Generalized cluster complexes via quiver representations

DOI 10.1007/s10801-007-0074-3

Generalized cluster complexes via quiver representations

Bin Zhu

Received: 1 August 2006 / Accepted: 6 April 2007 / Published online: 1 June 2007

© Springer Science+Business Media, LLC 2007

Abstract We give a quiver representation theoretic interpretation of generalized cluster complexes defined by Fomin and Reading. Usingd-cluster categories defined by Keller as triangulated orbit categories of (bounded) derived categories of repre- sentations of valued quivers, we define ad-compatibility degree(− −)on any pair of “colored” almost positive real Schur roots which generalizes previous definitions on the noncolored case and call two such roots compatible, provided that their d- compatibility degree is zero. Associated to the root systemΦ corresponding to the valued quiver, using this compatibility relation, we define a simplicial complex which has colored almost positive real Schur roots as vertices andd-compatible subsets as simplices. If the valued quiver is an alternating quiver of a Dynkin diagram, then this complex is the generalized cluster complex defined by Fomin and Reading.

Keywords Colored almost positive real Schur root·Generalized cluster complex· d-cluster category·d-cluster tilting object·d-compatibility degree

Mathematics Subject Classification (2000) 05A15·16G20·16G70·17B20

1 Introduction

Generalized cluster complexes associated to finite root systems are introduced by Fomin and Reading [12]. They have some nice properties, see [2] and references

Supported by the NSF of China (Grants 10471071) and by the Leverhulme Trust through the network ‘Algebras, Representations and Applications’.

B. Zhu (

Department of Mathematical Sciences, Tsinghua University, 100084 Beijing, People’s Republic of China

e-mail: bzhu@math.tsinghua.edu.cn

therein. They are a generalization of cluster complexes (so-called generalized asso- ciahedra) associated to the same root systems introduced in [14,15]. Cluster com- plexes describe the combinatorial structure of cluster algebras introduced by Fomin–

Zelevinsky [13] in order to give an algebraic and combinatorial framework for the canonical basis, see [11] for a nice survey on this combinatorics and also cluster com- binatorics of root systems. In [22], Marsh, Reineke and Zelevinsky use “decorated”

quiver representations and tilting theory to give a quiver interpretation of cluster com- plexes. This connection between tilting theory and cluster combinatorics leads Buan, Marsh, Reineke, Reiten and Todorov [6] to introduce cluster categories for a categori- cal model for cluster algebras, see also [9] for typeAn. Cluster categories are the orbit categoriesD/τ⁻¹[1]of derived categories of hereditary categories arising from the action of subgroupτ⁻¹[1]of the automorphism group. They are triangulated cate- gories [19] and now they have become a successful model for acyclic cluster algebras [5,7,8], see also the surveys [4,24] and references therein for recent developments and background of cluster tilting theory.

d-cluster categoriesD/τ⁻¹[d],as a generalization of cluster categories, were in- troduced by Keller [19] and Thomas [25] ford∈N. They are studied by Keller and Reiten [20], Palu [1,23]; see also [3] for a geometric description ofd-cluster cate- gories of typeA_n.d-cluster categories are triangulated categories with Calabi–Yau dimensiond+1. Whend=1, the cluster categories are recovered.

The aim of this paper is to give not only a quiver representation theoretic interpre- tation of all key ingredients in defining generalized cluster complexes usingd-cluster categories, but also a generalization of generalized cluster complexes to infinite root systems (compare Remark 3.13 in [12], where the authors asked whether there was such an extension). For the simply-laced Dynkin case, Thomas [25] gives a realiza- tion of generalized cluster complexes by defining thed-cluster categories.

The paper is organized as follows: In the first two parts, we recall the well-known facts ond-cluster categories and (generalized) cluster complexes of finite root sys- tems. In particular, we recall and generalize the BGP-reflection functors for cluster categories [26,27] tod-cluster categories. In the third part, we prove some properties ofd-cluster tilting objects, including that any basicd-cluster tilting object contains exactlynindecomposable direct summands. In the final section, for any root system Φ, using ad-cluster categoryCd(H), we define ad-compatibility degree on any pair of colored almost positive real Schur roots. Using thed-compatibility degree, we de- fine a generalized cluster complex associated toΦ, which has colored almost positive real Schur roots as the vertices, and any subset forms a face if and only if any two el- ements of this subset ared-compatible. This simplicial complex is isomorphic to the cluster complex ofd-cluster categoryCd(H). IfΦ is a finite root system, and if we takeH0to be the category of representations of an alternating quiver corresponding toΦ, then our generalized cluster complex is the usual generalized cluster complex ^d(Φ)defined by Fomin and Reading [12].

2 Basics ond-cluster categories

In this section, we collect some basic materials and fix the notation which we will use later on.

A valued graph(Γ ,d)is a finite set of vertices 1, . . . , n, together with nonnegative integersd_ij for all pairsi, j∈Γ such thatd_ii=0 and there exist positive integers {ε_i}i∈Γ satisfying

d_ijε_j=d_{j i}ε_i for alli, j∈Γ .

A pair{i, j}of vertices is called an edge of(Γ ,d)ifd_ij =0.An orientationΩ of a valued graph(Γ ,d)is given by prescribing for each edge{i, j}of(Γ ,d)an order (indicated by an arrowi→j). For simplicity, we denote a valued graph byΓ and a valued quiver by(Γ , Ω).

