Unfolding of acyclic sign-skew-symmetric cluster algebras and applications to positivity and F -polynomials
aa r X i v : . [ m a t h . R T ] N ov UNFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRASAND APPLICATIONS TO POSITIVITY AND F -POLYNOMIALS MIN HUANG FANG LI
Abstract.
In this paper, we build the unfolding approach from acyclic sign-skew-symmetric ma-trices of finite rank to skew-symmetric matrices of infinite rank, which can be regard as an im-provement of that in the skew-symmetrizable case. Using this approach, we give a positive answerto the problem by Berenstein, Fomin and Zelevinsky in [6] which asks whether an acyclic sign-skew-symmetric matrix is always totally sign-skew-symmetric. As applications, the positivity forcluster algebras in acyclic sign-skew-symmetric case is given; further, the F -polynomials of clusteralgebras are proved to have constant term 1 in acyclic sign-skew-symmetric case. Contents
1. Introduction and preliminaries 22. Unfolding and totality of acyclic sign-skew-symmetric matrices 52.1. Notions of covering and unfolding 52.2. Strongly almost finite quiver from an acyclic sign-skew-symmetric matrix 72.3. Statement of unfolding theorem and totality of acyclic sign-skew-symmetric matrices 113. Global dimensions of K -linear categories 113.1. Terminologies in categories 113.2. Global dimension and cluster tilting subcategory 133.3. Global dimension, nilpotent element and commutator subgroup 144. The category C Q arising from the preprojective algebra Λ Q C Q mod Λ Q with an (infinite) group action. 275.1. Maximal Γ-stable rigid subcategories and mutations 285.2. Global dimension of maximal Γ-stable rigid subcategories T and categories C h ( T ) 306. Proof of the unfolding theorem of acyclic sign-skew-symmetric matrices 357. Applications to positivity and F-polynomials 417.1. The surjective algebra morphism π F -polynomials 44Appendix A. A = U for acyclic sign-skew-symmetric cluster algebras 45References 46 Mathematics Subject Classification(2010) : 13F60, 05E15. 05E40
Keywords : cluster algebra, acyclic sign-skew-symmetric matrix, unfolding method, positivity conjecture, F -polynomial. Date : version of November 10, 2018. Introduction and preliminaries
Cluster algebras are commutative algebras that were introduced by Fomin and Zelevinsky [18]in order to give a combinatorial characterization of total positivity and canonical bases in algebraicgroups. The theory of cluster algebras is related to numerous other fields including Lie theory,representation theory of algebras, the periodicity issue, Teichm¨ u ller theory and mathematics physics.A cluster algebra is defined starting from a totally sign-skew-symmetric matrix. Skew-symmetricand skew-symmetrizable matrices are always totally sign-skew-symmetric, but other examples oftotally sign-skew-symmetric matrices are rarely known. A class of non-skew-symmetric and non-skew-symmetrizable 3 × cluster variables . Different cluster variables are related by a iterated procedure calledmutations. By construction, cluster variables are rational functions. In [18], Fomin and Zelevinsky infact proved that they are Laurent polynomials of initial cluster variables, where it was conjectured thecoefficients of these Laurent polynomials are non-negative, which is called the positivity conjecture.Since the introduction of cluster algebras by Fomin and Zelevinsky in [18], positivity conjecture isa hot topic. Positivity was proved in the acyclic skew-symmetric case in [32], in the case for thecluster algebras arising from surfaces in [37], in the skew-symmetric case in [34] and in the skew-symmetrizable case in [26]. However, the positivity for the totally sign-skew-symmetric case neverbe studied so far before this work.The concept of unfolding of a skew-symmetrizable matrix was introduced by Zelevinsky and theauthors of [17] and appeared in [17] firstly. Demonet [15] proved that any acyclic skew-symmetrizablematrix admits an unfolding. The idea based on unfolding is to use skew-symmetric cluster algebrasto characterize skew-symmetrizable cluster algebras. Note that in [15] and [17], all cluster algebrasare of finite rank.Galois covering theory was introduced by Bongartz and Gabriel for skeletal linear categories in [8].This theory was extended to arbitrary linear categories by Bautista and Liu in [4]. The basic ideaof Galois covering theory is to use infinite dimensional algebras to characterize finite dimensionalalgebras.In this paper, we combine the ideas of unfolding of a skew-symmetrizable matrix and Galoiscovering theory successfully to introduce the unfolding theory of a sign-skew-symmetric matrix, seeDefinition 2.4. For this, we have to deal with skew-symmetric matrices of infinite rank . Moreover,given an acyclic sign-skew-symmetric matrix B , we introduce a general construction of the unfolding( Q ( B ) , Γ). Our main result about unfolding theory is that
Theorem 2.16. If B ∈ M at n × n ( Z ) is an acyclic sign-skew-symmetric matrix, then ( Q ( B ) , Γ) constructed from B in Construction 2.6 is an unfolding of B . Following this result, we will prove or answer affirmatively some conjectures and problems oncluster algebras given by Fomin and Zelevinsky,etc, as we list as follows.(I) Berenstein, Fomin and Zelevinsky give an open problem in [6] in 2005 that.
Problem (1.28, [6] ). Is any acyclic seed totally mutable? In other words, is any acyclic sign-skew-symmetric integer matrix totally sign-skew-symmetric?
We will give a positive answer to this problem:
Theorem 2.17.
The acyclic sign-skew-symmetric matrices are always totally sign-skew-symmetric.
NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 3 (II) Fomin and Zelevinsky conjectured in [18] that,
Positivity Conjecture, [18] . Let A (Σ) be a cluster algebra over ZP . For any cluster X , and anycluster variable x , the Laurent polynomial expansion of x in the cluster X has coefficients which arenonnegative integer linear combinations of elements in P . Combing the positivity theorem on skew-symmetric cluster algebras in [34] by Lee and Schiffler,we will prove that,
Theorem 7.8.
The positivity conjecture is true for acyclic sign-skew-symmetric cluster algebras. (III) Fomin and Zelevinsky conjectured that,
Conjecture 5.4, [20] . In a cluster algebra, each F -polynomial has constant term . Using Theorem 2.16, we will verify this conjecture affirmatively for all acyclic sign-skew-symmetriccluster algebras.
Theorem 7.13.
In an acyclic sign-skew-symmetric cluster algebra, each F -polynomial has constantterm . Here we give the outline of our technique for the major target.We prove the main theorem (Theorem 2.16) about unfolding following the strategy of [15]. Pre-cisely, for an arbitrary acyclic matrix B , we construct an infinite quiver Q ( B ), briefly written as Q , with a group Γ action. To check ( Q, Γ) as an unfolding of B , it suffices to prove that Q has noΓ-2-cycles and Γ-loops, and then we can do orbit mutations on Q (see Definition 2.1), moreover, toshow the preservation of the property “ no Γ - -cycles and Γ -loops ”.In order to prove this above fact, we build a cluster-tilting subcategory T of a Frobienus 2-Calabi-Yau category C Q (Corollary 4.9) such that the Gabriel quiver of T is Q , where T is the subcategoryof the stable category C Q of C Q corresponding to T . By Proposition 6.4, proving this fact can bereplaced by proving that there are no Γ-2-cycles and Γ-loops in the cluster-tilting subcategoriesobtained by a sequence of mutations from T , which can be verified by using (2)(4)(5) in Theorem6.7 step by step.By Theorem 2.16, using Lemma 2.5, we will prove that acyclic sign-skew-symmetric matrices arealways totally sign-skew-symmetric (Theorem 2.17), as the positive answer of a problem asked byBerenstein, Fomin and Zelevinsky.Due to ( Q, Γ) as the unfolding of B , we can deduce a surjective algebra homomorphism π : A ( e Σ) →A (Σ), where e Σ is the seed associated with Q and Σ is the seed associated with B , see Theorem 7.5(3).The key of our method is that proving the preservation of the positivity conjecture and theconjecture on F -polynomials under this surjective homomorphism. Then using the known conclusionsfor A ( e Σ) in the skew-symmetric case by Lee and Schiffler [34] and Zelevinsky, etc [16] respectively,we obtain that the positivity conjecture holds in the acyclic sign-skew-symmetric case (Theorem 7.8)and the F -polynomials have constant terms 1 in the same case (Theorem 7.13).The paper is organized as follows. In Section 2, we firstly give the definition of covering andunfolding of a sign-skew-symmetric matrix (Definition 2.4), and then construct a strongly finitequiver Q with a group Γ action from an acyclic sign-skew-symmetric matrix B (Construction 2.6).We give the statement of the unfolding theorem, see Theorem 2.16. Based on this result, we provethat acyclic sign-skew-symmetric matrix is always totally sign-skew-symmetric (Theorem 2.17). InSection 3, 4, 5, the preparations are given for the proof of Theorem 2.16. Precisely, in Section3, we study the global dimension of K -linear categories and prove that a functorially finite rigidsubcategory of a strongly almost finite category, which has global dimension less than 3 and satisfiescertain conditions, is a cluster tilting subcategory, see Theorem 3.2. Lenzing’s Theorem ([35], Satz 5) MIN HUANG FANG LI is generalized for locally bounded categories (Theorem 3.9). In Section 4, using the theory of colimitsof additive categories (Proposition 4.5), we improve on the construction in [22] for a strongly almostfinite quiver Q to construct a strongly almost finite 2-Calabi-Yau Frobenius category C Q (Corollary4.9). For a strongly almost finite 2-Calabi-Yau Frobenius category, we discuss the mutation ofmaximal Γ-rigid subcategories and their global dimensions in Section 5 (Proposition 5.9 and 5.13).We prove Theorem 2.16 in Section 6. In Section 7, using the Theorem 2.16, we prove that there isa surjective algebra homomorphism π from the cluster algebra associated with Q to that associatedwith B (Theorem 7.5). Relying on the surjective map π , we prove the positivity (Theorem 7.8) andthe F -polynomials have constant 1 (Theorem 7.13) for acyclic sign-skew-symmetric cluster algebras.In this paper, we will always assume the background field K is an algebraic closure field withcharacteristic 0.The original definition of cluster algebra given in [18] is in terms of exchange pattern. We recallthe equivalent definition in terms of seed mutation in [19]; for more details, refer to [18, 19].An n × n integer matrix A = ( a ij ) is called sign-skew-symmetric if either a ij = a ji = 0 or a ij a ji < ≤ i, j ≤ n .An n × n integer matrix A = ( a ij ) is called skew-symmetric if a ij = − a ji for all 1 ≤ i, j ≤ n .An n × n integer matrix A = ( a ij ) is called D -skew-symmetrizable if d i a ij = − d j a ji for all1 ≤ i, j ≤ n , where D =diag( d i ) is a diagonal matrix with all d i ∈ Z ≥ .Let e A be an ( n + m ) × n integer matrix whose principal part, denoted by A , is the n × n submatrixformed by the first n -rows and the first n -columns. The entries of e A are written by a xy , x ∈ e X and y ∈ X . We say e A to be sign-skew-symmetric (respectively, skew-symmetric , D - skew-symmetrizable ) whenever A possesses this property.For two ( n + m ) × n integer matrices A = ( a ij ) and A ′ = ( a ′ ij ), we say that A ′ is obtained from A by a matrix mutation µ i in direction i, ≤ i ≤ n , represented as A ′ = µ i ( A ), if(1) a ′ jk = ( − a jk , if j = i or k = i ; a jk + | a ji | a ik + a ji | a ik | , otherwise . We say A and A ′ are mutation equivalent if A ′ can be obtained from A by a sequence of matrixmutations.It is easy to verify that µ i ( µ i ( A )) = A . The skew-symmetric/symmetrizable property of matricesis invariant under mutations. However, the sign-skew-symmetric property is not so. For this reason,a sign-skew-symmetric matrix A is called totally sign-skew-symmetric if any matrix, that ismutation equivalent to A , is sign-skew-symmetric.Give a field F as an extension of the rational number field Q , assume that u , · · · , u n , x n +1 , · · · , x n + m ∈ F are n + m algebraically independent over Q for a positive integer n and a non-negative in-teger m such that F = Q ( u , · · · , u n , x n +1 , · · · , x n + m ), the field of rational functions in the set e X = { u , · · · , u n , x n +1 , · · · , x n + m } with coefficients in Q .A seed in F is a triple Σ = ( X, X fr , e B ), where(a) X = { x , · · · x n } is a transcendence basis of F over the fraction field of Z [ x n +1 , · · · , x n + m ], whichis called a cluster , whose each x ∈ X is called a cluster variable (see [19]);(b) X fr = { x n +1 , · · · , x n + m } are called the frozen cluster or, say, the frozen part of Σ in F ,where all x ∈ X fr are called stable (cluster) variables or frozen (cluster) variables ;(c) e X = X ∪ X fr is called a extended cluster ;(d) e B = ( b xy ) x ∈ e X,y ∈ X = ( B T B T ) T is a ( n + m ) × n matrix over Z with rows and columns indexed NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 5 by X and e X , which is totally sign-skew-symmetric. The n × n matrix B is called the exchangematrix and e B the extended exchange matrix corresponding to the seed Σ.Let Σ = ( X, X fr , e B ) be a seed in F with x ∈ X , the mutation µ x of Σ at x is defined satisfying µ x (Σ) = ( µ x ( X ) , X fr , µ x ( e B )) such that(a) The adjacent cluster µ x ( X ) = { µ x ( y ) | y ∈ X } , where µ x ( y ) is given by the exchangerelation (2) µ x ( y ) = Q t ∈ f X,btx> t btx + Q t ∈ f X,btx< t − btx x , if y = x ; y, if y = x. This new variable µ x ( x ) is also called a new cluster variable .(b) µ x ( e B ) is obtained from B by applying the matrix mutation in direction x and then relabelingone row and one column by replacing x with µ x ( x ).It is easy to see that the mutation µ x is an involution, i.e., µ µ x ( x ) ( µ x (Σ)) = Σ.Two seeds Σ ′ and Σ ′′ in F are called mutation equivalent if there exists a sequence of mutations µ y , · · · , µ y s such that Σ ′′ = µ y s · · · µ y (Σ ′ ). Trivially, the mutation equivalence gives an equivalencerelation on the set of seeds in F .Let Σ be a seed in F . Denote by S the set of all seeds mutation equivalent to Σ. In particular,Σ ∈ S . For any ¯Σ ∈ S , we have ¯Σ = ( ¯ X, X fr , e ¯ B ). Denote X = ∪ ¯Σ ∈S ¯ X . Definition 1.1.
Let Σ be a seed in F . The cluster algebra A = A (Σ) , associated with Σ , is definedto be the Z [ x n +1 , · · · , x n + m ] -subalgebra of F generated by X . Σ is called the initial seed of A . This notion of cluster algebra was given in [18][19], where it is called the cluster algebra ofgeometric type as a special case of general cluster algebras.2.
Unfolding and totality of acyclic sign-skew-symmetric matrices
Firstly, we recall some terminologies of (infinite) quivers in [5]. Let Q be an (infinite) quiver.We denote by Q the set of vertices of Q ; by Q the set of arrows of Q . For i ∈ Q , denote by i + (respectively, i − ) the set of arrows starting (respectively, ending) in i . Say Q to be locally finite if i + and i − are finite for any i ∈ Q . For i, j ∈ Q , let Q ( i, j ) stands for the set of paths from i to j in Q . We say that Q is interval-finite if Q ( i, j ) is finite for any i, j ∈ Q . One calls Q stronglylocally finite if it is locally finite and interval-finite. In this paper, we call Q strongly almostfinite if it is strongly locally finite and has no infinity paths.Note that a strongly almost finite quiver Q is always acyclic since it is interval finite.2.1. Notions of covering and unfolding. .For a locally finite quiver Q , we can associate an (infinite) skew-symmetric row and column finitematrix e B Q = ( b ij ) i,j ∈ Q as follows.Assume Q has no 2-cycles and no loops. So, if there are arrows from i to j for i, j ∈ Q , there areno arrows from j to i . Thus, we define b ij to be the number of arrows from i to j and let b ji = − b ij .Trivially, b ii = 0 for i ∈ Q . Then e B Q = ( b ij ) i,j ∈ Q is obtained. It is easy to see that e B Q is askew-symmetric row and column finite matrix, Q and e B Q determinate uniquely each other.In case of no confusion, for convenient, we also denote the quiver Q as ( b ij ) i,j ∈ Q , or say, directlyreplace Q by the matrix e B Q .A (rank infinite) seed Σ( Q ) = ( e X, e B Q ) is associated to Q , with cluster e X = { x i | i ∈ Q } andexchange matrix e B Q = ( b ij ) i,j ∈ Q . MIN HUANG FANG LI
Let Q be a locally finite quiver Q with an action of a group Γ (maybe infinite). For a vertex i ∈ Q , as in [15], a Γ -loop at i is an arrow from i → h · i for some h ∈ Γ, a Γ - -cycle at i is a pairof arrows i → j and j → h · i for some j / ∈ { h ′ · i | h ′ ∈ Γ } and h ∈ Γ. Denote by [ i ] the orbit set of i under the action of Γ. Say Q has no Γ -loops (Γ - -cycles , respectively) at [ i ] if Q has no Γ-loops(Γ-2-cycles, respectively) at any i ′ ∈ [ i ]. Definition 2.1.
Let Q = ( b ij ) be a locally finite quiver with an group Γ action on it. Denote [ i ] = { h · i | h ∈ Γ } the orbit of vertex i ∈ Q . Assume that Q admits no Γ -loops and no Γ - -cyclesat [ i ] , we define an adjacent quiver Q ′ = ( b ′ i ′ j ′ ) i ′ ,j ′ ∈ Q from Q to be the quiver by following:(1) The vertices are the same as Q ,(2) The arrows are defined as b ′ jk = − b jk , if j ∈ [ i ] or k ∈ [ i ] ,b jk + P i ′ ∈ [ i ] | b ji ′ | b i ′ k + b ji ′ | b i ′ k | , otherwise.Denote Q ′ as e µ [ i ] ( Q ) and call e µ [ i ] the orbit mutation at direction [ i ] or at i under the action Γ . Inthis case, we say that Q can do orbit mutation at [ i ] . Since b jk = − b kj for all j, k ∈ Q , it is easy to see b ′ jk = − b ′ kj for all j, k ∈ Q ′ , that is, Q ′ = ( b ′ i ′ j ′ ) i ′ ,j ′ ∈ Q as matrix is skew-symmetric. Fact 2.2.
We can write that e µ [ i ] ( Q ) = ( Q i ′ ∈ [ i ] µ i ′ )( Q ) . Remark 2.3. (1) Since Q is locally finite, the sum in the definition of b ′ jk is finite and so is well-defined.(2) e µ [ i ] ( Q ) is also locally finite, since the finiteness of valency of any vertex does not changedunder orbit mutation by definition.(3) The conditions no Γ -loops and no Γ - -cycles become to no loops and no -cycles when Γ is atrivial group, which is necessary in the definition of cluster algebras. Such conditions are crucial inthe sequel, e.g. for Lemma 2.5 and Theorem 7.5 which is the key for the following applications.(4) If Q has no Γ -loops and Γ - -cycles at [ i ] , it is easy to see that e µ [ i ] ( Q ) has no Γ -loops at [ i ] . Note that if Γ is the trivial group { e } , then the definition of orbit mutation of a quiver is the sameas that of quiver mutation (see [18][19]) or say e µ [ i ] = µ i in this case. Definition 2.4. (i) For a locally finite quiver Q = ( b ij ) i,j ∈ Q with a group Γ (maybe infinite) action,let Q be the orbit sets of the vertex set Q under the Γ -action. Assume that n = | Q | < + ∞ and Q has no Γ -loops and Γ - -cycles.Define a sign-skew-symmetric matrix B ( Q ) = ( b [ i ][ j ] ) to Q satisfying (1) the size of the matrix B ( Q ) is n × n ; (2) b [ i ][ j ] = P i ′ ∈ [ i ] b i ′ j for [ i ] , [ j ] ∈ Q .(ii) For an n × n sign-skew-symmetric matrix B , if there is a locally finite quiver Q with a group Γ such that B = B ( Q ) as constructed in (i), then we call ( Q, Γ) a covering of B .(iii) For an n × n sign-skew-symmetric matrix B , if there is a locally finite quiver Q with an actionof group Γ such that ( Q, Γ) is a covering of B and Q can do arbitrary steps of orbit mutations, then ( Q, Γ) is called an unfolding of B ; or equivalently, B is called the folding of ( Q, Γ) . In the above definition, the part (iii) is a generalization of the notion of unfolding in the skew-symmetrizable case (see [17]) to the sign-skew-symmetric case.
NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 7
For j ′ ∈ [ j ], there is σ ∈ Γ with σ ( j ) = j ′ and then b i ′ j = b σ ( i ′ ) j ′ . Since the restriction of σ on [ i ]is a bijection, we have also b [ i ][ j ] = P i ′ ∈ [ i ] b σ ( i ′ ) j ′ for [ i ] , [ j ] ∈ Q . It means in (i), the expression (2) iswell-defined.Because Q has no Γ-2-cycles, fixed j , all b i ′ j have the same sign or equal 0 when i ′ runs overthe orbit [ i ]. In the case b i ′ j = 0 for all i ′ ∈ [ i ], then also b j ′ i = − b ij ′ = 0 for all j ′ ∈ [ j ] via σ ( j ) = j ′ . Thus, b [ i ][ j ] = 0 = b [ j ][ i ] . Otherwise, there is i ∈ [ i ] with b i j = 0, then b ji = − b i j = 0.Thus, b [ j ][ i ] = P j ′ ∈ [ j ] b j ′ i = 0 and b [ i ][ j ] = P i ′ ∈ [ i ] b i ′ j = 0 have the opposite sign. It follows that in (i), B = B ( Q ) = ( b [ i ][ j ] ) is indeed sign-skew-symmetric. Lemma 2.5. If ( Q, Γ) is a covering of B and Q can do orbit mutation at [ i ] such that e µ [ i ] ( Q ) hasno Γ - -cycles for some vertex i , then ( e µ [ i ] ( Q ) , Γ) is a covering of µ [ i ] ( B ) .Proof. Denote the matrix µ [ i ] ( B ) by ( b ′ [ j ][ k ] ). We have b ′ [ j ][ k ] = ( − b [ j ][ k ] , if [ j ] = [ i ] or [ k ] = [ i ] ,b [ j ][ k ] + | b [ j ][ i ] | b [ i ][ k ] + b [ j ][ i ] | b [ i ][ k ] | , otherwise.If [ j ] = [ i ] or [ k ] = [ i ], then we have b ′ [ j ][ k ] = − b [ j ][ k ] = − P j ′ ∈ [ j ] b j ′ k = P j ′ ∈ [ j ] b ′ j ′ k ; otherwise, for any i ′ ∈ [ i ], we have | P j ′ ∈ [ j ] b j ′ i | = | P j ′ ∈ [ j ] b j ′ i | . Since Q has no Γ-2-cycles at [ i ], we have | P i ′ ∈ [ i ] b i ′ k | = P i ′ ∈ [ i ] | b i ′ k | and | P j ′ ∈ [ j ] b j ′ i ′ | = P j ′ ∈ [ j ] | b j ′ i ′ | , then b ′ [ j ][ k ] = b [ j ][ k ] + | b [ j ][ i ] | b [ i ][ k ] + b [ j ][ i ] | b [ i ][ k ] | = P j ′ ∈ [ j ] b j ′ k + | P j ′∈ [ j ] b j ′ i | P i ′∈ [ i ] b i ′ k + P j ′∈ [ j ] b j ′ i | P i ′∈ [ i ] b i ′ k | = P j ′ ∈ [ j ] ( b j ′ k + P i ′ ∈ [ i ] | b j ′ i ′ | b i ′ k + b j ′ i ′ | b i ′ k | ) = P j ′ ∈ [ j ] b ′ j ′ k . Therefore, we have b ′ [ j ][ k ] = P j ′ ∈ [ j ] b ′ j ′ k for [ j ] , [ k ].In addition, since Q has no Γ-2-cycles, it is easy to see that e µ [ i ] ( Q ) admits no Γ-loops.Since e µ [ i ] ( Q ) has no Γ-2-cycles, we have b ′ j ′ k b ′ j ′′ k ≥ b ′ jk ′ b ′ jk ′′ ≥ j ′ , j ′′ ∈ [ j ], k ′ , k ′′ ∈ [ k ]. Moreover, since b j ′ k ′ = − b k ′ j ′ for all j ′ ∈ [ j ], k ′ ∈ [ k ], we have that b ′ [ j ][ k ] b ′ [ k ][ j ] =( P j ′ ∈ [ j ] b ′ j ′ k )( P k ′ ∈ [ k ] b ′ k ′ j ) ≤
0. Then it is easy to see that b ′ [ j ][ k ] b ′ [ k ][ j ] = 0 if and only if b ′ [ j ][ k ] = b ′ [ k ][ j ] = 0.This means µ [ i ] ( B ) is sign-skew-symmetric. It follows that ( e µ [ i ] ( Q ) , Γ) is a covering of µ [ i ] ( B ). (cid:3) This result means the preservability of covering under orbit mutations.2.2.
