complete tilting object in C\mathcal{C} with complements MM and M∗M^{*}, respectively. In [6, Theorem 1.3], the authors proved that we can pass from one algebra to the other by using mutation.

The next theorem shall be fundamental to develop some results of this paper.
Theorem 1.11. [5] Let TT be a tilting object in C\mathcal{C} and we denote by GG the functor Hom⁡C(T,−)\operatorname{Hom}_{\mathcal{C}}(T,-) : C→ mod Γ\mathcal{C} \rightarrow \bmod \Gamma. Then, the functor Gˉ:C/add⁡(τT)→ mod Γ\bar{G}: \mathcal{C} / \operatorname{add}(\tau T) \rightarrow \bmod \Gamma (induced by GG ) is an equivalence.

It follows from the above theorem that an indecomposable projective Γ\Gamma-module PuP_{u} is of the form Hom⁡C(T,Tu)\operatorname{Hom}_{\mathcal{C}}\left(T, T_{u}\right), where TuT_{u} is an indecomposable summand of TT. Moreover, it is known that the indecomposable injective Γ\Gamma-module IuI_{u}, which is the injective cover of the simple Su=top⁡PuS_{u}=\operatorname{top} P_{u}, is of the form Hom⁡C(T,τ2Tu)\operatorname{Hom}_{\mathcal{C}}\left(T, \tau^{2} T_{u}\right).

Furthermore, the Auslander-Reiten sequences in mod Γ≃C/add⁡(τT)\bmod \Gamma \simeq \mathcal{C} / \operatorname{add}(\tau T) are induced by the AuslanderReiten triangles in C\mathcal{C}. We can deduce that the irreducible morphisms in mod Γ\bmod \Gamma which do not factor through add⁡(τT)\operatorname{add}(\tau T) are induced by irreducible morphisms in C\mathcal{C}.

Consequently, a path of irreducible morphisms between indecomposable modules in mod Γ\bmod \Gamma is induced by a path of irreducible morphism between indecomposable objects in C\mathcal{C}, and both have the same length.

Finally, we recall the following important results useful for our further considerations.
Proposition 1.12. Let Tˉ\bar{T} be a cluster tilted object in C\mathcal{C} with complements MM and M∗M^{*}. We consider Γ=End⁡C(T)op \Gamma=\operatorname{End}_{\mathcal{C}}(T)^{\text {op }} and Γ′=End⁡C(T′)op \Gamma^{\prime}=\operatorname{End}_{\mathcal{C}}\left(T^{\prime}\right)^{\text {op }} with T=Tˉ⊕MT=\bar{T} \oplus M and T′=Tˉ⊕M∗T^{\prime}=\bar{T} \oplus M^{*}. Then,
(a) The Γ\Gamma-module Hom⁡C(T,τM∗)\operatorname{Hom}_{\mathcal{C}}\left(T, \tau M^{*}\right) is simple. Moreover, Hom⁡C(T,τM∗)≃top⁡Px\operatorname{Hom}_{\mathcal{C}}\left(T, \tau M^{*}\right) \simeq \operatorname{top} P_{x}, where Px=P_{x}= Hom⁡C(T,M)\operatorname{Hom}_{\mathcal{C}}(T, M).
(b) The Γ′\Gamma^{\prime}-module Hom⁡C(T′,τM)\operatorname{Hom}_{\mathcal{C}}\left(T^{\prime}, \tau M\right) is simple. Moreover, Hom⁡C(T′,τM)≃top⁡Py′\operatorname{Hom}_{\mathcal{C}}\left(T^{\prime}, \tau M\right) \simeq \operatorname{top} P_{y}^{\prime}, where Py′=P_{y}^{\prime}= Hom⁡C(T′,M∗)\operatorname{Hom}_{\mathcal{C}}\left(T^{\prime}, M^{*}\right).
Theorem 1.13. Let Tˉ\bar{T} be an almost complete tilting object in C\mathcal{C} with complements MM and M∗M^{*}. We consider Γ=End⁡C(T)op \Gamma=\operatorname{End}_{\mathcal{C}}(T)^{\text {op }} and Γ′=End⁡C(T′)op \Gamma^{\prime}=\operatorname{End}_{\mathcal{C}}\left(T^{\prime}\right)^{\text {op }} as above. Let SxS_{x} and Sy′S_{y}^{\prime} be the simples modules top⁡(Hom⁡C(T,M))\operatorname{top}\left(\operatorname{Hom}_{\mathcal{C}}(T, M)\right) and top⁡(Hom⁡C(T′,M∗))\operatorname{top}\left(\operatorname{Hom}_{\mathcal{C}}\left(T^{\prime}, M^{*}\right)\right), respectively. Then, there is an equivalence

θ: mod Γ/add⁡Sx→ mod Γ′/add⁡Sy′\theta: \bmod \Gamma / \operatorname{add} S_{x} \rightarrow \bmod \Gamma^{\prime} / \operatorname{add} S_{y}^{\prime}

Remark 1.14. We consider Γ\Gamma and Γ′\Gamma^{\prime} to be the cluster tilted algebras as in the above theorem. Let TaT_{a} be a direct summand of Tˉ\bar{T} and we denote by Pa=Hom⁡C(T,Ta)P_{a}=\operatorname{Hom}_{\mathcal{C}}\left(T, T_{a}\right) and Ia=Hom⁡C(T,τ2Ta)I_{a}=\operatorname{Hom}_{\mathcal{C}}\left(T, \tau^{2} T_{a}\right) the indecomposable projective and injective Γ\Gamma-modules corresponding to the vertex a∈QΓa \in Q_{\Gamma}, respectively. We denote by Pa′=Hom⁡C(T′,Ta)P_{a}^{\prime}=\operatorname{Hom}_{\mathcal{C}}\left(T^{\prime}, T_{a}\right) and Ia′=Hom⁡C(T′,τ2Ta)I_{a}^{\prime}=\operatorname{Hom}_{\mathcal{C}}\left(T^{\prime}, \tau^{2} T_{a}\right) the indecomposable projective and injective Γ′\Gamma^{\prime}-modules corresponding to the vertex a∈QΓ′a \in Q_{\Gamma^{\prime}}, respectively.

