Factorial Cluster Algebras

Christof Geiß, Bernard Leclerc, and Jan Schröer

Received: October 13, 2011

Communicated by Edward Frenkel

Abstract. We show that cluster algebras do not contain non-trivial units and that all cluster variables are irreducible elements. Both statements follow from Fomin and Zelevinsky’s Laurent phenomenon. As an application we give a criterion for a cluster algebra to be a factorial algebra. This can be used to construct cluster algebras, which are isomorphic to polynomial rings. We also study various kinds of upper bounds for cluster algebras, and we prove that factorial cluster algebras coincide with their upper bounds.

2010 Mathematics Subject Classification: Primary 13F60; Secondary 13F15, 17B37

Contents

1. Introduction and main results  250
2. Invertible elements in cluster algebras  255
3. Irreducibility of cluster variables  257
4. Factorial cluster algebras  258
5. The divisibility group of a cluster algebra  260
6. Examples of non-factorial cluster algebras  261
7. Examples of factorial cluster algebras  264
8. Applications  271
References  273

1. Introduction and main results

1.1. Introduction. The introduction of cluster algebras by Fomin and Zelevinsky [FZ1] triggered an extensive theory. Most results deal with the combinatorics of seed and quiver mutations, with various categorifications of cluster algebras, and with cluster phenomena occuring in various areas of mathematics, like representation theory of finite-dimensional algebras, quantum groups and Lie theory, Calabi-Yau categories, non-commutative Donaldson-Thomas invariants, Poisson geometry, discrete dynamical systems and algebraic combinatorics.

On the other hand, there are not many results on cluster algebras themselves. As a subalgebra of a field, any cluster algebra A is obviously an integral domain. It is also easy to show that its field of fractions Frac(A) is isomorphic to a field K(x_{1}, …, x_{m}) of rational functions. Several classes of cluster algebras are known to be finitely generated, e.g. acyclic cluster algebras [BFZ, Corollary 1.21] and also a class of cluster algebras arising from Lie theory [GLS2, Theorem 3.2]. Berenstein, Fomin and Zelevinsky gave an example of a cluster algebra which is not finitely generated. (One applies [BFZ, Theorem 1.24] to the example mentioned in [BFZ, Proposition 1.26].) Only very little is known on further ring theoretic properties of an arbitrary cluster algebra A. Here are some basic questions we would like to address:

• Which elements in A are invertible, irreducible or prime?

• When is A a factorial ring?

• When is A a polynomial ring?

In this paper, we work with cluster algebras of geometric type.

1.2. Definition of a cluster algebra. In this section we repeat Fomin and Zelevinsky’s definition of a cluster algebra.

A matrix A = (a_{ij}) ∈ M_{n,n}(ℤ) is skew-symmetrizable (resp. symmetrizable) if there exists a diagonal matrix D = Diag(d_{1}, …, d_{n}) ∈ M_{n,n}(ℤ) with positive diagonal entries d_{1}, …, d_{n} such that DA is skew-symmetric (resp. symmetric), i.e. d_{i}a_{ij} = −d_{j}a_{ji} (resp. d_{i}a_{ij} = d_{j}a_{ji}) for all i, j.

Let m,n and p be integers with

m ≥ p ≥ n ≥ 1 and m > 1.

Let B = (b_{ij}) ∈ M_{m,n}(ℤ) be an (m × n)-matrix with integer entries. By B^{∘} ∈ M_{n,n}(ℤ) we denote the principal part of B, which is obtained from B by deleting the last m − n rows.

Let Δ(B) be the graph with vertices 1, …, m and an edge between i and j provided b_{ij} or b_{ji} is non-zero. We call B connected if the graph Δ(B) is connected.

Throughout, we assume that K is a field of characteristic 0 or K = ℤ. Let F := K(X_{1}, …, X_{m}) be the field of rational functions in m variables.

A seed of F is a pair (x, B) such that the following hold:

(i) B ∈ M_{m,n}(ℤ), (ii) B is connected, (iii) B^{∘} is skew-symmetrizable, (iv) x = (x_{1}, …, x_{m}) is an m-tuple of elements in F such that x_{1}, …, x_{m}

are algebraically independent over K.

