Ringel-Hall Algebras of Duplicated Tame Hereditary Algebras
aa r X i v : . [ m a t h . R T ] J a n Ringel-Hall Algebras of Duplicated TameHereditary Algebras ⋆ Hongchang Dong, Shunhua Zhang
School of Mathematics, Shandong University, Jinan, 250100, P.R.China
Abstract.
Let A be a tame hereditary algebra over a finitefield k with q elements, and A be the duplicated algebra of A .In this paper, we investigate the structure of Ringel-Hall algebra H ( A ) and of the corresponding composition algebra C ( A ). As anapplication, we prove the existence of Hall polynomials g MXY for any A -modules M, X and Y with X and Y indecomposable if A is atame quiver k -algebra, then we also obtain some Lie subalgebrasinduced by A . Key words : duplicated algebra; Ringel-Hall algebra; Hall polyno-mial; Lie subalgebra
The duplicated algebras are interesting algebras that have been introduced recentlyin the context of cluster categories. In particular, it is the interesting theory of
MSC(2000): 16G10, 17B37 ⋆ Supported by the NSF of China (Grant No. 10771112) and NSF of Shandong Province(Grant No. Y2008A05).Email addresses: [email protected](H.Dong), [email protected](S.Zhang) g MXY for any A -modules M, X and Y with X and Y indecomposable(Theorem 4.8) when A is a tame quiver algebra. As an application, in section 5we also obtain some Lie subalgebras induced by duplicated tame quiver algebras.Section 2 is devoted to some notations and definitions needed for our research. Let A be a finite dimensional algebra over a field k . We denote by A -mod thecategory of finitely generated left A -modules, and A -ind a full subcategory of A -modcontaining exactly one representative of each isomorphism class of indecomposable A -modules. Given a class C of A -modules, we denote by add C the subcategory of A -mod whose objects are the direct summands of finite direct sums of modules in C .We denote by Γ A the Auslander-Reiten quiver of A and by τ the Auslander-Reitentranslation of A . We refer to [ARS, DR, R1] for further notations and definitionsin representation theory.Let M and N be indecomposable A -modules. A path from M to N in A -ind isa sequence of non-zero morphisms M = M f −→ M f −→ · · · f t −→ M t = N with all M i in A -ind. Following [R1], we denote the existence of such a path by2 ≤ N . We say that M is a predecessor of N (or that N is a successor of M ).More generally, if S and S are two sets of modules, we write S ≤ S if everymodule in S has a predecessor in S , every module in S has a successor in S , nomodule in S has a successor in S and no module in S has a predecessor in S .The notation S < S stands for S ≤ S and S ∩ S = ∅ . Given a finite set M , we denote its cardinality by | M | . In the sequel, we alwaysassume that k is a finite field with q elements, that is | k | = q , and assume that A is a finite-dimensional tame hereditary algebra over k .Let M, N , N be finite dimensional A -modules. We denote by G MN N the num-ber of submodules L of M with the property that L ≃ N and M/L ≃ N . By [R2]the Ringel-Hall algebra H ( A ) is a free abelian group with a basis { u [ M ] } [ M ] in-dexed by the isomorphism classes of finite (left) A -modules with the multiplicationdefined by u [ N ] · u [ N ] = X [ M ] G MN N u [ M ] Note that we only deal with finite sum since A is a finite ring. We denote by C ( A ) the subalgebra of H ( A ) generated by simple A -modules which is calledcomposition algebra.From now on, we always assume that A is a tame hereditary k -algebra and that A is the duplicated algebra of A , see [ABST1]. Then A = ( A DA A ) is the matrixalgebra, we see that A contains two copies of A given respectively by eAe and by e ′ Ae ′ , where e = ( 1 00 0 ), and e ′ = ( 