Following the equivalence of Theorem 1.13, note that it is not hard to see that θ(Pa)=Pa′\theta\left(P_{a}\right)=P_{a}^{\prime} and θ(Ia)=Ia′\theta\left(I_{a}\right)=I_{a}^{\prime}.
Notation 1.15. For a better understanding of the results, when we consider the cluster tilted algebras Γ≃kQΓ/IΓ\Gamma \simeq k Q_{\Gamma} / I_{\Gamma} and Γ′≃kQΓ′/IΓ′\Gamma^{\prime} \simeq k Q_{\Gamma^{\prime}} / I_{\Gamma^{\prime}}, in order to compute the nilpotency indices of ℜ( mod Γ)\Re(\bmod \Gamma) and ℜ( mod Γ′)\Re\left(\bmod \Gamma^{\prime}\right), we denote such values by ru=mu+nur_{u}=m_{u}+n_{u}, for each u∈QΓu \in Q_{\Gamma}, and by rv′=mv′+nv′r_{v}^{\prime}=m_{v}^{\prime}+n_{v}^{\prime} for each v∈QΓ′v \in Q_{\Gamma^{\prime}}, as defined in 1.5 .

2. The main results

In this section, we compute the nilpotency index of the radical of the module category of a representation-finite cluster tilted algebra.

Let HH be a hereditary algebra and Γ=End⁡C(T)op \Gamma=\operatorname{End}_{\mathcal{C}}(T)^{\text {op }} the cluster tilted algebra, where TT is a tilting object in the cluster category C=D/F,D\mathcal{C}=\mathcal{D} / F, \mathcal{D} is the bounded derived category of mod H\bmod H and F=τD−1[1]F=\tau_{\mathcal{D}}^{-1}[1].

It is well known that Γ\Gamma is representation-finite if and only if HH so is. In this case, H≃kΔH \simeq k \Delta with Δˉ\bar{\Delta} a Dynkin graph and the Auslander-Reiten quiver, Γ(D)\Gamma(\mathcal{D}), of D\mathcal{D} is isomorphic to ZΔ\mathbb{Z} \Delta.

Proposition 2.1. Let HH be a representation-finite hereditary algebra. Then the Auslander-Reiten quiver Γ(Db( mod H))\Gamma\left(\mathcal{D}^{b}(\bmod H)\right) is a component with length.

Proof. We analyze the orbit graph of Γ(Db( mod H))\Gamma\left(\mathcal{D}^{b}(\bmod H)\right). Since HH is a representation-finite hereditary algebra, we have that H≃kΔH \simeq k \Delta, with Δˉ\bar{\Delta} a Dynkin graph. Moreover, Γ(Db( mod H))≃ZΔ\Gamma\left(\mathcal{D}^{b}(\bmod H)\right) \simeq \mathbb{Z} \Delta. It is clear that the orbit graph of Γ(Db( mod H))\Gamma\left(\mathcal{D}^{b}(\bmod H)\right) is isomorphic to Δˉ\bar{\Delta}, which is of tree type. Therefore, Γ(Db( mod H))\Gamma\left(\mathcal{D}^{b}(\bmod H)\right) is a simply connected translation quiver and therefore Γ(Db( mod H))\Gamma\left(\mathcal{D}^{b}(\bmod H)\right) is a component with length.

In the next result, we give a relationship between morphisms of the categories mod Γ\bmod \Gamma and mod Γ′\bmod \Gamma^{\prime}.
Proposition 2.2. Let C\mathcal{C} be the cluster category of a hereditary algebra HH and let Tˉ\bar{T} be an almost complete tilting object in C\mathcal{C} with complements MM and M∗M^{*}. Consider Γ=End⁡C(T)op \Gamma=\operatorname{End}_{\mathcal{C}}(T)^{\text {op }} and Γ′=\Gamma^{\prime}= End⁡C(T′)op \operatorname{End}_{\mathcal{C}}\left(T^{\prime}\right)^{\text {op }} with T=Tˉ⊕MT=\bar{T} \oplus M and T′=Tˉ⊕M∗T^{\prime}=\bar{T} \oplus M^{*}. Let SxS_{x} and Sy′S_{y}^{\prime} be the simple tops of Hom⁡C(T,M)\operatorname{Hom}_{\mathcal{C}}(T, M) and Hom⁡C(T′,M∗)\operatorname{Hom}_{\mathcal{C}}\left(T^{\prime}, M^{*}\right), respectively, and let θ: mod Γ/add⁡Sx→ mod Γ′/add⁡Sy′\theta: \bmod \Gamma / \operatorname{add} S_{x} \rightarrow \bmod \Gamma^{\prime} / \operatorname{add} S_{y}^{\prime} be the equivalence of Theorem 1.13.

Consider a morphism f:X→Yf: X \rightarrow Y with X,Y∈ind⁡ΓX, Y \in \operatorname{ind} \Gamma. Then, ff is an irreducible morphism in mod Γ\bmod \Gamma which does not factor through add SxS_{x} if and only if θ(f)\theta(f) is an irreducible morphism in mod Γ′\bmod \Gamma^{\prime} which does not factor through add Sy′S_{y}^{\prime}.