Let(Γ , Ω)be a valued quiver. We always assume that the valued quiver(Γ , Ω) contains no oriented cycles. Such orientationΩis called admissible. LetKbe a field and M=(F_i,_iM_j)_i,j∈Γ a reduced K-species of (Γ , Ω); that is, for all i, j ∈Γ,

iM_j is an F_i−F_j-bimodule, where F_i andF_j are division rings which are finite- dimensional vector spaces overKand dim(iM_j)_Fj =d_ij and dimKF_i=ε_i. We de- note byHthe category of finite-dimensional representations of(Γ , Ω,M). It is a hereditary Abelian category [10]. Let Φ be the root system of the Kac–Moody Lie algebra corresponding to the graphΓ. We assume thatP1, . . . , P_nare nonisomorphic indecomposable projective representations inH,E1, . . . , E_nare simple representa- tions with dimension vectorsα1, . . . , αn, andα1, . . . , αn are simple roots inΦ.We useD(−)to denote HomK(−, K),which is a duality ofH.

Denote byD=D^b(H)the bounded derived category ofHwith shift functor[1]. 2.1 d-cluster categories

The derived category Dhas Auslander–Reiten triangles, and the Auslander–Reiten translateτis an automorphism ofD. Fix a positive integerdand denoteF_d=τ⁻¹[d];

it is an automorphism ofD. Thed-cluster category ofH is defined in [19,25]:

We denote byD/F_dthe corresponding factor category. The objects are by defini- tion theF_d-orbits of objects inD, and the morphisms are given by

Hom_D/Fd(X,Y ) =

i∈Z

Hom_D

X, F_dⁱY .

HereXandY are objects inD, andXandY are the corresponding objects inD/Fd

(although we shall sometimes write such objects simply asXandY).

Definition 2.1 ([19,25]) The orbit categoryD/Fdis called thed-cluster category of H(or of(Γ , Ω)), which is denoted byCd(H), sometimes denoted byCd(Ω).

By [19] the d-cluster category is a triangulated category with shift functor [1] which is induced by the shift functor inD, the projectionπ:D−→D/F is a triangle functor. Whend=1, this orbit category is called the cluster category ofH, denoted byC(H)(sometimes denoted byC(Ω)).

His a full subcategory ofDconsisting of complexes concentrated in degree 0, then passing toCd(H)by the projectionπ,His a (possibly, not full) subcategory of Cd(H). For anyi∈Z, we use(H)[i]to denote the copy ofHunder theith shift[i]as a subcategory ofCd(H).In this way, we have that(indH)[i] = {M[i] |M∈indH}.

For any objectMinCd(H), addMdenotes the full subcategory ofCd(H)consisting of direct summands of direct sums of copies ofM.

For X, Y ∈ Cd(H), we will use Hom(X, Y ) to denote the Hom-space Hom_Cd(H)(X, Y ) in the d-cluster category Cd(H) throughout the paper. Define Extⁱ(X, Y )to be Hom(X, Y[i]).

We summarize some known facts aboutd-cluster categories [6,19].

Proposition 2.2 (1)Cd(H)has Auslander–Reiten triangles and Serre functorΣ= τ[1], whereτ is the AR-translate inCd(H), which is induced from AR-translate inD.

(2)Cd(H)is a Calabi–Yau category of CY-dimensiond+1.

(3)Cd(H)is a Krull–Remark–Schmidt category.

(4) indCd(H)=_i=d−1

i=0 (indH)[i] ∪ {P_j[d] |1≤j≤n}.

Proof (1) This is Proposition 1.3 of [6] and Corollary 1 in Sect. 8.4 of [19].

(2) It is proved in Corollary 1 in Sect. 8.4 of [19].

(3) This is proved in Proposition 1.2 of [6].

(4) The proof ford=1 is given in Proposition 1.6 of [6], which can be modified

for the generald.

From Proposition2.2 we define the degree for every indecomposable object in Cd(H)as follows:

Definition 2.3 For any indecomposable objectX∈Cd(H), we call the nonnegative integer min{k∈Z_≥0|X∼=M[k]inCd(H)for someM∈indH}the degree of X, denoted by degX.

By Definition2.3any indecomposable objectXof degreekis isomorphic toM[k] inCd(H), whereMis an indecomposable representation inH; 0≤degX≤d,Xhas degree d if and only if X∼=P[d] in Cd(H)for some indecomposable projective object P ∈H; andX has degree 0 if and only if X∼=M[0] in Cd(H)for some indecomposable objectM∈H. HereM[0]means regarding the objectMofHas a complex concentrated in degree 0.

2.2 BGP-reflection functors

If T is a tilting object in H, then the endomorphism algebra A=End_H(T ) is called a tilted algebra. The tilting functor Hom_H(T ,−) induces the equivalence RHom(T ,−):D^b(H)→ D^b(A), where RHom(T ,−) is the derived functor of Hom_H(T ,−).

Any standard triangle functorG:D^b(H)→D^b(H)induces a well-defined func- torG˜ :Cd(H)−→Cd(H)with the following commutative diagram [19,26]:

D^b(H) −−−−→^G D^b(H)

⏐⏐ ⏐⏐

Cd(H) −−−−→^G˜ Cd(H) The following result is proved in [26,27].

