On the generalized cluster algebras of geometric types

We develop and prove the analogs of some results shown in \cite{bfz} concerning lower and upper bounds of cluster algebras to the generalized cluster algebras of geometric types. We show that lower bounds coincide with upper bounds under the conditions of acyclicity and coprimality. Consequently, we obtain the standard monomial bases of these generalized cluster algebras. Moreover, in the appendix, we prove that an acyclic generalized cluster algebra is equal to the corresponding upper generalized cluster algebra without the assumption of the existence of coprimality.


Background
Fomin and Zelevinsky invented the concept of cluster algebras [3] [4] in order to create an algebraic framework for studying total positivity in algebraic groups and canonical bases in quantum groups. As a natural generalization, Chekhov and Shapiro introduced the generalized cluster algebras which arise from the Teichmuller spaces of Riemann surfaces with orbifold points [2]. The main difference between cluster algebras and generalized cluster algebras is that the binomial exchange relations for cluster variables of cluster algebras are replaced by the multinomial exchange relations for those cluster variables of generalized cluster algebras. In [2], Chekhov and Shapiro have shown that the generalized cluster algebras possess the remarkable Laurent phenomenon. Many other important properties of cluster algebras are also shown to hold in the generalized cluster algebras such as finite type classification, g-vectors and F-polynomials [2] [7].
Motivated by the Laurent phenomenon established in [3], Berenstein, Fomin and Zelevinsky in [1] introduced the notion of upper cluster algebras. They proved that for a cluster algebra possessing an acyclic and coprime seed, its lower bound coincides with its upper bound, and hence coincides with itself. The standard monomial bases of this kind of cluster algebras can then be naturally constructed.
Gekhtman, Shapiro and Vainshtein [5] proved that upper generalized cluster algebras over certain rings retain all properties of classical upper cluster algebras, and under certain coprimality conditions coincide with the intersection of rings of Laurent polynomials in a finite collection of clusters.
The aim of this paper is to continually investigate the structure of generalized cluster algebras. By using the methods developed in [1], we prove that the conditions of acyclicity and coprimality close the gap between lower bounds and upper bounds associated to generalized cluster algebras as the extension of the similar results of classical cluster algebras. Consequently, we obtain the standard monomial bases of these generalized cluster algebras. It would be desirable to apply the results that we obtain in this paper to construct some good bases of the corresponding generalized cluster algebras in the future work.
In appendix, we extend Muller's results on acyclic and locally acyclic cluster algebras to acyclic generalized cluster algebras. By using the same arguments as given in [6], we prove that acyclic generalized cluster algebras coincide with their upper generalized cluster algebras without the assumption of the existence of coprimality.

Preliminaries
First, let us recall the definition of the generalized cluster algebras of geometric types (see [2] and [5]).
In the following, we use [i, j] to denote the set {i, i + 1, . . . , j − 1, j} for integers i < j. Let m and n be positive integers with m ≥ n. An n × n matrix B is called skew-symmetrizable if there exists a diagonal matrix D = diag( d 1 , d 2 , . . . , d n ), where d i are positive integers for all i ∈ [1, n], such that DB is skew-symmetric. Let ex be a subset of [1, m] with |ex| = n. Let B = (b ij ) be an m × n integer matrix such that B has the n × n skew-symmetrizable submatrix B with rows labeled by ex. The matrix B is called the principal part of B. For i ∈ ex, let d i be a positive integer such that d i divides b ji for any j ∈ [1, m], and let β ji := b ji d i . Let us denote by Q(x 1 , x 2 , . . . , x m ) the function field of m variables over Q with a transcendence basis {x 1 , x 2 , . . . , x m }. Let the coefficient group P be the multiplicative free abelian group generated by {x i | i ∈ [1, m] − ex} and ZP be the integer group ring. For each i ∈ ex, we will denote by ρ i : which is called the exchange relation; Note that µ i is an involution. The generalized seed ( y, ε, A) is said to be mutationequivalent to ( x, ρ, B), if ( y, ε, A) can be obtained from ( x, ρ, B) by a sequence of seed mutations. Definition 2.3. For an initial generalized seed ( x, ρ, B), the generalized cluster algebra A( x, ρ, B) is the ZP-subalgebra of Q(x 1 , . . . , x m ) generated by all cluster variables from all generalized seeds which are mutation-equivalent to ( x, ρ, B). The integer n is the rank of A( x, ρ, B).
For each i ∈ ex, we denote by Definition 2.4. The generalized seed ( x, ρ, B) is called coprime if P i and P j are coprime for any two different i, j ∈ ex.
Let ( x, ρ, B) be a generalized seed. The direct graph Γ( x, ρ, B) is defined as follows: (1) its vertices are i ∈ ex; (2) a pair (i, j) is a direct edge of Γ( x, ρ, B) if and only if b ij > 0. The following definition is a natural generalization of [1, Definition 1.14].
Definition 2.6. Let ( x, ρ, B) be a generalized seed. A standard monomial in {x i , x ′ i | i ∈ ex} is a monomial such that it does not have any factor of the form x i x ′ i for any i ∈ ex.
In order to define the upper bounds and lower bounds, we write ex = {i 1 , . . . , i n }.
Definition 2.7. For a generalized seed ( x, ρ, B), the upper bound is defined by and the lower bound The following definition is a generalization of [6, Section 3.1].
Definition 2.9. Let ( x, ρ, B) be a generalized seed and i ∈ ex. A new generalized seed ( x † , ρ † , B † ) is defined as follows: The generalized seed ( x † , ρ † , B † ) is called the freezing of ( x, ρ, B) at x i . The freezing of A( x, ρ, B) at x i is defined to be the generalized cluster algebra A( x † , ρ † , B † ), which is the ZP † -subalgebra of Q(x 1 , . . . , x m ) generated by all cluster variables from the generalized seeds which are mutation-equivalent to ( x † , ρ † , B † ).
The freezing at x i is compatible with the mutation in direction j for i = j, so we have the following result.
Proof. The proof is straightforward, so we omit the details.
The freezing of ( x, ρ, B) at {x j 1 , . . . , x j k } ⊂ x is the generalized seed obtained from ( x, ρ, B) by iterated freezing at each cluster variable in {x j 1 , . . . , x j k } in any order. Note that the freezing at {x j 1 , . . . , x j k } is compatible with the mutation in direction k for k / ∈ {j 1 , . . . , j k }. In the same manner, the generalized cluster algebra