Strongly almost finite quiver from an acyclic sign-skew-symmetric matrix. .Let
P, Q be two quivers.(1) If there is a full subqiver P ′ of P , a full subquiver Q ′ of Q and a quiver isomorphism ϕ : P ′ → Q ′ , then we say P and Q to be glueable along ϕ .(2) Let P and Q respectively be corresponding to the skew-symmetric matrices B B B B ! and B ′ B ′ B ′ B ′ ! , where the row/column indices of B and B ′ be respectively corresponding tothe vertices of P ′ and Q ′ . Since ϕ is an isomorphism, we have B = B ′ .(3) Denote by P ` ϕ Q the quiver corresponding to the skew-symmetric matrix B B B ′ B ′ B B ′ B .We call P ` ϕ Q the gluing quiver of P and Q along ϕ . MIN HUANG FANG LI
Roughly speaking, P ` ϕ Q is obtained through gluing P and Q along the common full subquiver(up to isomorphism).For example, for P ′ = Q ′ the arrow α : 1 → P and Q , we can say P ` ϕ Q to be thegluing quiver of P and Q along ϕ = Id α .For any n × n sign-skew-symmetric matrix B , we associate a (simple) quiver ∆( B ) with vertices1 , · · · , n such that for each pair ( i, j ) with b ij >
0, there is exactly one arrow from vertex i to vertex j . Trivially, ∆( B ) has no loops and no 2-cycles.The matrix B is said in [33] to be acyclic if ∆( B ) is acyclic and to be connected if ∆( B ) isconnected.In the sequel, we will always assume that B is connected and acyclic. Construction 2.6.
Let B = ( b ij ) ∈ M at n × n ( Z ) be an acyclic matrix. We construct the (infinite)quiver Q ( B ) (written as Q briefly if B is fixed) via the following steps inductively. • For each i = 1 , · · · , n , define a quiver Q i like this: Q i has n P j =1 | b ji | + 1 vertices with onevertex labeled by i and | b ji | ( j = i ) various vertices labeled by j . If b ji > , there is an arrowfrom each vertex labeled by j to the unique vertex labeled by i ; if b ji < , there is an arrowfrom the unique vertex labeled by i to each vertex labeled by j . Note that if b ji = 0 , thenthere is no vertex labelled by j . • Let Q (1) = Q . Call the unique vertex label by the “old” vertex, and the other vertices the“new” vertices. • For a “new” vertex in Q (1) which is labelled by i , Q i and Q (1) share a common arrow,denoted as α , which is the unique arrow incident with the “new” vertex labelled by i . Thenthe gluing quiver Q i ` Id α Q (1) is obtained along the common arrow α . For another “new”vertex labelled by i in Q (1) , we get similarly Q i ` Id α ( Q i ` Id α Q (1) ) , where α is theunique arrow incident with the “new” arrow labelled by i . Using the above procedure stepby step for all “new” vertices in Q (1) , we obtain finally a quiver, denoted as Q (2) . Clearly, Q (1) is a subquiver of Q (2) . We call the vertices of Q (1) as the “old” vertices of Q (2) and theother vertices of Q (2) as its “new” vertices. • Inductively, by the same procedure for obtaining Q (2) from Q (1) , we can construct Q ( m +1) from Q ( m ) via a series of gluing for any m ≥ . Similarly, we call the vertices of Q ( m ) asthe “old” vertices of Q ( m +1) and the other vertices of Q ( m +1) as its “new” vertices. • Finally, we define the (infinite) quiver Q = Q ( B ) = S + ∞ m =1 Q m , since Q m is always a fullsubquiver of Q m +1 for any m . Define Γ = { h ∈ AutQ : if h · a s = a t for a s , a t ∈ Q , then a s , a t have the same label } . Trivially,Γ is the maximal subgroup of AutQ whose action on Q preserves the labels of vertices of Q .For h ∈ Γ, if there is v ∈ Q such that h · v = v , we call v a fixed point under h . In this case wesay h to have fixed points. Example 2.7.
Let B = − − − . Then Q = Q (1) , Q , Q and Q (2) are shown in theFigure 1. By the construction of Q ( B ) as above, we have that Observation 2.8. (1) As starting of this program, the quiver Q (1) can be chosen to be any quiver Q i for any i in this construction. NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 9
13 22 211 333 32 1122 3333331 113 12 Q (1) = Q Q Q Q (2) Figure 1 (2) The underlying graph of Q = Q ( B ) is a tree, thus is acyclic.(3) For each vertex a ∈ Q labeled by i , the subquiver of Q formed by all arrows incident with a is isomorphic to Q i .(4) For any m ∈ N , and any automorphism σ of Q ( m ) which preserve the labels of vertices, thenthere exists h ∈ Γ such that h | Q ( m ) = σ and h has the same order with σ , where h | Q ( m ) means therestriction of h to Q ( m ) .(5) In most of cases, Q are infinite quivers with infinite number of vertices. However, in some spe-cial cases, Q may be finite quivers. For example, when B is a skew-symmetric matrix correspondingto a finite quiver Q ′ of type A , we have Q ( B ) = Q ′ a finite quiver. In the sequel of this section, all Q are just the quiver Q ( B ) defined in Construction 2.6. Here wegive some results on Q which will be useful in the following context. Lemma 2.9.
For a ∈ Q , define Γ a := { h ∈ Γ | h · a = a } . For any finite subquiver Q ′ of Q , thereexist h , · · · , h s ∈ Γ such that { h | Q ′ : h ∈ Γ a } = { h | Q ′ , · · · , h s | Q ′ } for any h ∈ Γ a , where h | Q ′ means the restriction of h to Q ′ .Proof. Since Q is strongly almost finite and Q ′ is finite, { h · Q ′ | h ∈ Γ a } is a finite set, written as { Q h i , · · · , Q h t i } . For each i = 1 , · · · , t , let S i = { f ∈ Iso ( Q ′ , Q h i i ) : f preserves the labels of vertices } .It is clear that S i is a finite set since Q ′ is finite, then we write that S i = { σ i , · · · , σ ik } . Since σ ij ∈ Iso ( Q ′ , Q h i i ) preserves the labels of vertices, by Observation 2.8 (3), we can lift σ ij to an iso-morphism h ij ∈ Γ such that h ij | Q ′ = σ ij . For any h ∈ Γ a , we have h · Q ′ = Q h i i for some i = 1 , · · · , t ,then h | Q ′ : Q ′ → Q h i i is a quiver isomorphism preserving the labels of vertices. Thus, h | Q ′ = σ ij forsome j . Therefore, we have { h | Q ′ : h ∈ Γ a } ⊆ { h ij | Q ′ : i = 1 , · · · , t ; j = 1 , · · · , k } . Then the resultfollows. (cid:3) Example 2.10.
Let B = − − − . Then Q = 2 1 o o / / , Q = 1 / / / / and Q = 1 / / o o . We have Q ( B ) to be the following quiver: · · · / / o o o o / / o o o o / / · · · Proposition 2.11. If B is an acyclic sign-skew-symmetric matrix, then Q ( B ) is a strongly almostfinite quiver. Proof.
By Observation 2.8 (3), for each vertex a of Q which is labeled by i , the subquiver of Q formed by all arrows incident with a is isomorphic to Q i . Thus, Q is locally finite. Moreover, foreach vertex a and b , we have | Q ( a, b ) | ≤ Q has an infinite path a a · · · a n +1 · · · . Since all vertices are labeled by 1 , · · · , n ,there are a s and a t admitting the same label, denoted as i . Thus, the path from a s to a t correspondsto a cycle in ∆( B ). It contradicts to the fact B is acyclic. (cid:3) Lemma 2.12.
Keep the forgoing notations. For any h ∈ Γ and finite connected subquiver Q ′ of Q such that h · Q ′ = Q ′ , then there exist fixed points in Q ′ under the action of h .Proof. By Observation 2.8 (2), the underlying graph of Q is acyclic, so the underlying graph of Q ′ is acyclic. Assume that h has no fixed points. Let a ∈ Q ′ such that the distance between a and h · a is minimal and let w be a walk connecting a and h · a with the minimal length, where the distancemeans the shortest length of walks between a and h · a . Such a always exist since Q ′ is connected.For any i ∈ N , we have h i · w and h i +1 · w share the only common vertex h i +1 · a by the choice of w and the fact that the underlying graph of Q ′ is acyclic. Applying h, h , · · · , we get an infinite walk · · · ( h · w )( h · w ) w . However, since Q ′ is a finite quiver, the infinite walk · · · ( h · w )( h · w ) w containsat least a cycle as graph. It contradicts to the fact the underlying graph of Q ′ is acyclic. The resultfollows. (cid:3) Lemma 2.13.
Assume Q ′ is a finite connected subquiver of Q . If h ∈ Γ has fixed points, then thereexists h ′ ∈ Γ of finite order such that h ′ · a = h · a for every vertex a of Q ′ .Proof. By Observation 2.8 (1), Q (1) can be chosen to any Q i in the construction of Q , so we mayassume that h fixes a vertex labelled by 1. As Q ′ is finite, there exists an m ∈ N such that Q ′ is asubquiver of the finite quiver Q ( m ) . It is clear that h | Q ( m ) has finite order, where h | Q ( m ) means therestriction of h to Q ( m ) . Applying Observation 2.8 (4), there exists an h ′ ∈ Γ with finite order suchthat h ′ | Q ( m ) = h | Q ( m ) . Our result follows. (cid:3) Lemma 2.14. If h ∈ Γ has no fixed points, then h n has no fixed points for any n ∈ Z .Proof. Assume that h n has a fixed point x ∈ Q . By the construction of Q , there exists an uniquewalk ( x , x , · · · , x m , h · x ) from x to h · x with x i = x j for i = j . Applying h i to the walkfor any i = 1 , · · · , n −
1, we get a walk ( h i · x , · · · , h i +1 · x ). Therefore, we obtained a walk( x , x , · · · , h · x , h · x , · · · , h n · x = x ). Since the underlying graph of Q has no cycles, and x i = x j so h k · x i = h k · x j for i = j , therefore, we have h k +1 · x = h k · x m or h k +1 · x = h k · x n − for some k . Thus, we have h · x = x m or h · x = x m − . In case h · x = x m , since the underlyinggraph of Q has no cycles and ( x , x , · · · , h · x , h · x , · · · , h n · x = x ) is a walk, we have h · x = x ,which contradicts to h has no fixed points. In case h · x = x m − , then ( x , · · · , x m − , h · x ) is awalk, which contradicts to ( x , x , · · · , x m , h · x ) is the unique walk from x to h · x . Therefore,our result follows. (cid:3) Proposition 2.15. If B ∈ M at n × n ( Z ) is an acyclic sign-skew-symmetric matrix, then ( Q ( B ) , Γ) constructed from B in Construction 2.6 is a covering of B .Proof. Denote B = ( b ij ) and Q ( B ) = Q = ( e b a i a j ). Since the vertices of Q has n labels and Γpreserves the labels, the number of orbits of vertices under the action of Γ is n . It means that B ( Q ),given in Definition 2.4, is an n × n matrix; denote B ( Q ) = ( b ′ [ a i ][ a j ] ). Assume that a j is labelled by j , by Observation 2.8 (3) and the construction of Q j , we have b ′ [ a i ][ a j ] = P a ′ i ∈ [ a i ] e b a ′ i a j = b ij . Thus, wehave B ( Q ) = B , which means that ( Q, Γ) is a covering of B . (cid:3) NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 11
Statement of unfolding theorem and totality of acyclic sign-skew-symmetric matri-ces. .Following Proposition 2.15, we give the main result about the existence of unfolding of B asfollows, which is indeed important for the major consequences in this paper. Theorem 2.16. ( Unfolding Theorem ) If B ∈ M at n × n ( Z ) is an acyclic sign-skew-symmetricmatrix, then ( Q ( B ) , Γ) constructed from B in Construction 2.6 is an unfolding of B . By Definition 2.4 and Proposition 2.15, it suffices to prove that e µ [ i s ] · · · e µ [ i ] ( Q ) has no Γ-2-cycles for any sequence ([ i ] , · · · , [ i s ]) of orbit mutations. However, for this aim, we will makethe preparations in Sections 3, 4, 4, 6, and then finish the proof of Theorem 2.16 at the end ofSection 6.As a direct consequence of Theorem 2.16, we have Theorem 2.17.
Any acyclic sign-skew-symmetric matrix B ∈ M at n × n ( Z ) is always totally sign-skew-symmetric.Proof. It suffices to prove that µ [ i t ] · · · µ [ i ] ( B ) is sign-skew-symmetric for any sequence ([ i ] , · · · , [ i t ]).By Lemma 2.5 and Theorem 2.16, there exists a ( Q, Γ) such that ( e µ [ i t ] e µ i [ t − · · · e µ [ i ] ( Q ) , Γ) is acovering of µ [ i t ] µ [ i t − ] · · · µ [ i ] ( B ). Thus, by the definition of covering, our result follows. (cid:3) According to this conclusion, we know that any acyclic sign-skew-symmetric matrix can alwaysbe an exchange matrix for a cluster algebra.3.
Global dimensions of K -linear categories Terminologies in categories. .Now we recall some standard terminologies in categories. All subcategories are assumed to befully faithful if there is no other statement.
Definition 3.1. (p.27, [31] ) Let A be an additive category. A pair ( f, g ) of composable morphisms → X f → Y g → Z → is exact if f is a kernel of g and g is a cokernel of f . Let E be a class ofexact pairs closed under isomorphism and satisfies the following axioms Ex0, Ex , Ex and Ex op .The admissible epimorphism mentioned in these axioms are by definition the second components g of ( f, g ) ∈ E . The first components f are admissible monomorphisms . Ex : id X is an admissible epimorphism for any X ∈ A . Ex : The composition of two admissible epimorphism is an admissible epimorphism. Ex : For each h ∈ Hom A ( Z ′ , Z ) and each admissible epimorphism g ∈ Hom A ( Y, Z ) , there isa pullback square Y ′ g ′ / / h ′ (cid:15) (cid:15) Z ′ h (cid:15) (cid:15) Y g / / Z, where g ′ is an admissible epimorphism. Ex op : For each h ∈ Hom A ( X ′ , X ) and each admissible monomorphism f ∈ Hom A ( X, Y ) ,there is a pushout square X f / / h (cid:15) (cid:15) Y h ′ (cid:15) (cid:15) X ′ f ′ / / Y ′ , where f ′ is an admissible monomorphism.Then ( A , E ) (or A for shortly) is called an exact category .An exact sequence in an exact category is said to be an admissible exact sequence . Given an exact category A . An object P is called projective if 0 → Hom A ( P, X ) → Hom A ( P, Y ) → Hom A ( P, Z ) → → X → Y → Z →
0. Dually, the injective objects in exact category A is defined. P is called projective-injective if P is bothprojective and injective. We say that A has enough projectives if for each X ∈ A , there existsan admissible short exact sequence 0 → Y → P → X → P is projective and Y ∈ A . Dually, A has enough injectives is defined. An exact category A is called Frobenius if it has enoughproject/injective objects and the set of project objects is coincide to that of injective objects. See[27]. We call a subcategory T of C a co-generator subcategory if there exists T ∈ T and anadmissible short exact sequence 0 → X → T → Y → X ∈ C . Dually, we can define the generator subcategory of C .A Hom -finite exact K -linear category A is called 2 -Calabi-Yau in sense that Ext A ( X, Y ) ∼ = DExt A ( Y, X ) naturally for every pair
X, Y ∈ A , where D is the K -linear dual. See [30].Recall that a K -linear Krull-Schmidt category C is called Hom -finite if dim K Hom C ( X, Y ) < ∞ for any X, Y ∈ C . We say a K -linear Krull-Schmidt category C to be strongly almost finite ifit is Hom -finite and there are only finite indecomposable objects Y (up to isomorphism) such that Hom C ( X, Y ) = 0 or Hom C ( Y, X ) = 0 for any indecomposable object X .A K -linear category C is called locally bounded in [8] if it satisfies:(a) each object is indecomposable,(b) distinct objects are non-isomorphic,(c) P Y ∈C ( dim K Hom C ( X, Y ) + dim K Hom C ( Y, X )) ≤ ∞ .Note that the condition (a) is replaced in [8] by that End C ( X ) is local for X ∈ C .Assume C is either a strongly almost finite category or a locally bounded category. A representa-tion of C over K is a covariant functor F : C → modK . A representation F is called representable if it is naturally isomorphic to Hom C ( X, − ) for some X ∈ C ; F is called finitely presented ifthere is an exact sequence Hom C ( X, − ) → Hom C ( Y, − ) → F → X, Y ∈ C . Denote by mod C the category consisting of all finitely presented representations of C . It is well known from [30]that mod C is an abelian category whose projective objects are just finite direct sum of representablerepresentations of C . Denote by proj ( mod C ) the projective subcategory of mod C .When C is a strongly almost finite category, let S X be the simple representation of C via S X ( Y ) = Hom C ( X, Y ) /rad ( X, Y ) for any indecomposable object X of C , where rad is the radicalof category C .When C is a locally bounded category, let S X be the simple representation of C via S X ( X ) = End C ( X ) /J and S X ( Y ) = 0 if Y = X , for any object X of C , where J is the Jacobson radical of End C ( X ).For C a strongly almost finite category or a locally bounded category, like as module categories ofalgebras, we define the projective dimension of S X , denoted as pd.dim ( S X ), to be that pd.dim ( S X ) = inf { n | → F n · · · → F → F → S X → , F i ∈ proj ( mod C ) } ;and define the global dimension of C , denoted as gl.dim ( C ), to be that gl.dim ( C ) = sup { pd.dim ( S X ) | X is an indecomposable object of C} . NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 13
For a strongly almost finite category C , a morphism f ∈ C ( X, Y ) is called right minimal ifit has not a direct summand of the form T → = T ∈ C . Dually, the left minimal morphism can be defined. For a subcategory D of C , f : X → Y is called a right D -approximation of Y ∈ C if X ∈ D and C ( − , X ) C ( − ,f ) −→ C ( − , Y ) → D .Moreover, we call that f is a minimal right D -approximation if it is right minimal. D is said tobe a contravariantly finite subcategory of C if any Y ∈ C has a right D -approximation. Dually,a (minimal) left D -approximation and a covariantly finite subcategory can be defined. Acontravariantly and covariantly finite subcategory is called functorially finite . For a subcategory X of C , denote X ⊥ = { T ∈ C | Ext C ( X , T ) = 0 } and ⊥ X = { T ∈ C | Ext C ( T, X ) = 0 } ; X is called rigid if Ext C ( X , X ) = 0; and X is said to be a cluster tilting subcategory if X is functorially finiteand X = X ⊥ = ⊥ X . See [3], [30].Assume C is a strongly almost finite category, for any subcategory T of C , add ( T ) denotes thesmallest additive full subcategory of C containing T , that is, the full subcategory of C whose objectsare the direct sums of direct summands of objects in T . Let Add ( C ) denote the class of all full K -linearsubcategories of C , where the subcategories are stable under isomorphisms and direct summands. Itis clear that T ∈
Add ( C ) if and only if T = add ( T ).3.2. Global dimension and cluster tilting subcategory. .In this part, we will prove that a functorially finite rigid subcategory of a strongly almost finitecategory which has global dimension less than 3 and satisfies certain conditions is a cluster tiltingsubcategory. This result generalizes Theorem 3.30 in [15] and Theorem 5.1 (3) in [29].We have the following theorem.
Theorem 3.2.