Proposition 2.4 IfG:D^b(H)→D^b(H)is a triangle equivalence, thenG˜ is also an equivalence of triangulated categories.

Let k be a vertex in the valued quiver (Γ , Ω); the reflection of(Γ , Ω) atk is the valued quiver(Γ , s_kΩ), wheres_kΩ is the orientation ofΓ obtained fromΩ by reversing all arrows starting or ending at k. The corresponding category of repre- sentations of (Γ , s_kΩ,M)is denoted simply by s_kH. If k is a sink in the valued quiver(Γ , Ω), thenk is a source of(Γ , s_kΩ), and the reflection of (Γ , s_kΩ) atk is(Γ , Ω). Letkbe a sink in(Γ , Ω). ThenPk is a simple projective representation, and T = ⊕j=kPj ⊕τ⁻¹Pk is a tilting representation in H[24]. The tilting func- torS_k⁺=Hom_H(T ,−)is a so-called BGP-reflection functor, and its derived functor RHom(T ,−)is a triangle equivalence fromD^b(H)toD^b(s_kH), which is also de- noted byS_k⁺.Similarly, one has BGP-reflection functorsS_k⁻for sourcesk.

Definition 2.5 The induced functorsS_k⁺:Cd(H)−→Cd(skH)for sinkskandS_k⁻: Cd(H)−→Cd(skH)for sources k are called BGP-reflection functors of d-cluster categories.

Remark 2.6 Whend=1, BGP-reflection functors are discussed in [26].

We remind the reader that H (or H) is the category of representations of the valued quiver(Γ , Ω)((Γ , skΩ), respectively); the Pi (respectively, the P_i) are the indecomposable projective representations inH(respectively, H), and theEi (re- spectively, theE_i) are the corresponding simple representations which are the tops of theP_i (respectively, theP_i) fori=1, . . . , n.

We recall from Proposition2.2and Definition2.3that any indecomposable ob- jectY inCd(H)is isomorphic toX[i],whereX∈indH,andiis the degree ofY. Keeping this notation, we have the following proposition which gives the images of indecomposable objects inCd(H)under the BGP-reflection functorS_k⁺.

Proposition 2.7 Letkbe a sink of the valued quiver(Γ , Ω)andY an indecompos- able object inCd(H)with degreei. ThenY∼=X[i]for an indecomposable represen- tationXinH, and

S_k⁺(X[i])=

⎧⎪

⎪⎪

⎨

⎪⎪

⎪⎩

P_k[d] ifX∼=P_k(∼=E_k)andi=0, E_k[i−1] ifX∼=P_k(∼=E_k)and 0< i≤d, P_j[d] ifX∼=PjPkandi=d, S_k⁺(X)[i] otherwise.

Proof The statement in the proposition was proved in [26,27] whend=1. The proof for the cased >1 is the same as there. We give a sketch of the proof for the conve- nience of readers. The BGP-reflection functorS_k⁺:H−→s_kHinduces a triangle equivalenceD^b(H)−→D^b(s_kH),denoted also by S⁺_k. It induces an equivalence indD^b(H)−→indD^b(skH). For any indecomposable objectX[i] ∈indD^b(H),it is not hard to show thatS_k⁺(X[i])=S_k⁺(X)[i]forXPk (note thatPk=Ek, since

kis a sink in(Γ , Ω)), andS_k⁺(P_k[i])=E_k[i−1]fori∈Z (cf. [26] or [27]). SinceE_k is an injective representation inskH, we haveτ P_k[i] =E_k[i−1]inD^b(skH). Now passing to thed-cluster categoryCd(H)(which is an orbit category of the derived categoryD^b(H)), we get the images of indecomposable objects ofCd(H)underS_k⁺

as stated in the proposition.

3 Cluster combinatorics of root systems

For a valued graphΓ, we denote byΦ=Φ⁺∪Φ⁻the set of roots of the correspond- ing Kac–Moody Lie algebra.

Definition 3.1 (1) The set of almost positive roots is Φ_≥−1=Φ⁺∪ {−αi|i=1, . . . n}.

(2) Denote byΦ_≥−^re ₁the subset ofΦ_≥−1consisting of the positive real roots to- gether with the negatives of the simple roots.

WhenΦis of finite type,Φ_≥−1=Φ_≥−^re ₁.

Definition 3.2 Lets_ibe the Coxeter generator of the Weyl group ofΦcorresponding toi∈Γ0. We call the following map the “truncated simple reflection” σ_i ofΦ_≥−1

[14]:

σ_i(α)=

α, α= −α_j, j=i, s_i(α), otherwise.

It is easy to see thatσi is an automorphism ofΦ_≥−^re ₁. 3.1 Cluster complexes of finite root systems

In this first paragraph, we do not assume thatΓ is a Dynkin diagram (i.e., of finite type). Leti₁, . . . , i_nbe an admissible ordering ofΓ with respect toΩ, i.e.,i_tis a sink with respect tos_it−1· · ·s_i2s_i1Ωfor any 1≤t≤n.DenoteR_Ω=σ_in· · ·σ_i1.This is an automorphism ofΦ_≥−1 and does not depend on the choice of admissible ordering of Γ with respect to Ω.It is the automorphism induced by the Auslander–Reiten translationτ inC(H)(cf. [26,27]).