For a seed (x, B), the matrix B is the exchange matrix of (x, B). We say that B has maximal rank if rank(B) = n.

Given a seed (x, B) and some 1 ≤ k ≤ n we define the mutation of (x, B) at k as

μ_{k}(x, B) := (x′, B′),

where B′ = (b′_{ij}) is defined as

b′_{ij} := \begin{cases}−b_{ij} & if i = k or j = k, \\ b_{ij} + \frac{|b_{ik}|b_{kj} + b_{ik}|b_{kj}|}{2} & otherwise, \end{cases}

and x′ = (x′_{1}, …, x′_{m}) is defined as

x′_{s} := \begin{cases}x_{k}^{−1} ∏_{b_{ik}>0} x_{i}^{b_{ik}} + x_{k}^{−1} ∏_{b_{ik}<0} x_{i}^{−b_{ik}} & if s = k, \\ x_{s} & otherwise. \end{cases}

The equality

(1)  x_{k}x′_{k} = ∏_{b_{ik}>0} x_{i}^{b_{ik}} + ∏_{b_{ik}<0} x_{i}^{−b_{ik}}

is called an exchange relation. We write

μ_{(x,B)}(x_{k}) := x′_{k}

and

μ_{k}(B) := B′.

It is easy to check that (x′, B′) is again a seed. Furthermore, we have μ_{k}μ_{k}(x, B) = (x, B).

Two seeds (x, B) and (y, C) are mutation equivalent if there exists a sequence (i_{1}, …, i_{t}) with 1 ≤ i_{j} ≤ n for all j such that

μ_{i_{t}} ⋯ μ_{i_{2}}μ_{i_{1}}(x, B) = (y, C).

In this case, we write (y, C) ∼ (x, B). This yields an equivalence relation on all seeds of F. (By definition (x, B) is also mutation equivalent to itself.)

For a seed (x, B) of F let

X_{(x,B)} := ⋃_{(y,C)∼(x,B)}{y_{1}, …, y_{n}},

where the union is over all seeds (y, C) with (y, C) ∼ (x, B). By definition, the cluster algebra A(x, B) associated to (x, B) is the L-subalgebra of F generated by X_{(x,B)}, where

L := K[x_{n+1}^{±1}, …, x_{p}^{±1}, x_{p+1}, …, x_{m}]

is the localization of the polynomial ring K[x_{n+1}, …, x_{m}] at x_{n+1} ⋯ x_{p}. (For p = n we set x_{n+1} ⋯ x_{p} := 1.) Thus A(x, B) is the K-subalgebra of F generated by

{x_{n+1}^{±1}, …, x_{p}^{±1}, x_{p+1}, …, x_{m}} ∪ X_{(x,B)}.

The elements of X_{(x,B)} are the cluster variables of A(x, B).

We call (y, C) a seed of A(x, B) if (y, C) ∼ (x, B). In this case, for any 1 ≤ k ≤ n we call (y_{k}, μ_{(y,C)}(y_{k})) an exchange pair of A(x, B). Furthermore, the m-tuple y is a cluster of A(x, B), and monomials of the form y_{1}^{a_{1}}y_{2}^{a_{2}} ⋯ y_{m}^{a_{m}} with a_{i} ≥ 0 for all i are called cluster monomials of A(x, B).

Note that for any cluster y of A(x, B) we have y_{i} = x_{i} for all n + 1 ≤ i ≤ m. These m − n elements are the coefficients of A(x, B). There are no invertible coefficients if p = n.

Clearly, for any two seeds of the form (x, B) and (y, B) there is an algebra isomorphism η: A(x, B) → A(y, B) with η(x_{i}) = y_{i} for all 1 ≤ i ≤ m, which respects the exchange relations. Furthermore, if (x, B) and (y, C) are mutation equivalent seeds, then A(x, B) = A(y, C) and we have K(x_{1}, …, x_{m}) = K(y_{1}, …, y_{m}).