Let C be a Frobenius, -Calabi-Yau strongly almost finite K -category. Assume that C is a extension closed subcategory of an abeian category D . Let T be a functorially finite subcategoryof C . If T is a rigid subcategory of C such that gl.dim ( T ) ≤ , and T contains all projective-injectiveobjects of C , then T is a cluster-tilting subcategory of C .Proof. For any X ∈ C such that Ext C ( X, T ) = 0 for all T ∈ T , take a projective resolution of X , P : P → P f −→ P f −→ P f −→ X → . Applying
Hom C ( − , T ), we get a sequence of T representations,0 → Hom C ( X, − ) → Hom C ( P , − ) → Hom C ( P , − ) → Hom C ( P , − ) → Hom C ( P , − ) . Denote by H the homology group of the above sequence at Hom C ( P , − ).We first prove that Hom C ( X, − ) is a projective object in mod T . Denote X = ker ( f ) , X = ker ( f ) and X = ker ( f ), note that X , X , X are in D . Since P → X →
0, we have 0 → Hom C ( X , − ) → Hom C ( P , − ). As gl.dim ( mod T ) ≤ Hom C ( P , − ) is projective by P ∈ T .Thus, we have pd.dim ( Hom C ( X , − )) ≤
2. Furthermore, we have the exact sequence0 → Hom C ( X , − ) → Hom C ( P , − ) → Hom C ( X , − ) g → H → . Since pd.dim ( Hom C ( X , − )) ≤ pd.dim ( H ) ≤ gl.dim ( mod T ) ≤
3, we get pd.dim ( ker ( g )) ≤
2. Thus, we have pd.dim ( Hom C ( X , − )) ≤
1. Applying
Hom C ( − , T ) to 0 → X → P → X → → Hom C ( X, − ) → Hom C ( P , − ) → Hom C ( X , − ) → Ext C ( X, T ) = 0. Therefore, pd.dim ( Hom C ( X, − )) = 0 which means Hom C ( X, − ) is a projectiveobject in mod T . Now we show X ∈ T . Since Hom C ( X, − ) is a projective object in mod T , there exists a T ∈ T such that Hom C ( X, − ) isomorphic to Hom C ( T, − ) as representations of T . Assume f ∈ Hom C ( X, T )corresponds to id T ∈ Hom C ( T, T ). To prove X ∈ T , it suffices to prove X f ∼ = T . For any object Y ∈ C , take an injective resolution of Y , 0 → Y → I → I . Applying Hom C ( X, − ) and Hom C ( T, − )on it respectively, we get the following commuting diagram,0 / / Hom C ( T, Y ) / / Hom C ( f,Y ) (cid:15) (cid:15) Hom C ( T, I ) / / Hom C ( f,I ) (cid:15) (cid:15) Hom C ( T, I ) Hom C ( f,I ) (cid:15) (cid:15) / / Hom C ( X, Y ) / / Hom C ( X, I ) / / Hom C ( X, I ) . Since I , I ∈ T , Hom C ( f, I ) and Hom C ( f, I ) are isomorphisms. Then Hom C ( f, Y ) : Hom C ( T, Y ) → Hom C ( X, Y ) is also an isomorphism. It is easy to see that
Hom C ( f, − ) : Hom C ( T, − ) → Hom C ( X, − )is a functor isomorphism over C . Therefore, by Yoneda embedding Lemma, X f ∼ = T .Therefore, since C is 2-Calabi-Yau and T is functorially finite, we know that T is a clustersubcategory of C . (cid:3) Global dimension, nilpotent element and commutator subgroup. .In this part, we will generalize Lenzing’s Theorem [35] to locally bounded categories.We recall some terminologies in [35] firstly. Let A be an abelian category and B be a subcategoryof A which is closed under extension. B [ X ] is defined as the category whose objects are pairs ( E, f )for objects E ∈ B and f ∈ Hom ( E, E ), whose morphisms from (
E, f ) to (
F, g ) are the morphisms u : E → F satisfying gu = uf . We call that 0 → f ′ → f → f ′′ → B [ X ] if there exists acommutative diagram: 0 / / E ′ / / f ′ (cid:15) (cid:15) E / / f (cid:15) (cid:15) E ′′ / / f ′′ (cid:15) (cid:15) / / / / E ′ / / E / / E ′′ / / , where 0 → E ′ → E → E ′′ → B . Definition 3.3. (Definition 1, [35] ) A picture
T r : Obj ( B [ X ]) → G from the class of objects of B [ X ] to a commutative group G is called trace picture if it satisfies:(1) If → f ′ → f → f ′′ → is exact, then T r ( f ) = T r ( f ′ ) + T r ( f ′′ ) ,(2) For any f, f ′ ∈ Hom ( E, E ) , we have T r ( f + f ′ ) = T r ( f ) + T r ( f ′ ) . Recall in [35] that f : E → E ∈ B is called B -nilpotent if there exists a filtration E = E ⊇ E ⊇ · · · ⊇ E s = 0 such that E i , E i /E i +1 are objects of B and f ( E i ) ⊆ E i +1 for i = 0 , · · · , s . Lemma 3.4. (Satz 1, [35] ) If
T r : Obj ( B [ X ]) → G is a trace picture and f : E → E is B -nilpotent,then T r ( f ) = 0 . Let D be a locally bounded category. Let V = L X,Y ∈D Hom D ( X, Y ) and [
V, V ] be the K -subspaceof V generated by f g − gf for all f, g ∈ V . Let P = add ( { Hom D ( X, − ) | X ∈ D} ) be the subcategoryof mod D . Then P contains all projective objects in mod D .For any X ∈ D and P X = Hom D ( X, − ) ∈ P ( D ), by Yoneda Lemma, we have a natural isomor-phism End ( P X ) ϕ ∼ = End D ( X ). For f ∈ End ( P X ), define T r ( f ) = ϕ ( f ) + [ V, V ] ∈ V / [ V, V ]. Lemma 3.5. (1) Let X i ∈ D ( i = 1 , · · · , n ) be pairwise non-isomorphic objects and P = n L i =1 P X i . If g = ( g ij ) ∈ End ( P ) , then T r ( g ) = n P i =1 T r ( g ii ) . NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 15 (2) Let P = m L i =1 P i , P ′ = n L i =1 P ′ i be objects in P and f = ( f ij ) i ∈ [1 ,n ] ,j ∈ [1 ,m ] : P → P ′ , f ′ =( g ′ ij ) i ∈ [1 ,m ] ,j ∈ [1 ,n ] : P ′ → P be morphisms. Then T r ( f f ′ ) = T r ( f ′ f ) .Proof. (1) For any 1 ≤ s, t ≤ n , let f s,t = ( f ij ) ∈ End ( P ) with f st = g st and f ij = 0 for all ( i, j ) =( s, t ). When s = t , we have ϕ ( f st ) = id X t ϕ ( f st ) − ϕ ( f st ) id X t ∈ [ V, V ], so
T r ( f s,t ) = T r ( f st ) = 0.Thus, T r ( f s,t ) = ( T r ( f ss ) , if s = t, , otherwise. Therefore, the result follows.(2) According to (1), we have T r ( f f ′ ) = n P i =1 m P j =1 T r ( a ji a ′ ij ) and T r ( f ′ f ) = m P j =1 n P i =1 T r ( a ′ ij a ji ).Moreover, as a ji a ′ ij − a ′ ij a ji ∈ [ V, V ], we have
T r ( f f ′ ) = T r ( f ′ f ). (cid:3) Corollary 3.6.
Let P = m L i =1 P i , P ′ = m L i =1 P ′ i be objects in P with morphisms f = ( a ij ) i ∈ [1 ,m ] ,j ∈ [1 ,m ] : P → P , f ′ = ( a ′ ij ) i ∈ [1 ,m ] ,j ∈ [1 ,m ] : P ′ → P ′ . If there exists an isomorphism g = ( b ij ) i ∈ [1 ,m ] ,j ∈ [1 ,m ] : P → P ′ such that f ′ g = gf , then T r ( f ) = T r ( f ′ ) .Proof. Since f ′ g = gf and g is isomorphic, we have f = g − f ′ g . Therefore, by Lemma 3.5, T r ( f ) = T r ( g − f ′ g ) = T r ( g − gf ′ ) = T r ( f ′ ). (cid:3) After the above preparations, we can prove that
T r is a trace picture.
Theorem 3.7.
T r is a trace picture from
Obj ( P [ X ]) to V / [ V, V ] .Proof. (1) For f, f ′ ∈ End ( P ), we have T r ( f + f ′ ) = ϕ ( f + f ′ ) = ϕ ( f ) + ϕ ( f ′ ) = T r ( f ) + T r ( f ′ ).(2) For 0 → f ′ → f → f ′′ →
0, by the definition, we have the commutative diagram:0 / / P ′ x / / f ′ (cid:15) (cid:15) P y / / f (cid:15) (cid:15) P ′′ / / f ′′ (cid:15) (cid:15) / / P ′ x / / P y / / P ′′ / / . Since P ′′ is projective, 0 → P ′ → P → P ′′ → c ∈ Hom ( P, P ′ ) , d ∈ Hom ( P ′′ , P ) such that cx = id P ′ , yd = id P ′′ and ( x, d ) : P ′ ⊕ P ′′ → P is isomorphic with inverse( c, y ) T . Therefore, the following diagram is commutative:0 / / P ′ ( id, T / / id (cid:15) (cid:15) P ′ ⊕ P ′′ (0 ,id ) / / ( x,d ) (cid:15) (cid:15) P ′′ / / id (cid:15) (cid:15) / / P ′ x / / f ′ (cid:15) (cid:15) P y / / f (cid:15) (cid:15) P ′′ / / f ′′ (cid:15) (cid:15) / / P ′ x / / id (cid:15) (cid:15) P y / / ( c,y ) T (cid:15) (cid:15) P ′′ / / id (cid:15) (cid:15) / / P ′ ( id, T / / P ′ ⊕ P ′′ (0 ,id ) / / P ′′ / / . Thus, by Lemma 3.5 and Corollary 3.6,
T r ( f ) = T r (( c, y ) T f ( x, d )) = T r ( cf x cf dyf x yf d ! ) = T r ( cf x ) + T r ( yf d ) = T r ( f ′ ) + T r ( f ′′ ) . (cid:3) Theorem 3.8.
Keeps the forgoing notations. If D has finite global dimension, then T r : Obj ( P [ X ]) → V / [ V, V ] can be extended to a trace picture T r : Obj ( mod D [ X ]) → V / [ V, V ] . Proof.
The proof is the same as that of Satz 4 in [35] since D has finite global dimension. (cid:3) Theorem 3.9.
Keeps the notations as above. Let D be a locally bounded category. Assume that D has finite global dimension. If a ∈ V is nilpotent, then a ∈ [ V, V ] .Proof. We follow the strategy of the proof of Satz 5 in [35]. In detail, on one hand, by the definition,we have
T r ( ϕ − ( a )) = a + [ V, V ]. On the other hand, we have the following filtration P X ⊇ ϕ − ( a ) P X ⊇ · · · ⊇ ϕ − ( a m ) = 0 for some m since a is nilpotent. Thus, ϕ − ( a ) is mod D -nilpotent, byLemma 3.4, we have T r ( ϕ − ( a )) = 0. Therefore, we have a + [ V, V ] = 0, equivalently, a ∈ [ V, V ]. (cid:3) Remark 3.10. If D has finite objects, then Theorem 3.9 is Satz 5 of [35] . The category C Q arising from the preprojective algebra Λ Q In this section, we will use the colimit of categories to obtain a Frobenius 2-Calabi-Yau stronglyalmost finite category C Q from a strongly almost finite quiver Q . This is available for the sequel. Itmay be regarded as a generalization of the results in [23].Note that a strongly almost finite quiver Q is always acyclic since it is interval finite.4.1. Colimit of categories. .We study the existence and the properties of colimit of categories in this subsection.
Definition 4.1. ( [42] , Definition 2.6.13) Let A i , i ∈ N be additive categories. Assume that F ij : A i → A j are additive functors for all i ≤ j satisfying (a) F ii = id A i the identity functor for all i ∈ N ; (b) F kj F ji = F ki for all i ≤ j ≤ k .We say an additive category A to be the colimit of ( A i , F ji ) if(1) there exist additive functors F i : A i → A for all i ∈ N and natural equivalence η ji : F i → F j F ji for all i ≤ j satisfying η ii = id F i , and F i η ji / / id Fi (cid:15) (cid:15) F j F jiη kj ( F ji ) (cid:15) (cid:15) F i η ki / / F k F ki , are commutative for i ≤ j ≤ k ;(2) for any additive category B with additive functors G i : A i → B and natural equivalence ϕ ji : G i → G j F ji for all i ≤ j satisfying ϕ ii = id G i , and the commutative diagrams: G i ϕ ji / / id Gi (cid:15) (cid:15) G j F jiϕ kj ( F ji ) (cid:15) (cid:15) G i ϕ ki / / G k F ki , for i ≤ j ≤ k , there exists an unique additive functor (up to natural equivalence) ξ : A → B suchthat G i = ξF i for all i ∈ N and ϕ ji = ξη ji for all i ≤ j .In this case, we denote lim → A i = A or lim → A i = ( A , F i ) . Lemma 4.2.
Let C i , i ∈ N be Frobenius categories and F ji : C i → C j be exact functors for i ≤ j ,which satisfies:(a) Hom C i ( X, Y ) F ji ∼ = Hom C j ( F ji ( X ) , F ji ( Y )) are naturally isomorphic for all i ≤ j and X, Y ∈ C i , NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 17 (b) F ii = id C i for all i ∈ N , and(c) F ki = F kj F ji for all i ≤ j ≤ k .Then,(1) C = lim → C i exists;(2) if C has enough projective and injective objects, then C is a Frobenius category.Proof. (1) Firstly, we construct an additive category C from C i , i ∈ N .The objects of C consists of all of the forms [ X i ] for any X i ∈ C i , where it means X i ∈ C i whenwe write [ X i ] ∈ C .Now we define the homomorphisms and their compositions. For any two objects [ X i ] , [ Y j ] ∈ C ,We define Hom C ([ X i ] , [ Y j ]) := Hom C max { i,j } ( F max { i,j } i ( X i ) , F max { i,j } j ( Y j )) . Trivially, f : [ X i ] → [ Y i ] means just the morphism f : X i → Y i .For f ∈ Hom C ([ X i ] , [ Y j ]) and g ∈ Hom C ([ Y j ] , [ Z k ]), let l = max { i, j, k } . Define(3) g ◦ f := F lmax { j,k } ( g ) F lmax { i,j } ( f ) . Then, g ◦ f ∈ Hom C l ( F li ( X i ) , F lk ( Z k )) by( a ) = Hom C max { i,k } ( F max { i,k } i ( X i ) , F max { i,k } k ( Z k )) = Hom C ([ X i ] , [ Z k ]).For any f ∈ Hom C ([ X i ] , [ Y j ]), g ∈ Hom C ([ Y j ] , [ Z k ]), h ∈ Hom C ([ Z k ] , [ U m ]), we have h ◦ ( g ◦ f ) = h ◦ ( F lj,k ( g ) F li,j ( f )) = F max { l,m } max { k,m } ( h ) F max { l,m } l ( F lmax { j,k } ( g ) F lmax { i,j } ( f ))= F max { i,j,k,m } max { k,m } ( h ) F max { i,j,k,m } max { j,k } ( g ) F max { i,j,k,m } max { i,j } ( f ) . Similarly, ( h ◦ g ) ◦ f = F max { i,j,k,m } max { k,m } ( h ) F max { i,j,k,m } max { j,k } ( g ) F max { i,j,k,m } max { i,j } ( f ) . Hence, h ◦ ( g ◦ f ) = ( h ◦ g ) ◦ f always holds.By the definition of composition, it is easy to see the composition is bilinear. Thus, C is an additivecategory.Now we claim C = lim → C i .For any i ∈ N , define a functor F i : C i → C like this: F i ( X i ) = [ X i ] for all X i ∈ C i and F i ( f ) = f for all f : X i → Y i . It is clear that F i is an additive functor. For any f : X i → Y i in C i and j ≥ i ,we have an isomorphism id [ F ji ( X i )] : [ X i ] → [ F ji ( X i )]. Moreover, it is easy to verify the commutativediagram: [ X i ] id [ Fji ( Xi )] / / f (cid:15) (cid:15) [ F ji ( X i )] F ji ( f ) (cid:15) (cid:15) [ Y i ] id [ Fji ( Yi )] / / [ F ji ( Y i )] . Thus, we have the natural isomorphism of functors η ji : F i → F j F ji such that ( η ji ) X i = id [ F ji ( X i )] foreach X i ∈ C i . Then ( η ii ) X i = id [ X i ] for all i and X i ∈ C i . It is easy to see that ( η kj ) F ji ( X i ) ( η ji ) X i =( η ki ) X i for all i ≤ j ≤ k and X i ∈ C i .Assume for an additive category D , there are functors G i : C i → D and natural equivalences δ ji : G i → G j F ji for i ≤ j , satisfying ( δ ii ) X i = id G i ( X i ) and ( δ kj ) F ji ( X i ) ( δ ji ) X i = ( δ ki ) X i for all i ≤ j ≤ k and X i ∈ C i . We need to find an unique functor H : C → D such that G i = HF i for all i ∈ N and δ ji = Hη ji for all i ≤ j .We define the functor H : C → D satisfying H ([ X i ]) := G i ( X i ) for all X i ∈ C i and for allmorphisms f : [ X i ] → [ Y j ], defining that H ( f ) := ( G j ( f )( δ ji ) X i , if i ≤ j, ( δ ij ) − Y j G i ( f ) , otherwise. Then H ( id [ X i ] ) = G i ( id X i ) = id G i ( X i ) = id H ([ X i ]) . For f : [ X i ] → [ Y j ] and g : [ Y j ] → [ Z k ],by (3), we have g ◦ f := F lmax { j,k } ( g ) F lmax { i,j } ( f ) for l = max { i, j, k } . Since δ lmax { i,j } is a naturalequivalence, we obtain( δ lmax { i,j } ) F max { i,j } j ( Y j ) G max { i,j } ( f ) = ( G l F lmax { i,j } )( f )( δ lmax { i,j } ) F max { i,j } i ( X i ) . If i ≤ j ≤ k , we have H ( g ◦ f ) = H ( gF kj ( f )) = G k ( gF kj ( f ))( δ ki ) X i = G k ( g ) G k ( F kj ( f ))( δ ki ) X i ; H ( g ) H ( f ) = ( G k ( g )( δ kj ) Y j )( G j ( f )( δ ji ) X i ) = G k ( g ) G k ( F kj ( f ))( δ kj ) F ji ( X i ) ( δ ji ) X i = G k ( g ) G k ( F kj ( f ))( δ ki ) X i = H ( g ◦ f ) . If i ≤ k ≤ j , we have H ( g ◦ f ) = H (( F jk ) − ( gf )) = G k (( F jk ) − ( gf ))( δ ki ) X i ; H ( g ) H ( f ) = (( δ jk ) − Z k G j ( g ))( G j ( f )( δ ji ) X i ) = ( δ jk ) − Z k G j ( gf )( δ jk ) F ki ( X i ) ( δ ki ) X i , Since δ jk is a natural transformation, we have G j ( gf )( δ jk ) F ki ( X i ) = ( δ jk ) Z k G k (( F jk ) − ( gf )). Thus H ( g ◦ f ) = H ( g ) H ( f ).In the other cases, we can similarly prove H ( g ◦ f ) = H ( g ) H ( f ). Thus, H is a functor.For any X i , Y i ∈ C i and f : X i → Y i , we have HF i ( X i ) = H ([ X i ]) = G i ( X i ) and HF i ( f ) = H ( f ) = G i ( f )( δ ii ) X i = G i ( f ), then G i = HF i for all i ∈ N .If there is another functor H : C → D satisfying G i = H F i for all i ∈ N , then H ([ X i ]) = H F i ( X i ) = G i ( X i ) = H ([ X i ]) for each [ X i ] ∈ C .Let f : [ X i ] → [ Y j ] ∈ C . Since H η ts = δ ts , ( H η ts ) X s = H ( id [ F ts ( X s )] ) = ( δ ts ) X s for s ≤ t . Then H ( id − F ts ( Y s )] ) = ( δ ts ) − Y s for s ≤ t . If i ≤ j , then f : [ X i ] id [ Fji ( Xi )] → [ F ji ( X i )] f → [ Y j ]. Thus, H ( f ) = H ( f id [ F ji ( X i )] ) = H ( f ) H ( id [ F ji ( X i )] ) = G j ( f )( δ ji ) X i = H ( f ) . Similarly, if i ≥ j , f : [ X i ] f → [ F ij ( Y j )] id − Fij ( Yj )] → [ Y j ], then H ( f ) = ( δ ji ) − Y j G i ( f ) = H ( f ). Hence H areuniquely determined.(2) Define the admissible short exact sequences in C are sequences isomorphic to the form 0 → [ X i ] f → [ Y i ] g → [ Z i ] →
0, where 0 → X i f → Y i g → Z i → C i .Now we prove that it gives an exact structure on C .For admissible short exact sequence 0 → [ X i ] f → [ Y i ] g → [ Z i ] → h : [ Y i ] → [ C j ] suchthat h ◦ f = 0. According to construction, 0 → X i f → Y i g → Z i → C i . We have h : F max { i,j } i ( Y i ) → F max { i,j } j ( C j ) and h ◦ f = hF max { i,j } i ( f ) = 0. Since F max { i,j } i is exact, 0 → F max { i,j } i ( X i ) F max { i,j } i ( f ) −→ F max { i,j } i ( Y i ) F max { i,j } i ( g ) −→ F max { i,j } i ( Z i ) → C max { i,j } , and as C max { i,j } is exact, we have h = h ′ F max { i,j } i ( g )for h ′ : F max { i,j } i ( Z i ) → F max { i,j } j ( C j ). Thus, h = h ′ ◦ g in C . Therefore, g is the cokernel of f in C .Dually, f is the kernel of g in C . Ex C i are exact.For Ex
1, let f : [ X i ] → [ Y i ] and f : [ Y i ] → [ Z j ] be admissible epimorphisms. We have f : X i → Y i , f : F max { i,j } i ( Y i ) → F max { i,j } j Z j are admissible epimorphisms in C i and C max { i,j } , and f ◦ f = f F max { i,j } i ( f ). As F max { i,j } i is exact, then F max { i,j } i ( f ) is an admissible epimorphismin C max { i,j } . Thus, f F max { i,j } i ( f ) is an admissible epimorphism in C max { i,j } . Therefore, f ◦ f isadmissible epimorphism in C . NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 19
For Ex
2, for each h : [ Z ′ j ] → [ Z i ] and each admissible epimorphism g : [ Y i ] → [ Z i ]. We have h : F max { i,j } j Z ′ j → F max { i,j } i ( Z i ) and g : Y i → Z i is an admissible epimorphism in C i , thus F max { i,j } i ( g ) : F max { i,j } i ( Y i ) → F max { i,j } i ( Z i ) is an admissible epimorphism in C max { i,j } . Since C max { i,j } is exact,there exists the following pullback square Y ′ max { i,j } g ′ / / h ′ (cid:15) (cid:15) F max { i,j } j ( Z ′ j ) h (cid:15) (cid:15) F max { i,j } i ( Y i ) F max { i,j } i ( g ) / / F max { i,j } i ( Z i )in C max { i,j } such that h ′ is an admissible epimorphism. Thus, we have the following pullback square[ Y ′ max { i,j } ] g ′ / / h ′ (cid:15) (cid:15) [ Z ′ j ] h (cid:15) (cid:15) [ Y i ] g / / [ Z i ]in C such that h ′ is an admissible epimorphism in C . Thus, Ex Ex op .Therefore, C is an exact category.Finally, we claim that C is Frobenius. We show that for P i ∈ C i , [ P i ] ∈ C is projective if and onlyif F ji ( P i ) is projective in C j for all j ≥ i .Assume F ji ( P i ) is projective in C j for all j ≥ i . For any admissible short exact sequence 0 → [ X j ] f → [ Y j ] g → [ Z j ] →
0, then 0 → X j f → Y j g → Z j → C j . Since F max { i,j } j is exact, we have 0 → F max { i,j } j ( X j ) → F max { i,j } j ( Y j ) → F max { i,j } j ( Z j ) → C max { i,j } . As F max { i,j } i ( P i ) is projective in C max { i,j } , so Hom C max { i,j } ( F max { i,j } i ( P i ) , F max { i,j } j ( Y j )) → Hom C max { i,j } ( F max { i,j } i ( P i ) , F max { i,j } j ( Z j )) → Hom C ([ P i ] , [ Y j ]) → Hom C ([ P i ] , [ Z j ]) → P i ] is projective in C .Conversely, assume [ P i ] is projective. For any j ≥ i and admissible short exact sequence 0 → X j f → Y j g → Z j → C j , then 0 → [ X j ] f → [ Y j ] g → [ Z j ] → C . As [ P i ] is projective, so Hom C ([ P i ] , [ Y j ]) → Hom C ([ P i ] , [ Z j ]) → Hom C j ( F ji ( P i ) , Y j ) → Hom C j ( F ji ( P i ) , Z j ) → F ji ( P i ) is projective in C j .Dually, for I i ∈ C i , [ I i ] ∈ C is injective if and only if F ji ( I i ) is injective in C j for all j ≥ i .Furthermore, since C i are Frobenius for all i , it follows that C is Frobenius. (cid:3) We call the family { F ji } i ∈ N satisfying the conditions of this lemma a system of exact functors for C i , i ∈ N .In this proof of Lemma 4.2, for any i ∈ N , the additive functor F i : C i → C satisfying F i ( X i ) = [ X i ]for all X i ∈ C i and F i ( f ) = f for all f : X i → Y i , is called the embedding functor from C i to C . Remark 4.3. (1) In Lemma 4.2, the condition (a) ensures that F ji ( X ) ∼ = F ji ( Y ) if and only if X ∼ = Y . More precisely, if F ji ( X ) ∼ = F ji ( Y ) , by (a), then for any M ∈ C i , we have the naturalisomorphism Hom C i ( X, M ) ∼ = Hom C i ( Y, M ) . Then by Yoneda Lemma, we have X ∼ = Y .(2) For every i ≤ j , since F ji is exact and additive, we have a group homomorphism F ji : Ext C i ( X, Y ) → Ext C j ( F ji ( X ) , F ji ( Y )) . (3) If C i are abelian categories satisfying the same conditions, then we can also prove that lim → C i exists and is an abelian category. In the above remark, if F ji are isomorphisms and C i are 2-Calabi-Yau categories, we say that F ji : C i → C j is compatible with the 2-Calabi-Yau structures if the following diagram is commutative, Ext C i ( X i , Y i ) ⋍ / / F ji (cid:15) (cid:15) DExt C i ( Y i , X i ) Ext C j ( F ji ( X i ) , F ji ( Y i )) ⋍ / / DExt C j ( F ji ( Y i ) , F ji ( X i )) . DF ji O O Lemma 4.4.