In the rest of this subsection, we always assume thatΓ is a valued Dynkin graph, which is not necessarily connected. Fomin and Zelevinsky [15] associate a nonnega- tive integer(αβ), known as the compatibility degree, to each pairα, β of almost positive roots.

This is defined in the following way: LetΩ0denote one of the alternating orien- tations of Γ, and Γ⁺ (respectively, Γ⁻) the set of sinks (respectively, sources) of (Γ , Ω0). Define

τ_±=

i∈Γ^±

σ_i.

ThenR_Ω0=τ₋τ₊, which is simply denoted byR.

Denote byn_i(β)the coefficient ofα_i in the expansion ofβin terms of the simple rootsα₁, . . . , α_n. Then()is uniquely defined by the following two properties:

(∗) (−αi β)=max

[β:αi],0 , (∗∗) (τ_±ατ_±β)=(αβ), for anyα, β∈Φ_≥−1and anyi∈Γ.

Two almost positive rootsα, βare called compatible if(αβ)=0.

The cluster complex(Φ)associated to the finite root systemΦis defined in [14].

Definition 3.3 The cluster complex(Φ)is a simplicial complex on the ground set Φ_≥−1. Its faces are mutually compatible subsets ofΦ_≥−1.The facets of(Φ)are called the (root-)clusters associated toΦ.

3.2 Generalized cluster complexes of finite root systems

At the beginning of this subsection, we assume thatΓ is an arbitrary valued graph, which is not necessarily connected, except where we express specifically. As before, Φ denotes the set of roots of the corresponding Lie algebra, andΦ_≥−1denotes the set of almost positive roots. Fix a positive integerd; for anyα∈Φ⁺, following [12], we callα¹, . . . , α^dthed “colored” copies ofα.

Definition 3.4 ([12]) The set of colored almost positive roots is Φ_≥−^d ₁=

αⁱ:α∈Φ>0, i∈ {1, . . . , d}

∪

(−αi)¹:1≤i≤n .

WhenΓ is a Dynkin graph, the root systemΦ of the corresponding Lie algebra is finite. In this case, the generalized cluster complex^d(Φ)is defined on the ground setΦ_≥−^d ₁and using the binary compatibility relation onΦ_≥−^d ₁. This binary compat- ibility relation is a natural generalization of binary compatibility relation onΦ_≥−1, which we now recall from [12].

For a rootβ∈Φ_≥−1, lett (β)denote the smallestt such thatR^t(β)is a negative root.

Definition 3.5 ([12]) Two colored rootsα^k, β^l∈Φ_≥−^d ₁are called compatible if and only if one of the following conditions is satisfied:

(1) k > l. t (α)≤t (β), and the rootsR(α) and β are compatible (in the original

“non-colored” sense).

(2) k < l. t (α)≥t (β),and the rootsαandR(β)are compatible.

(3) k > l. t (α) > t (β),and the rootsαandβare compatible.

(4) k < l. t (α) < t (β),and the rootsαandβare compatible.

(5) k=l.And the rootsαandβ are compatible.

Now we are ready to recall the definition of generalized cluster complex^d(Φ) for a finite root systemΦ.

Definition 3.6 ([12])^d(Φ)hasΦ_≥−^d ₁as the set of vertices, its simplices are mutu- ally compatible subsets ofΦ_≥−^d ₁.The subcomplex of^d(Φ)which hasΦ_>0^d as the set of vertices is denoted by^d₊(Φ)

Now we generalize the definition ofRd[12] for a finite root system to an arbitrary root system.

Definition 3.7 Let(Γ , Ω)be a valued quiver. Forα^k∈Φ_≥−^d ₁,we set R_d,Ω

α^k

=

α^k+1 ifα∈Φ>0andk < d, (RΩ(α))¹ otherwise.

Remark 3.8 If(Γ , Ω0)is a valued Dynkin graph with an alternating orientation, then the automorphismRofΦ_≥−1defined by Fomin and Zelevinsky [14] isRΩ₀;hence, R_d,Ω0 is the usual one (Rd) defined by Fomin and Reading [12].

Theorem 3.9 ([12]) Let Φ be a finite root system. The compatibility relation on Φ_≥−^d ₁has the following properties:

(1) α^k is compatible withβ^lif and only ifRd(α^k)is compatible withRd(β^l).

(2) (−αi)¹is compatible withβ^lif and only ifni(β)=0.

Moreover, conditions 1–2 uniquely determine this relation.

Now we generalize the “truncated simple reflections” ofΦ_≥−1to the colored al- most positive roots. LetΦbe an arbitrary root system (not necessarily of finite type).

Definition 3.10 Lets_kbe the Coxeter generator of the Weyl group ofΦcorrespond- ing tok∈Γ0. We define the following mapσk,dofΦ_≥−^d ₁:

σk,d(αⁱ)=

⎧⎪

⎪⎨

⎪⎪

⎩

α_k^d ifi=1 andα= −αk, α_kⁱ⁻¹ if 1< i≤dandα=α_k, (−αj)¹ ifi=1 andα= −αj, j=k, (s_k(α))ⁱ otherwise.

σ_k,d is a bijection ofΦ_≥−^d ₁.We call it ad-truncated simple reflection ofΦ_≥−^d ₁.

4 d-cluster tilting ind-cluster categories

LetCd(H)be ad-cluster category of typeH, whereHis the category of represen- tations of the valued quiver (Γ , Ω). It is a Calabi–Yau triangulated category with CY-dimensiond+1.