0 00 1 ). We denote the first one by A = A andthe second one by A ′ = A . Accordingly, Q ′ A denotes the quiver of A ′ , x ′ the vertexof Q ′ A cerresponding to x ∈ ( Q A ) , and e ′ x the corresponding idempotent. Let S x , P x , I x denote respectively the corresponding simple, indecomposable projectiveand indecomposable injective module in A -mod corresponding to x ∈ ( Q A S Q ′ A ) .For simpleness, we write Q A = { , · · · , n } and Q ′ A = { ′ , · · · , n ′ } .Recall from [ABST2] that the Auslander-Reiten quiver of Γ A can be described3s follows. It starts with the Auslander-Reiten quiver of A = A . Then projective-injective modules start to appear, such projective-injective module has its soclecorresponding to a simple A -module, and its top corresponding to a simple A -module. Next occurs a part denoted by A -ind where indecomposables contain atsame time simple composition factors from simple A -modules, and simple composi-tion factors from simple A -modules. When all projective-injective modules whosesocle corresponding to simple A -modules have appeared, we reach the projective A -modules and thus the Auslander-Reiten quiver of A .From the description above, we can divide the Auslander-Reiten-quiver Γ A into7 parts, denoted by P , R , X , R , X , R , I respectively, where P (resp. I )is the preprojective (resp. preinjective) component of Γ A (resp. Γ A ); X and X are forms of translation quiver of Z Q A ; R , R and R are the same types oftubes since A is tame type.Let M be an A -module. We denote by Ω − i ( M ) the i th cosyzygy of M andby Ω i ( M ) the i th syzygy of M respectively. Let L A be the left part of A -mod.By definitions in [HRS], L A is the full subcategory of A -mod consisting of allindecomposable A -modules such that if L is a predecessor of M , then the projectivedimension pd L of L is at most one.It is well known that gl . dim A , the global dimension of A , is 3. We denote byΣ the set of all non-isomorphic indecomposable projective A -modules, and writeΣ k = Ω − k Σ = { Ω − k X | X ∈ Σ } for 1 ≤ k ≤ In this section, we mainly investigate the structure of Ringel-Hall algebra of du-plicated tame hereditary algebras, the decomposition of the composition algebra,indecomposable elements in the composition algebra, and prove that the excep-tional elements can be written as skew communicators.4et A -ind be the indecomposable A -modules with the composition factorshaving both simple A -modules and A -modules. That is A -ind = A -ind S A -ind S A -ind. Note that add( A -ind) is an exact category which is closed underextensions, we can define the corresponding Ringel-Hall algebra which is denotedby H ( A ). Theorem 3.1. H ( A ) = H ( A ) H ( A ) H ( A ). Proof.
Let M be an A -module. Suppose that [ M ] = [ M ] ⊕ [ M ] ⊕ [ M ], where M ∈ add ( A − ind), and M i ∈ A i − mod with i = 0 , g M ⊕ M M M = 1, g MM M ⊕ M = 1, and u [ M ] u [ M ] u [ M ] = u [ M ] u [ M ] ⊕ [ M ] = u [ M ] ⊕ [ M ] ⊕ [ M ] = u [ M ] . ✷ Let C ( A ) = C ( A ) T H ( A ) and C ( A ) (resp. C ( A )) be the subalgebra of C ( A ) generated by the simple modules S , · · · , S n (resp. S ′ , · · · , S n ′ ). Note that C ( A ) = H ( A ) T C ( A ) and that C ( A ) = H ( A ) T C ( A ), by using Theorem3.1, we have the following. Corollary 3.2. C ( A ) = C ( A ) C ( A ) C ( A ).Let M be an indecomposable A -module. M is said to be exceptional if Ext iA ( M, M ) =0 for all i > Theorem 3.3.
Let A be the duplicated tame hereditary algebra A and M bean indecomposable A -module. Then u [ M ] ∈ C ( A ) if and only if M is an exceptional A -module . Proof.
Assume that M is an exceptional A -module. Case I. If M ∈ A − ind or M ∈ A − ind, then u [ M ] ∈ C ( A ) by [PZ1, SZ2]. Case II.