Proof. Let Γ\Gamma and Γ′\Gamma^{\prime} be the cluster tilted algebras as above. Let XX and YY be indecomposable modules in mod Γ\bmod \Gamma and let f:X→Yf: X \rightarrow Y be a non-zero morphism such that ff does not factor through add SxS_{x}. By the equivalence of the Theorem 1.13, we have that θ(f):θ(X)→θ(Y)\theta(f): \theta(X) \rightarrow \theta(Y) is a non-zero morphism and moreover θ(f)\theta(f) does not factor through add Sy′S_{y}^{\prime}

Assume that ff is irreducible. We prove that θ(f)\theta(f) so is. In fact, assume that θ(f)\theta(f) is a section. Then there exists a morphism f′~:θ(Y)→θ(X)\widetilde{f^{\prime}}: \theta(Y) \rightarrow \theta(X) such that f′~θ(f)=1θ(X)\widetilde{f^{\prime}} \theta(f)=1_{\theta(X)}. Moreover, f′~\widetilde{f^{\prime}} does not factor through add Sy′S_{y}^{\prime} because θ(f)\theta(f) neither does. Then, there is a morphism f′:Y→Xf^{\prime}: Y \rightarrow X such that f′~=θ(f′)\widetilde{f^{\prime}}=\theta\left(f^{\prime}\right). Therefore,

θ(1X)=1θ(X)=θ(f′)θ(f)=θ(f′f)\theta\left(1_{X}\right)=1_{\theta(X)}=\theta\left(f^{\prime}\right) \theta(f)=\theta\left(f^{\prime} f\right)

and since θ\theta is a faithful functor, then 1X=f′f1_{X}=f^{\prime} f, which is a contradiction to the fact that ff is not a section. Thus, we prove that θ(f)\theta(f) is not a section.

Analogously, we can prove that θ(f)\theta(f) is not a retraction.
Now, assume that the is a Γ′\Gamma^{\prime}-module Z~\widetilde{Z} and that there are morphisms g~:θ(X)→Z~\widetilde{g}: \theta(X) \rightarrow \widetilde{Z} and h~:Z~→θ(Y)\widetilde{h}: \widetilde{Z} \rightarrow \theta(Y), such that θ(f)=h~g~\theta(f)=\widetilde{h} \widetilde{g}. Since θ(f)\theta(f) does not factor though add Sy′S_{y}^{\prime}, we infer that neither do the morphisms g~\widetilde{g} y h~\widetilde{h}. By Theorem 1.13, there exist Z∈ mod ΓZ \in \bmod \Gamma and morphisms g:X→Zg: X \rightarrow Z and h:Z→Yh: Z \rightarrow Y which do not factor trough add SxS_{x} such that g~=θ(g)\widetilde{g}=\theta(g) and h~=θ(h)\widetilde{h}=\theta(h). Then, θ(f)=θ(h)θ(g)=θ(hg)\theta(f)=\theta(h) \theta(g)=\theta(h g) and consequently f=hgf=h g. Since ff is an irreducible morphism, then gg is a section (and therefore θ(g)\theta(g) also does) or hh is a retraction (and therefore θ(h)\theta(h) also does). Thus, θ(f)\theta(f) is an irreducible morphism.

The converse follows by considering θ′\theta^{\prime} the quasi-inverse equivalence of θ\theta.
Our next goal is to prove that the nilpotency index is invariant under mutation.
Following the above notation, we donote by aa the vertex of QΓQ_{\Gamma} and of QΓ′Q_{\Gamma^{\prime}}, which come from TaT_{a}, a direct indecomposable summand of Tˉ\bar{T}, and we denote by xx ( yy, respectively) the vertex of QΓQ_{\Gamma} (QΓ′\left(Q_{\Gamma^{\prime}}\right., respectively) which come from MM, the summand of TT ( M∗M^{*}, the summand of T′T^{\prime}, respectively).

We start with some lemmas in order to prove one of the main theorems of this section.

Lemma 2.3. Let C\mathcal{C} be a cluster category of a representation-finite hereditary algebra HH, and let Tˉ\bar{T} be an almost complete tilting object in C\mathcal{C} with complements MM and M∗M^{*}. Consider Γ=End⁡C(Tˉ⊕\Gamma=\operatorname{End}_{\mathcal{C}}(\bar{T} \oplus M)op≃kQΓ/IΓM)^{o p} \simeq k Q_{\Gamma} / I_{\Gamma} and Γ′=End⁡C(Tˉ⊕M∗)op≃kQΓ′/IΓ′\Gamma^{\prime}=\operatorname{End}_{\mathcal{C}}\left(\bar{T} \oplus M^{*}\right)^{o p} \simeq k Q_{\Gamma^{\prime}} / I_{\Gamma^{\prime}} the cluster tilted algebras. Then, for all indecomposable summand TaT_{a} of Tˉ\bar{T}, we have that ra=ra′r_{a}=r_{a}^{\prime}.

Proof. Let Γ\Gamma and Γ′\Gamma^{\prime} be cluster tilted algebras as in the statement, and let TaT_{a} be an indecomposable summand of Tˉ\bar{T}. Consider Pa,SaP_{a}, S_{a} and IaI_{a} the projective, simple and injective Γ\Gamma-modules, respectively, corresponding to the vertex a∈QΓa \in Q_{\Gamma}, and Pa′,Sa′P_{a}^{\prime}, S_{a}^{\prime} and Ia′I_{a}^{\prime} the projective, simple and injective Γ′\Gamma^{\prime} modules, respectively, corresponding to the vertex a∈QΓ′a \in Q_{\Gamma^{\prime}}. Let rar_{a} and ra′r_{a}^{\prime} be the bounds defined in Notation 1.15. We prove that ra=ra′r_{a}=r_{a}^{\prime}.