Definition 4.1 (1) An objectXinCd(H)is called exceptional if Extⁱ(X, X)=0 for any 1≤i≤d.

(2) An object X is called a d-cluster tilting object if it satisfies the property:

Y ∈add(X)if and only if Extⁱ(X, Y )=0 for 1≤i≤d.

(3) An object X is called almost complete tilting if there is an indecomposable objectY such thatX⊕Y is ad-cluster tilting object. Such an indecomposable object Y is called a complement ofX.

Proposition 4.2 (1) For an objectXinH,Xis exceptional inHi.e., Ext¹_H(X, X)= 0 if and only ifX[0]is exceptional inCd(H).

(2) Any indecomposable exceptional objectXinCd(H)is of the formM[i]with Mbeing an exceptional representation inHand 0≤i≤d−1 or of the formPj[d] for some 1≤j≤n. In particular, ifΓ is a Dynkin graph, then any indecomposable object inCd(H)is exceptional.

(3) Suppose thatd >1. Then End_Cd(H)Xis a division algebra for any indecom- posable exceptional objectX.

(4) Suppose thatd >1. LetP be a projective representation in HandX a rep- resentation inH. Then, for any−d≤i≤d, Ext¹(P , X[i])=0 except possibly for i∈ {−1, d−1, d}.

Proof (1) LetX∈Hbe exceptional. We will prove that Extⁱ(X, X)=0 for anyi∈ {1, . . . , d}. By definition we have that Extⁱ(X, X)=

k∈ZExtⁱ_D(X, τ^−kX[kd])= Extⁱ_D(X, X)⊕Extⁱ_D(X, τ X[−d]). In this sum, the first summand Extⁱ_D(X, X)= 0,∀i≥1,while the second summand Extⁱ_D(X, τ X[−d])∼=Hom_D(X, τ X[i−d]), which is zero wheni < d and is isomorphic to Ext¹_D(X, X)=0 wheni=d.This proves thatX is exceptional inCd(H).The proof for the converse directly follows from the definition.

(2) The statements follow from Proposition2.2(4) and Definition4.1, also using Part 1 and the fact that the shift is an autoequivalence.

(3) Let X be an indecomposable exceptional representation in H,and suppose that d >1. From the definition of the orbit category It follows that End_Cd(H )X∼=

m∈ZHom_D(X, τ^−mX[dm])∼=End_HX. The last isomorphism holds due to the facts: Hom_D(X, τ^mX[−md])∼=Hom_D(X[md], τ^mX)∼=Ext¹_D(τ^m−1X, X[md])= 0 for any positive integer m and Hom_D(X, τ^−mX[md])∼=Hom_D(τ^mX, X[md]), which is also zero, sincemd >1 (we use the assumptiond >1 here) for any positive integerm. Then End_{Cd(H )}Xis a division algebra, since End_HXis a division algebra.

Since any indecomposable exceptional objectMinCd(H )is some shiftX[i]of an indecomposable exceptional representationXinH,End_{Cd(H )}M=End_{Cd(H )}X[i] ∼= End_{Cd(H )}Xis a division algebra.

(4) Suppose that d >1. Let P be a projective representation in H and X a representation in H. Then, for any −d≤i≤d, Ext¹(P , X[i])= k∈ZExt¹_D(P , τ^−kX[dk+i])∼=Ext¹_D(P , τ X[−d +i])⊕Ext¹_D(P , X[i]). Now if i= −1, d−1, d,then Ext¹_D(P , τ X[−d+i])=0=Ext¹_D(P , X[i]). Then, for any

−d≤i≤d, Ext¹(P , X[i])=0 except fori= −1, d−1,andd. Remark 4.3 Any basic (i.e., multiplicity-free) exceptional object contains at most (d+1)nnonisomorphic indecomposable direct summands.

Proof LetXbe a basic exceptional object inCd(H). Then any indecomposable di- rect summand of X is exceptional; hence, by Proposition4.2(2), we write M as M=_k=d

k=0

i∈IkM_i,k[k] with M_i,k being an indecomposable exceptional repre- sentation. Therefore,

i∈IkM_i,k is an exceptional object in hereditary categoryH;

hence, the number of direct summands is at mostn, i.e.,|Ik| ≤n. Then the number of indecomposable direct summands ofMis at most(d+1)n.

For any pair of objectsT , XinCd(H), due to the Calabi–Yau property ofCd(H), we have that Extⁱ(X, T )=0 for 1≤i≤dif and only if Extⁱ(T , X)=0 for 1≤i≤d. Hence, by Remark4.3and Definition4.1,T is ad-cluster tilting object inCd(H)if and only if addT is a maximald-orthogonal subcategory ofCd(H)in the sense of [17]: i.e., addT is contravariantly finite and covariantly finite inCd(H)and satisfies the following property:X∈addT if and only if Extⁱ(X, T )=0 for 1≤i≤d if and only if Extⁱ(T , X)=0 for 1≤i≤d. In the following, we will prove that any basicd- cluster tilting object contains exactlynindecomposable direct summands. First of all, we recall some results from [17] which hold in any(d+1)-Calabi–Yau triangulated category.