1.3. Trivial cluster algebras and connectedness of exchange matrices. Note that we always assume m > 1. For m = 1 we would get the trivial cluster algebra A(x, B) with exactly two cluster variables, namely x_{1} and x′_{1} := μ_{(x,B)}(x_{1}) = x_{1}^{−1}(1 + 1). In particular, for K ≠ ℤ, both cluster variables are invertible in A(x, B), and A(x, B) is just the Laurent polynomial ring K[x_{1}^{±1}].

Furthermore, for any seed (x, B) of F the exchange matrix B is by definition connected. For non-connected B one could write A(x, B) as a product A(x_{1}, B_{1}) × A(x_{2}, B_{2}) of smaller cluster algebras and study the factors A(x_{i}, B_{i}) separately. The connectedness assumption also ensures that there are no exchange relations of the form x_{k}x′_{k} = 1 + 1.

1.4. The Laurent phenomenon. It follows by induction from the exchange relations that for any cluster y of A(x, B), any cluster variable z of A(x, B) is of the form

z = \frac{f}{g} ,

where f, g ∈ ℕ[y_{1}, …, y_{m}] are integer polynomials in the cluster variables y_{1}, …, y_{m} with non-negative coefficients. For any seed (x, B) of F let

ℒ_{x} := K[x_{1}^{±1}, …, x_{n}^{±1}, x_{n+1}^{±1}, …, x_{p}^{±1}, x_{p+1}, …, x_{m}]

be the localization of K[x_{1}, …, x_{m}] at x_{1}x_{2} ⋯ x_{p}, and let

ℒ_{x,ℤ} := ℤ[x_{1}^{±1}, …, x_{n}^{±1}, x_{n+1}, …, x_{m}]

be the localization of ℤ[x_{1}, …, x_{m}] at x_{1}x_{2} ⋯ x_{n}. We consider ℒ_{x} and ℒ_{x,ℤ} as subrings of the field F. The following remarkable result, known as the Laurent phenomenon, is due to Fomin and Zelevinsky and is our key tool to derive some ring theoretic properties of cluster algebras.

Theorem 1.1 ([FZ1, Theorem 3.1], [FZ2, Proposition 11.2]). For each seed (x, B) of F we have

A(x, B) ⊆ \overline{A}(x, B) := ⋂_{(y,C)∼(x,B)} ℒ_{y}

and

X_{(x,B)} ⊂ ⋂_{(y,C)∼(x,B)} ℒ_{y,ℤ}.

The algebra \overline{A}(x, B) is called the upper cluster algebra associated to (x, B), compare [BFZ, Section 1].

1.5. Upper bounds. For a seed (x, B) and 1 ≤ k ≤ n let (x_{k}, B_{k}) := μ_{k}(x, B). Berenstein, Fomin and Zelevinsky [BFZ] called

U(x, B) := ℒ_{x} ∩ ⋂_{k=1}^{n} ℒ_{x_{k}}

the upper bound of A(x, B). They prove the following:

Theorem 1.2 ([BFZ, Corollary 1.9]). Let (x, B) and (y, C) be mutation equivalent seeds of F. If B has maximal rank and p = m, then U(x, B) = U(y, C). In particular, we have \overline{A}(x, B) = U(x, B).

For clusters y and z of A(x, B) define

U(y, z) := ℒ_{y} ∩ ℒ_{z}.

1.6. Acyclic cluster algebras. Let (x, B) be a seed of F with B = (b_{ij}). Let Σ(B) be the quiver with vertices 1, …, n, and arrows i → j for all 1 ≤ i, j ≤ n with b_{ij} > 0, compare [BFZ, Section 1.4]. So Σ(B) encodes the sign-pattern of the principal part B^{∘} of B.

The seed (x, B) and B are called acyclic if Σ(B) does not contain any oriented cycle. The cluster algebra A(x, B) is acyclic if there exists an acyclic seed (y, C) with (y, C) ∼ (x, B).

