THE CLUSTER CHARACTER FOR CYCLIC QUIVERS MINGDINGANDFANXU 0 1 Abstract. We define an analogue of the Caldero-Chapoton map ([CC]) for 0 theclustercategoryoffinitedimensionalnilpotentrepresentationsoveracyclic 2 quiver. We prove that it is a cluster character (in the sense of [Pa]) and satisfiessomeinductiveformulasforthemultiplicationbetweenthegeneralized n clustervariables(theimagesofobjectsoftheclustercategoryunderthemap). a Moreover,weconstructaZ-basisforthealgebrasgeneratedbyallgeneralized J clustervariables. 5 2 Contents ] T 1. Introduction 1 R 2. The cluster character for cyclic quivers 2 . 3. Inductive multiplication formulas 6 h 4. A Z-basis for cyclic quivers 8 t a Acknowledgements 10 m References 10 [ 1 v 0 1. Introduction 6 3 ClusteralgebraswereintroducedbyS.FominandA.Zelevinsky[FZ]inorderto 4 developacombinatorialapproachto study problemsoftotalpositivityin algebraic . 1 groups and canonical bases in quantum groups. The link between cluster algebras 0 andrepresentationtheoryofquiverswerefirstrevealedin[MRZ]. In[BMRRT],the 0 authors introduced the cluster category of an acyclic quiver Q (a quiver without 1 oriented cycles) as the categorification of the corresponding cluster algebras. In : v order to show that a cluster category categorifiesthe involving cluster algebra,the i X Caldero-Chapoton map was defined by P. Caldero and F. Chapoton in [CC]. Let C(Q) be the cluster category associated to an acyclic quiver Q (a quiver without r a oriented cycles). The Caldero-Chapotonmap of an acyclic quiver Q is a map XQ :obj(C(Q))→Q(x ,··· ,x ). ? 1 n The map was extensively defined by Y. Palu for a Hom-finite 2-Calabi-Yau trian- gulated categories with a cluster tilting object ([Pa]). As in [Ke], the cluster category can be defined for any small hereditary abelian categorywithfinitedimensionalHom-andExt-spaces. Itisinterestingtostudythe cluster categorieswithout cluster tilting objects andthe involving cluster algebras. For example, the cluster category of a 1-Calabi-Yau abelian category contains no cluster tilting objects (even no rigid objects). Keywordsandphrases: cyclicquiver,clusteralgebra,Z-basis. FanXuwassupportedbyAlexandervonHumboldtStiftungandwasalsopartiallysupported bythePh.D.ProgramsFoundation ofMinistryofEducationofChina(No. 200800030058). 1 2 MINGDINGANDFANXU In this paper, we will focus on the simplest example of the cluster category without cluster tilting objects: the cluster category of a cyclic quiver. We first define an analogue of the Caldero-Chapoton map for a cyclic quiver. We prove a multiplication formula analogous to the cluster multiplication theorem for acyclic cluster algebras ([CK], [XX]). As a corollary, the map is a cluster character in the sense of [Pa]. Let A be the cyclic quiver with r vertices and C(A ) be its r r cluster category. Let AHebe the subalgebra of Q(x ,··· ,x ) generatedeby {X | 1 r M M ∈ modCA } and EH be the subalgebra of AH generated by {X | M ∈ r M modCA ,Exte1 (M,M)=0}.WecallthealgebraEHtheclusteralgebraofA . r e r e C(An) e We show that AH coincides with EH and construct a Z-basis of EH. 2. The cluster character for cyclic quivers Let k = C be the field of complex number. Throughout the rest part of this paper, we fix a cyclic quiver Q = A , i.e, a quiver