Theorem 4.4 (Iyama) LetXbe an almost complete tilting object inCd(H)andX₀ a complement ofX. Then there ared+1 triangles:

(∗) Xi+1 g_i

−→Bi f_i

−→Xi σ_i

−→Xi+1[1],

where f_i is the minimal right addX-approximation of X_i and g_i minimal left addX-approximation of X_i+1, allX_i are indecomposable and complements of X, i=0, . . . , d.

For the convenience of readers, we sketch the proof; for details, see [18].

Proof We suppose thatd >1; the same statement ford=1 was proved in [6]. For the complementX₀ ofX, we consider the minimal right addX-approximationf₀: B0→X0 of X0, extendf0 to the triangleX1

g0

→B0 f0

→X0 σ0

→X1[1]. It is easy to see that X₁ is indecomposable,g₀ is the minimal left addX-approximation ofX₁, andX⊕X₁is an exceptional object inCd(H)(cf. [6]). From Theorem 5.1 in [18]

it follows thatX⊕X₁is ad-cluster tilting object. Continuing this step, one can get complementsX₁, . . . , X_d+1with trianglesX_i+1→^gi B_i→^fi X_i→^σi X_i[1]for 0≤i≤d, wheref_i (gi) is the minimal right (left, resp.) addX-approximation of X_i (X_i+1,

resp.), andX⊕X_i is ad-cluster tilting object.

Corollary 4.5 With the notation of Theorem4.4, we have thatσ_d[d]σ_d−1[d−1] · · · σ1[1]σ0=0.In particular, Hom(Xi, Xj[j−i])=0 andXiXj,∀0≤i < j≤d.

Proof From Theorem 4.4 we have that σ₀ =0, since the triangle (∗) at i=0 in Theorem 4.4 is nonsplitting. Suppose that σ_d[d]σ_d−1[d −1] · · ·σ₁[1]σ₀ =0;

then σ_d−1[d −1] · · ·σ₁[1]σ₀ : X₀ → X_d[d] factors through f_d[d] : B_d[d] → X_d[d],since we have a triangleX_d+1[d]^g−→^d[d]B_d[d]^f−→^d[d]X_d[d]^σ−→^d[d]X_d+1[d+1].

Since Hom(X0, B_d[d])=Ext^d(X₀, B_d)=0, σ_d−1[d −1] · · ·σ₁[1]σ₀=0. Simi- larly, σ_d−2[d −2] · · ·σ1[1]σ0 =0 and, finally, σ0=0, a contradiction. Now we prove the final statement: we have thatσj−1[j−1] · · ·σi∈Hom(Xi, Xj[j−i])and σj−1[j−1] · · ·σi=0.Otherwiseσj−1[j −1] · · ·σi=0, and henceσd−1[d−1] · · · σ₁[1]σ₀=0, a contradiction. Now suppose that X_i ∼=X_j for some i < j. Then Ext^k(X_i, X_j)=0 for 1≤k≤d, a contradiction. ThenX_iX_j.

Now we state our main result of this section.

Theorem 4.6 Any basicd-cluster tilting object in Cd(H)contains exactlyn inde- composable direct summands.

To prove the theorem, we need some technical lemmas.

Lemma 4.7 Letd >1,and letX=M[i], Y=N[j]be indecomposable objects of degreesi, j,respectively, inCd(H). Suppose that Hom(X, Y )=0.Then one of the following holds:

(1) We havei=jorj−1 (provided thatj≥1).

(2) We havei=0,i=d (andM=P )ord−1 (provided thatj =0).

Proof Letd >1. Firstly we note that, for any indecomposable objectX∈Cd(H), 0 ≤ degX ≤ d, degX =d if and only if X = P_i[d] for an indecomposable projective representation Pi. This implies that −d ≤degY −degX ≤d for in- decomposable objects X, Y ∈ Cd(H). Let X = M[i], Y = N[j] be indecom- posable objects of degrees i, j, respectively, in Cd(H). We have Hom(X, Y )∼= Hom(M, N[j−i])=

k∈ZHom_D(M, τ^−kN[j−i+kd])=Hom_D(M, τ N[j−i−d])⊕ Hom_D(M, N[j−i])⊕Hom_D(M, τ⁻¹N[j−i+d]). The last equality holds due to

−d≤j−i≤d, and Hom_D(M, τ^−kN[j−i+kd])=0 fork= −1,0,1. We divide the calculation of Hom(X, Y )into three cases:

(1) The case−d < j−i < d.We have that Hom(X, Y )∼=Hom_D(M, N[j−i])⊕ Hom_D(M, τ⁻¹N[j−i+d]). The first summand is zero whenj−i=0,1, while the second is zero whend+j −i=1 (equivalently,d+j −i >1,since 0<

d+j−i <2d).

(2) The case j −i = −d. Then j =0, i =d (M =P ). Then Hom(X, Y )= Hom_D(P , τ⁻¹N ).

(3) The case j −i =d. Then j =d (N =P ), i = 0. Then Hom(X, Y ) = Hom_D(M, τ P )⊕Hom_D(M, P[d])⊕Hom_D(M, τ⁻¹P[2d])=0.

Therefore, if Hom(X, Y )=0, then Hom(M[0], N[j −i])=0. Proof of (1). Sup- pose that j ≥1. Then combining with Case 3, we have that −d < j −i < d.