Keeps the condition of Lemma 4.2. Assume that for all i ≤ j , Ext C i ( X, Y ) F ji ∼ = Ext C j ( F ji ( X ) , F ji ( Y )) and C i are -Calabi-Yau categories. If F ji : C i → C j are compatible with the -Calabi-Yau structures, then C is a -Calabi-Yau category.Proof. Let [ X i ] , [ Y j ] ∈ C . For any ξ : 0 → [ Y j ] → [ Z k ] → [ X i ] → ∈ Ext C ([ X i ] , [ Y j ]) and η :0 → [ X i ] → [ Z ′ s ] → [ Y j ] → ∈ Ext C ([ Y j ] , [ X i ]). According to the construction of C , for any t ≥ max { i, j, k, s } , e ξ : 0 → F tj ( Y j ) → F tk ( Z k ) → F ti ( X i ) → ∈ Ext C t ( F ti ( X i ) , T tj ( Y j )) and e η :0 → F ti ( X i ) → F ts ( Z ′ s ) → F tj ( Y j ) → ∈ Ext C t ( F tj ( Y j ) , F ti ( X i )). Sine C t is 2-Calabi-Yau, there is anon-degenerate bilinear form γ t : Ext C t ( X, Y ) × Ext C t ( Y, X ) → K .Define a bilinear map γ : Ext C ([ X i ] , [ Y j ]) × Ext C ([ Y j ] , [ X i ]) → K, ( ξ, η ) → γ t ( e ξ, e η ) . Firstly, we show that γ is well-defined. Assume that t ′ is the other integer such that t ′ ≥ max { i, j, k, s } . Without loss of generality, we may assume that t ′ ≥ t . Then, F t ′ t ( e ξ ) ∈ Ext C t ′ ( F t ′ i ( X i ) , F t ′ j ( Y j )) and F t ′ t ( e η ) ∈ Ext C t ′ ( F t ′ j ( Y j ) , F t ′ i ( X i )) . Since F t ′ t is compatible with the 2-Calabi-Yau structure, we have γ t ′ ( F t ′ t ( e ξ ) , F t ′ t ( e η )) = γ t ( e ξ, e η ). Forany ξ = ξ ′ , which means the admissible exact sequences ξ and ξ ′ are isomorphic. We can choose t bigger enough, such that e ξ and e ξ ′ are isomorphic as admissible exact sequences in C t , by thewell-defined of γ t , we have γ ( ξ, η ) = γ t ( e ξ, e η ) = γ t ( e ξ ′ , e η ) = γ ( ξ ′ , η ). Dually, if η = η ′ , we have γ ( ξ, η ) = γ ( ξ, η ′ ).It remains to prove that γ is non-degenerate. Assume that there is a ξ ∈ Ext C ([ X i ] , [ Y j ]) such that γ ( ξ, η ) = 0 for all η ∈ Ext C ([ Y j ] , [ X i ]). Thus, we have γ t ( e ξ, e η ) = 0 for all e η ∈ Ext C t ([ F tj ( Y j )] , [ F ti ( X i )]) =0. Since γ t is non-degenerate, we have e ξ = 0, it implies ξ = 0. So, γ is non-degenerate.By the non-degenerate bilinear form γ , we have isomorphisms Ext C ([ Y j ] , [ X i ]) ∼ = DExt C ([ X i ] , [ Y j ])for all [ X i ] , [ Y j ] ∈ C . It is easy to see that such isomorphisms are functorial since γ is constructed by { γ t } and { γ t } can induce functorial isomorphisms. (cid:3) For exact categories A and B with an exact functor π : A → B , let M be an additive subcategoryof B . Define C M := π − ( M ), that is, the subcategory of A generated by all objects whose imagesunder π are in M . Proposition 4.5.
Let C i and D i , i ∈ N be exact categories, and exact functors π i : C i → D i . Assumethe families { F ji } i ∈ N and { G ji } i ∈ N are respectively systems of exact functors for C i and D i , i ∈ N satisfying π j F ji = G ji π i for all i ≤ j . Let C = lim → C i and D = lim → D i as constructed in Lemma 4.2with the embedding functors F i : C i → C and G i : D i → D . Let M i , i ∈ N be additive subcategories NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 21 of D i . Assume that S := { [ M i ] ∈ D | i ∈ N and M i ∈ M i satisfying G ji ( M i ) ∈ M j , ∀ j ≥ i } 6 = ∅ . Let M be the additive subcategory of D generated by S . Then the following statements hold:(1) There is a π : C → D such that πF i = G i π i for all i .(2) If all C M i are extension closed subcategories of C i , then C M is an extension closed subcategoryof C .(3) If all C M i are closed under factor, then C M is closed under factor.(4) Under the conditions of (2) and (3), assume that all C M i , i ∈ N are -Calabi-Yau Frobeniuscategories. Denote by I i the class consisting of all injective objects of C M i satisfying F ji ( I i ) ∈ I j for all j ≥ i and I i ∈ I i , and by I the subcategory consisting of all objects of C M isomorphic to [ I i ] for all I i ∈ I i . If I is a generator and co-generator subcategory of C M and F ji : Ext C i ( X, Y ) → Ext C j ( F ji ( X ) , F ji ( Y )) are group isomorphisms for all i ≤ j and compatible with the -Calabi-Yaustructures, then C M is also a -Calabi-Yau Frobenius category whose projective-injective objects justform I .Proof. (1) By Lemma 4.2, there exist F i : C i → C , G i : D i → D and natural functor isomorphisms δ ji : G i → G j G ji such that ( δ ii ) Y i = id G i ( Y i ) and ( δ kj ) G ji ( Y i ) ( δ ji ) Y i = ( δ ki ) Y i for all i ≤ j ≤ k and Y i ∈ D i .Since G ji π i = π j F ji , we have the natural functor isomorphism δ ji π i : G i π i → G j G ji π i = ( G j π j ) F ji for i ≤ j . For any i ≤ j ≤ k and X i ∈ C i , we have ( δ ii π i ) X i = id G i π i ( X i ) and ( δ kj π j ) F ji ( X i ) ( δ ji π i ) X i =( δ kj G ji π i ) X i ( δ ji π i ) X i = ( δ ki π i ) X i . Thus, by the universal property of colimit, there is a functor π : C → D such that πF i = G i π i .(2) For any [ X i ] , [ Z k ] ∈ C M and short exact sequence 0 → [ X i ] → [ Y j ] → [ Z k ] → C , then thereexists a short exact sequence 0 → X ′ s → Y ′ s → Z ′ s → C s such that 0 → [ X i ] → [ Y j ] → [ Z k ] → → [ X ′ s ] → [ Y ′ s ] → [ Z ′ s ] →
0. (Since F ji ( j ≥ i ) are exact, we may assume that s ≥ i, j, k .) Thus, X ′ s ∼ = F si ( X i ), Y ′ s ∼ = F sj ( Y j ) and Z ′ s ∼ = F sk ( Z k ). Since [ X i ] , [ Z k ] ∈ C M , so F ts ( X ′ s ) = F ts F si ( X i ) = F ti ( X i ) ∈ C M t and F ts ( Z ′ s ) = F ts F sk ( Z k ) = F tk ( Z k ) ∈ C M t for all t ≥ s , and since C M t isextension closed, we have F ts ( Y ′ s ) ∈ C M t for all t ≥ s . Therefore, [ Y ′ s ] ∼ = [ F sj ( Y j )] ∼ = [ Y j ] ∈ C M . Thus, C M is extension closed.(3) For any [ X i ] ∈ C M and surjective morphism [ X i ] f → [ Y j ] in C . For any s ≥ max ( i, j ), then F si ( X i ) ∈ C M s and F si ( X i ) f → F sj ( Y j ) is surjective in C s , and since C M s is closed under factor. Thus, F sj ( Y j ) ∈ C M s . Hence, [ F max ( i,j ) j ( Y j )] ∼ = [ Y j ] ∈ C M . Therefore, C M is closed under factor.(4) By Lemma 4.4, C is a 2-Calabi-Yau category, and by (2) C M is closed under extension. Thus, C M is 2-Calabi-Yau.Firstly, we show any object [ I i ] ∈ I is injective in C M . For any exact sequence 0 → [ X j ] → [ Y j ] → [ Z j ] →
0, then 0 → X j → Y j → Z j → C M j . If i ≤ j , since F ji ( I i ) isinjective in C M j , we have 0 → Hom C j ( Z j , F ji ( I i )) → Hom C j ( Y j , F ji ( I i )) → Hom C j ( X j , F ji ( I i )) → → Hom C ([ Z j ] , [ I i ]) → Hom C ([ Y j ] , [ I i ]) → Hom C ([ X j ] , [ I i ]) → i ≥ j , since F ij is exact, 0 → F ij ( X j ) → F ij ( Y j ) → F ij ( Z j ) → C i . As I i is injectivein C i , we have 0 → Hom C i ( F ij ( Z j ) , I i ) → Hom C i ( F ij ( Y j ) , I i ) → Hom C i ( F ij ( X j ) , I i ) → → Hom C ([ Z j ] , [ I i ]) → Hom C ([ Y j ] , [ I i ]) → Hom C ([ X j ] , [ I i ]) → I i ] isinjective.Now we prove that any injective object [ X i ] is in I . Since I is a co-generator subcategory of C M , there exists [ I j ] ∈ I and exact sequence 0 → [ X i ] → [ I j ] → [ Y k ] →
0. By (3), C M is closedunder factor, [ Y k ] ∈ C M . As [ X i ] is injective in C M , we have the exact sequence is split. Thus,[ I j ] ∼ = [ X i ] ⊕ [ Z k ] ∼ = [ F max ( i,k ) i ( X i ⊕ Z k )]. Let s = max ( i, j, k ). Since Hom C ([ I j ] , [ F max ( i,k ) i ( X i ⊕ Z k )]) = Hom C s ( F sj ( I j ) , F si ( X i ) ⊕ F sk ( Z k )) . Thus, F sj ( I j ) ∼ = F si ( X i ) ⊕ F sk ( Z k ) in C s . For any t ≥ s ,since Hom C s ( X, Y ) ∼ = Hom C t ( F ts ( X ) , F ts ( Y )) for all X, Y ∈ C s , we have F tj ( I j ) ∼ = F ti ( X i ) ⊕ F tk ( Z k ) in C t , moreover, as [ I j ] ∈ I , F ti ( I j ) is injective in C M t . Thus, F ti ( X i ) is injective in C M t . Therefore,[ X i ] ∼ = [ F ts ( X i )] ∈ I .Therefore, I is the subcategory formed by all injective objects of C M . As C M is 2-Calabi-Yau,[ I ] ∈ C M is projective if and only if [ I ] is injective. Moreover, I is the generator and co-generatorsubcategory, then C M has enough projective and injective objects. The result follows. (cid:3) Strongly almost finite Frobenius -Calabi-Yau category C Q . .In this part, we use Proposition 4.5 to construct a strongly almost finite Frobenius 2-Calabi-Yaucategory C Q via a strongly almost finite quiver Q . Moreover, we study some properties of C Q .Assume that Q is a strongly almost finite quiver. Let Q be the double quiver of Q , more precisely,it is obtained by adding an new arrow α ∗ : j → i whenever there is an arrow α : i → j . For KQ the path algebra associated with Q , let I be the idea of KQ generated by P s ( α )= i α ∗ α − P t ( α )= i αα ∗ for all i ∈ Q . Denote by Λ Q the quotient algebra of KQ/ I . As like as preprojective algebras froma finite quiver, we call Λ Q the preprojective algebra of Q , although in this case the algebra Λ Q may not possess identity when Q has infinite vertices. Thanks to KQ as a subalgebra of Λ Q , wedenote π : mod Λ Q → modKQ the restriction functor. For any additive subcategory M of modKQ ,let C M := π − ( M ).Recall that, for any finite acyclic quiver Q ′ , Ringel [41] defined a category C Q ′ (1 , τ Q ′ ) as follows.The objects are of the form ( X, f ), where X ∈ modKQ ′ and f ∈ Hom KQ ′ ( X, τ Q ′ ( X )). Here τ Q ′ is the Auslander-Reiten translation in modKQ ′ . The homomorphism from ( X, f ) to ( X ′ , f ′ ) is a KQ ′ -module homomorphism g : X → X ′ such that the following diagram X f / / g (cid:15) (cid:15) τ Q ′ ( X ) τ Q ′ ( g ) (cid:15) (cid:15) X ′ f ′ / / τ Q ′ ( X ′ )commutes. Ringel (Theorem B , [41]) proved that C Q ′ (1 , τ Q ′ ) isomorphic to mod Λ Q ′ . More precisely,there exists an isomorphic functor Φ Q ′ : mod Λ Q ′ → C Q ′ (1 , τ Q ′ ) such that Φ Q ′ ( X ) = ( Y, f ) implies π Q ′ ( X ) = Y . We extend such construction to strongly almost finite quiver Q to obtain the category C Q (1 , τ ), briefly written as C (1 , τ ), where τ is the Auslander-Reiten translation in modKQ . Theorem 4.6.
Keep the forgoing notations. Let Q be a strongly almost finite quiver. Then C (1 , τ ) is categorically isomorphic to mod Λ Q with an isomorphic functor Φ : mod Λ Q → C (1 , τ ) satisfying π ( X ) = Y for Φ( X ) = ( Y, f ) .Proof. For any Y = (( Y i ) i ∈ Q , ( Y α ) α ∈ Q ) ∈ modKQ , we have τ ( Y ) = (( τ ( Y ) i ) i ∈ Q , ( τ ( Y ) α ) α ∈ Q ).Since Y and τ ( Y ) are finite dimensional, we can find a finite subquiver Q ′ of Q such that Y, τ ( Y ) ∈ modKQ ′ under the restriction of KQ -action. According the proof of Theorem B’, [41], we can definelinear maps g α : τ ( Y ) t ( α ) → Y s ( α ) for all α ∈ Q ′ inductively on the given partial order of vertices of Q ′ due to the orientation of arrows.First, for all α ∈ Q ′ with t ( α ) = i a sink (i.e. a minimal vertex on the partial order), we cangive the exact sequence0 → τ ( Y ) i ( g α ) t ( α i −→ M t ( α )= i Y s ( α ) ( Y α ) ⊤ t ( α i −→ Y i . NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 23 to obtain g α for all α with t ( α ) = i . Note in this sequence ( g α ) t ( α )= i means to arrange it ina row and ( Y α ) ⊤ t ( α )= i means to arrange it in a column such that the operations can be made. Inthe next sequence the description is similar.Next, using the following exact sequence:0 → τ ( Y ) i (( g β ) t ( β )= i , ( τ ( Y ) γ )) s ( γ )= i −→ M t ( β )= i Y s ( β ) ⊕ M s ( γ )= i τ ( Y ) t ( γ ) ( Y β ) ⊤ t ( β )= i ( g γ ) ⊤ s ( γ )= i −→ Y i , for any i ∈ Q ′ we can decide all g β with t ( β ) = i inductively by those g γ with s ( γ ) = i . Note thatsuch exact sequence always exists by Theorem B’, [41] since Q ′ is finite acyclic. For all α ∈ Q \ Q ′ ,set g α = 0. Then, we obtain ( g α ) α ∈ Q on Q ′ .Use this ( g α ) α ∈ Q above from Q ′ , we can construct, relying on g α , the functors Ψ : C (1 , τ ) → mod Λ Q and Φ : mod Λ Q → C (1 , τ ) as follows.Precisely, given ( Y, f ) ∈ C (1 , τ ), define Ψ( Y, f ) = (( X i ) i ∈ Q , ( X α ) α ∈ Q ) as follows: X i = Y i foreach i ∈ Q and X α = Y α , X α ∗ = g α f t ( α ) for each α ∈ Q . As in the proof of Theorem B’, [41], wecan prove that X ∈ mod Λ Q . Conversely, given X ∈ mod Λ Q , there exist uniquely f = ( f i ) i ∈ Q suchthat X α ∗ = g α f t ( α ) for each α ∈ Q . Let Y = π ( X ) for π the restriction functor from mod Λ Q to modKQ . Then f : Y → τ ( Y ) is a modKQ homomorphism, and define Φ( X ) = ( Y, f ).It can be proved similarly as Theorem B’ of [41] that ΦΨ = id C (1 ,τ ) and ΨΦ = id mod Λ Q . Therefore,our result follows. (cid:3) Let Q be a strongly almost finite quiver. Following [23], for any a ≤ b ∈ N and i ∈ Q , let I i bethe indecomposable representation of KQ corresponding to vertex i , define I i, [ a,b ] = b M j = a τ j ( I i ) , and e i, [ a,b ] = : I i, [ a,b ] → τ ( I i, [ a,b ] ) . Let Q ′ be a finite acyclic quiver, recall that an object M of modKQ ′ is called terminal (2.2.,[23]) if the following hold:(1) M is pre-injective;(2) If X is an indecomposable module of KQ ′ such that Hom ( M, X ) = 0, then X ∈ add ( M );(3) I i ∈ add ( M ) for all indecomposable injective KQ ′ -modules I i .If M is a terminal module, define t i ( M ) := max { j ≥ | τ j ( I i ) ∈ add ( M ) \ { }} . See [23].Note that if Q ′ is not type A with linear orientation, then M = DKQ ′ ⊕ τ ( DKQ ′ ) is a terminalmodule of KQ ′ with t i ( M ) = 1 for all i ∈ Q ′ .If Q ′ is an acyclic finite quiver with n vertices, denote π Q ′ : mod Λ Q ′ → modKQ ′ be the restrictionfunctor, where Λ Q ′ is the preprojective algebra of KQ ′ . Assume M is a terminal module of KQ ′ .Let C ′ add ( M ) := π − Q ′ ( add ( M )). We obtain the following: Theorem 4.7. (Theorem 2.1., Lemma 5.5., Lemma 5.6., [23] )(1) C ′ add ( M ) is a Frobenius -Calabi-Yau category with n indecomposable C ′ add ( M ) -projective-injectiveobjects { ( I i, [0 ,t i ( M )] , e i, [0 ,t i ( M )] ) | i = 1 , · · · , n } ; (2) C ′ add ( M ) is closed under extension;(3) C ′ add ( M ) is closed under factor. Theorem 4.8.
Let Q be a strongly almost finite quiver. Denote the vertices of Q by , , · · · , and Q [ i ] be the sub-quiver of Q generated by vertices , · · · , i for any i ∈ N . Let G ji : modKQ [ i ] → modKQ [ j ] bethe embedding functors for all i ≤ j ∈ N . Assume that M i are terminal modules of KQ [ i ] and N is theadditive subcategory of modKQ generated by all objects M ∈ add ( M i ) satisfying G ji ( M ) ∈ add ( M j ) for all j ≥ i . Denote by I i the subcategory of injective objects in C ′ add ( M i ) and by I the additivesubcategory of C N generated by all objects I ∈ I i satisfying G ji ( I ) ∈ I j for all j ≥ i . Assume that I is a generator and co-generator subcategory of C N . Then,(1) C N is closed under extension and under factor respectively;(2) C N is a Frobenius -Calabi-Yau category whose projective-injective subcategory is just I .Proof. Let Λ i := Λ Q [ i ] be preprojective algebras of KQ [ i ] and F ji : mod Λ i → mod Λ j be the embed-ding functors for all i ≤ j ∈ N . It is easy to see that the conditions of Lemma 4.2 are satisfied for mod Λ i and F ji . Therefore, C := lim → mod Λ i exists.Now we prove that mod Λ Q is equivalent to C . Let e H : C → mod Λ Q be the functor such that e H ([ X i ]) = X i for each [ X i ] ∈ C and e H ′ : mod Λ Q → C be the functor such that e H ′ ( X i ) = [ X i ] foreach X i ∈ mod Λ Q , where i is the least number such that X i ∈ mod Λ Q [ i ] . It is clear that e H and e H ′ are equivalent functors between mod Λ Q and C .Similarly, lim → modKQ [ i ] exists, denoted by D , and modKQ is equivalent to D . Moreover, H : D → modKQ via [ X i ] X i is an equivalence functor.It is clear that π j F ji = G ji π i . Thus, according to Proposition 4.5 (1), there is a functor e π suchthat e πF i = G i π i for all i . It is easy to verify the following commutative diagram: C e H / / e π (cid:15) (cid:15) mod Λ Qπ (cid:15) (cid:15) D H / / modKQ. Denote by M the additive subcategory of D generated by all objects [ M i ] with M i ∈ M i and G ji ( M i ) ∈ M j for all j ≥ i . Thus, H induces an equivalence between M and N . By the abovecommutative diagram, e H induces an equivalence between C M and C N .(1) By Theorem 4.7, C ′ add ( M i ) are closed subcategories of mod Λ Q [ i ] under extension. Using Propo-sition 4.5 (2), we have C M is a closed subcategory of C under extension. Thus, C N is an extensionclosed subcategory of mod Λ Q .By Theorem 4.7, C ′ add ( M i ) are closed under factor. Using Proposition 4.5 (3), we have C M is closedunder factor. Thus, C N is an extension closed subcategory of mod Λ Q .(2) Let e I be the subcategory of C M which contains all objects [ I ] with I ∈ I i and G ji ( I ) ∈ I j forall j ≥ i . Then e H induces equivalence between e I and I . Since I is a generator and co-generatorsubcategory of C N , e I is a generator and co-generator subcategory of C M . By the proof of Theorem3 of [24], F ji are compatible with the 2-Calabi-Yau structures. Therefore, by Proposition 4.5 (4), ourresult follows. (cid:3) From the given quiver Q above, let M be the additive subcategory of modKQ generated by I j and τ ( I j ) for all injective objects I j and j ∈ Q . Then, we will always denote C Q := C M . Corollary 4.9.
Let us keep the forgoing notations. Assume that Q is a strongly almost finite quiverbut not a linear quiver of type A . Then, NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 25 (1) C Q is a -Calabi-Yau Frobenius category;(2) The projective-injective subcategory is { ( I j, (0 , , e j, [0 , ) | j ∈ Q } .Proof. (1) When Q is finite, the result follows immediately by Theorem 4.7. So, we can assume that Q is an infinite quiver. Following this, since Q is not a linear quiver of type A , we can renumberthe vertices of Q such that Q [ i ] is not of type A i with linear orientation for each i ∈ N . Let M i := DKQ [ i ] ⊕ τ ( DKQ [ i ] ). Thus, M i is a terminal module of KQ [ i ] with t j ( M i ) = 1 for all j ∈ Q [ i ]0 . As Q is strongly almost finite, by the similar proof of Corollary 8.4 and Corollary 8.8 of[23], we have { ( I j, (0 , , e j, [0 , ) | j ∈ Q } is a co-generator and generator subcategory of C Q . Thus, C Q satisfies the condition of Theorem 4.8. Therefore, our result follows.(2) It follows by Theorem 4.8 (2). (cid:3) We say that a group Γ acts on a quiver Q if there is a group homomorphism ϕ : Γ → Aut ( Q ),where Aut ( Q ) is the automorphism of Q as a directed graph. In this case, we have h · i = ϕ ( h )( i )for i ∈ Q . Definition 4.10.