Assume that M ∈ A − ind. First of all, we suppose that M is aprojective-injective A -module.If M ∈ L A , then top M = S i ′ and rad M ∈ A − mod. Note that rad M is a5reinjectve A -module and u [rad M ] ∈ C ( A ), we have the following: u [ M ] = u [ S i ′ ] u [rad M ] − u [rad M ] u [ S i ′ ] ∈ C ( A ) . If M
6∈ L A , then Soc M = S i and M/ Soc M ∈ A − mod. In this case, one caneasy to see that M/ Soc M is a preprojectve A -module and u [ M/ Soc M ] ∈ C ( A ), wehave the following: u [ M ] = u [ M/ Soc M ] u [ S i ] − u ] S i ] u ] M/ Soc M ] ∈ C ( A ) . Finally, we can assume that M ∈ A − ind and M is not a projective-injective A -module. Read from the Auslander-Reiten quiver of A and by using Theorem 9.1in [PZ1] and Theorem 1 in [SZ2], we know that u [ M ] ∈ C ( A ).Conversely, let M be an indecomposable A -module and u [ M ] ∈ C ( A ). We wantto prove that M is an exceptional A -module. If M ∈ A − ind or M ∈ A − ind,then M is an exceptional A -module follows from [ZZ] if A is a tame quiver algebraand follows from [SZ2] when A is a non-simply-laced tame hereditary algebra.Now assume that M ∈ A − ind and we may assume that M is not a projective-injective A -module. It is easy to read from the Auslander-Reiten quiver of A that M is in a full subquiver of Γ A which is isomorphic to Γ D ( A ) . Then M is anexceptional A -module follows from [ZZ, SZ2] again. This completes the proof. ✷ Example 3.4.
Let A be the duplicated tame quiver algebra of type e D . Thatis, A = k f D /I , and f D is the following quiver,2 2 ′ ւ տ ւ f D : 1 ⇔ ⇔ ′ ⇔ ′ ′ տ ւ տ ′ . Then the indecomposable projective-injective A -modules are represented bytheir Loewy series as the following, P ′ = 1 ′ , P ′ = 2 ′ ′ , P ′ = 3 ′ ′ , P ′ = 4 ′ ′ , P ′ = 5 ′ ′ .
6e should mention that the minimal positive imaginary root of k e D is δ =(2 , , , ,
1) and every indecomposable A -module M which belongs to R , R or R with l ( M ) ≥ l ( M ) is the length of M .According to Theorem 3.3, for any indecomposable A -module M , u [ M ] belongsto C ( A ) if and only if M belongs to P , X , X , I or to R , R , R with l ( M ) < B be a k -algebra, and x, y ∈ B ,and c, d ∈ k ∗ = k \{ } . The element cxy − dyx is called a skew commutator of x and y . Let X = { x , · · · , x n } be a set of B . Define the sets X i inductively: Let X = X . Let X i be the set of all skew commutators of arbitrary two differentelements in S j
Let A be a tame hereditary algebra over k and A be the du-plicated algebra of A . Let M be a non-simple indecomposable A -module. Thenthe element u [ M ] ∈ C ( A ) can be written as an iterated skew commutator of theisoclasses of simple A -modules . Proof.