We want to prove that if j −i =0,1, then Hom(X, Y )=0, and this will fin- ish the proof of (1). Under the conditionj −i=0,1, from Case 1 we have that Hom(X, Y )∼=Hom_D(M, τ⁻¹N[j−i+d]), which is zero ford+j−i=1. But if d+j−i=1, i.e.,i=d,thenM=P andj=1. Then Hom_D(M, τ⁻¹N[j−i+d])= Hom_D(P , τ⁻¹N[1])=0. We have finished the proof of (1).

Proof of (2) Suppose thatj =0.Then−d≤j−i≤0. From Cases 1–2 it follows thati=0, i=d (M=P ),ori=d−1.This finishes the proof of (2).

Lemma 4.8 Ifd >2, then Ext²(M[i], N[i])=0 for objectsM, N∈Hand anyi.

Proof It is sufficient to prove that Ext²(M[0], N[0])=0. From the definition of the orbit categoryD/τ⁻¹[d]we have that

Ext²

M[0], N[0]

=Hom

M[0], N[2]

=

k∈Z

Hom_D

M, τ^−kN[kd+2]

,

where each summand Hom_D(M, τ^−kN[kd+2]) equals 0, since kd+2≥2 or kd+2≤ −1 by the conditiond >2.Hence, Ext²(M[0], N[0])=0.

Lemma 4.9 Letd >1 andM, N∈H. Then Ext¹(M[0], N[0])∼=Ext¹_H(M, N ). Fur- thermore, any non-split triangle betweenM[0]andN[0]inCd(H)is induced from a non-split exact sequence betweenMandN inH.

Proof Under the condition d > 1, it is easy to see that Ext¹(M[0], N[0]) =

k∈ZExt¹_D(M, τ^−kN[2k])=Ext¹_D(M, N ) =Ext¹_H(M, N ). This proves the first statement. Since H⊂Cd(H) is a (not necessarily full) embedding and any exact short sequence in Hinduces a triangle in Cd(H), the final statement then follows

from the first statement.

Proof (of Theorem4.6) We assume thatd >1,since it was proved in [6] ford=1.

LetM=

i∈IM_i[k_i]be ad-cluster tilting object inCd(H), where allM_i are in- decomposable representations in H, 0≤k_i ≤d (when k_i =d, M_i is projective).

One can assume that one ofk_i is 0, otherwise one can replaceMby a suitable shift of M. Denote ν(M)=max{|k_i −k_j| | ∀i, j}. We prove that|I| =nby induction on ν(M), where|I| denotes the cardinality ofI. If ν(M)=0, i.e., k_i =0 for all i, then

i∈IM_i[0]is ad-cluster tilting object inCd(H)and hence a tilting object inH. Then|I| =n. Now assume thatν(M)=m >0. Without loss of generality, we assume that k1= · · · =k_t =mand k_j < m for j > t. From the complement X0=M1[k1]ofX=M\M1[k1](here we useX\X1to denote a complement ofX1

inX for a direct summandX1ofX), by Theorem4.4, we have at leastd+1 com- plementsX_j,j =0, . . . , d,which form the triangles(∗)in Theorem4.4. In these triangles, it is easy to see thatf_i =0 if and only ifB_i =0 if and only if g_i =0.

We will prove that there is at least one of complementsXj with smaller degree than m. At first, we prove this statement for the special casem=1. We claim that the degree ofX1is 0 or 1 in this case. OtherwiseX1=P[d]for some indecomposable projective representation P or X1=Y[d −1]for some indecomposable represen- tation Y. Write X0 as Z[1], where Z is an indecomposable representation in H. IfX1=P[d], then Hom(X1, X0[d])=Hom(P[d], X0[d])∼=Hom(P , Z[1])=0, a contradiction to the fact that Hom(X1, X0[d])∼=Hom(X0, X1[1])is not zero by The- orem4.4or Corollary4.5. IfX1=Y[d−1], thenX1has degree 1 whend=2, and Hom(X1, X0[d])=Hom(Y[d−1], Z[d+1])∼=Ext²(Y, Z)=0 by Lemma4.8when d >2, which also contradicts to the fact that Hom(X1, X₀[d])∼=Hom(X0, X₁[1])is not zero. This proves the statement thatX₁ has degree 0 or 1. Now if there are no complementsX_j ofXwith degree 0, then allX_j have degree 1. We prove that any three successive complements, sayX₀, X₁, X₂, cannot have the same degree. If all

degrees ofX_i, i=0,1,2,are the same, we can assume that allX_i have degree 0. By Lemma4.9, we have non-split short exact sequences inH:

0−→X1−→B0−→X0−→0, 0−→X2−→B1−→X1−→0.

From the first short exact sequence we have Ext¹_H(X0, X1) 0. Applying Hom_H(X0,−)to the second exact sequence, we have the long exact sequence

· · · →Ext¹_H(X0, X2)→Ext¹_H(X0, B1)→Ext¹_H(X0, X1)

→Ext²_H(X0, X2)→Ext²_H(X0, B1).

SinceX⊕X₀is ad-cluster tilting object inCd(H)andB₁∈addX, Ext¹(X₀, B₁)=0.

Hence we have that Ext¹_H(X₀,B₁)=0 by Lemma4.9. It follows that Ext¹_H(X₀,X₁)=0, since Ext²_H(X₀, X₂)=0 due toHbeing hereditary. It is a contradiction. This finishes the proof form=1.