Let fa:Pa→Iaf_{a}: P_{a} \rightarrow I_{a} be a non-zero morphism in mod Γ\bmod \Gamma that factors trough SaS_{a}. By Lemma 1.7, we have that dp⁡(fa)=ra\operatorname{dp}\left(f_{a}\right)=r_{a}. Therefore, by Proposition 1.3, we can write the morphism faf_{a} as follows

fa=Σi=1sgifif_{a}=\Sigma_{i=1}^{s} g_{i} f_{i}

for some s≥1s \geq 1, where fi∈ℜΓ(Pa,Xi)f_{i} \in \Re_{\Gamma}\left(P_{a}, X_{i}\right), with Xi∈ind⁡ΓX_{i} \in \operatorname{ind} \Gamma, and gig_{i} is a finite sum of composition of ra−1r_{a}-1 irreducible morphisms between indecomposable modules, for i=1,…,si=1, \ldots, s.

Let SxS_{x} be the simple top of the projective Γ\Gamma-module Px=Hom⁡C(T,M)P_{x}=\operatorname{Hom}_{\mathcal{C}}(T, M). Since Sa≠SxS_{a} \neq S_{x}, neither fif_{i} nor gig_{i} factor through add SxS_{x}, because Hom⁡Γ(Pa,Sx)=0=Hom⁡Γ(Sx,Ia)\operatorname{Hom}_{\Gamma}\left(P_{a}, S_{x}\right)=0=\operatorname{Hom}_{\Gamma}\left(S_{x}, I_{a}\right). Then, by the equivalence θ: mod Γ/add⁡Sx→ mod Γ′/add⁡Sy\theta: \bmod \Gamma / \operatorname{add} S_{x} \rightarrow \bmod \Gamma^{\prime} / \operatorname{add} S_{y} defined above, we have that θ(fa)=Σi=1sθ(gi)θ(fi)\theta\left(f_{a}\right)=\Sigma_{i=1}^{s} \theta\left(g_{i}\right) \theta\left(f_{i}\right) is a nonzero morphisms, where each θ(fi)∈ℜΓ′(θ(Pa),θ(Xi))\theta\left(f_{i}\right) \in \Re_{\Gamma^{\prime}}\left(\theta\left(P_{a}\right), \theta\left(X_{i}\right)\right). Moreover, by Proposition 2.2, each θ(gi)\theta\left(g_{i}\right) is a finite sum of composition of ra−1r_{a}-1 irreducible morphisms between indecomposable modules. Then, θ(fa)∈ℜΓ∘ra(θ(Pa),θ(Ia))\theta\left(f_{a}\right) \in \Re_{\Gamma^{\circ}}^{r_{a}}\left(\theta\left(P_{a}\right), \theta\left(I_{a}\right)\right), that is, θ(fa)∈ℜΓ∘ra(Pa′,Ia′)\theta\left(f_{a}\right) \in \Re_{\Gamma^{\circ}}^{r_{a}}\left(P_{a}^{\prime}, I_{a}^{\prime}\right). By Lemma 1.7 we have that ra≤ra′r_{a} \leq r_{a}^{\prime}.

Similarly, we can prove that ra′≤rar_{a}^{\prime} \leq r_{a}. Hence, ra=ra′r_{a}=r_{a}^{\prime} as we wish to prove.
Lemma 2.4. Let C\mathcal{C} be the cluster category of a representation-finite hereditary algebra HH. let Tˉ\bar{T} be an almost complete tilting object in C\mathcal{C} with complements MM and M∗M^{*}. Consider Γ=End⁡C(Tˉ⊕M)op≃\Gamma=\operatorname{End}_{\mathcal{C}}(\bar{T} \oplus M)^{o p} \simeq kQΓ/IΓk Q_{\Gamma} / I_{\Gamma} and Γ′=End⁡C(Tˉ⊕M∗)op≃kQΓ′/IΓ′\Gamma^{\prime}=\operatorname{End}_{\mathcal{C}}\left(\bar{T} \oplus M^{*}\right)^{o p} \simeq k Q_{\Gamma^{\prime}} / I_{\Gamma^{\prime}} the cluster tilted algebras. Let xx and yy be the vertices of QΓQ_{\Gamma} and QΓ′Q_{\Gamma^{\prime}}, respectively which come from the summands MM of TT and M∗M^{*} of T′T^{\prime}, respectively. Then rx=ry′r_{x}=r_{y}^{\prime}.

Proof. Let Γ≃kQΓ/IΓ\Gamma \simeq k Q_{\Gamma} / I_{\Gamma} and Γ′≃kQΓ′/IΓ′\Gamma^{\prime} \simeq k Q_{\Gamma^{\prime}} / I_{\Gamma^{\prime}} be the cluster tilted algebras defined as above.
To prove that rx=ry′r_{x}=r_{y}^{\prime}, we shall prove the fact that nx=my′n_{x}=m_{y}^{\prime} and mx=ny′m_{x}=n_{y}^{\prime}, where rx=nx+mxr_{x}=n_{x}+m_{x} and ry′=ny′+my′r_{y}^{\prime}=n_{y}^{\prime}+m_{y}^{\prime} are the bounds defined in Notation 1.15.

Consider a non-zero morphism fx:Px→Sxf_{x}: P_{x} \rightarrow S_{x} and the irreducible morphism ix:rad⁡Px→Pxi_{x}: \operatorname{rad} P_{x} \rightarrow P_{x}. Since the cluster tilted algebra Γ\Gamma is representation-finite then by [10, Theorem A] we know that the right degree of ixi_{x} is finite and more precisely it is nxn_{x}. Therefore, by the dual result of [10, Proposition 3.4], we have that dp⁡(fx)=nx\operatorname{dp}\left(f_{x}\right)=n_{x}. Hence, by Proposition 1.3 we know that there is a non-zero path of irreducible morphisms between indecomposable modules of length nxn_{x} in mod Γ\bmod \Gamma as follows

φx:Px→h1X1→h2X2⟶⋯⟶Xnx−1→hnxSx\varphi_{x}: P_{x} \xrightarrow{h_{1}} X_{1} \xrightarrow{h_{2}} X_{2} \longrightarrow \cdots \longrightarrow X_{n_{x}-1} \xrightarrow{h_{n_{x}}} S_{x}