Lower bounds and upper bounds
We follow the arguments in [1] to prove that lower bounds and upper bounds coincide under the assumptions of acyclicity and coprimality. For simplicity, in the following we assume that ex = [1, n]. Without loss of generality, we can assume that b lk ≥ 0 for 1 ≤ k < l ≤ n if the generalized seed ( x, ρ, B) is acyclic.
. Proof. By using the same technique as in [1], we can prove the statement and we write the proof down here for the readers' convenience.
For any a = (a 1 , . . . , a n ) ∈ Z n , we denote by . . x an n by x a . Let " ≺ " denote the lexicographic order on Z n which induces the lexicographic order on the Laurent monomials as Note that x Using the assumption that b lk ≥ 0 for l > k, it follows that x (1) i = x i and the lexicographically first monomial which appears in x . We then conclude that the lexicographically first monomial that appears in x (a ′ ) i is preceded by the one in x (a) if a ≺ a ′ . Therefore the standard monomials in {x 1 , x ′ 1 , . . . , x n , x ′ n } are linearly independent. The following two lemmas are generalizations of [1,Lemma 4.1,4.2]. The proofs of them are quite similar and so are omitted. ) satisfies one of the following conditions: i . Both are algebra homomorphisms and consequently, n ] is also an algebra homomorphism.
Given y ∈ ZP[x ±1 1 , . . . .x ±1 n ], as in [1, Definition 6.3], we also define the leading term LT(y) of y with respect to x 1 to be the sum of Laurent monomials with the smallest power of x 1 , which are obtained from the Laurent expression of y with non-zero coefficient.
The following results parallel to [1, Lemma 6.2, 6.4, 6.5] can be obtained similarly. Proof. By suitable and non-trivial modifications to the proof of [1, Lemma 6.6], we can prove the statement. By a direct calculation, for j ∈ [2, n] one can show that In order to prove the converse inclusion, it suffices to show that x −1 j ∈ Im(f ) for each j ∈ b jr l r , and l j − 1 ≥ 0, we have that M/M j ∈ M. By the induction hypothesis, we obtain that M/M j ∈ Im(f ).
, which implies M j ∈ Im(f ). It follows that M = (M/M j ) · M j ∈ Im(f ). Now assume that j ∈ J. Using the fact that b lk ≥ 0 for 1 ≤ k < l ≤ n, it follows that By multiplying two sides of this equation by M/M j , we have that By applying the facts that l j > 0, j ∈ J and (3.5), we know that there exists the smallest integer h ∈ [2, j − 1] such that b jh l h > 0 which implies b jh > 0, l h > 0 and It is clearly that deg(M ′ ) ≥ 1. In order to prove M/M ′ ∈ M, we need verify that and  x c k k ∈ Im(f ), we only need to show that c k ≥ 0 for all k ∈ [2, n]. Note that 0 ≤ r ≤ d j − 1. Then we have: (1) When k ∈ [2, h−1], we have β kj ≤ 0 and c k = 0+(−b kj )−0+rβ kj = (r−d j )β kj ≥ 0; (2) When k = h, since b jh > 0, we obtain β hj < 0. It follows that The proof is completed.
In order to prove Theorem 3.8, we need the following two lemmas.
Proof. For each j ∈ J, we can write z in the form z . . , x ±1 n ] and c j,a j = 0. If a j ≥ 0 then we have that z ∈ ZP[x ±1 2 , . . . , x ±1 j−1 , x j , x ±1 j+1 , . . . , x ±1 n ]. Now assume that a j < 0. The fact zP 1 ∈ Im(f ) implies that c k x k . . , x ±1 n ] for k < 0. Since P 1 and P j are coprime, we have that The proof is completed. The following result shows that the upper bound of the generalized cluster algebra of rank 2 is equal to the corresponding lower bound.
Lemma 3.7. Suppose that n = 2. If the generalized seed ( x, ρ, B) is coprime, then . Note that the element y can be written as and c a = 0. By Lemma 3.4 (3), if the leading term LT(y) = f (c a )x a 1 ∈ ZP[x 1 , x ±1 2 ], i.e., a > 0, then y ∈ ZP[x 1 , x 2 , x ′ 2 ]. Suppose that a ≤ 0. We will prove (3.9) by induction on −a. It is enough to find an element y ′ ∈ ZP[x 1 , x ′ 1 , x 2 , x ′ 2 ] such that LT(y) = LT(y ′ . Hence the absolute value of the power of x 1 in y − y ′ is strictly less than −a. We see . . , x n , x n ] = Im(f ). Thus z ∈ Im(f ) and there exists some z 1 ∈ ZP[x 2 , x ′ 2 , . . . , x n , x ′ n ] such that z = f (z 1 ). Let y 1 = z 1 (x ′ 1 ) |a| . Then LT(y 1 ) = LT(y). This completes the proof. The following result follows immediately from Theorem 3.1 and Theorem 3.8. Corollary 3.9. If the generalized seed ( x, ρ, B) is acyclic and coprime, then the standard monomials in {x 1 , x ′ 1 , . . . , x n , x ′ n } form a ZP-basis of A( x, ρ, B). For convenience, the basis consisting of the standard monomials is called the standard monomial basis.