with oriented cycles, where r e Q ={1,2,··· ,r}: 0 qqqqqqq88 r PPPPPPP(( 1oo ··· oo r−1 We denote by modkQ the category of finite-dimensional nilpotent representations ofkQ. Letτ betheAuslander-Reitentranslationfunctor. LetE ,··· ,E besimple 1 r modules of the vertices 1,··· ,r, respectively. Set dimE = s for i = 1,··· ,r. We i i have τE = E ,··· ,τE = E The Auslander-Reiten quiver of kQ is a tube of 2 1 1 r rank r with E ,··· ,E lying the mouth of the tube. For 1 ≤ i ≤ r, we denote 1 r by E = E , and by E [n] the unique nilpotent indecomposable representation i i+r i with socle E and length n. Set E [0] = 0 for i = 1,··· ,r. We note that that any i i indecomposable kQ-module is of the form E [j] for i = 1,...,r and j ∈ N⊔{0}. i Let x = {x |i ∈ Q } is a family indeterminates over Z and set x = x for i 0 i i+mr 1≤i≤r,m∈Z . Here we denote by Z =Z⊔{0}. ≥0 ≥0 Bydefinition,theclustercategoryC =C(Q)istheorbitcategoryDb(modkQ)/τ◦ [−1]. It is a triangulatedcategoryby [Ke, Theorem9.9]. Different fromthe cluster categoryofanacyclicquiver,thesetofobjectsinC coincideswiththesetofobjects in modkQ. Also, for any two indecomposable objects M,N ∈C, we have Ext1(M,N)=Ext1 (M,N)⊕Ext1 (N,M). C kQ kQ It is possible that both of two terms in the right side don’t vanish. We denote by h−,−i the Euler form on modkQ, i.e., for any M,N in modkQ, hdimM,dimNi:=dim Hom (M,N)−dim Ext1 (M,N). k kQ k kQ It is well defined. Thus according to the Caldero-Chapoton map, we can similarly define a map on modkQ X :obj(modkQ)−→Z[x±1] ? by mapping M to XM =Xχ(Gre(M)) Y x−i he,sii−hsi,dimM−ei e i∈Q0 where Gr (M) is the e-Grassmannian of M, i.e., the variety of finite-dimensional e nilpotent submodules of M with dimension vector e, and set X = 1. Here, we 0 need not assume that M is indecomposable by the following Proposition 2.2. The fraction X is called a generalized cluster variable. M THE CLUSTER CHARACTERS FOR CYCLIC QUIVERS 3 Proposition 2.1. With the above notation, we have n−1 x x x x l+n l+n l+r−1 l+r−1 XEl[n] = x +X x x + x . l l+k−1 l+k l+n−1 k=1 for n∈N and l=1,··· ,r. Proof. It is known that all submodules of E[n] are E [0],E [1],··· ,E[n]. Set l l l l d =dimE [j]. By definition, i,j i n XEl[n] =X Y x−i hdl,k,di,1i−hdi,1,dl+k,n−ki. k=0i∈Q0 By the definition of the Euler form, we have − d ,d − d ,d =−dim Hom(E [k],E )+dim Ext1(E [k],E ) (cid:10) l,k i,1(cid:11) (cid:10) i,1 l+k,n−k(cid:11) k l i k l i −dim Hom(E ,E [n−k])+dim Ext1(E ,E [n−k]). k i l+k k i l+k If k =0, then x−hdl,k,di,1i−hdi,1,dl+k,n−ki = xl+n. Qi∈Q0 i xl If k =n, then x−hdl,k,di,1i−hdi,1,dl+k,n−ki = xl+r−1. Qi∈Q0 i xl+n−1 If 0i, then 1)for k ≥j−i, we have X X =X X + Ei[k] Ej[mr+l] Ei[j−i−1] Ek+i+1[mr+l+j−k−i−1] X X , Ei[mr+l+j−i] Ej[k+i−j] 2)for k < j − i and i ≤ l + j − r ≤ k + i − 1, we have X X = Ei[k] Ej[mr+l] X X +X X , Ej[(m+1)r+k+i−j] Ei[l+j−r−i] Ej[(m+1)r+i−j−1] El+j+1[k+r+i−l−j−1] 3)for other conditions, i.e, there are no extension between E [k] and E [mr+l], i j we have X X =X . Ei[k] Ej[mr+l] Ei[k]⊕Ej[mr+l] Proof. We only prove (1) and (2) is totally similar to (1). 