By the equivalence defined in Theorem 1.11, it is induced by a non-zero path of also nxn_{x} irreducible morphisms between indecomposable objects in the cluster category C\mathcal{C}, such that it does not factor trough add τT\tau T

φ~x:M→h~1X~1→h~2X~2⟶⋯⟶X~nx−1→h~nxτM∗\widetilde{\varphi}_{x}: M \xrightarrow{\widetilde{h}_{1}} \widetilde{X}_{1} \xrightarrow{\widetilde{h}_{2}} \widetilde{X}_{2} \longrightarrow \cdots \longrightarrow \widetilde{X}_{n_{x}-1} \xrightarrow{\widetilde{h}_{n_{x}}} \tau M^{*}

where Px=Hom⁡C(T,M),Sx=Hom⁡C(T,τM∗)P_{x}=\operatorname{Hom}_{\mathcal{C}}(T, M), S_{x}=\operatorname{Hom}_{\mathcal{C}}\left(T, \tau M^{*}\right) and Xi=Hom⁡C(T,X~i)X_{i}=\operatorname{Hom}_{\mathcal{C}}\left(T, \widetilde{X}_{i}\right) for 1≤i≤nx−11 \leq i \leq n_{x}-1.

On the other hand, if we consider a non-zero morphism gy′:Sy′→Iy′g_{y}^{\prime}: S_{y}^{\prime} \rightarrow I_{y}^{\prime} in mod Γ′\bmod \Gamma^{\prime}, by Theorem A and Proposition 3.4 in [10], we have that dp⁡(gy′)=my′\operatorname{dp}\left(g_{y}^{\prime}\right)=m_{y}^{\prime}. Hence, with an analogous analysis to the previous one, there exists a non-zero path ψy′\psi_{y}^{\prime} of my′m_{y}^{\prime} irreducible morphisms between indecomposable modules from Sy′S_{y}^{\prime} to Iy′I_{y}^{\prime} in mod Γ′\bmod \Gamma^{\prime}. Moreover, such a path is induced by a non-zero path ψ~y′\widetilde{\psi}_{y}^{\prime}, from τM\tau M to τ2M∗\tau^{2} M^{*}, of my′m_{y}^{\prime} irreducible morphisms between indecomposable modules in the cluster category C\mathcal{C} and such that it does not factor trough add T′T^{\prime}

ψ~y′:τM→Y′~1→Y′~2→…→Y′~my−1→τ2M∗\widetilde{\psi}_{y}^{\prime}: \tau M \rightarrow \widetilde{Y^{\prime}}_{1} \rightarrow \widetilde{Y^{\prime}}_{2} \rightarrow \ldots \rightarrow \widetilde{Y^{\prime}}_{m_{y}-1} \rightarrow \tau^{2} M^{*}

because Sy′=Hom⁡C(T′,τM)S_{y}^{\prime}=\operatorname{Hom}_{\mathcal{C}}\left(T^{\prime}, \tau M\right) and Iy′=Hom⁡C(T′,τ2M∗)I_{y}^{\prime}=\operatorname{Hom}_{\mathcal{C}}\left(T^{\prime}, \tau^{2} M^{*}\right).
We also have that φ~x∈Hom⁡C(M,τM∗)\widetilde{\varphi}_{x} \in \operatorname{Hom}_{\mathcal{C}}\left(M, \tau M^{*}\right), with φ~x≠0\widetilde{\varphi}_{x} \neq 0, where

Hom⁡C(M,τM∗)=Hom⁡D(F−1M,τM∗)⊕Hom⁡D(M,τM∗)\operatorname{Hom}_{\mathcal{C}}\left(M, \tau M^{*}\right)=\operatorname{Hom}_{\mathcal{D}}\left(F^{-1} M, \tau M^{*}\right) \oplus \operatorname{Hom}_{\mathcal{D}}\left(M, \tau M^{*}\right)

and ψ~y′∈Hom⁡C(τM,τ2M∗)\widetilde{\psi}_{y}^{\prime} \in \operatorname{Hom}_{\mathcal{C}}\left(\tau M, \tau^{2} M^{*}\right), with ψ~y′≠0\widetilde{\psi}_{y}^{\prime} \neq 0, where

Hom⁡C(τM,τ2M∗)=Hom⁡D(F−1τM,τ2M∗)⊕Hom⁡D(τM,τ2M∗)\operatorname{Hom}_{\mathcal{C}}\left(\tau M, \tau^{2} M^{*}\right)=\operatorname{Hom}_{\mathcal{D}}\left(F^{-1} \tau M, \tau^{2} M^{*}\right) \oplus \operatorname{Hom}_{\mathcal{D}}\left(\tau M, \tau^{2} M^{*}\right)

In both cases, only one of the summands is non-zero since HH is representation-finite.
Hence, if Hom⁡D(F−1M,τM∗)≠0\operatorname{Hom}_{\mathcal{D}}\left(F^{-1} M, \tau M^{*}\right) \neq 0, then Hom⁡D(F−1τM,τ2M∗)≠0\operatorname{Hom}_{\mathcal{D}}\left(F^{-1} \tau M, \tau^{2} M^{*}\right) \neq 0 because

Hom⁡D(F−1M,τM∗)=Hom⁡D(τM[−1],τM∗)≃Hom⁡D(τ2M[−1],τ2M∗)≃Hom⁡D(F−1τM,τ2M∗)\begin{aligned} \operatorname{Hom}_{\mathcal{D}}\left(F^{-1} M, \tau M^{*}\right) & =\operatorname{Hom}_{\mathcal{D}}\left(\tau M[-1], \tau M^{*}\right) \\ & \simeq \operatorname{Hom}_{\mathcal{D}}\left(\tau^{2} M[-1], \tau^{2} M^{*}\right) \\ & \simeq \operatorname{Hom}_{\mathcal{D}}\left(F^{-1} \tau M, \tau^{2} M^{*}\right) \end{aligned}