Appendix A. Upper generalized cluster algebras
In this appendix, we prove that the an acyclic generalized cluster algebra coincides with its corresponding upper cluster algebra without the assumption of the existence of coprimality. We mimic the proof of the main results given in [6]. For simplicity, we assume that ex = [1, n].
Definition A.1. Let ( x, ρ, B) be a generalized seed. The upper generalized cluster algebra of A( x, ρ, B) is defined as where the generalized seed ( y, ε, A) is mutation-equivalent to ( x, ρ, B).
Obviously, the upper generalized cluster algebra U ( x, ρ, B) is contained in the corresponding upper bound.
is the freezing of ( x, ρ, B) at {x i 1 , . . . , x i k }, then we have that Proof. By the definitions of the freezing of generalized cluster algebras and the upper generalized cluster algebras, the first and the third inclusions are immediate. By the Laurent phenomenon A( x, ρ, B) ⊆ U( x, ρ, B), we obtain the second inclusion.
Definition A.4. Let A( x, ρ, B) be the generalized cluster algebra. Let {A i | i ∈ I} be the set such that A i are the cluster localizations of A( x, ρ, B). For each prime ideal p of A( x, ρ, B), if there exists some i ∈ I such that pA i A i , then the set called a cover of A( x, ρ, B). x k−1 x k+1 = 1 + x k if k is odd, 1 + hx k + x 2 k if k is even.
By direct calculation, we have all cluster variables in A(x, ρ, B) as follows: 2 , x 7 = x 1 and x 8 = x 2 . Let A i denote the freezing of A(x, ρ, B) at x i for i = 1, 2. Then we have that . Since the ideal (x 1 , x 2 ) = A(x, ρ, B), we conclude that {x 1 , x 2 } p for any prime ideal p of A(x, ρ, B). It follows that {A 1 , A 2 } is a cover of A(x, ρ, B).
Lemma A.6. If A( x, ρ, B) has a cover {A i | i ∈ I} and each A i has a cover {A ij | j ∈ J i }, then {A ij | i ∈ I, j ∈ J i } is a cover of A. Namely, the covers are transitive.
Proof. Let p be a prime ideal of A( x, ρ, B), there exists some A i such that pA i A i . Since pA i is a proper ideal of A i , there exists a maximal ideal m in A i such that pA i ⊆ m. By the definition of A ij , we have m ⊆ mA ij . There exists some A ij such that mA ij A ij . Therefore pA ij ⊆ m ⊆ mA ij A ij . This completes the proof.
Lemma A.7. Let {A i | i ∈ I} be a cover of the generalized cluster algebra A( x, ρ, B). For each i ∈ I, let U i denote the upper generalized cluster algebra of A i . We have (1) A( x, ρ, B) = i∈I A i ; (2) If A i = U i for all i ∈ I, then A( x, ρ, B) = U( x, ρ, B).
Proof. The proof of the lemma is quite similar to which used in [6, Proposition 2, Lemma 2], so is omitted.
A generalized cluster algebra A( x, ρ, B) is called the isolated generalized cluster algebra if the principal part B = 0.
Proof. Proving the proposition uses the same ideas in [6,Proposition 3].