Let C be a strongly almost finite Frobenius exact category and Γ be an (infinite)group. We call that C admits a Γ -action if there are categorical isomorphisms ϕ ( h ) : C → C for h ∈ Γ satisfying(a) ϕ ( e ) = id C , where e is the identity element of Γ and id C is the identity functor;(b) ϕ ( hh ′ ) = ϕ ( h ) ϕ ( h ′ ) for any pair h, h ′ ∈ Γ . We write ϕ ( h )( X ) as h · X briefly for any X ∈ C . Definition 4.11. (Definition 3.17, [15] ) Let X ∈ C . An epimorphism f : X → Y is said to be a left rigid quasi-approximation of X if the following conditions are satisfied: • add ( { h · Y | h ∈ Γ } ) is rigid; • Y has an injective envelope, without direct summand in add ( { h · X | h ∈ Γ } ) ; • If add ( { h · Z | h ∈ Γ } ) is rigid, then every morphism from X to Z without invertible matrixcoefficient, factorizes through f . Let Q be a strongly almost finite quiver which admits a group Γ action, it is easy to see Γ actsin C Q . Assume that Q is not a linear quiver of type A . We will show that every projective object in C Q has a left rigid quasi-approximation.We first recall a crucial lemma of how to judge a projective object has a left rigid quasi-approximation. Lemma 4.12. (Lemma 3.20, [15] ) Let D be an abelian category endowed with an action of Γ . Let C be an exact full subcategory of D . Let P be a projective indecomposable object of C . If • every monomorphism of C to a projective object is admissible, • every monomorphism of C from P to an indecomposable object is admissible, • P has a simple socle S in D , • add ( { h · S | h ∈ Γ } ) is rigid in D , • the cokernel of S ֒ → P is in C ,then this cokernel is a left rigid quasi-approximation of P in C .Proof. The proof of (Lemma 3.20, [15]) for finite group can be transferred to that for infinite groupΓ. (cid:3)
The following corollary is inspired by Corollary of 4.42, [15].
Corollary 4.13.
Let Q be a strongly almost finite quiver with a group Γ -action. Q is not a linearquiver of type A . Then all projective objects of C Q have left rigid quasi-approximations.Proof. Let P = ( I i, (0 , , e i, (0 , ) be a projective object in C Q . By Theorem 4.8 (2), C Q is closed underfactor, so all every monomorphism of C Q to a projective object is admissible. Since C Q is Frobenius,every monomorphism of C Q from P to an indecomposable object is split, so it is admissible. ByLemma 8.1, [23], P has simple socle ( S i , Q has no Γ-loops and Γ-2-cycles, add ( { h · S | h ∈ Γ } ) is rigid. In addition, C Q is closed under factor, the cokernel of S ֒ → P is in C Q .Applying Lemma 4.12, our result follows. (cid:3) For a Krull-Schmidt K -category T , we define the Gabriel quiver Q ( T ) of T as follows: the verticesare the equivalent classes of indecomposable objects. For two vertices [ T ] and [ T ], the number ofarrows from [ T ] to [ T ] equal to the dimension of Irr ( T , T ) := rad ( T , T ) /rad ( T , T ) in thecategory T , where rad is the radical of T . Note that if T = add ( n L i =1 T i ) with { T i | i = 1 , · · · , n } arepairwise not isomorphic, then the Gabriel quiver of T is the same as the Gabriel quiver of the finitedimensional algebra End T ( n L i =1 T i ).Let T = add { ( I i, [0 , , e i, [0 , ) , ( I i, [1 , , e i, [1 , ) | i ∈ Q } as an subcategory of C Q . Lemma 4.14. (Lemma 6.3, [23] ) Let Q be a strongly almost finite quiver which is not a linear quiverof type A . For any ( X, f ) ∈ C Q , we have(1) Hom KQ ( τ ( I i ) , X ) ∼ = → Hom C Q (( I i, [0 , , e i, [0 , ) , ( X, f )) : g ( τ − ( f ) τ − ( g ) , g ) ;(2) { g ∈ Hom KQ ( τ ( I i ) , X ) | f g = 0 } ∼ = → Hom C Q (( I i, [1 , , e i, [1 , ) , ( X, f )) : g g ;(3) { g ∈ Hom KQ ( I i , X ) | f g = 0 } ∼ = → Hom C Q (( I i, [0 , , e i, [0 , ) , ( X, f )) : g g . Lemma 4.15.
Keep the notations as above. Let Q be a strongly almost finite quiver which is not alinear quiver of type A .(1) For any object C in C Q , there are only finite indecomposable objects T (up to isomorphism)in T such that Hom C Q ( T, C ) = 0 or Hom C Q ( C, T ) = 0 . In particular, T is functorially finite.(2) T is a maximal rigid subcategory of C Q .(3) The Gabriel quiver of T is equal to the Gabriel quiver of { τ ( I i ) , I i | i ∈ Q } by adding an arrowfrom [ I i ] to [ τ ( I i )] for each i ∈ Q , more precisely, the vertex [( I i, (0 , , e i, (0 , )] in the Gabriel quiverof T corresponds to the vertex [ I i ] in the Gabriel quiver of { τ ( I i ) , I i | i ∈ Q } and [( I i, (1 , , e i, (1 , )] to I i .(4) The Gabriel quiver of T is isomorphic to Q , where T is the subcategory of the stable category C Q of C Q generated by all objects X ∈ T .Proof. (1) Since C ∈ C Q is finite dimensional, there are only finite vertices i ∈ Q such that C i = 0.As Q is strongly almost finite, there are only finite indecomposable objects T in T such that T i = 0for each i ∈ Q with C i = 0. Thus, our result follows.(2) By Lemma 7.1, [23], since Q is strongly almost finite, we have T is a rigid subcategory. Nowwe show it is actually maximal rigid.For any i ∈ Q , it is easy to verify that0 → ( I i, [0 , , e i, [0 , ) (1 , t → ( I i, [0 , , e i, [0 , ) (0 , → ( I i, [1 , , e i, [1 , ) → X, f ) ∈ C Q , since ( I i, [0 , , e i, [0 , ) is projective-injective, applying Hom C Q ( − , ( X, f ))to the exact sequence, we get
Hom C Q (( I i, [0 , , e i, [0 , ) , ( X, f )) → Hom C Q (( I i, [0 , , e i, [0 , ) , ( X, f )) → Ext C Q (( I i, [1 , , e i, [1 , ) , ( X, f )) → . NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 27
Thus, using Lemma 4.14,
Ext C Q (( I i, [1 , , e i, [1 , ) , ( X, f )) = 0 if and only if
Hom KQ ( τ ( I i ) , X ) → { g ∈ Hom KQ ( I i , X ) | f g = 0 } : g τ − ( f g )is surjective. Therefore, ( X, f ) ∈ T ⊥ if and only if Hom KQ ( τ ( I i ) , X ) → { g ∈ Hom KQ ( I i , X ) | f g = 0 } : g τ − ( f g )is surjective for any i ∈ Q . By the definition of C Q , we may assume that X = I ⊕ τ ( I ′ ).Moreover, since Hom KQ ( I, τ ( I ′ )) = 0, in fact f : τ ( I ′ ) → τ ( I ). Since Hom KQ ( I i , τ ( I ′ )) = 0, { g ∈ Hom KQ ( I i , X ) | f g = 0 } = Hom KQ ( I i , X ) = Hom KQ ( I i , I ). Thus, we have Hom KQ ( τ ( I ) , I ⊕ τ ( I ′ )) → Hom KQ ( I, I ) : g τ − ( f g )is surjective. Therefore, we have I = 0; or I ′ = I and then f = 1 : τ ( I ) → τ ( I ). In both cases, wehave ( X, f ) ∈ T . We have that T is a maximal rigid subcategory.(3) Since we have Lemma 4.14, the proof is similar as Lemma 7.6 of [23].(4) It follows by (3). (cid:3) Remark 4.16. (i) The (2) and (3) in Lemma 4.14 are respectively the generalizations of Corollary7.4 and Lemma 7.6 in [23] .(ii) Let P C Q be the subcategory of Hom C Q generated by all projective objects. Then C Q = C Q / P C .Since P C Q is a subcategory of T , we have T = T / P C Q . Subcategory of mod Λ Q with an (infinite) group action. Assume C is a strongly almost finite subcategory with a group Γ action. Let Γ ′ = ( h ) be thesubgroup of Γ generated by h ∈ Γ. Define a category C h as follows: the objects are X ∈ C and Hom C h ( Y, X ) := L h ′ ∈ Γ ′ Hom C ( Y, h ′ · X ) for X, Y ∈ C h . For all f : X i → h ′ · X j and g : X j → h ′′ · X k ,their composition g ◦ f is defined to be h ′ ( g ) f : X i → h ′ h ′′ · X k . Clearly, C h is well-defined as a K -linear category. For any subcategory of T of C , denote by C h ( T ) the minimal subcategory of C h which contains all objects T of T . Note that C h is not a Krull-Schmidt category in general.Let Q be a strongly almost finite quiver and Γ be an (infinite) subgroup of Aut ( Q ). In this section,we study a strongly almost finite 2-Calabi-Yau Frobenius subcategory C of mod Λ Q which is stableunder the action of group Γ. Since Γ is a subgroup of Aut ( Q ), it is clear the action of Γ on C is exact , that is, ϕ ( h ) are exact functors for all h ∈ Γ. We set some assumptions to the action of Γ on C : • for each indecomposable X ∈ C and h ∈ Γ, there exists a h ′ ∈ Γ such that h ′ · X ∼ = h · X and C h ′ is Hom -finite. • every projective object in C has a left rigid quasi-approximation.In this section, we study some useful properties of such subcategory C of mod Λ Q given above.However, for its applications in Section 6, indeed, we will only consider C = C Q as its special kind. Infact, for M the additive subcategory of modKQ generated by I j and τ ( I j ) for all injective modules I j ( j ∈ Q ), we have C Q = π − ( M ) which is stable under the action of Γ since M is stable underthe action of Γ. By Corollary 4.9, C Q is a strongly almost finite 2-Calabi-Yau Frobenius category.Using the action of Γ on Q , we give precisely the action of Γ on C Q satisfying that for each X = ( X i , f α ) i ∈ Q ,α ∈ Q ∈ C Q , h ∈ Γ, define(4) h · X = ( X h · i , f h · α ) i ∈ Q ,α ∈ Q where Q is the double quiver of Q . Maximal Γ -stable rigid subcategories and mutations. .In this subsection, we study the mutation of maximal Γ-stable rigid subcategories of C . We followthe strategy of Laurent Demonet [15] through replacing finite group by infinite group. So, in thispart, the definitions and results are parallel with that in [15]. Definition 5.1. (Definition 2.32, [15] ) A subcategory T of C is said to be Γ -stable if h · X ∈ T forevery X ∈ T and h ∈ Γ . Denote by
Add ( C ) Γ the family of all Γ-stable subcategories of Add ( C ). Definition 5.2. (Definition 3.22, [15] ) Let
T ∈
Add ( C ) Γ and let X ∈ T be indecomposable. A Γ -loop of T at X is an irreducible morphism X → h · X for some h ∈ Γ . A Γ - -cycle at X is apair of irreducible morphisms X → Y and Y → h · X for some h ∈ Γ . It can be seen that the proofs of these results below can be verified similarly as in [15]. Hence,these proofs are omitted here.
Lemma 5.3. (Lemma 3.6, [15] ) Let
T ∈
Add ( C ) Γ be a rigid subcategory of C and X ∈ C such that add ( { h · X | h ∈ Γ } ) is rigid. If → X f → T g → Y → is an admissible short exact sequence and f is a left T -approximation, then the category add ( T ∪ { h · Y | h ∈ Γ } ) is rigid. Although the description of the above lemma is different with Lemma 3.6 in [15], but its proof isexactly parallel with the proof of Lemma 3.6 in [15].
Lemma 5.4. (Proposition 3.7, [15] ) Let
T ∈
Add ( C ) Γ be a rigid subcategory of C and X ∈ C beindecomposable such that X / ∈ T . Suppose that T contains all projective objects of C , and add ( T ∪{ h · X | h ∈ Γ } ) is rigid. If X has minimal left and right T -approximations, then up to isomorphisms,there exist two admissible short exact sequences: → X f → T g → Y → and → Y ′ f ′ → T ′ g ′ → X → such that(1) f and f ′ are minimal left T -approximations;(2) g and g ′ are minimal right T -approximations;(3) add ( T ∪ { h · Y | h ∈ Γ } ) and add ( T ∪ { h · Y ′ | h ∈ Γ } ) are rigid;(4) Y / ∈ T and Y ′ / ∈ T ;(5) Y and Y ′ are indecomposable;(6) add ( { h · X | h ∈ Γ } ) ∩ add ( { h · Y | h ∈ Γ } ) = 0 and add ( { h · X | h ∈ Γ } ) ∩ add ( { h · Y ′ | h ∈ Γ } ) = 0 . In the case Γ is an infinite group given in Definition 4.10, the proof of this lemma is exactlyparallel with that of Proposition 3.7 in [15].In order to be used in the sequel, write µ rX ( T , X ) = add ( T ∪ { h · Y | h ∈ Γ } ) and µ lX ( T , X ) = add ( T ∪ { h · Y ′ | h ∈ Γ } ),where µ rX and µ lX are called respectively the right and left mutation of T at X , ( X, Y ) and (
X, Y ′ )are called respectively a right exchange pair and left exchange pair .Recall that in [15], two indecomposable objects X, Y ∈ C are called neighbours if they satisfyone of the following equivalent conditions:(5) dimExt C ( h · X, Y ) = ( , if h · X ∼ = X, , otherwise. NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 29 (6) dimExt C ( X, h · Y ) = ( , if h · Y ∼ = Y , , otherwise. Proposition 5.5. (Proposition 3.14, [15] ) Let
X, Y ∈ C be neighbours such that add ( { h · X | h ∈ Γ } ) and add ( { h · Y | h ∈ Γ } ) are rigid. Let → X f → W g → Y → be a non-splitting admissible shortexact sequence (which is unique up to isomorphism because X and Y are neighbours). Then,(1) add ( { h · X, h · M | h ∈ Γ } ) and add ( { h · Y, h · M | h ∈ Γ } ) are rigid;(2) X, Y / ∈ add { h · M | h ∈ Γ } ;(3) if there is a Γ -stable subcategory T ∈
Add ( C ) Γ such that add ( T , { h · X | h ∈ Γ } ) and add ( T , { h · Y | h ∈ Γ } ) are maximal Γ -stable rigid, then f is a minimal left T -approximation and g is a minimalright T -approximation. Corollary 5.6. (Corollary 3.15, [15] ) Let
T ∈
Add ( C ) Γ and X ∈ C be an indecomposable objectsuch that X / ∈ T and add ( T , { h · X | h ∈ Γ } ) is maximal Γ -stable rigid. Assume that T contains allprojective objects and X has left and right T -approximation. Then, the following are equivalent:(1) There exists an indecomposable object Y ∈ C such that µ lX ( add ( T , { h · X | h ∈ Γ } )) = add ( T , { h · Y | h ∈ Γ } ) and X, Y are neighbours.(2) There exists an indecomposable object Y ′ ∈ C such that µ rX ( add ( T , { h · X | h ∈ Γ } )) = add ( T , { h · Y ′ | h ∈ Γ } ) and X, Y ′ are neighbours.In this case, we have Y ∼ = Y ′ and denote that µ X ( add ( T , { h · X | h ∈ Γ } )) = µ lX ( add ( T , { h · X | h ∈ Γ } )) = µ rX ( add ( T , { h · X | h ∈ Γ } )) , then µ Y ( µ X ( add ( T , { h · X | h ∈ Γ } ))) = add ( T , { h · X | h ∈ Γ } ) . Note that the final formula in this corollary can be regarded as the analogue of the involution ofa mutation of matrix.
Lemma 5.7. (Lemma 3.25 [15] ) Let T ′ ∈ Add ( C ) Γ and X be indecomposable in C such that add ( T ′ ∪{ h · X | h ∈ Γ } ) is rigid. Assume that T ′ contains all projective objects and X / ∈ T ′ has left andright minimal T ′ -approximations. Let ( X, Y ) be a left (respectively, right) exchange pair associatedto T ′ such that add ( T ′ ∪ { h · X | h ∈ Γ } ) = T . The following are equivalent:(1) T has no Γ -loops at X .(2) For any indecomposable X ′ ∈ { h · X | h ∈ Γ } , every non-invertible morphism from X to X ′ factorizes through T ′ .(3) X and Y are neighbours. In the case Γ is infinite, the proofs of Proposition 5.5, Corollary 5.6 and Lemma 5.7 are parallelrespectively with that of Proposition 3.14, Corollary 3.15 and Lemma 3.25 in [15].In summary, we have:
Lemma 5.8.
Let T ′ ∈ Add ( C ) Γ and X be indecomposable in C such that T := add ( T ′ ∪{ h · X | h ∈ Γ } ) is maximal Γ -stable rigid. Assume that T ′ contains all projective objects and X / ∈ T ′ has left andright minimal T ′ -approximations. If T has no Γ -loops at X , then(1) there exists an indecomposable object Y ∈ C such that ( X, Y ) are neighbours,(2) µ X ( T ) = add ( T ′ , { h · Y | h ∈ Γ } ) is Γ -stable rigid. Global dimension of maximal Γ -stable rigid subcategories T and categories C h ( T ) . In this subsection, we study the global dimension of maximal Γ-stable rigid subcategories and thecategories C h ( T ). Fix h ∈ Γ such that C h is Hom -finite, denote Γ ′ = ( h ) the subgroup of Γ generatedby h .Let T be a Γ-stable rigid subcategory of C which contains all projective objects. From nowon, we always assume that T = add { h · n L i =1 X i | h ∈ Γ } for indecomposable objects X i such that X i / ∈ { h · X j | h ∈ Γ } for i = j until the end the this subsection. Proposition 5.9. (Proposition 3.26, [15] ) Keep the forgoing notations. Suppose that T = add { h · n L i =1 X i | h ∈ Γ } is a maximal Γ -stable rigid subcategory of C h ( T ) without Γ -loops. Then, gl.dim ( mod T ) ( ≤ , if any object of T is projective , = 3 , otherwise.Proof. The proof in the case of a finite group is suitable to Γ as an infinite group. (cid:3)
For X ∈ C h ( T ) which is also indecomposable in T , denote A := End C h ( T ) ( X ) the endomorphismalgebra of X in C h ( T ) and(7) J = J ( A ) := { ( f h ′ ) h ′ ∈ Γ ′ ∈ A | f h ′ is not an isomorphism for any h ′ ∈ Γ ′ } . Lemma 5.10.
Keep the notations as above. Assume that C h is Hom -finite. Then J is the Jacobsonradical of A .Proof. Since A = End C ( T ) ( X ) is local, J is an ideal of the algebra A . By assumption, C h is Hom -finite, then
End C h ( T ) ( X ) = L h ′ ∈ Γ ′ Hom C ( X, h ′ · X ) is finite dimensional for any X ∈ C . Followingthis, we know that either (a) Γ ′ = ( h ) is a finite group, or (b) Γ ′ is an infinite group but the subgroupΓ ′ X := { h ′ ∈ Γ ′ | h ′ · X ∼ = X } is a trivial group. Otherwise, if Γ ′ is an infinite group but Γ ′ X is anon-trivial subgroup of Γ ′ , then Γ ′ X is also infinite and thus, End C h ( X ) is infinite dimensional, whichis a contradiction.(a) In case Γ ′ is a finite group. Assume that h has finite order m . Now we show that J is anilpotent ideal of A . Any f = ( f , · · · , f m − ) ∈ J determinate a morphism F = ( F ij ) i,j ∈ [1 ,m ] ∈ End C ( m − L k =0 h k · X ) with F ij = h i − · f j − i , where the plus j − i is in Z m and f i : X → h i · X . Accordingto the composition in C h ( T ), it is easy to see that f k is the first row of F k for any k ∈ N . Thus,to prove that f is nilpotent, it suffices to prove that F is nilpotent. It suffices to show that alleigenvalues of F as a linear map are equal to 0 since K is algebraical closed. For any λ ( = 0) ∈ K ,since f i are non-isomorphic and End C ( X ) is local, we have λ − F ii are isomorphic and − F ij ( i = j )are non-isomorphic. Since End C ( X ) is local, an isomorphic endomorphism plus a non-isomorphicendomorphism is isomorphic. Therefore, we can do invertible row and column operations on λ − F to obtain an isomorphic endomorphism. Thus, we have λ − F is isomorphic for any λ = 0, whichimplies F is nilpotent and f is nilpotent. Therefore, J is a nilpotent ideal of A . In addition, A isfinite dimensional, J is included in the Jacobson radical of A .To show J is the Jacobson radical of A , it suffices to prove that A/J is semi-simple.Let Γ ′ X = { h ′ ∈ Γ ′ | h ′ · X ∼ = X } . Since Γ ′ is a cyclic group, we may assume that Γ ′ X = ( h k ) forsome k ∈ N . Denote the order of h k be s . Fixed an isomorphism ϕ : X → h k · X such that ϕ s − id X is nilpotent. (We can always choose such ϕ since for any isomorphism ϕ : X → h k · X , assume that ϕ s − λ id X is nilpotent, then we can set ϕ = s √ λ ϕ .) Let ϕ i = ϕ i : X → h ki · X for i = 1 , · · · , s . NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 31
Let π : End C ( X ) → End C ( X ) /M ∼ = → K be the canonical surjective algebra homomorphism, where M is the unique maximal ideal of End C ( X ).We define an algebra homomorphism θ : A/J → K [Γ ′ X ] like this: for any f : X → h i · X (0 ≤ i < sk ), define θ ( f ) = 0 if h i / ∈ Γ ′ X , θ ( f ) = π ( ϕ − ik f ϕ s ) h i if h i ∈ Γ ′ X . Now we prove that θ defines an algebra isomorphism from A/J to K [Γ ′ X ].It is clear that θ is a well defined map, since if f is non-isomorphic, θ ( f ) = 0. For f : X → h ki · X (0 ≤ i < s ) and f ′ : X → h kj · X (0 ≤ j < s ), denote θ ( f ) = λ i h ki and θ ( f ′ ) = λ j h kj . Thus, we have ϕ − i f ϕ s = λ i id X + g i and ϕ − j f ′ ϕ s = λ j id X + g j for nilpotent homomorphisms g i and g j . Considerthe following commutative diagram: X / / ϕ s (cid:15) (cid:15) X / / ϕ i (cid:15) (cid:15) X ϕ t (cid:15) (cid:15) X f / / h ki · X h ki · f ′ / / h ki + kj · X, where t ∈ [1 , s ] such that t ≡ i + j ( mod s ). Denote ϕ − t ( h ki · f ′ ) ϕ i = ϕ − t ( h ki · f ′ ) ϕ i = λid X + g for anilpotent homomorphism g . Therefore, θ ( f ′ ◦ f ) = π ( ϕ − t ( h ki · f ′ ) f ϕ s ) = λλ i h ki + kj . Thus, to show θ ( f ◦ f ′ ) = θ ( f ) θ ( f ), it suffices to show that λ = λ j . Consider the following commutative diagram: X ϕ i / / (cid:15) (cid:15) h ki X h ki · ϕ s / / (cid:15) (cid:15) h ki · X h ki · f ′ (cid:15) (cid:15) X ϕ i / / h ki X h ki · ϕ j / / h ki + kj · X, since ϕ − j f ′ ϕ s = λ j id X + g j , we have( h ki · ϕ − j )( h ki · f ′ )( h ki · ϕ s ) = h ki · ( λ j id X ) + h ki · g j = λ j id h ki · X + h ki · g j . Therefore, ϕ − i ( h ki · ϕ − j )( h ki · f ′ )( h ki · ϕ s ) ϕ i = ϕ − i ( λ j id h ki · X + h ki · g j ) ϕ i = λ j id X + ϕ − i ( h ki · g j ) ϕ i . According to our construction, ϕ s − id X is nilpotent, so ϕ − i ( h ki · ϕ − j )( h ki · f ′ ) ϕ i − λ j id X = ϕ − ( i + j )1 ( h ki · f ′ ) ϕ i − λ j id X is nilpotent. Moreover, i + j ≡ t ( mods ) and ϕ s − id X = ϕ s − id X is nilpotent. Thus, ϕ − t ( h ki · f ′ ) ϕ i − λ j id X is nilpotent. Since ϕ − t ( h ki · f ′ ) ϕ i − λid X is nilpotent, we obtain λ = λ j . Therefore, θ is an algebra homomorphism. Since θ ( ϕ i ) = π ( ϕ − i ϕ i ϕ s ) h ki = h ki for all i = 0 , · · · , s −
1. Thus, θ issurjective. It is clear that dim ( A/J ) ≤ s and dim ( K [Γ ′ X ]) = s . Therefore, we obtain θ is an algebraisomorphism.Since char ( K ) = 0, A/J ∼ = K [Γ ′ X ] is semi-simple. In addition, J is nilpotent and A is finitedimensional, we get J is the Jacobson radical of A .(b) In case Γ ′ is an infinite group but Γ ′ X = { h ′ ∈ Γ ′ | h ′ · X ∼ = X } is a trivial group. Since End C ( X ) is local, J is the unique maximal ideal of A . Thus, J is the Jacobson radical of A . (cid:3) By the definition, in order to calculate the global dimension of C h ( T ), we have to calculate theprojective dimension of any simple representations S X , where S X is the simple top of indecomposableprojective representation Hom C h ( X, − ) of C h ( T ) for an indecomposable object X . We divide thediscussion into two parts: (I) X is not projective as an object of C , (II) X is projective as an objectof C .(I) In the case that X is not projective as an object of C . Let
X, Y be neighbours and 0 → X → T → Y → h ′ ∈ Γ ′ , applying Hom C ( − , h ′ · X ) to the admissible short exact sequence, we getthe exact sequence: Hom C ( X, h ′ · X ) δ h ′ −→ Ext C ( Y, h ′ · X ) → Ext C ( Y, h ′ · T ) = 0 , with δ h ′ as the connecting morphism, where the last equality is from the rigidity of µ X ( T ) and Y, h ′ · T ∈ µ X ( T ). Thus, M h ′ ∈ Γ ′ Hom C ( X, h ′ · X ) ( δ h ′ ) h ′∈ Γ ′ −→ M h ′ ∈ Γ ′ Ext C ( Y, h ′ · X ) → . For any f : X → h ′ · X and ξ ∈ Ext C ( Y, h ′′ · X ), define f ξ via the following push out diagram:(8) ξ : 0 / / h ′′ · X / / h ′′ · f (cid:15) (cid:15) Z / / (cid:15) (cid:15) Y / / id Y (cid:15) (cid:15) f ξ : 0 / / h ′′ h ′ · X / / Z ′ / / Y / / , where Z ′ = ( h ′′ h ′ · X ) Q ( h ′′ · X ) Z . Lemma 5.11.