Therefore, the path in (2) is induced by a path of irreducible morphisms between indecomposable modules of length nxn_{x} from τM[−1]\tau M[-1] to τM∗\tau M^{*} in D\mathcal{D}, and the path in (3) is induced by a path of my′m_{y}^{\prime} irreducible morphisms between indecomposable modules from τ2M[−1]\tau^{2} M[-1] to τ2M∗\tau^{2} M^{*} in D\mathcal{D}. Moreover, since Hom⁡D(τM[−1],τM∗)≃Hom⁡D(τ2M[−1],τ2M∗)\operatorname{Hom}_{\mathcal{D}}\left(\tau M[-1], \tau M^{*}\right) \simeq \operatorname{Hom}_{\mathcal{D}}\left(\tau^{2} M[-1], \tau^{2} M^{*}\right) and Γ(D)\Gamma(\mathcal{D}) is a translation quiver with length, then nx=my′n_{x}=m_{y}^{\prime}.

Now, if Hom⁡D(M,τM∗)≠0\operatorname{Hom}_{\mathcal{D}}\left(M, \tau M^{*}\right) \neq 0, with the same argument as before we can conclude that nx=my′n_{x}=m_{y}^{\prime}.
Analogously, considering a non-zero morphism gx:Sx→Ixg_{x}: S_{x} \rightarrow I_{x} in mod Γ\bmod \Gamma and a non-zero morphism fy′:Py′→Sy′f_{y}^{\prime}: P_{y}^{\prime} \rightarrow S_{y}^{\prime} in mod Γ′\bmod \Gamma^{\prime}, with a similar analysis as above, we conclude that mx=ny′m_{x}=n_{y}^{\prime}. Thus, rx=mx+nx=ny′+my′=ry′r_{x}=m_{x}+n_{x}=n_{y}^{\prime}+m_{y}^{\prime}=r_{y}^{\prime}.

Now, we are in position to show one Theorem A.
Theorem 2.5. Let C\mathcal{C} be the cluster category of a representation-finite hereditary algebra HH. Consider Γ=End⁡C(Tˉ⊕M)op\Gamma=\operatorname{End}_{\mathcal{C}}(\bar{T} \oplus M)^{o p} and Γ′=End⁡C(Tˉ⊕M∗)op\Gamma^{\prime}=\operatorname{End}_{\mathcal{C}}(\bar{T} \oplus M^{*})^{o p} the cluster tilted algebras, where Tˉ\bar{T} is an almost complete tilting object in C\mathcal{C} with complements MM and M∗M^{*}. Let rΓr_{\Gamma} and rΓ′r_{\Gamma^{\prime}} be the nilpotency indices of ℜ( mod Γ)\Re(\bmod \Gamma) and ℜ( mod Γ′)\Re\left(\bmod \Gamma^{\prime}\right), respectively. Then, rΓ=rΓ′r_{\Gamma}=r_{\Gamma^{\prime}}.
Proof. Let Γ≃kQΓ/IΓ\Gamma \simeq k Q_{\Gamma} / I_{\Gamma} and Γ′≃kQΓ′/IΓ′\Gamma^{\prime} \simeq k Q_{\Gamma^{\prime}} / I_{\Gamma^{\prime}} be the cluster tilted algebras defined as above. Since HH is a representation-finite algebra, then so are Γ\Gamma and Γ′\Gamma^{\prime}. We denote by rΓr_{\Gamma} and rΓ′r_{\Gamma^{\prime}}, the nilpotency indices of ℜ( mod Γ)\Re(\bmod \Gamma) and ℜ( mod Γ′)\Re\left(\bmod \Gamma^{\prime}\right), respectively. We prove that rΓ=rΓ′r_{\Gamma}=r_{\Gamma^{\prime}}. In fact, we know that

rΓ=max⁡{ru∣u∈(QΓ)0}+1=max⁡{ru∣Tu∈ind⁡(add⁡T)}+1r_{\Gamma}=\max \left\{r_{u} \mid u \in\left(Q_{\Gamma}\right)_{0}\right\}+1=\max \left\{r_{u} \mid T_{u} \in \operatorname{ind}(\operatorname{add} T)\right\}+1

and

rΓ′=max⁡{rv′∣v∈(QΓ′)0}+1=max⁡{rv′∣Tv∈ind⁡(add⁡T′)}+1r_{\Gamma^{\prime}}=\max \left\{r_{v}^{\prime} \mid v \in\left(Q_{\Gamma^{\prime}}\right)_{0}\right\}+1=\max \left\{r_{v}^{\prime} \mid T_{v} \in \operatorname{ind}\left(\operatorname{add} T^{\prime}\right)\right\}+1

By Lemma 2.3 and Lemma 2.4, we have that

rΓ=max⁡{ru∣Tu∈ind⁡(add⁡T)}+1=max⁡{ra∣Ta∈ind⁡(add⁡Tˉ),rx}+1=max⁡{ra′∣Ta∈ind⁡(add⁡Tˉ),ry′}+1=max⁡{rv∣Tv∈ind⁡(add⁡T′)}+1=rΓ′\begin{aligned} r_{\Gamma} & =\max \left\{r_{u} \mid T_{u} \in \operatorname{ind}(\operatorname{add} T)\right\}+1 \\ & =\max \left\{r_{a} \mid T_{a} \in \operatorname{ind}\left(\operatorname{add} \bar{T}\right), r_{x}\right\}+1 \\ & =\max \left\{r_{a}^{\prime} \mid T_{a} \in \operatorname{ind}\left(\operatorname{add} \bar{T}\right), r_{y}^{\prime}\right\}+1 \\ & =\max \left\{r_{v} \mid T_{v} \in \operatorname{ind}\left(\operatorname{add} T^{\prime}\right)\right\}+1 \\ & =r_{\Gamma^{\prime}} \end{aligned}