Now suppose thatm >1. We will prove that there is at least one of complements Xjwith smaller degree thanm. We divide the proof into two cases:

Case 1. All mapsfi (equivalentlygi) are nonzero. Now we assume that there are no complements ofXwith smaller degree thanm. Then by Lemma4.7the degrees of allX_iarem. Ifd >2, then Ext²(X0, X2)=0 by Lemma4.8, a contradiction to Corol- lary4.5. Ifd=2, then the same proof as above shows that Ext¹(X0, X1)=0,which contradicts to Corollary4.5. Therefore, there is a complement ofXwith smaller de- gree thanm.

Case 2. There are some i such that f_i =0 (equivalently g_i =0). Then X_i ∼= X_i+1[1]for suchi. It follows thatX_i+1has smaller degree thanX_iifX_i has a strictly positive degree. Therefore, we have a complement ofX, sayX_s, such that the de- greek₁ofXs is smaller thanm=k1. Now we replaceX byX=(X\X0)⊕Xs, which is, by Theorem 4.4, a d-cluster tilting object inCd(H)containing |I| inde- composable direct summands. The number of indecomposable direct summands of Xwith the (maximal) degreem(=ν(M))ist−1. We repeat the step for the com- plement M2[k2] of almost complete tilting objectX\M2[k2], getting a d-cluster tilting object X containing |I|indecomposable direct summands, and the number of indecomposable direct summands ofXwith the (maximal) degreem(=ν(M))is t−2. Repeating such a stept times, one can get a (basic)d-cluster tilting objectT containing|I|indecomposable direct summands andν(T ) < ν(M).By induction,T contains exactlynindecomposable direct summands. Then|I| =n.

Remark 4.10 Theorem 4.6is proved by Thomas [25] for a simply-laced Dynkin quiver(Γ , Ω0), using the fact that indD^b(K) ≈Z for a Dynkin quiver. This fact does not hold for non-Dynkin quivers. Our proof is more categorical.

Denote by E(H)the set of isomorphism classes of indecomposable exceptional representations inH. The setE(Cd(H))of isoclasses of indecomposable exceptional objects in Cd(H) is the (disjoint) union of subsets E(H)[i], i=0,1, . . . , d −1, with{P_j[d]|1≤j ≤n}. A subsetMofE(Cd(H))is called exceptional if, for any

X, Y ∈M, Extⁱ(X, Y )=0 for all i=1, . . . , d. Denote by E₊(Cd(H)) the sub- set of E(Cd(H)) consisting of all indecomposable exceptional objects other than P1[d], . . . , P_n[d].

Now we are ready to define a simplicial complex associated to thed-cluster cat- egoryCd(H), which is a generalization of the classical cluster complexes of cluster categories [6,24,26].

Definition 4.11 The cluster complex^d(H)ofCd(H)is a simplicial complex which has E(Cd(H)) as the set of vertices and has exceptional subsets in Cd(H) as its simplices. The positive part ^d₊(H) is the subcomplex of ^d(H) on the subset E+(Cd(H)).

By the definition, the facets (maximal simplices) are exactly thed-cluster tilting subsets (i.e., the sets of indecomposable objects ofCd(H)(up to isomorphism) whose direct sum is ad-cluster tilting object).

Proposition 4.12 (1)^d(H)and^d₊(H)are pure of dimensionn−1.

(2) For any sink (or source)k, the BGP-reflection functorS˜_k⁺(resp.S˜_k⁻) induces an isomorphism between^d(H)and^d(s_kH). In particular, ifΓ is a Dynkin diagram andΩandΩare two orientations ofΓ, then^d(H)and^d(H)are isomorphic.

Proof (1) From Theorem 4.6it follows that any d-cluster tilting subset contains exactly n elements. Hence^d(H)is pure of dimension n−1. Now suppose that M= ⊕ⁿ_i=⁻₁¹Mi is an exceptional object in Cd(H) and that none of theMi are iso- morphic to Pj[d] for any j. In the proof of Theorem 4.6, we proved that not all complements of an almost complete tilting objects have the same degrees. ThenM has a complement inE₊(Cd(H)). This proves that^d₊(H)is pure of dimensionn−1.

(2) Since S˜⁺_k is a triangle equivalence from the d-cluster category Cd(H) to Cd(s_kH), it sends (indecomposable) exceptional objects to (indecomposable) excep- tional objects. Thus it induces an isomorphism from^d(H)to^d(s_kH). The second statement follows from the first statement together with the fact that, for two orien- tationsΩ, Ω of a Dynkin graphΓ, there is a admissible sequence with respect to sinksi1, . . . , i_nsuch thatΩ=s_in· · ·s_i1Ω.

5 Cluster combinatorics ofd-cluster categories

We now define a mapγ_H^d from indCd(H)toΦ_≥−^d ₁. Note that any indecomposable objectXof degreeiinCd(H)has the formM[i]withM∈indH, and ifi=d,then M=Pj, an indecomposable projective representation.

Definition 5.1 Let γ_H^d be defined as follows. Let M[i] ∈indCd(H), where M∈ indH andi∈ {1, . . . , d}(note that ifi=d,thenM=P_j for somej). We set

γ_H^d(M[i])=

(dimM)ⁱ⁺¹ ifM[i] ∈indH[i]for some 0≤i≤d−1; (−αj)¹ ifM[i] =Pj[d].