Keep the notations as above. Assume that C h is Hom -finite and X is indecomposablenon-projective in C . Then,(1) L h ′ ∈ Γ ′ Ext C ( Y, h ′ · X ) admits a left A -module structure defined via the expansion of the actiongiven in the above (8).(2) A = L h ′ ∈ Γ ′ Hom C ( X, h ′ · X ) δ −→ L h ′ ∈ Γ ′ Ext C ( Y, h ′ · X ) → , where δ = ( δ h ′ ) h ′ ∈ Γ ′ . Then δ isan A -module homomorphism and ker ( δ ) is the Jacobson radical of A , where A is viewed as a leftregular A -module.Proof. (1) It is clear that id X ξ = ξ for all ξ ∈ Ext C ( Y, h ′ · X ). For any f : X → h ′ · X , f ′ : X → h ′′ · X and ξ ∈ Ext C ( Y, h ′′′ · X ), we have the following commutative diagram: ξ : 0 / / h ′′′ · X / / h ′′′ · f ′ (cid:15) (cid:15) Z / / (cid:15) (cid:15) Y / / id Y (cid:15) (cid:15) f ′ ξ : 0 / / h ′′′ h ′′ · X / / h ′′′ h ′′ · f (cid:15) (cid:15) Z ′ / / (cid:15) (cid:15) Y / / id Y (cid:15) (cid:15) f ( f ′ ξ ) : 0 / / h ′′′ h ′′ h ′ · X / / Z ′′ / / Y / / , where Z ′ = ( h ′′′ h ′′ · X ) ` h ′′′ · X Z and Z ′′ = ( h ′′′ h ′′ · Z ) ` ( h ′′′ h ′′ · X ) Z ′ . Thus, Z ′′ = ( h ′′′ h ′′ · Z ) ` ( h ′′′ h ′′ · X ) Z ′ = ( h ′′′ h ′′ · Z ) ` ( h ′′′ h ′′ · X ) ( h ′′′ h ′′ · X ) ` h ′′′ · X Z = ( h ′′′ h ′′ · Z ) ` h ′′′ · X Z . Therefore, wehave f ( f ′ ξ ) = ( f f ′ ) ξ . Our result follows.(2) According to Lemma 5.10, J = { ( f h ′ ) h ′ ∈ Γ ′ ∈ A | f h ′ is not an isomorphism } .For any h ′ ∈ Γ ′ and f : X → h ′ · X , we have δ h ′ ( f ) ∈ Ext C ( Y, h ′ · X ) by the following diagram:0 / / X / / f (cid:15) (cid:15) T / / (cid:15) (cid:15) Y / / id Y (cid:15) (cid:15) δ h ′ ( f ) : 0 / / h ′ · X / / T ′ / / Y / / , where T ′ = ( h ′ · X ) ` X T . NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 33
For any h ′′ ∈ Γ ′ and f ′ : X → h ′′ · X , we have the commutative diagram:0 / / X / / f (cid:15) (cid:15) T / / (cid:15) (cid:15) Y / / id Y (cid:15) (cid:15) δ h ′ ( f ) : 0 / / h ′ · X / / h ′ · f ′ (cid:15) (cid:15) T ′ / / (cid:15) (cid:15) Y / / id Y (cid:15) (cid:15) f ′ ( δ h ′ ( f )) : 0 / / h ′ h ′′ · X / / T ′′ / / Y / / , where T ′ = ( h ′ · X ) ` X T and T ′′ = ( h ′ h ′′ · X ) ` ( h ′ · X ) T ′ . Thus, T ′′ = ( h ′ h ′′ · X ) ` X T . Therefore,we get f ′ ( δ h ′ ( f )) = δ h ′ ( f ′ ◦ f ). Hence, δ = ( δ h ′ ) h ′ ∈ Γ ′ is an A -module homomorphism.Now we prove that ker ( δ ) = J , where J is the Jacobson radical of A .Give any non-isomorphic morphism f h ′ : X → h ′ · X . If h ′ · X = X , then Ext C ( Y, h ′ · X ) = 0.Hence, δ h ′ ( f h ′ ) = 0. If h ′ · X ∼ = X , then h ′ ∈ Γ ′ X has finite order m . Thus, f mh ′ ∈ Hom C ( X, X ).Since
Hom C ( X, X ) is local, and f h ′ is non-isomorphic, so f mh ′ is nilpotent. Therefore, f h ′ is nilpotent.Moreover, since dimExt C ( Y, h ′ · X ) = 1, δ ( f h ′ ) = δ h ′ ( f h ′ ) = 0. Thus, J ⊆ ker ( δ ).For any isomorphic morphism f h ′ : X → h ′ · X , we have the commutative diagram:0 / / X / / f h ′ (cid:15) (cid:15) T / / (cid:15) (cid:15) Y / / id Y (cid:15) (cid:15) δ h ′ ( f h ′ ) : 0 / / h ′ · X / / T ′ / / Y / / . If δ h ′ ( f h ′ ) = 0, then δ h ′ ( f h ′ ) = 0 is split. Further, since f h ′ is isomorphic, 0 → X → T → Y → δ ( f h ′ ) = δ h ′ ( f h ′ ) = 0. Hence, ker ( δ ) ⊆ J .Then, we obtain ker ( δ ) = J . (cid:3) (II) In the case that X is projective as an object in C .By the assumption of the beginning of this section, let X ։ Y be the left rigid quasi-approximationof X . Denote by M the cokernel of L h ′ ∈ Γ ′ Hom C ( Y, h ′ · X ) → L h ′ ∈ Γ ′ Hom C ( X, h ′ · X ). Thus, we obtainthe exact sequence L h ′ ∈ Γ ′ Hom C ( Y, h ′ · X ) → L h ′ ∈ Γ ′ Hom C ( X, h ′ · X ) ε → M → . Lemma 5.12.
Keep the notations as above. Assume that C h ( T ) is Hom -finite and X is an inde-composable projective object in C . Then M given above is an A -module and M ∼ = A/J , where J isthe Jacobson radical of A .Proof. According to Lemma 5.10, J = { ( f h ′ ) h ′ ∈ Γ ′ ∈ A | f h ′ is not an isomorphism } . It suffices toprove that the kernel of ε equals to J . It is clear that ker ( ε ) = J by the definition of left rigidquasi-approximation. (cid:3) Now we can already calculate the global dimension of C h ( T ). Proposition 5.13.
Keep the forgoing notations. Suppose that T = add { h · n L i =1 X i | h ∈ Γ } , withindecomposables X i ( i = 1 , · · · , n ) , is a maximal Γ -stable rigid subcategory of C without Γ -loops.Assume C h ( T ) is Hom -finite for h ∈ Γ , then gl.dim ( C h ( T )) ( ≤ , if all objects in T are projective , = 3 , otherwise. Proof.
The X i ( i = 1 , · · · , n ) are also all indecomposables of C h ( T ). So, gl.dim ( C h ( T )) = max { pd.dim ( S X i ) | i =1 , · · · , n } , where S X i : C h ( T ) → K − vec with S X i ( X ) = Hom C h ( T ) ( X i , X ) /J ( X i , X ). Then, wehave to compute pd.dim ( S X i ) for any fixed X i .Suppose firstly that X i is not projective. Denote T ′ ( i ) = add ( { h ′ · X j | h ′ ∈ Γ , j = i } ) andΓ ′ = ( h ) ⊆ Γ. Since T has no Γ-loops, by Lemma 5.8, there exists Y ∈ C such that ( X i , Y ) areneighbours and µ X ( T ) = add ( T ′ ( i ) , { h ′ · X | h ′ ∈ Γ } ) is Γ-stable rigid. Moreover, using Corollary 5.6,we get two admissible short exact sequences,0 → X i → T ′ → Y → → Y → T ′′ → X i → T ′ , T ′′ ∈ T ′ ( i ) .For each h ′ ∈ Γ ′ and X ∈ T , applying Hom C ( − , h ′ · X ) to the above sequences, we get the twolong exact sequences,0 → Hom C ( Y, h ′ · X ) → Hom C ( T ′ , h ′ · X ) → Hom C ( X i , h ′ · X ) → Ext C ( Y, h ′ · X ) → Ext C ( X i , h ′ · X ) = 0 , and 0 → Hom C ( X i , h ′ · X ) → Hom C ( T ′′ , h ′ · X ) → Hom C ( Y, h ′ · X ) → Ext C ( X i , h ′ · X ) = 0 . Combing the two long exact sequences, and by the arbitrary of h ′ , we can obtain that(9)0 → Hom C h ( T ) ( X i , − ) → Hom C h ( T ) ( T ′′ , − ) → Hom C h ( T ) ( T ′ , − ) → Hom C h ( T ) ( X i , − ) → F → , where the functor F : C h ( T ) → K - vec with X L h ′ ∈ Γ ′ Ext C ( Y, h ′ · X ).For indecomposable X = X i in C h ( T ), we have F ( X ) = L h ′ ∈ Γ ′ Ext C ( Y, h ′ · X ) = 0 since µ X i ( T ) isΓ-stable rigid and then Y, h ′ · X ∈ µ X i ( T ). And, S X i ( X ) = Hom C h ( T ) ( X i , X ) /J ( X i , X ) = 0 sincein this case Hom C h ( T ) ( X i , X ) = J ( X i , X ) by [2]. Then F ( X ) = S X i ( X ).In the case X ∼ = X i , by Lemma 5.11, F ( X ) ∼ = F ( X i ) = L h ′ ∈ Γ ′ Ext C ( Y, h ′ · X i ) ∼ = End C h ( T ) ( X i ) /J .And, S X i ( X ) ∼ = S X i ( X i ) = Hom C h ( T ) ( X i , X i ) /J ( X i , X i ) = End C h ( T ) ( X i ) /J . Hence, also, F ( X ) = S X i ( X ).In summary, we have F ∼ = S X i . As a consequence from (9), proj.dim ( S X i ) ≤
3. Since X i / ∈ T ′ ( i ) and T ′′ ∈ T ′ ( i ) , we get Hom mod C h ( T ) ( Hom C h ( T ) ( T ′′ , − ) , S X i ) = 0 . Therefore,
Ext mod C h ( T ) ( S X i , S X i ) ∼ = Hom mod C h ( T ) ( Hom C h ( T ) ( X i , − ) , S X i ) = 0 . Thus, proj.dim ( S X i ) = 3 . Now, suppose that X i is projective, then X i is also injective in C Q . By the assumption at thebeginning of this section, let X i ։ Y be a left rigid quasi-approximation. According to the definition,the injective envelope of Y does not intersect with add ( { h ′ · X | h ′ ∈ Γ } ) and, by Lemma 2.38 of[15], there is an admissible short exact sequence0 → Y f → T ′ → Z → , where f is a left T ′ ( i ) -approximation. Using Lemma 5.3, add ( T , { h ′ · Z | h ′ ∈ Γ } ) is Γ-stable rigid.Moreover, since T is maximal Γ-stable rigid, we have Z ∈ T . Applying Hom C ( − , h ′ · X ) to theadmissible sequence, we obtain0 → Hom C ( Z, h · X ) → Hom C ( T ′ , h · X ) → Hom C ( Y, h · X ) → Ext C ( Z, h · X ) = 0 . Thus, 0 → Hom C h ( T ) ( Z, − ) → Hom C h ( T ) ( T ′ , − ) → Hom C h ( T ) ( Y, − ) → . Let G : C h ( T ) → K - vec be the functor satisfying that G ( X ) = coker ( M h ′ ∈ Γ ′ Hom C ( Y, h ′ · X ) → M h ′ ∈ Γ ′ Hom C ( X i , h ′ · X )) . NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 35
Thus, 0 → Hom C h ( T ) ( Y, − ) → Hom C h ( T ) ( X i , − ) → G → . Therefore, we get(10) 0 → Hom C h ( T ) ( Z, − ) → Hom C h ( T ) ( T ′ , − ) → Hom C h ( T ) ( X i , − ) → G → . As X i ։ Y is a left rigid quasi-approximation, we get G ( X ) = 0 = S X i ( X ) when X = X i in C h ( T ). By Lemma 5.12, we have G ( X i ) ∼ = A/J = S X i ( X i ). Thus, G ∼ = S X i . As a consequence of(10), pd.dim ( S X i ) = pd.dim ( G ) ≤ (cid:3) Proof of the unfolding theorem of acyclic sign-skew-symmetric matrices
In this section, we use the previous preparation to prove that any acyclic sign-skew-symmetriccluster algebra has an unfolding corresponding to a skew-symmetric cluster algebra. As application,we have given a positive answer for the problem by Berenstein, Fomin, Zelevinsky on the totality ofacyclic sign-skew-symmetric matrices in Theorem 2.17.Let B ∈ M at n × n ( Z ) be an acyclic sign-skew-symmetric matrix and Q be the strongly almostfinite quiver constructed in section 2.2, Γ be the group acts on Q . Let C Q be the strongly almostfinite Frobenius 2-Calibi-Yau category associated to Q which has been defined in section 4 beforeCorollary 4.9. The action of Γ on C Q is exactly given from the action of Γ acts on Q as shown in (4).In this section, we firstly focus on the action of Γ on the C Q . We prove that the action satisfiesthe assumptions at the beginning of section 4. By Corollary 4.12, every projective object in C Q hasa left rigid quasi-approximation. It remains to prove that for an indecomposable X ∈ C Q and h ∈ Γ,there exists a h ′ ∈ Γ such that h ′ X ∼ = hX and C Qh ′ is Hom -finite. To see this, we need the followingpreparation.Assume that h ∈ Γ has no fixed points, for any indecomposable object X ∈ C Q , set Q X to be thesub-quiver of Q generated by all vertices a such that X ( a ) = 0, and Γ ′ X := { h ′ ∈ Γ ′ | h ′ X ∼ = X } . It iseasy to see that there is a group homomorphism ω : Γ ′ X → Aut ( Q X ) , h ′ → h ′ | Q X , where h ′ | Q X meansthe restriction of h ′ to Q X . Since h has no fixed points, by Lemma 2.14, we have ker ( ω ) = { e } .Thus, Γ ′ X is a finite group. Lemma 6.1.
The category C Qh is Hom -finite if one of two conditions holds: (i) the order of h isfinite, (ii) h ∈ Γ acts on Q has no fixed points.Proof. If the order of h is finite, then Γ ′ = ( h ) is finite. Since C Q is Hom -finite, C Qh is a Hom -finitecategory.If h has no fixed points. We may assume that X is indecomposable. To show Hom C Qh ( Y, X ) isfinite dimensional. It suffices to prove there are only finite h ′ ∈ Γ ′ such that Q h ′ ( X ) has commonvertices with Q Y .Otherwise, since Q Y is a finite quiver, there are infinite h ′ ∈ Γ ′ such that Q h ′ · X contains a vertexof Q Y , assume to be 1 ′ . Assume that 1 ′ is a common vertex of Q h ′ · X for h ′ ∈ S for S ⊆ Γ ′ . Thus, { h ′− · | h ′ ∈ S } ⊆ Q X . Moreover, as Q X is a finite quiver, there are infinite h ′ ∈ S such that h ′− · ′ have the same image. Without loss of generality, assume that h ′− · ′ = 1 for a vertex1 ∈ Q X and for all h ′ ∈ S . For a vertex 2 ∈ Q X with an arrow α : 1 → h ′ · ′ to h ′ · h ′ ∈ S . Since Q isstrongly almost finite, there are only finite arrows starting from 1 ′ . Thus, there is a infinite subset S ⊆ S such that h ′ · h ′ ∈ S . Since X in indecomposable, Q X isconnected. Inductively, there is a infinite subset S n ⊆ S n − ⊆ Γ ′ , such that h ′ · a have common imagefor all h ′ ∈ S n and vertices a ∈ Q X . Thus, pick h ′ = h ′′ ∈ S n , we have h ′ | Q X = h ′′ | Q X . Therefore, we have h ′ h ′′− | Q X = id Q X , which contradicts to Lemma 2.12 which says that any element in Γ ′ hasno fixed points.Thus, in both cases, C Qh is Hom -finite. (cid:3)
Proposition 6.2.
Let X ∈ C Q be indecomposable and h ∈ Γ . Then, there exists an h ′ ∈ Γ such that h ′ · X ∼ = h · X and C Qh ′ is Hom -finite.Proof. If h has no fixed points, then according to Lemma 6.1, C Qh is Hom -finite. In this case, choose h ′ = h . Otherwise, if h has fixed points, by Lemma 2.13, there exists h ′ ∈ Γ such that the order of h ′ is finite and h · X ∼ = h ′ · X . By Lemma 6.1, C Qh ′ is Hom -finite. (cid:3)
Until now, we have proved that the action of Γ on C Q satisfying the assumptions at the beginningof section 4.A subcategory S of C Q is called finitely non-Hom-orthogonal in C Q if for each X ∈ C Q , thereare only finite indecomposable objects T (up to isomorphism) in S such that Hom C Q ( X, T ) = 0 or Hom C Q ( T, X ) = 0. It is easy to see a finitely non-Hom-orthogonal subcategory is functorially finite. Lemma 6.3.
Let T = add ( { hX i | i = 1 , · · · , n, h ∈ Γ } ) be a maximal Γ -stable rigid subcategory of C Q with each X i indecomposable, containing all projective-injective objects. If T is finitely non-Hom-orthogonal in C Q , then µ X i ( T ) is also finitely non-Hom-orthogonal in C Q for each non-projective-injective indecomposable X i ∈ T .In particular, µ X i ( T ) is functorially finite.Proof. By Lemma 5.8, there exists indecomposable Y i such that ( X i , Y i ) are neighbours. By Corollary5.6, there exist two admissible short exact sequences 0 → X i → T → Y i → → Y i → T ′ → X i → T, T ′ ∈ T . By the assumption of the conditions on T , to prove our result, it suffices toprove that there are only finite indecomposable h · Y i (up to isomorphism), h ∈ Γ such that
Hom C Q ( h · Y i , X ) = 0 or Hom C Q ( X, h · Y i ) = 0 for each X ∈ C Q . Since Hom C Q ( h · Y i , X ) ⊆ Hom C Q ( h · T, X )and
Hom C Q ( X, h · Y i ) ⊆ Hom C Q ( X, h · T ′ ), it suffices to prove that there are only finite h (up to theisomorphic of h · T and h · T ′ ) such that Hom C Q ( h · T, X ) = 0 or Hom C Q ( X, h · T ′ ) = 0.We only prove that there are only finite h (up to the isomorphic of hT ) such that Hom C Q ( hT, X ) =0. The case for Hom C Q ( X, h · T ′ ) = 0 can be proved dually.Assume that T = m L i =1 T n i i for indecomposable objects T i . Thus, Hom C Q ( h · T, X ) = 0 if and onlyif Hom C Q ( h · T i , X ) = 0 for some i . Since T i ∈ T , there are only finite h (up to the isomorphic of h · T i ) such that Hom C Q ( h · T i , X ) = 0, we may assume that Hom C Q ( h j · T i , X ) = 0 for j = 1 , · · · , s .Now we show there are only finite h (up to the isomorphic of h · T ) such that h · T i ∼ = h j · T i for each j = 1 , · · · , s . If we can do this, our result follows according to the above discussion. Foreach j = 1 , · · · , s , we may assume that h j = e without loss of generality. For each h ∈ Γ such that h · T i ∼ = T i , since T i is indecomposable, by Lemma 2.12, h has fixed points. By Observation 2.8 (1),we may assume that h fix a vertex labelled by 1. Then it follows by Lemma 2.9 immediately thatthere are only finite h (up to the isomorphic of h · T ) such that h · T i ∼ = T i . (cid:3) Proposition 6.4.
Let us keep the forgoing notations. Suppose that T is a maximal Γ -stable rigidsubcategory which has no Γ -loops. Assume Q ′ is the Gabriel quiver of T and Q ′′ is the Gabriel quiverof µ X i ( T ) for a non-projective object X i . Then Q ′′ = e µ [ i ] ( Q ′ ) . Proof.