According to Theorem 3.3, we know that M is an exceptional A -module.If M ∈ A − ind or M ∈ A − ind, then M is an iterated skew commutator ofthe isoclasses of simple A -modules by Theorem 2.1 in [PZ2].Now, Let M ∈ A − ind and M be a projective-injective A -module.If M ∈ L A , according to the proof of Theorem 3.3, we can write u [ M ] as follow-ing: u [ M ] = u [ S i ′ ] u [rad M ] − u [rad M ] u [ S i ′ ] ∈ C ( A ) , rad M ∈ A − mod is a preinjectve A -module and top M = S i ′ . By using theTheorem 2.1 in [PZ2], u [rad M ] is an iterated skew commutator of the isoclasses ofsimple A -modules, hence u [ M ] can be written as an iterated skew commutator ofthe isoclasses of simple A -modules. 7f M
6∈ L A , then Soc M = S i is a simple A -module and M/ Soc M is a prepro-jectve A -module. By using Theorem 2.1 in [PZ2] again, u [ M/ Soc M ] can be writtenas an iterated skew commutator of the S ′ , · · · , S n ′ .Note that u [ M ] = u [ M/ Soc M ] u [ S i ] − u [ S i ] u [ M/ Soc M ] ∈ C ( A ), thus u [ M ] is an iteratedskew commutator of the isoclasses of simple A -modules.Finally, we can assume that M ∈ A − ind and M is not a projective-injective A -module. It follows from the Auslander-Reiten quiver of A that M is in a fullsubquiver of Γ A which is isomorphic to Γ D ( A ) . Since M is an exceptional A -module, we know that u [ M ] is an iterated skew commutator of the isoclasses ofsimple A -modules by using Theorem 2.1 in [PZ2]. The proof is completed. ✷ In this section, we always assume that A is a tame quiver algebra over k and A be the duplicated algebra of A , and we will prove that some Hall polynomials forduplicated tame hereditary algebras exist. Note that we can, in this case, dividethe Auslander-Reiten quiver Γ A into 7 parts, denoted by P , R , X , R , X , R , I respectively.Let E be a field extension of k . For any k -space V , we denote by V E the E -space V ⊗ k E ; then, of course, A E naturally becomes an E -algebra. If S is asimple A -module, according to Theorem 7.5 in [La], we know that S E is the simple A E -module. For any M ∈ A -mod, E is called M -conservative for A if for anyindecomposable summand N of M , (End N/ rad End N ) E is a field. Under fieldisomorphism, we putΩ M = { E | E is a finite field extension of k and E is M − conservative for A } . Note that Ω M is an infinite set, since M has only finitely indecomposable sum-mands. By [SZ1], we say that Hall polynomials exist for A , if for any M , N , N ∈ A -mod, there exists a polynomial g MN N ∈ Z [ x ] and an infinite subset Ω MN N
8f Ω M ⊕ N ⊕ N , such that for any E ∈ Ω MN N , g MN N ( | E | ) = G M E N E N E . Such a polynomial g MN N is called a Hall polynomial of A . Remark.
When A is a representation-finite algebra, the above definition is thesame as in [R5].The following results were proved in [SZ1] for tame quiver algebras, and weobserve that they are also true for duplicated tame hereditary algebras, we refer to[SZ1] for details. Lemma 4.1.
Assume that u [ M ] = u [ M ] + u [ M ] in H ( A ) . Then G LMN = G LM N + G LM N for any A -modules L and N . Lemma 4.2.
Given M , N ∈ A - mod , then there exists a nonnegative integer h ( M, N ) such that | Hom A E ( M E , N E ) | = | E | h ( M,N ) for any E ∈ Ω M ⊕ N . Lemma 4.3.
Let N , N , · · · , N t be simple A -modules except at most only one.Then there exists the Hall polynomial g MN N ··· N t for all M ∈ A - mod . Lemma 4.4.
Let M , N , L be A -modules with N ∈ P , X , X or I . Thenthe Hall polynomials g MNL and g MLN exist.
Proof.