proving the result.
For the convenience if the reader we state [20, Theorem 4.11].
Theorem 2.6. Let H=kΔH=k \Delta be a representation-finite hereditary algebra and let rHr_{H} be the nilpotency index of ℜ( mod H)\Re(\bmod H). Then the following conditions hold.
(a) If Δˉ=An\bar{\Delta}=A_{n}, then rH=nr_{H}=n, for n≥1n \geq 1.
(b) If Δˉ=Dn\bar{\Delta}=D_{n}, then rH=2n−3r_{H}=2 n-3, for n≥4n \geq 4.
© If Δˉ=E6\bar{\Delta}=E_{6}, then rH=11r_{H}=11.
(d) If Δˉ=E7\bar{\Delta}=E_{7}, then rH=17r_{H}=17.
(e) If Δˉ=E8\bar{\Delta}=E_{8}, then rH=29r_{H}=29.

The next corollary shall be important to prove Theorem B, and follows from [5] and [6].
Corollary 2.7. Let Δ\Delta be a connected and acyclic quiver. The classes of quivers obtained of Δ\Delta by mutations coincide with the classes of quivers of the cluster tilted algebras of type Δ\Delta. Moreover, if Δ\Delta is of Dynkin type then there is a finite number of the mentioned classes.

Now, we are in conditions to present Theorem B.
Theorem 2.8. Let Δ\Delta be a Dynkin quiver and let Γ\Gamma be a cluster tilted algebra of type Δˉ\bar{\Delta}. Let rΓr_{\Gamma} be the nilpotency index of ℜ( mod Γ)\Re(\bmod \Gamma). Then the following conditions hold.
(a) If Δˉ=An\bar{\Delta}=A_{n}, then rΓ=nr_{\Gamma}=n for n≥1n \geq 1.
(b) If Δˉ=Dn\bar{\Delta}=D_{n}, then rΓ=2n−3r_{\Gamma}=2 n-3 for n≥4n \geq 4.
© If Δˉ=E6\bar{\Delta}=E_{6}, then rΓ=11r_{\Gamma}=11.
(d) If Δˉ=E7\bar{\Delta}=E_{7}, then rΓ=17r_{\Gamma}=17.
(e) If Δˉ=E8\bar{\Delta}=E_{8}, then rΓ=29r_{\Gamma}=29.

Proof. Let Γ≃kQΓ/IΓ\Gamma \simeq k Q_{\Gamma} / I_{\Gamma} be a cluster tilted algebra of type Δˉ\bar{\Delta}, where Δ\Delta is a Dynkin quiver and let HH be the hereditary algebra H=kΔH=k \Delta. Since HH is representation-finite, then so is Γ\Gamma.

Let rHr_{H} and rΓr_{\Gamma} the nilpotency indices of ℜ( mod H)\Re(\bmod H) and ℜ( mod Γ)\Re(\bmod \Gamma), respectively. We claim that rΓ=rHr_{\Gamma}=r_{H}. In fact, by Corollary 2.7, we can change the algebra Γ\Gamma into the algebra HH by a finite sequence of mutations of the quiver QΓQ_{\Gamma}. By Theorem 2.5, we have that rΓ=rHr_{\Gamma}=r_{H} where rHr_{H} takes the values given in Theorem 2.6. Hence we prove the statement.

In [18], C. Ringel proved that all self-injective cluster tilted algebras are representation-finite. Furthermore, the author showed that this particular algebras are cluster tilted algebras of type DnD_{n}, with n≥3n \geq 3 (considering D3=A3D_{3}=A_{3} ).

The next result follows immediately from Theorem 2.8.
Corollary 2.9. Let Γ\Gamma be a self-injective cluster tilted algebra. Then, the nilpotency index of ℜ( mod Γ)\Re(\bmod \Gamma) is 2n−32 n-3, where nn is the number of vertices of the quiver QΓQ_{\Gamma}.

We end up this section with a remark on Coxeter numbers. We refer the reader to [13] for a detailed account on Root Systems and Coxeter Groups.

Remark 2.10. It is known that the theory of cluster algebras has many connections with different areas in mathematics. In particular, there exists a connection with Root Systems and with Coxeter Groups.

An element in a Coxeter group WW is called a coxeter element if it is the product of all simple reflections and moreover its order is called the coxeter number of WW. On the other hand, the coxeter number is related with the highest root in its corresponding root system.

For a finite irreducible Coxeter group WW, there is a corresponding root system Φ\Phi of Dynkin type Δ\Delta. Now, if Γ\Gamma is a cluster tilted algebra of Δ\Delta type then it is known that the cardinal of the set of positive roots of Φ\Phi coincide with the cardinal of ind Γ\Gamma. Moreover, the coxeter number of WW is exactly one more than the nilpotency index of ℜ( mod Γ)\Re(\bmod \Gamma).

3. ON COMPOSITION OF IRREducIBLE MORPHISMS

In this section, we establish the relationship between the composition of irreducible morphisms between indecomposable modules and the power of the radical where it belongs.

We start with the following proposition.
Proposition 3.1. Let Γ\Gamma be a representation-finite cluster tilted algebra. Let MM and NN be indecomposable Γ\Gamma-modules such that Irr⁡Γ(M,N)≠0\operatorname{Irr}_{\Gamma}(M, N) \neq 0. Then, dim⁡k(Hom⁡Γ(M,N))=1\operatorname{dim}_{k}\left(\operatorname{Hom}_{\Gamma}(M, N)\right)=1. In particular, ℜΓ2(M,N)=0\Re_{\Gamma}^{2}(M, N)=0.
Proof. Let Γ\Gamma be a representation-finite cluster tilted algebra. Then Γ=End⁡C(T)op\Gamma=\operatorname{End}_{\mathcal{C}}(T)^{o p}, where C=D/F\mathcal{C}=\mathcal{D} / F is the cluster category of a representation-finite hereditary algebra HH and TT a tilting object in C\mathcal{C}.