Since Q is strongly almost finite, the proof is parallel with the proof of Theorem 7.8, [22].In fact, we only need to replace the matrix S in section 7.1 of [22] to S = ( s jk ) j,k ∈ Q , where s jk = ( − δ jk + | b jk |− b jk , if j ∈ [ i ] ,δ jk , otherwise. (cid:3) NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 37
This result means the preservability of Gabriel quivers under orbit mutations.
Proposition 6.5. ( [15] , Proposition 3.31; [7] , P463) Let Q be a strongly almost finite quiver and I be an admissible ideal of KQ . Denote by J the ideal of KQ generated by all of its arrows. Let i, j ∈ Q . Then, dime j ( I/ ( IJ + JI )) e i = dimExt modKQ/I ( S i , S j ) for S i , S j simple KQ/I -modulessupported on vertices i and j .Proof. Although the corresponding result in ([7], P463) deals with the algebra
KQ/I in the case offinite dimension from a finite quiver, we can see its proof can be applied to
KQ/I similarly when Q is strongly almost finite. Precisely, since Q is strongly almost finite, for any simple modules S i , S j and a projective resolution of P → P → P → P → S i → S i in modKQ/I , we can choosea finite acyclic subquiver Q ′ of Q such that S i , S j and P k are all in modKQ ′ for k = 0 , , , e j KQ ′ e i = e j KQe i holds. In this case, we denote I ′ = KQ ′ ∩ I . Then we can regard P k , k = 0 , , , S i , S j as simple in modKQ ′ /I ′ . Thus, following the definition of the Ext -group,we obtain Ext modKQ/I ( S i , S j ) = Ext modKQ ′ /I ′ ( S i , S j ).Since I ′ = I ∩ KQ ′ , we have e j I ′ e i = e j ( I ∩ KQ ′ ) e i = e j Ie i ∩ e j KQ ′ e i = e j Ie i ∩ e j KQe i = e j Ie i .Clearly, e j I ′ J ′ e i ⊆ e j IJe i . Write P s k s e j a s b s e i ∈ e j IJe i ⊆ e j KQe i with 0 = k s ∈ K and a s ∈ e j Ie s , b s ∈ e s Je i , a s b s = 0. Since e j KQe i = e j KQ ′ e i , any vertex s through a path in Q from i to j is avertex of Q ′ and the paths from i to s and from s to j are in Q ′ . Hence, we have a s , b s ∈ KQ ′ . Thus, a s ∈ KQ ′ ∩ I = I ′ , b s ∈ KQ ′ ∩ J = J ′ , then P s k s e i a s b s e j ∈ e j I ′ J ′ e i , this implies e j IJe i ⊆ e j I ′ J ′ e i .Hence, e j IJe i = e j I ′ J ′ e i . Similarly, e j JIe i = e j J ′ I ′ e i . Then e j ( IJ + JI ) e i = e j ( I ′ J ′ + J ′ I ′ ) e i .Therefore, e j ( I ′ / ( I ′ J ′ + J ′ I ′ )) e i = e j ( I/ ( IJ + JI )) e i . Applying the result in [7] for KQ ′ /I ′ , we have dimExt modKQ/I ( S i , S j ) = dimExt modKQ ′ /I ′ ( S i , S j ) = dime j ( I ′ / ( I ′ J ′ + J ′ I ′ )) e i = dime j ( I/ ( IJ + JI )) e i . (cid:3) For a strongly almost finite quiver Q , we can view KQ as a Krull-Schmidt K -category whoseobjects are vertices of Q and their all finite direct sums L i ∈ Q i n i (here n i ∈ N ) and whose morphisms Hom KQ ( M i ∈ Q i n i , M j ∈ Q j m j ) := M i,j ∈ Q KQ ( i, j ) n i m j for any objects L i ∈ Q i n i , L j ∈ Q j m j of KQ . Trivially, vertices of Q are just all indecomposable objectsin KQ . Any admissible ideal I of the algebra KQ can be view as a two-sided ideal in KQ as category.Thus, KQ/I can be also regarded as a quotient category.
Theorem 6.6. (Categorical Gabriel’s Theorem) Let A be a Hom -finite Krull-Schmidt category and Q be its Gabriel quiver. If Q is strongly almost finite, then there exists an admissible ideal I of KQ such that A is categorically equivalent to KQ/I .Proof.
The following proof is parallel with the Gabriel Theorem for basic algebras, see Chapter 2,Theorem 3.7 of [3]. According to the definition of Gabriel quiver, we can write the set of indecom-posable objects (up to isomorphism) of A as { i | i ∈ Q } , and any path p : i → j in Q correspondsto a morphism p : i → j . Define an additive functor F : KQ → A as follows: F ( i ) = i and F ( p ) = p for any i, j ∈ Q and path p : i → j . For paths p : i → j and p : j → k in Q , obviously p p = p p . Thus, we have F ( p p ) = p p = F ( p ) F ( p ). Its additivity is defined via linearexpansions. Therefore, F is an additive functor.Since Q is strongly almost finite, we know that Q ( i, j ) is finite for any i, j ∈ Q . Then thereexists m ∈ N such that rad m A ( i, j ) = 0. For any morphism f : i → j of A , we may assume that f ∈ rad n A ( i, j ) \ rad n +1 A ( i, j ) for some n ∈ N , and decompose f = f n · · · f for irreduciblemorphisms f k = (cid:16) f kts (cid:17) t ∈ Λ k +1 ,s ∈ Λ k ∈ Hom A ( L s ∈ Λ k i s , L t ∈ Λ k +1 i t ), 1 ≤ k ≤ n + 1, where Λ k are finiteindex sets. In particular, | Λ | = | Λ n +1 | = 1. So, write Λ = { s } and Λ n +1 = { s n +1 } . Then f = (cid:16) f nts (cid:17) t ∈ Λ n +1 ,s ∈ Λ n · · · (cid:16) f ts (cid:17) t ∈ Λ ,s ∈ Λ is exactly a 1 × f into two parts, i.e. f = X ( s n , ··· ,s ) ∈ Λ ′ f ns n +1 s n · · · f s s + X ( s n , ··· ,s ) ∈ Λ ′′ f ns n +1 s n · · · f s s , where Λ ′ = { ( s n , · · · , s ) | s k ∈ Λ k , f ks k +1 s k are irreducible for all k } , and Λ ′′ = { ( s n , · · · , s ) | s k ∈ Λ k } \ Λ ′ . Thus, f − P ( s n , ··· ,s ) ∈ Λ ′ f ns n +1 s n · · · f s s ∈ rad n +1 A ( i, j ).We claim that there exists β ∈ Hom KQ ( i, j ) such that f − F ( β ) ∈ rad n +1 A ( i, j ). In fact, for any k , by the definition of Q , the set B i sk i sk +1 = { F ( α s k ) + rad A ( i s k , i s k +1 ) | α s k ∈ Q ( i s k , i s k +1 ) } is abasis of rad A ( i s k , i s k +1 ) /rad A ( i s k , i s k +1 ), where Q ( i s k , i s k +1 ) denotes the set of arrows from i s k to i s k +1 . Then there exist a s k ∈ K such that(11) f ks k +1 s k − X a s k F ( α s k ) ∈ rad A ( i s k , i s k +1 ) . Therefore, for any ( s n , · · · , s ) ∈ Λ ′ , we have n Q k =1 f ks k +1 s k − F ( n Q k =1 P a s k α s k )= n Q k =1 ( f ks k +1 s k − P a s k F ( α s k ) + P a s k F ( α s k )) − n Q k =1 ( P a s k F ( α s k ))= n Q k =1 ( f ks k +1 s k − P a s k F ( α s k )) + E + n Q k =1 ( P a s k F ( α s k )) − n Q k =1 ( P a s k F ( α s k ))= n Q k =1 ( f ks k +1 s k − P a s k F ( α s k )) + E ∈ rad n +1 A ( i, j ) , by (11) and since in the expansion E each monomial includes at least once a factor in the form f ks k +1 s k − P a s k F ( α s k ). Thus, we have f − F ( P ( s n , ··· ,s ) ∈ Λ ′ Q k = n P a s k α s k ) = f − P ( s n , ··· ,s ) ∈ Λ ′ F ( Q k = n P a s k α s k ) ∈ rad n +1 A ( i, j ) . Hence, the claim follows.Using the fact that rad m A ( i, j ) = 0 and the above claim inductively, there exists g ∈ Hom KQ ( i, j )such that F ( g ) = f . Therefore, F ( i, j ) : Hom KQ ( i, j ) → Hom A ( i, j ) is surjective for all i, j ∈ Q .It is easy to see that F makes a bijection from Ob ( KQ ) to Ob ( A ) / ∼ =, where ∼ = means theisomorphisms of objects. Let I be the ideal of KQ generated by ker ( F ( i, j )) for all i, j ∈ Q . Then, Hom
KQ/I ( i, j ) ∼ = Hom A ( i, j ) for all i, j ∈ Q . Thus, as categories, we have KQ/I ∼ = A .It remains to prove that I is admissible. Denote rad KQ the radical of KQ as a category, whichis generated by all arrows in Q . For any s, t ∈ Q and P i ∈ Q a i id i + P α l ∈ Q b l α l + y ∈ ker ( F ( s, t )) with y ∈ rad KQ ( s, t ) and a i , b l ∈ K , it suffices to prove a i =0 and b l = 0 for all i, l .In fact, P a i id i + P α l ∈ Q b l F ( α l ) + F ( y ) = 0 as morphism in A . Thus, P a i id i = − P α l ∈ Q b l F ( α l ) − F ( y ) ∈ rad A ( s, t ). Since A is Hom -finite, we have − P α l ∈ Q b l F ( α l ) − F ( y ) is nilpotent. Thus, a i = 0for all i . Then, P α l ∈ Q b l F ( α l ) = − F ( y ) ∈ rad ( A )( s, t ). Hence, ( P α l ∈ Q b l F ( α l )) + rad ( A )( s, t ) =0 holds in rad A ( s, t ) /rad A ( s, t ). By the definition of the Gabriel quiver Q , we have { F ( α l ) + rad A ( s, t ) | α l ∈ Q } are linearly independent. Therefore, b l = 0 for all l . (cid:3) Note that an admissible ideal I of the algebra KQ is just an admissible ideal I (that is, which isincluded in rad KQ ) of KQ as a category. NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 39
Theorem 6.7.
Keep the forgoing notations. Suppose that T = add ( { h · n L i =1 X i | h ∈ Γ } ) ∈ Add ( C Q ) Γ ,with X i indecomposable, is a maximal Γ -stable rigid subcategory of C Q without Γ -loops, T containsall projective-injective objects. If T is finitely non-Hom-orthogonal, then:(1) gl.dim ( mod T ) = 3 ;(2) T is a cluster tilting subcategory of C Q ;(3) For all simple T -modules S and S ′ such that add ( L h ∈ Γ h · S ) = add ( L h ∈ Γ h · S ′ ) , we have Ext mod T ( S, S ′ ) = Ext mod T ( S, S ′ ) = 0 ;(4) T has no Γ - -cycles;(5) If X i is non-projective for some i ∈ { , · · · , n } , then µ X i ( T ) has no Γ -loops and moreover itis a cluster tilting subcategory (certainly is maximal rigid).Proof. (1) It follows by Proposition 5.9;(2) It follows from Theorem 3.2 and (1);(3) By the relation between S and S ′ , we know they are in an orbit under action of Γ. So, fromthe fact T has no Γ-loops, we get Ext mod T ( S, S ′ ) = 0.Concerning Ext mod T ( S, S ′ ), assume S = S X i for some X i ∈ T , set with T ′ ( i ) = add ( { h · n L j = i X j | h ∈ Γ } ). If X i ∈ T is not projective, S = S X i has the following projective resolution, given in the proofof Proposition 5.13:0 → Hom C Q ( X i , − ) → Hom C Q ( T ′′ , − ) → Hom C Q ( T ′ , − ) → Hom C Q ( X i , − ) → S → T ′ , T ′′ ∈ T ′ ( i ) . Since add ( T ′ ( i ) ) ∩ add ( { h · X i | h ∈ Γ } ) = 0, we have Hom mod T ( Hom C ( T ′′ , − ) , S ′ ) =0, therefore, Ext mod T ( S, S ′ ) = 0 . If X i ∈ T is projective, then due to the proof of Proposition 5.13, S = S X i has the projectiveresolution: 0 → Hom C Q ( Z, − ) → Hom C Q ( T ′ , − ) → Hom C Q ( X i , − ) → S → T ′ ∈ T ′ ( i ) and Z ∈ T , and there is an admissible epimorphism T ′ ։ Z .If Z has a direct summand h · X i for some h ∈ Γ, then there is an admissible epimorphism Z ։ h · X i . Therefore, T ′ → Z → h · X i is an admissible epimorphism. Since h · X i is projective, then T ′ ։ h · X i is split. It contradicts to the fact T ′ ∈ T ′ ( i ) . So, Z has no direct summand isomorphic to h · X i for any h ∈ Γ. Thus, add ( Z ) ∩ add ( { h · X i | h ∈ Γ } ) = 0. Then Hom mod T ( Hom C ( Z, − ) , S ′ ) = 0.Therefore, Ext mod T ( S, S ′ ) = 0 . (4) Assume that T admits a Γ-2-cycle. Without loss of generality, assume there are irreduciblemorphisms b : X i → X j , a : X j → h · X i for some i = j and h ∈ Γ. By Proposition 6.2, we mayassume that C Qh ( T ) is Hom -finite. By the definition of J in (7) and the irreducibility of a, b , we have a ◦ b ∈ J ( End C Qh ( X i )). Then using Lemma 5.10, a ◦ b is nilpotent. According to Theorem 3.9, weobtain a ◦ b = m P i =1 λ i [ u i , v i ] as morphisms in C Qh ( T ) since gl.dim ( C Qh ( T )) ≤ u i , v i are compositions of irreducible morphisms in T and [ u i , v i ] = u i ◦ v i − v i ◦ u i . In general,we can write that a ◦ b = λ [ a, b ] + m P i ≥ λ i [ u i , v i ], and { u i , v i } 6 = { a, b } for i ≥
2. By the compositionof morphisms in C Qh ( T ), a ◦ b = ab (as the composition in T ), b ◦ a = ( h · b ) a .If λ = 1, then ab = a ◦ b = id h · X i a ◦ bid X i = − λ ( − λ id h · X i b ◦ aid X i + P i ≥ id h · X i [ u i , v i ] id X i )= − λ P i ≥ id h · X i [ u i , v i ] id X i . which is an equality as morphisms in T . Let Q T be the Gabriel quiver of T . Then by Theorem6.6, T ∼ = KQ T /I for an admissible ideal I . Hence, ab − − λ P i ≥ id h · X i [ u i , v i ] id X i ∈ I \ { } in KQ T , where { } is deleted according to that { u i , v i } 6 = { a, b } for i ≥
2. Thus, we have ¯0 = ab − − λ P i ≥ id h · X i [ u i , v i ] id X i + e h · X i ( IJ + JI ) e X i ∈ e h · X i ( I/ ( IJ + JI )) e X i for J the ideal of KQ T generated by all arrows of Q T . Therefore, e h · X i ( I/ ( IJ + JI )) e X i = ¯0.If λ = 1, ( h · b ) a = b ◦ a = id h · X j a ◦ bid X j = − X i ≥ id h · X j [ u i , v i ] id X j which is also an equality as morphisms in T . Similarly, we have e h · X j ( I/ ( IJ + JI )) e X j = ¯0.In both cases, by Proposition 6.5, we obtain Ext mod T ( S X i , S h · X i ) = 0 or Ext mod T ( S X j , S h · X j ) =0. But, it contradicts to (3).(5) By Proposition 6.4, we have Q ′ = e µ i ( Q ), where Q is the Gabriel quiver of T and Q ′ is theGabriel quiver of µ X i ( T ). Moreover, by (4), Q has no Γ-2-cycles, then Q ′ has no Γ-loops, as well as µ X i ( T ).Recall that in[30], given a functorially finite subcategory D of a triangulated category A , a pairof subcategories ( X , Y ) of A is called a D -mutation pair if D ⊆ X ⊆ µ ( Y ; D ) := add ( { T | T → D b → Y, Y ∈ Y , D ∈ D , b is a right D -approximation } ) , and D ⊆ Y ⊆ µ − ( X ; D ) := add ( { T | X a → D → T, X ∈ X , D ∈ D , a is a left D -approximation } ) . Since T is finitely non-Hom-orthogonal, the subcategory T ′ ( i ) is finitely non-Hom-orthogonal. Hence, T ′ ( i ) is a finitely non-Hom-orthogonal subcategory of C Q , then it is functorially finite. According to thedefinition of µ X i ( T ), in the triangulated category C Q , it is easy to see that T ′ ( i ) ⊆ T ⊆ µ ( µ X i ( T ); T ′ ( i ) )and T ′ ( i ) ⊆ µ X i ( T ) ⊆ µ − ( T ; T ′ ( i ) ). Thus, ( T , µ X i ( T )) is a T ′ ( i ) -mutation pair. By Theorem 5.1 of [30], T is a cluster tilting subcategory of C Q if and only if µ X i ( T ) is so. By Lemma II.1.3 of [9], T ( µ X i ( T ))is a cluster tilting subcategory of C Q if and only if T ( µ X i ( T )) is a cluster tilting subcategory of C Q .Thus, µ X i ( T ) is a cluster tilting subcategory of C Q since T is such one of C Q . (cid:3) Note that the (1)-(4) of the above theorem is inspired by Theorem 3.33, [15], but they have variouscircumstances.We are now already to give the proof of the main result Theorem 2.16:
Proof of Theorem 2.16:
For any sequence ([ i ] , · · · , [ i s ] , [ i s +1 ]) of orbits, by Remark 2.3 (3), if e µ [ i s ] · · · e µ [ i ] ( Q ) has no Γ-2-cycles and no Γ-loops, then e µ [ i s +1 ] e µ [ i s ] · · · e µ [ i ] ( Q ) has no Γ-loops. Hence, following Definition 2.4and Proposition 2.15, it suffices to prove the claim that Q has no Γ-2-cycles and no Γ-loops and e µ [ i s ] · · · e µ [ i ] ( Q ) has no Γ-2-cycles for any sequence ([ i ] , · · · , [ i s ]) of orbits.In the case Q is not a linear quiver of type A . From Construction 2.6, it is easy to see that Q hasno Γ-2-cycles and no Γ-loops. By Lemma 4.15 (4), the Gabriel quiver of T is isomorphic to Q . Sincethe Gabriel quiver of µ X is · · · µ X i ( T ) is a subquiver of the Gabriel quiver of µ X is · · · µ X i ( T ), thenby Proposition 6.4, it suffices to prove that µ X is · · · µ X i ( T ) has no Γ-2-cycles, where X i j are theindecomposable objects in C corresponding to vertices i j for all 1 ≤ j ≤ s . This follows inductivelyby Theorem 6.7 and Lemma 6.3. Also, we know that the Gabriel quiver of µ X is · · · µ X i ( T ) is NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 41 isomorphic to e µ [ i s ] · · · e µ [ i ] ( Q ), where i j are the vertices in Q correspondence to X i j for 1 ≤ j ≤ s .Thus, the above claim holds.In the case Q is a linear quiver of type A , since Q is strongly almost finite and the it has noinfinite path, thus it is a finite quiver. Moreover, it is clear that Γ = { e } . In this case, our resultfollows at once. (cid:3) Applications to positivity and F-polynomials
In this section, we apply the unfolding theorem (Theorem 2.16) to study positivity and F-polynomials of an acyclic sign-skew-symmetric cluster algebra A (Σ) in a uniform method. Thatis, in order to prove a property for A (Σ), one can firstly try to check this property for the skew-symmetric cluster algebra A ( e Σ) of infinite rank, where e Σ is the unfolding of Σ; secondly, try to givea surjective algebra morphism π : A ( e Σ) → A (Σ) which preserves such property.7.1. The surjective algebra morphism π . .Let B ∈ M at n × n ( Z ) be sign-skew-symmetric, B ′ ∈ M at m × n ( Z ) and e B = BB ′ ! ∈ M at ( n + m ) × n ( Z ).Assume that the exchange matrix B ∈ M at n × n ( Z ) is acyclic. Denote B ′′ = B − B ′ T B ′ ! . Since B is acyclic, B ′′ is acyclic. Let ( Q ′′ , Γ) be an unfolding of B ′′ . Let e Q be the quiver obtained from Q ′′ by freezing all vertices of Q ′′ corresponding to B ′ . Denote by Q the subquiver of e Q generatedby the exchangeable vertices. Clearly, ( Q, Γ) is just the unfolding of B . In fact, it is easy to verifythat Q is a covering of B and e µ [ i s ] · · · e µ [ i ] ( Q ) is a subquiver of e µ [ i s ] · · · e µ [ i ] ( e Q ) for any sequences([ i ] , · · · , [ i s ]) of orbits.Let e Σ = Σ( e Q ) = ( e X, e Y , e Q ) be the seed associated with e Q , where e X = { x u | u ∈ Q } , e Y = { y v | v ∈ e Q \ Q } . Let Σ = Σ( e B ) = ( X, Y, e B ) be the seed associated with e B , where X = { x [ i ] | i = 1 , · · · , n } , Y = { y [ j ] | j = 1 , · · · , m } . It is clear that there is a surjective algebra homomorphism:(12) π : Q [ x ± i , y j | i ∈ Q , j ∈ e Q \ Q ] → Q [ x ± i ] , y [ j ] | ≤ i ≤ n, ≤ j ≤ m ] with x i x [ i ] , y j y [ j ] where [ i ], [ j ] is the orbits of i , j respectively under the action of Γ.For any cluster variable x u ∈ e X , define e µ [ i ] ( x u ) = µ u ( x u ) if u ∈ [ i ]; otherwise, e µ [ i ] ( x u ) = x u if u [ i ]. Formally, we define e µ [ i ] ( e X ) = { e µ [ i ] ( x ) | x ∈ e X } and e µ [ i ] ( e X ± ) = { e µ [ i ] ( x ) ± | x ∈ e X } . Lemma 7.1.
Keep the notations as above. Assume that B is acyclic. If [ i ] is an orbit of verticeswith i ∈ Q , then we have(1) e µ [ i ] ( x j ) is a cluster variable of A ( e Q ) for any j ∈ Q ,(2) e µ [ i ] ( e X ) is algebraic independent over Q [ y j | j ∈ e Q \ Q ] .Proof. (1) It follows immediately by the definition.(2) By the definition, we need to prove any finite variables in e µ [ i ] ( e X ) is algebraic independent.It suffices to show that any finite variables in e µ [ i ] ( e X ) are in a cluster of A ( e Σ). Assume that { z , · · · , z s } ⊆ e µ [ i ] ( e X ), by definition of e µ [ i ] ( e X ), there exist s vertices i t , t = 1 , · · · , s in [ i ] suchthat z t = µ i t ( x t ) for t = 1 , · · · , s . Denote I = S t =1 , ··· ,s { i t } , since e Q has no Γ-loops, we have z t = µ i t ( x t ) = Q i ′ ∈ I µ i ′ ( x t ), then { z , · · · , z s } ⊆ Q i ′ ∈ I µ i ′ ( e X ). Our result follows. (cid:3) By Lemma 7.1, e µ [ i ] ( e Σ) := ( e µ [ i ] ( e X ) , e Y , e µ [ i ] ( e Q )) is a seed. Thus, we can define e µ [ i s ] e µ [ i s − ] · · · e µ [ i ] ( x )and e µ [ i s ] e µ [ i s − ] · · · e µ [ i ] ( e X ) and e µ [ i s ] e µ [ i s − ] · · · e µ [ i ] ( e Σ) for any sequence ([ i ] , [ i ] , · · · , [ i s ]) of orbits in Q . Lemma 7.2.