By duality, we only need to prove the existence of the Hall polynomial g MNL . Note that for any indecomposable A -module X ∈ P , X , X or I whichis exceptional, according to Theorem 3.3, we know that u [ X ] ∈ C ( A ). Thereforewe can assume that u [ N ] = X i , ··· ,i t ∈{ , ··· ,n, ′ , ··· ,n ′ } a i ··· i t u [ s i ] · · · u [ s it ] , where a i ··· i t ∈ Z . By using Lemma 4.1, we have G MNL = X i , ··· ,i t ∈{ , ··· ,n, ′ , ··· ,n ′ } a i ··· i t G MS i ··· S it L . g MS i ··· S it L ∈ Z [ x ] such that thereexists an infinite subset Ω MNL of Ω M ⊕ N ⊕ L and for any E ∈ Ω MNL G M E S Ei ··· S Eit L E = g MS i ··· S it L ( | E | ) . Let g MNL = P i , ··· ,i t ∈{ , ··· ,n, ′ , ··· ,n ′ } a i ··· i t g MS i ··· S it L ∈ Z [ x ]. For any E ∈ Ω MNL , we havethat G M E N E L E = g MNL ( | E | ), that is, g MNL is the Hall polynomial of A . This completesthe proof. ✷ Lemma 4.5.
Let M , N and L be A -modules with N and L indecomposable.If N, L ∈ R , N, L ∈ R or N, L ∈ R , then the Hall polynomial g MNL exists.
Proof.
We may assume that M is an extension of L by N , since otherwise wemay take g MNL = 0. Therefore we have a short exact sequence 0 → L → M → N →
0, it follows that
M, N, L belong to the same part of Γ A , that is, M, N, L ∈ R , M, N, L ∈ R or M, N, L ∈ R . By using the same method as Lemma 2.9 in[SZ1], we know that g MNL exists. This completes the proof. ✷ For any
M, N, L ∈ A − mod, we denote by Ext A ( N, L ) M the set of all exactsequences in Ext A ( N, L ) with middle term M . The following lemma was proved in[P, Rie]. Lemma 4.6.
For any A -modules M , N , L , G MNL = | Ext A ( N, L ) M | · | Aut A M || Aut A N | · | Aut A L | · | Hom A ( N, L ) | . Lemma 4.7.
Let
M, N and L be A -modules with N and L indecomposable.Assume that N ∈ R i , L ∈ R j with i = j ∈ { , , } . Then the Hall polynomial g MNL exists.
Proof.
We only need to consider the cases L ∈ R with N ∈ R , and L ∈ R with N ∈ R , since in other cases we have that Ext A ( N, L ) = 0, and byusing Lemma 4.6, the existence of the Hall polynomial g MNL follows.
Case I.
Let L ∈ R with N ∈ R . Assume that E ( L ) is the injective envelope10f L , then we have a short exact sequence( ∗ ) 0 → L → E ( L ) → Ω − L → , where E ( L ) is projective-injective A -module since L ∈ R , and Ω − L is an inde-composable A -module which belongs to R . Note that E ( L ) is a predecessor of N ,by applying Hom A ( N, − ) to ( ∗ ), we obtain that Hom A ( N, Ω − L ) ≃ Ext A ( N, L ).Hence dim k Ext A ( N, L ) = dim k Hom A ( N, Ω − L ) ≤ N and Ω − L are in-decomposable A -modules belonging to R . For any A -module M , according toLemma 4.2 and Lemma 4.6 we know that the Hall polynomial g MNL exists.
Case II.
Let L ∈ R with N ∈ R . By using the same method as in Case I,we can prove that the Hall polynomial g MNL exists. This completes the proof. ✷ Theorem 4.8.
Let X and Y be indecomposable A -modules. Then for any A -module M , there exists the Hall polynomial g MXY . Proof.