1) When k =1, by k+i≥r+j and 1≤j ≤i≤r =⇒i=r and j =1. Then by Proposition 2.2 and Corollary 2.5, we have X X =X +X . Er E1[mr+l] Er[mr+l+1] E2[mr+l−1] When k =2, by k+i≥r+j and 1≤j ≤i≤r =⇒i=r or i=r−1. For i=r =⇒j =1 or j =2: The case for i=r and j =1, we have X X Er[2] E1[mr+l] = (X X −1)X Er E1 E1[mr+l] = X (X +X )−X E1 Er[mr+l+1] E2[mr+l−1] E1[mr+l] = X X +(X +X )−X E1 Er[mr+l+1] E1[mr+l] E3[mr+l−2] E1[mr+l] = X X +X . E1 Er[mr+l+1] E3[mr+l−2] The case for i=r and j =2, we have X X Er[2] E2[mr+l] = (X X −1)X Er E1 E2[mr+l] = X (X +X )−X Er E1[mr+l+1] E3[mr+l−1] E2[mr+l] = X +(X +X X )−X Er[mr+l+2] E2[mr+l] Er E3[mr+l−1] E2[mr+l] = X +X X . Er[mr+l+2] Er E3[mr+l−1] For i=r−1=⇒j =1: X X Er−1[2] E1[mr+l] = (X X −1)X Er−1 Er E1[mr+l] = X (X +X )−X Er−1 Er[mr+l+1] E2[mr+l−1] E1[mr+l] = (X +X )+X X −X Er−1[mr+l+2] E1[mr+l] Er−1 E2[mr+l−1] E1[mr+l] = X +X X . Er−1[mr+l+2] Er−1 E2[mr+l−1] THE CLUSTER CHARACTERS FOR CYCLIC QUIVERS 7 Now, suppose it holds for k ≤n, then by induction we have X X Ei[n+1] Ej[mr+l] = (X X −X )X Ei[n] Ei+n Ei[n−1] Ej[mr+l] = X (X X )−X X Ei+n Ei[n] Ej[mr+l] Ei[n−1] Ej[mr+l] = X (X X +X X ) Ei+n Ei[(m+1)r+l+j−i] Ej[n+i−r−j] Ei[r+j−i−1] En+i+1[(m+1)r+l+j−n−i−1] −(X X +X X ) Ei[(m+1)r+l+j−i] Ej[n+i−r−j−1] Ei[r+j−i−1] En+i[(m+1)r+l+j−n−i] = X (X +X ) Ei[(m+1)r+l+j−i] Ej[n+i+1−r−j] Ej[n+i−r−j−1] +X (X +X ) Ei[r+j−i−1] En+i[(m+1)r+l+j−n−i] En+i+2[(m+1)r+l+j−n−i−2] −(X X +X X ) Ei[(m+1)r+l+j−i] Ej[n+i−r−j−1] Ei[r+j−i−1] En+i[(m+1)r+l+j−n−i] = X X +X X . Ei[(m+1)r+l+j−i] Ej[n+i+1−r−j] Ei[r+j−i−1] En+i+2[(m+1)r+l+j−n−i−2] 2) When k =1, by i≤l+j ≤k+i−1=⇒i≤l+j ≤i=⇒i=l+j. Then by by Proposition 2.2 and Corollary 2.5, we have X X =X X =X +X Ei Ej[mr+l] El+j Ej[mr+l] Ej[mr+l+1] Ej[mr+l−1] Whenk =2,byi≤l+j ≤k+i−1=⇒i≤l+j ≤i+1=⇒i=l+j ori+1=l+j: For i=l+j, we have X X Ei[2] Ej[mr+l] = X X El+j[2] Ej[mr+l] = (X X −1)X El+j El+j+1 Ej[mr+l] = (X +X )X −X Ej[mr+l+1] Ej[mr+l−1] El+j+1 Ej[mr+l] = X +X +X X −X Ej[mr+l+2] Ej[mr+l] El+j+1 Ej[mr+l−1] Ej[mr+l] = X +X X . Ej[mr+j+2] El+j+1 Ej[mr+l−1] For i+1=l+j, we have X X Ei[2] Ej[mr+l] = X X El+j−1[2] Ej[mr+l] = (X X −1)X El+j−1 El+j Ej[mr+l] = (X +X )X −X Ej[mr+l+1] Ej[mr+l−1] El+j−1 Ej[mr+l] = X X +(X +X )−X Ej[mr+l+1] El+j−1 Ej[mr+l] Ej[mr+l−2] Ej[mr+l] = X X +X . Ej[mr+l+1] El+j−1 Ej[mr+l−2] Suppose it holds for k≤n, then by induction we have X X Ei[n+1] Ej[mr+l] = (X X −X )X Ei[n] Ei+n Ei[n−1] Ej[mr+l] = (X X )X −X X Ei[n] Ej[mr+l] Ei+n Ei[n−1] Ej[mr+l] = (X X +X X )X Ej[mr+n+i−j] Ei[l+j−i] Ej[mr+i−j−1] El+j+1[n+i−l−j−1] Ei+n −(X X +X X ) Ej[mr+n+i−j−1] Ei[l+j−i] Ej[mr+i−j−1] El+j+1[n+i−l−j−2] = (X +X )X Ej[mr+n+i+1−j] Ej[mr+n+i−j−1] Ei[l+j−i] +(X +X )X El+j+1[n+i−l−j] El+j+1[n+i−l−j−2] Ej[mr+i−j−1] −(X X +X X ) Ej[mr+n+i−j−1] Ei[l+j−i] Ej[mr+i−j−1] El+j+1[n+i−l−j−2] = X X +X X . Ej[mr+n+i+1−j] Ei[l+j−i] Ej[mr+i−j−1] El+j+1[n+i−l−j] 3) It is trivial by the definition of the Caldero-Chapoton map. (cid:3) 8 MINGDINGANDFANXU 4. A Z-basis for cyclic quivers In this section, we will focus on studying the following set B(Q)={X |Ext1 (R,R)=0}. R kQ We prove that B(Q) is a Z-basis of the algebra AH(Q) generated by all these generalized cluster variables. We first give the following definition. Definition 4.1. For M,N ∈ modkQ with dimM = (m ,··· ,m ) and dimN = 1 r (s ,··· ,s ), we write dimM (cid:22)dimN if m ≤s for 1≤i≤r. Moreover, if there 1 r i i exists some i such that m

The cluster character for cyclic quivers

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