Keep the notations as above. Assume that B is acyclic. If a is a vertex of Q , thenany finite cluster variables in a cluster of A ( e µ [ a ] e Σ) is contained in a cluster of A ( e Σ) .Proof. Let ∆ and ∆ be two finite subsets of Q . For any finite cluster variables z j = µ i s · · · µ i ( e µ [ a ] x j ), j ∈ ∆ of A ( e µ [ a ] e Σ) and finite vertices ∆ of µ i s · · · µ i ( e µ [ a ] e Q ), we prove inductively on s that thereexists a finite set S ⊆ [ a ] such that (1) z j = µ i s · · · µ i ( Q a ′ ∈ [ S ′ ] µ a ′ x j ) for any j ∈ ∆ and finite set S ⊆ S ′ ⊆ [ a ]; (2) for any a j ∈ ∆ , the subquiver of µ i s · · · µ i ( e µ [ a ] e Q ) formed by the arrows incidentwith a j equals the subquiver of µ i s · · · µ i ( Q a ′ ∈ [ S ′ ] µ a ′ e Q ) formed by the arrows incident with a j .For s = 0, set S = S ∪ S , where S = { j ∈ [ a ] | j ∈ ∆ } and S = { c ∈ [ a ] | j incidents with or in ∆ } .For any finite set S ′ such that S ⊆ S ′ ⊆ [ a ], by the definition of z j = e µ [ a ] x j , it is easy to see that z j = Q a ′ ∈ [ S ′ ] µ a ′ x j , (1) holds; for any a j ∈ ∆ , denote e µ [ a ] ( e Q ) = ( b ′ ij ) and Q a ′ ∈ [ S ′ ] µ a ′ ( e Q ) = ( b ′′ ij ). If a j ∈ [ a ], since a j ∈ S ′ , we have b ′′ a j k = − b a j k = b ′ a j k . If a j [ a ], since all c ∈ [ a ] incident with a j arein S ′ , we have b ′′ a j k = b a j k + P a ′ ∈ [ a ] | b aja ′ | b a ′ k + b aja ′ | b a ′ k | = b ′ a j k . Thus, (2) holds.Assume that the statements hold for s −
1. For case s , set ∆ ′ = { j ∈ Q | j incidents with or in ∆ } and ∆ ′ = ∆ ∪ ∆ . Applying the assumption to ∆ ′ and ∆ ′ , there exists a finite subset S ′ ⊆ [ a ]such that (a) µ i s − · · · µ i ( e µ [ a ] x j ) = µ i s − · · · µ i ( Q a ′ ∈ [ S ′′ ] µ a ′ x j ) for any j incidents with or in ∆ andfinite set S ′ ⊆ S ′′ ⊆ [ a ] and (b) for any a ′ j ∈ ∆ ′ , the subquiver of µ i s − · · · µ i ( e µ [ a ] e Q ) formed bythe arrows incident with a ′ j equals the subquiver of µ i s − · · · µ i ( Q a ′ ∈ [ S ′ ] µ a ′ e Q ) formed by the arrowsincident with a ′ j . For any j ∈ ∆ , if j = i s , since ∆ ⊆ ∆ ′ , by (a), we have µ i s · · · µ i ( e µ [ a ] x j ) = µ i s − · · · µ i ( e µ [ a ] x j ) = µ i s − · · · µ i ( Y a ′ ∈ [ S ′′ ] µ a ′ x j ) = µ i s · · · µ i ( Y a ′ ∈ [ S ′′ ] µ a ′ x j );if j = i s , since j ∈ ∆ ′ , ∆ ′ , by (a), (b) and the definition of mutation of cluster variable, we have µ i s · · · µ i ( e µ [ a ] x j ) = µ i s · · · µ i ( Q a ′ ∈ [ S ′′ ] µ a ′ x j ) . (1) holds. For any a j ∈ ∆ , since ∆ ⊆ ∆ ′ , by (b), wehave the subquiver of µ i s · · · µ i ( e µ [ a ] e Q ) formed by the arrows incident with a j equals the subquiver of µ i s · · · µ i ( Q a ′ ∈ [ S ′ ] µ a ′ e Q ) formed by the arrows incident with a j . (2) holds. Therefore, the statementshold for all s ∈ N .In particular, for any finite cluster variables { z , · · · , z m } in a cluster of A ( e µ [ a ] e Σ), let ∆ be thevertices correspond to the cluster variables and ∆ = ∅ . Applying the statements to ∆ and ∆ , ourresult follows. (cid:3) Lemma 7.3.
Keep the above notations. Assume that B is acyclic. Then for any sequence of orbits ([ i ] , · · · , [ i s ]) in Q , we have(1) The cluster variables of A ( e µ [ i s ] · · · e µ [ i ] ( e Σ)) are the same as the cluster variables of A ( e Σ) ;(2) Any finite variables in e µ [ i s ] · · · e µ [ i ] ( e X ) is contained in a cluster of A ( e Σ) ;(3) Any variable in e µ [ i s ] · · · e µ [ i ] ( e X ) is a cluster variable of A ( e Σ) ;(4) Any monomial with variables in e µ [ i s ] · · · e µ [ i ] ( e X ) is a cluster monomial of A ( e Σ) .Proof. (1) Using Lemma 7.2, the cluster variables of A ( e µ [ i ] ( e Σ)) are the cluster variables of A ( e Σ).Since it is easy to see that e µ [ i ] e µ [ i ] ( e Σ) = e Σ, dually, we have the cluster variables of A ( e Σ) are the
NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 43 cluster variables of A ( e µ [ i ] ( e Σ)). Therefore, The cluster variables of A ( e µ [ i ] ( e Σ)) are the same as thecluster variables of A ( e Σ). Our result follows.(2) Since any finite variables of e µ [ i s ] · · · e µ [ i ] ( e X ) are in the initial cluster of A ( e µ [ i s ] · · · e µ [ i ] ( e Σ)),applying Lemma 7.2 step by step, our result follows.(3) It follows immediately by (2).(4) It follows immediately by (1). (cid:3)
Corollary 7.4.
Keep the notations as above. Assume that B is acyclic. Then for any sequence oforbits ([ i ] , · · · , [ i s ]) in Q , any cluster variable of A ( e Σ) can be expressed as a Laurent polynomial of e µ [ i s ] · · · e µ [ i ] ( e X ) with ground field Z [ y j | j ∈ e Q \ Q ] .Proof. By Lemma 7.3 (1), any cluster variable of A ( e Σ) is a cluster variable of A ( e µ [ i s ] · · · e µ [ i ] ( e Σ)).By Laurent Phenomenon in [18], our result follows. (cid:3)
Theorem 7.5.
Keep the notations as above with an acyclic sign-skew-symmetric matrix B and π defined in (12). Restricting π to A ( e Σ) , then π : A ( e Σ) → A (Σ) is a surjective algebra morphism satis-fying that π ( e µ [ j k ] · · · e µ [ j ] ( x a )) = µ [ j k ] · · · µ [ j ] ( x [ i ] ) ∈ A (Σ) and π ( e µ [ j k ] · · · e µ [ j ] ( e X )) = µ [ j k ] · · · µ [ j ] ( X ) for any sequences of orbits [ j ] , · · · , [ j k ] and any a ∈ [ i ] .Proof. For k = 1, by the definition of π and b [ j ][ i ] = P j ′ ∈ [ j ] b j ′ i , we have π ( e µ [ j ] ( x i )) = µ [ j ] ( x [ i ] ).Assume that the equation holds for every k < s . Since A ( e µ [ j ] ( e Σ)) ⊆ Q [ x ± i , y j | i ∈ Q , j ∈ e Q \ Q ]and A ( µ [ j ] (Σ)) ∼ = A (Σ), applying π to A ( e µ [ j ] ( e Σ)) and by induction, we have π ( e µ [ j k ] · · · e µ [ j ] ( x i )) = π ( e µ [ j k ] · · · e µ [ j ] ( e µ [ j ] x i )) = µ [ j k ] · · · µ [ j ] ( π ( e µ [ j ] x [ i ] )) = µ [ j k ] · · · ( µ [ j ] ( x [ i ] )) = x. Thus, π ( e µ [ j k ] · · · e µ [ j ] ( e X )) = µ [ j k ] · · · µ [ j ] ( X ).Now, to prove that π is a surjective algebra homomorphism, it suffices to show that π ( A ( e Σ)) ⊆A (Σ). For any cluster X ′ = µ [ j k ] · · · µ [ j ] ( X ) ∈ A (Σ), we have π ( e µ [ j k ] · · · e µ [ j ] ( e X )) = X ′ . Thus, π ( T e µ [ jk ] ··· e µ [ j ( e X ) Z [ y j | j ∈ e Q \ Q ][ e µ [ j k ] · · · e µ [ j ] ( e X ) ± ])= T µ [ jk ] ··· µ [ j ( X ) Z [ y [ j ] | ≤ j ≤ m ][ µ [ j k ] · · · µ [ j ] ( X ) ± ] = U (Σ) . Therefore, due to Lemma 7.3 (4) and Corollary A.8, we have π ( A ( e Σ)) ⊆ π ( U ( e Σ)) ⊆ π ( \ e µ [ jk ] ··· e µ [ j ( e X ) Z [ y j | j ∈ e Q \ Q ][ e µ [ j k ] · · · e µ [ j ] ( e X ) ± ]) = U (Σ) = A (Σ) . Our result follows. (cid:3)
Positivity conjecture.Conjecture 7.6. ( [18] , Positivity conjecture) Let A (Σ) be a cluster algebra over ZP . For any cluster X , and any cluster variable x , the Laurent polynomial expansion of x in the cluster X has coefficientswhich are nonnegative integer linear combinations of elements in P . By Fomin-Zelevinskys separation of addition formula (Theorem 3.7, [20]), to verify positivityconjecture, it suffices to prove that it holds for cluster algebras of geometric type. By Theorem 4.4of [28], see also Theorem 2.23 of [12], any cluster algebra of geometric type A (Σ ) is a rooted clustersubalgebra of a cluster algebra A (Σ) with coefficients Z via freezing some exchangeable variables.Thus, any cluster variable of A (Σ ) is a cluster variable of A (Σ) and a cluster of A (Σ ) correspondsto a cluster of A (Σ). Then the positivity of A (Σ) implies the positivity of A (Σ ). Thus, we onlyneed to prove the positivity of sign-skew-symmetric cluster algebras A (Σ) in the case A (Σ) is with coefficients Z , that is, the exchange matrix of A (Σ) is a square matrix, aswhich discussed in the foregoing. Theorem 7.7. (Theorem 7, [25] ) The positivity conjecture is true for skew-symmetric cluster alge-bras of infinite rank.
Theorem 7.8.
The positivity conjecture is true for acyclic sign-skew-symmetric cluster algebras.Proof.
By the remark at the beginning of this subsection, it suffices to prove that positivity holds forcluster algebras with coefficients Z . Let A (Σ) be an acyclic sign-skew-symmetric cluster algebra withcoefficients Z . For any cluster variable z = µ [ j k ] · · · µ [ j ] ( x [ i ] ) ∈ A (Σ) and cluster µ [ i t ] · · · µ [ i ] ( X ) of A (Σ), applying Theorem 7.5 to the coefficients Z case, we have π ( e z ) = z and π ( e µ [ i t ] · · · e µ [ i ] ( e X )) = µ [ i t ] · · · µ [ i ] ( X ), where e z = e µ [ j k ] · · · e µ [ j ] ( x i ). By Corollary 7.4 and Theorem 7.7, we have e z ∈ N [ e µ [ i t ] · · · e µ [ i ] ( e X ) ± ]. Thus, we obtain z ∈ N [ µ [ i t ] · · · µ [ i ] ( X ) ± ] by the definition of π in (12). (cid:3) F -polynomials. Let Σ = (
X, Y, B ) be a seed of rank n with principal coefficients and the coef-ficients variables are y , · · · , y n , that is, its extended exchange matrix e B = BI n ! ∈ M at n × n ( Z ),where I n is the n × n identity matrix. For each cluster variable x ∈ A (Σ), it can be expressed as aLaurent polynomial x = x ( x , · · · , x n , y , · · · , y n ) ∈ Z [ x ± , · · · , x ± n , y , · · · , y n ] by Laurent phenom-enon, whose F polynomial is defined as F ( x ) = x (1 , · · · , , y , · · · , y n ) ∈ Z [ y , · · · , y n ]. For details,see [20]. It is conjectured in [20] that Conjecture 7.9. ( [20] , Conjecture 5.4) In a cluster algebra with principal coefficients, each F -polynomial has constant term . In view of Proposition 5.3 in [20], this conjecture means that each F -polynomial F ( x ) has a uniquemonomial of maximal degree. Moreover, this monomial has coefficient 1, and it is divisible by all ofthe other occurring monomials. See Conjecture 5.5 of [20]. Remark 7.10.
The seeds defined in [20] are skew-symmetrizable with finite rank, it is easy to seethat the above definition and conjecture can also apply to cluster algebras with sign-skew-symmetricseeds with finite/infinite rank.
Theorem 7.11. ( [16] , Theorem 1.7) Conjectures 7.9 holds under the condition that B is skew-symmetric. Lemma 7.12.
Conjecture 7.9 holds for all skew-symmetric cluster algebras of infinite rank withprincipal coefficients.Proof.
Assume that Q is an infinite locally finite quiver and e Σ = (
X, Y, Q ) is the seed with prin-cipal coefficients associated with Q , where X = { x i | i ∈ Q } and Y = { y i | i ∈ Q } . For anycluster variable x = µ i s · · · µ i ( x i ) of A ( e Σ), denote Q ′ the subquiver of Q generated by the vertices { i , i , · · · , i s } . Assume that S be the set of vertices in Q which are incident with a vertex of Q ′ butnot belong to Q ′ . Denote Q ′′ be the subquiver generated by S ∪ Q ′ . Let Σ ′ be the seed obtained fromΣ by frozen x i , i ∈ S and deleted x i , i Q ′′ and y j , j Q ′ . Using the analogue result of Theorem4.4 of [28], we have A (Σ ′ ) is the rooted cluster subalgebra of A ( e Σ). Since { i , i , · · · , i s } = Q ′ , wehave x ∈ A (Σ ′ ). Let Σ ′′ = ( X ′′ , Y ′′ , Q ′′ ) be the seed with principal coefficients associated with Q ′′ ,by Theorem 4.4 of [28], A (Σ ′ ) is a rooted cluster subalgebra of A (Σ ′′ ), then x ∈ A (Σ ′′ ). By Theorem7.11, the F -polynomial has constant term 1. The result follows. (cid:3) NFOLDING OF ACYCLIC SIGN-SKEW-SYMMETRIC CLUSTER ALGEBRAS 45
From now on, assume that B ∈ M at n × n ( Z ) is an acyclic matrix and ( Q, Γ) is an unfolding of B .Let e Σ = Σ( Q ) = ( e X, e Y , Q ) and Σ = Σ( B ) = ( X, Y, B ) be the seeds with principal coefficients andextend matrix be the matrix associated with Q and B respectively, where e X = { x i | i ∈ Q } , e Y = { y i | i ∈ Q } and X = { x [ i ] | i = 1 , · · · , n } , Y = { y [ i ] | i = 1 , · · · , n } .Combing Theorem 7.11 and Theorem 7.5 in the principal coefficients case, we can deduce thatConjectures 7.9 hold for all acyclic cluster algebras. Theorem 7.13.
In an acyclic sign-skew-symmetric cluster algebra with principal coefficients, each F -polynomial has constant term .Proof. By Theorem 7.5 (1) and Proposition 7.3 (3), any cluster variable x ∈ A (Σ) can lift to a clustervariable e x ∈ A ( e Σ), that is π ( e x ) = x . Thus, we have π ( F ( e x )) = F ( x ), where F ( e x ) ( F ( x ) respectively)means the F -polynomial associated with e x ( x respectively). By Lemma 7.12, our result follows. (cid:3) Appendix A. A = U for acyclic sign-skew-symmetric cluster algebras Let Σ = (
X, Y, e B ) be a sign-skew-symmetric seed, where X = { x , · · · , x n } , Y = { y , · · · , y m } .In [36], the author considered the case that e B is skew-symmetrizable and the ground ring is Z [ y ± , · · · , y ± m ]. It can be seen that the approach to the result on A = U for acyclic e B in [36] canbe applied to the totally sign-skew-symmetric case. So, we will cite the result in [36] as that in theacyclic sign-skew-symmetric case.In this appendix, we consider the cluster algebra A (Σ) in the case that e B is sign-skew-symmetricand the ground ring is Z [ y , · · · , y m ]. We will see here that according to the difference of theground ring, it is only needed to do little modification in the definitions and proofs in [36]. Let e A (Σ) = A (Σ)[ y − , · · · , y − m ], then e A (Σ) is a cluster algebra with the ground ring Z [ y ± , · · · , y ± m ].For the seed Σ = ( X, Y, e B ) and a subset S ⊆ X , the seed Σ S, ∅ means the sub-seed obtained byfreezing the variables in S . For details, see [28]. Lemma A.1. ( [36] , Lemma 1) Let Σ = (
X, Y, e B ) be a seed, S ⊆ X . Let U (Σ) be the upper clusteralgebra of A (Σ) and U (Σ S, ∅ ) be the upper cluster algebra of A (Σ S, ∅ ) . Then A (Σ S, ∅ )[ x − | x ∈ S ] ⊆ A (Σ)[ x − | x ∈ S ] ⊆ U (Σ)[ x − | x ∈ S ] ⊆ U (Σ S, ∅ )[ x − | x ∈ S ] . Definition A.2. ( [36] , Definition 1) Let Σ = (
X, Y, e B ) is a seed, S ⊆ X . We say A (Σ S, ∅ ) to be a cluster localization of A (Σ) if A (Σ)[ x − | x ∈ S ] = A (Σ S, ∅ )[ x − | x ∈ S ] . Definition A.3.
For a cluster algebra A (Σ) , a set {A (Σ S i , ∅ ) } of cluster localizations of A (Σ) iscalled a cover of A (Σ) if, for every prime ideal P in A (Σ) , there exist some A (Σ S i , ∅ ) ( i ∈ I ) suchthat A (Σ S i , ∅ )[ x − | x ∈ S i ] P ( A (Σ S i , ∅ )[ x − | x ∈ S i ] . Proposition A.4. ( [36] , Proposition 2) If {A (Σ S i , ∅ ) } i ∈ I is a cover of A (Σ) , then A (Σ) = \ i ∈ I A (Σ S i , ∅ )[ x − | x ∈ S i ] . Lemma A.5. ( [36] , Lemma 2) Let {A (Σ S i , ∅ ) } i ∈ I be a cover of A (Σ) . If A (Σ S i , ∅ ) = U (Σ S i , ∅ ) foreach i ∈ I , then A (Σ) = U (Σ) .Proof. Since A (Σ S i , ∅ ) = U (Σ S i , ∅ ), we have A (Σ S i , ∅ )[ x − | x ∈ S i ] = U (Σ S i , ∅ )[ x − | x ∈ S i ]. ByLemma A.1, U (Σ) ⊆ U (Σ S i , ∅ )[ x − | x ∈ S i ] for all i . Then U (Σ) ⊆ T i ∈ I U (Σ S i , ∅ )[ x − | x ∈ S i ].Thus, by Proposition A.4, U (Σ) ⊆ \ i ∈ I U (Σ S i , ∅ )[ x − | x ∈ S i ] = \ i ∈ I A (Σ S i , ∅ )[ x − | x ∈ S i ] = A (Σ) ⊆ U (Σ) . (cid:3) In [36], A (Σ) is called isolated if the exchange matrix of Σ is zero. Proposition A.6. ( [36] , Proposition 3) Let A (Σ) be an isolated cluster algebra. Then A (Σ) = U (Σ) . Proposition A.7. ( [36] , Proposition 4) If A (Σ) is acyclic, then it admits a cover by isolated clusteralgebras. Corollary A.8. ( [36] , Corollary 2) If A (Σ) is acyclic, then A (Σ) = U (Σ) .Proof. It follows immediately by Proposition A.6, Proposition A.7 and Lemma A.5. (cid:3)
Theorem A.9.
An acyclic sign-skew-symmetric cluster algebra A (Σ) of infinite rank equals to itsupper cluster algebra U (Σ) .Proof. It is clear that A (Σ) ⊆ U (Σ) by the definition of upper cluster algebra;On the other side of the inclusion, for any f ∈ U (Σ) ⊆ Q [ X ± ], there exists a finite subset { x , · · · , x n } ⊆ X such that f = g ( x , ··· ,x n ) x a ··· x ann ∈ Q [ x ± , · · · , x ± n ] for some non-negative integer a i anda polynomial g . Let A (Σ ′ ) be the rooted cluster subalgebra of finite rank of A (Σ) which contains x , · · · , x n as initial cluster variables.Now we show that f ∈ U (Σ ′ ). For any cluster { x ′ , · · · , x ′ m } of A (Σ ′ ), there exists a clus-ter X ′ of A (Σ) such that { x ′ , · · · , x ′ m } ⊆ X ′ . Since f ∈ U (Σ) ⊆ Q [ X ′± ], f can be writtenin the form f = g ′ ( x ′ , ··· ,x ′ m , x ) x ′ b ··· x ′ mbm x b for some non-negative integer b i , a non-negative integer vector b , x = X ′ \{ x ′ , · · · , x ′ m } and a polynomial g ′ , where we may assume that g ′ and x ′ b · · · x ′ mb m x b are coprime. Since x , · · · , x n ∈ A (Σ ′ ), they can be represented as polynomials of x ′ , · · · , x ′ m ,then we have g ( x , ··· ,x n ) x a ··· x ann = h ( x ′ , ··· ,x ′ m ) h ′ ( x ′ , ··· ,x ′ m ) for two coprime polynomials h and h ′ . Thus, we have g ′ ( x ′ , ··· ,x ′ m , x ) x ′ b ··· x ′ mbm x b = h ( x ′ , ··· ,x ′ m ) h ′ ( x ′ , ··· ,x ′ m ) . Since ( g ′ , x ′ b · · · x ′ mb m x b ) = 1 and ( h, h ′ ) = 1, we obtain that g ′ = h and x ′ b · · · x ′ mb m x b = h ′ up to multiple with a non-zero element in Q . It follows that b = 0.Therefore, we get f = h ( x ′ , ··· ,x ′ m ) x ′ b ··· x ′ mbm . Then f ∈ Q [ x ′ ± , · · · , x ′ m ± ]. Thus, we obtain f ∈ U (Σ ′ ) sincethe cluster { x ′ , · · · , x ′ m } is arbitrary.Since A (Σ) is acyclic, A (Σ ′ ) is also acyclic. By Corollary A.8, we get f ∈ U (Σ ′ ) = A (Σ ′ ) ⊆ A (Σ).Therefore, we have U (Σ) ⊆ A (Σ). (cid:3) Acknowledgements:
This project is supported by the National Natural Science Foundation ofChina (No. 11671350 and No.11571173) and the Zhejiang Provincial Natural Science Foundation ofChina (No.LZ13A010001).
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E-mail address : [email protected] Fang LiDepartment of Mathematics, Zhejiang University (Yuquan Campus), Hangzhou, Zhejiang 310027, P.R.China
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