If one of the indecomposable A -modules X and Y belongs to P , X , X or I , by Lemma 4.4, we know that the Hall polynomial g MNL exists.If none of X and Y belongs to P , X , X or I , then X and Y must belongto R i or R j , where ( i, j ∈ { , , } ). In case i = j , then the existence of the Hallpolynomial g MXY follows from Lemma 4.5. If i = j , according to Lemma 4.7, wehave the Hall polynomial g MXY exists. The proof is completed. ✷ Remark. If A is a representation-finite hereditary k -algebra, then the dupli-cated algebra A is represented-direct, thus according to [R5], we know that theHall polynomial g MNL exists for any A -modules M, N, L . In this section, we also assume that A is a tame quiver algebra over k and A is theduplicated algebra of A , and we will investigate some Lie subalgebras induced by A which seem to have an independent interest.Let Ω be an infinite set of finite field extension of k up to isomorphism. Since11 is a tame quiver algebra, according to [CD] and Theorem 7.5 in [La], we knowthat E is S -conservative for any simple A -module S .Denote by H ( A, Ω) the subring of Q E ∈ Ω H ( A E ) generated by { ([ M E ]) E ∈ Ω | M ∈ A − mod } and q Ω = ( | E | u [0] ) E ∈ Ω . Denote by H ( A ) the quotient ring H ( A, Ω) / ( q Ω − H ( A, Ω), called the degenerate Ringel-Hall algebra of A . The subalgebra of H ( A ) generated by the simple A -modules, denoted by C ( A ) , is called the de-generate composition algebra of A .The following Lemma was proved in [R5]. Lemma 5.1.
Let
M, X, Y ∈ A - mod with X and Y indecomposable. For any E ∈ Ω M ⊕ X ⊕ Y , then (1) If M X ⊕ Y , then | E | − divides G M E X E Y E ;(2) If M ≃ X ⊕ Y . If X ≃ Y , then | E | − divides G M E X E Y E − ;If X Y , then | E | − divides G M E X E Y E − . Let L ( A ) = L N ∈ A − ind Z u [ N ] be the free Abel group with basis the set of isomor-phism classes determined by indecomposable A -modules. Theorem 5.2. L ( A ) is the Lie subalgebra of H ( A ) . Proof.
Assume that X and Y are indecomposable A -modules such that X Y .For any A -module M , according to Theorem 4.8, we know that the Hall polynomials g MXY and g MY X ∈ Z [ x ] exist, satisfying for any E ∈ Ω M ⊕ X ⊕ Y , g MXY ( | E | ) = G M E X E Y E and g MY X ( | E | ) = G M E Y E X E . By Lemma 5.1, in H ( A ) , u [ X ] · u [ Y ] = X Z ∈ A − ind g ZXY (1) u [ Z ] + u [ X ⊕ Y ] ,u [ Y ] · u [ X ] = X Z ∈ A − ind g ZY X (1) u [ Z ] + u [ X ⊕ Y ] , therefore [ u [ X ] , u [ Y ] ] = X Z ∈ A − ind ( g ZXY (1) − g ZY X (1)) u [ Z ] ∈ L ( A ) ,
12o we have that L ( A ) is the Lie subalgebra of H ( A ) . ✷ Let L ′ ( A ) be the Lie subalgebra of L ( A ) generated by the simple A -modules.According to [R5] and by using PBW-basis Theorem, we have the following. Proposition 5.3. C ( A ) ⊗ Z Q is the universal enveloping algebra of L ′ ( A ) ⊗ Z Q . Let L ( A ) be the Lie subalgebra of L ′ ( A ) generated by S , · · · , S n and L ( A )the Lie subalgebra of L ′ ( A ) generated by S ′ , · · · , S n ′ respectively. Then by [Rie]we have L ( A ) ∼ = L ( A ) as Lie subalgebras, which is also isomorphic to the positivepart of the corresponding affine Kac-Moody algebra of type A .We denote by Σ PI1 the set of indecomposable projective-injective A -moduleswhich are predecessors of Σ , and by Σ PI2 the set of indecomposable projective-injective A -modules which are successors of Σ . Note that Σ < Σ PI2 < Σ .Let Ξ = { X ∈ A − ind | Σ PI1 ≤ X ≤ Σ PI2 } . For any M ∈ Ξ which is notprojective-injective, reading from the Auslander-Reiten quiver of Γ A , we know thatΣ ≤ X ≤ τ − Σ . We denote by L (Ξ ) the free subgroup L N ∈ Ξ Z u [ N ] of L ( A )and by H (Ξ ) the subalgebra of H ( A ) generated by indecomposable A -modulesin Ξ . Let L ( A ) = L (Ξ ) T C ( A ) and C ( A ) = H (Ξ ) T C ( A ) . Theorem 5.4. (1) L ( A ) is a Lie subalgebra of L ′ ( A ) . In particular, L ′ ( A ) = L ( A ) ⊕ L ( A ) ⊕ L ( A ) . (2) C ( A ) ⊗ Z Q is the universal enveloping algebra of L ( A ) ⊗ Z Q . Proof: (2) is trivial, so we only need to prove (1). According to the Auslander-Reiten quiver Γ A , we know that add Ξ is closed under extensions, thus L (Ξ ) isa Lie subalgebra of L ( A ), hence L ( A ) is a Lie subalgebra of L ′ ( A ). The proofis completed. ✷ Let δ = ( a , · · · , a n ) be the minimal positive imaginary root of A and m = n P i =1 a i . Theorem 5.5.