Since Irr⁡Γ(M,N)≠0\operatorname{Irr}_{\Gamma}(M, N) \neq 0, then there exists an irreducible morphism f:M→Nf: M \rightarrow N. We claim that all the morphisms g:M→Ng: M \rightarrow N in mod Γ\bmod \Gamma are kk-linearly dependent with ff. In fact, suppose that there exists a non-zero morphisms g:M→Nkg: M \rightarrow N k-linearly independent with ff. Since Γ\Gamma is representationfinite, we know that dim⁡k(Irr⁡Γ(M,N))=1\operatorname{dim}_{k}\left(\operatorname{Irr}_{\Gamma}(M, N)\right)=1. Hence, gg is not irreducible. Then g∈ℜ2(M,N)g \in \Re^{2}(M, N). Moreover, there is a integer n≥2n \geq 2 such that g∈ℜn(M,N)\ℜn+1(M,N)g \in \Re^{n}(M, N) \backslash \Re^{n+1}(M, N). Therefore, there exist morphisms f~,g~:M~→N~\widetilde{f}, \widetilde{g}: \widetilde{M} \rightarrow \widetilde{N} in the cluster category C\mathcal{C} such that they do not factor through add⁡(τT)\operatorname{add}(\tau T). Furthermore, these morphisms are induced by morphisms in the derived category. Moreover, since Hom⁡C(M~,N~)=Hom⁡D(F−1M,N)⊕Hom⁡D(M,N)\operatorname{Hom}_{\mathcal{C}}(\widetilde{M}, \widetilde{N})=\operatorname{Hom}_{\mathcal{D}}\left(F^{-1} M, N\right) \oplus \operatorname{Hom}_{\mathcal{D}}(M, N), and only one of this summands are non-zero, we can deduce the existence of an irreducible morphism in Γ(D)\Gamma(\mathcal{D}) and a path of length nn, with n≥2n \geq 2, between the same objects, contradicting the fact that Γ(D)\Gamma(\mathcal{D}) is a quiver with length.

Therefore, there is not a morphism g:M→Ng: M \rightarrow N in mod Γ\bmod \Gamma linearly independent with ff. Then, dim⁡k(Hom⁡Γ(M,N))=1\operatorname{dim}_{k}\left(\operatorname{Hom}_{\Gamma}(M, N)\right)=1. Moreover, since ff is irreducible, we conclude that ℜΓ2(M,N)=0\Re_{\Gamma}^{2}(M, N)=0.
Theorem 3.2. Let Γ\Gamma be a representation-finite cluster tilted algebra. Consider the irreducible morphisms hi:Xi→Xi+1h_{i}: X_{i} \rightarrow X_{i+1}, with Xi∈ind⁡ΓX_{i} \in \operatorname{ind} \Gamma for 1≤i≤m1 \leq i \leq m. Then hm…h1∈ℜm+1(X1,Xm+1)h_{m} \ldots h_{1} \in \Re^{m+1}\left(X_{1}, X_{m+1}\right) if and only if hm…h1=0h_{m} \ldots h_{1}=0.
Proof. If hm…h1=0h_{m} \ldots h_{1}=0, then clearly we have that hm…h1∈ℜm+1(X1,Xm+1)h_{m} \ldots h_{1} \in \Re^{m+1}\left(X_{1}, X_{m+1}\right).
Conversely. Assume that hm…h1∈ℜm+1(X1,Xm+1)h_{m} \ldots h_{1} \in \Re^{m+1}\left(X_{1}, X_{m+1}\right) and hm…h1≠0h_{m} \ldots h_{1} \neq 0. Then, by [10, Theorem 5.1] there are irreducible morphisms fi:Xi→Xi+1f_{i}: X_{i} \rightarrow X_{i+1}, for 1≤i≤m1 \leq i \leq m such that fm…f1=0f_{m} \ldots f_{1}=0. By Proposition 3.1, we have that dim⁡k(Hom⁡A(Xi,Xi+1))=1\operatorname{dim}_{k}\left(\operatorname{Hom}_{A}\left(X_{i}, X_{i+1}\right)\right)=1, for each ii. Hence, fif_{i} and hih_{i} are kk-linearly dependent, that is, hi=λifih_{i}=\lambda_{i} f_{i} where λi\lambda_{i} is a non-zero element of kk. Thus, hm…h1=λfm…f1=0h_{m} \ldots h_{1}=\lambda f_{m} \ldots f_{1}=0, which is a contradiction to our assumption. Therefore, hm…h1=0h_{m} \ldots h_{1}=0 proving the result.

Remark 3.3. We observe that if we consider a cluster tilted algebra of type AnA_{n} or DnD_{n}, then the results of this article can be proven with the geometric approach developed for cluster categories and cluster tilted algebras of type AnA_{n} and DnD_{n} given in [7] and in [19], respectively. For a detail account on this approach see [14].

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Centro Marplatense de Investigaciones Matemáticas, Facultad de Ciencias Exactas y Naturales, Funes 3350, Universidad Nacional de Mar del Plata y CONICET 7600 Mar del Plata, Argentina E-mail address: claudia.chaio@gmail.com

Centro Marplatense de Investigaciones Matemáticas, Facultad de Ciencias Exactas y Naturales, Funes 3350, Universidad Nacional de Mar del Plata, 7600 Mar del Plata, Argentina

E-mail address: victoria.guazzelli@hotmail.com