Let M be an indecomposable A -module and l ( M ) be the length f M . Assume that m cannot divide l ( M ) , i.e., m l ( M ) , then u [ M ] belongs to C ( A ) ⊗ Z Q . Proof:
First we assume that M belongs to one of P , X , X , I . If M is nota projective-injective A -module, then u [ M ] ∈ C ( A ) ⊗ Z Q follows by Theorem 3.2in [SZ3].If M is projective-injective, then according to the proof of Theorem 3.3, weknow that u [ M ] ∈ C ( A ) ⊗ Z Q .Finally, we assume that M belongs to one of R , R , R , that is M belongsto one tube. Note that m cannot divide l ( M ), it follows that M must belong tonon-homogenous tube, then by Corollary 3.1 in [SZ3], we know that u [ M ] belongsto C ( A ) ⊗ Z Q . The proof is finished. ✷ Remark.
According to Theorem 3.3 and Theorem 5.5, there is a big differencebetween the indecomposable A -modules which belong to C ( A ) and those whichbelong to C ( A ) , see the following example. Example 5.6.
Let A be the duplicated tame quiver algebra of type e D . Thatis, A = k f D /I as in Example 3.4. According to Theorem 5.5, u [ M ] belongs to C ( A ) if and only if M belongs to P , X , X , I or to R , R , R with 6 l ( M ).The following example indicates that the converse of Theorem 5.5 does nothold. Example 5.7.
Let K be the Kronecker algebra and K be the duplicatedalgebra of K . We may assume that K = kQ K /I with Q K : 1 ⇔ ⇔ ′ ⇔ ′ .The indecomposable projective-injective A -modules are P ′ = 1 ′
221 and P ′ = 2 ′ ′ ′ C ( K ) be the degenerated composition algebra generated by simple K -modules S , S , S ′ , S ′ . Note that δ = (1 ,
1) is the minimal positive imaginaryroot of K and m = 2 in this case. l ( P ′ ) = 4 and u [ P ′ ] ∈ C ( A ) ⊗ Z Q since u [ P ′ ] = [ u [ S ′ ] , [ u [ S ] , u [ S ] u [ S ] ]].Let M be an indecomposable K -module. If l ( M ), the length of M , is a positive14ven number, then u [ M ] ∈ C ( A ) ⊗ Z Q if and only if M is P ′ or P ′ since otherwise M belongs to one of homogeneous tubes and in this case u [ M ] dose not belong to C ( A ) ⊗ Z Q .If l ( M ) is a positive odd number, then from the Auslander-Reiten quiver ofΓ K , we know that M belongs to one of components P , X , X , I . By usingTheorem 3.3, we have that u [ M ] ∈ C ( A ) ⊗ Z Q . Acknowledgments.
The authors would like to thank Wenxu Ge and HongboLv for many useful discussions.
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