1.7. Skew-symmetric exchange matrices and quivers. Let B = (b_{ij}) be a matrix in M_{m,n}(ℤ) such that B^{∘} is skew-symmetric. Let Γ(B) be the quiver with vertices 1, …, m and b_{ij} arrows i → j if b_{ij} > 0, and −b_{ij} arrows j → i if b_{ij} < 0. Thus given Γ(B), we can recover B. In the skew-symmetric case one often works with quivers and their mutations instead of exchange matrices.

1.8. Main results. For a ring R with 1, let R^{×} be the set of invertible elements in R. Non-zero rings without zero divisors are called integral domains. A non-invertible element a in an integral domain R is irreducible if it cannot be written as a product a = bc with b, c ∈ R both non-invertible. Cluster algebras are integral domains, since they are by definition subrings of fields.

Theorem 1.3. For any seed (x, B) of F the following hold:

(i) We have A(x, B)^{×} = {λx_{n+1}^{a_{n+1}} ⋯ x_{p}^{a_{p}} | λ ∈ K^{×}, a_{i} ∈ ℤ}.

(ii) Any cluster variable in A(x, B) is irreducible.

For elements a, b in an integral domain R we write a|b if there exists some c ∈ R with b = ac. A non-invertible element a in a commutative ring R is prime if whenever a|bc for some b, c ∈ R, then a|b or a|c. Every prime element is irreducible, but the converse is not true in general. Non-zero elements a, b ∈ R are associate if there is some unit c ∈ R^{×} with a = bc. An integral domain R is factorial if the following hold:

(i) Every non-zero non-invertible element r ∈ R can be written as a product r = a_{1} ⋯ a_{s} of irreducible elements a_{i} ∈ R.

(ii) If a_{1} ⋯ a_{s} = b_{1} ⋯ b_{t} with a_{i}, b_{j} ∈ R irreducible for all i and j, then s = t and there is a bijection π: {1, …, s} → {1, …, s} such that a_{i} and b_{π(i)} are associate for all 1 ≤ i ≤ s.

For example, any polynomial ring is factorial. In a factorial ring, all irreducible elements are prime.

Two clusters y and z of a cluster algebra A(x, B) are disjoint if {y_{1}, …, y_{n}} ∩ {z_{1}, …, z_{n}} = ∅.

The next result gives a useful criterion when a cluster algebra is a factorial ring.

Theorem 1.4. Let y and z be disjoint clusters of A(x, B). If there is a subalgebra U of A(x, B), such that U is factorial and

{y_{1}, …, y_{n}, z_{1}, …, z_{n}, x_{n+1}^{±1}, …, x_{p}^{±1}, x_{p+1}, …, x_{m}} ⊂ U,

then

U = A(x, B) = U(y, z).

In particular, A(x, B) is factorial and all cluster variables are prime.

We obtain the following corollary on upper bounds of factorial cluster algebras.

Corollary 1.5. Assume that A(x, B) is factorial.

(i) If y and z are disjoint clusters of A(x, B), then A(x, B) = U(y, z). (ii) For any (y, C) ∼ (x, B) we have A(x, B) = U(y, C).

In Section 7 we apply the above results to show that many cluster algebras are polynomial rings. In Section 8 we discuss some further applications concerning the dual of Lusztig’s semicanonical basis and monoidal categorifications of cluster algebras.

1.9. Factoriality and maximal rank. In Section 6.1 we give examples of cluster algebras A(x, B), which are not factorial. In these examples, B does not have maximal rank.

After we presented our results at the Abel Symposium in Balestrand in June 2011, Zelevinsky asked the following question:

Problem 1.6. Suppose (x, B) is a seed of F such that B has maximal rank. Does it follow that A(x, B) is factorial?

After we circulated a first version of this article, Philipp Lampe [La] discovered an example of a non-factorial cluster algebra A(x, B) with B having maximal rank. With his permission, we explain a generalization of his example in Section 6.2.

2. Invertible elements in cluster algebras

In this section we prove Theorem 1.3(i), classifying the invertible elements of cluster algebras.

The following lemma is straightforward and well-known.

Lemma 2.1. For any seed (x, B) of F we have

ℒ_{x}^{×} = {λx_{1}^{a_{1}} ⋯ x_{p}^{a_{p}} | λ ∈ K^{×}, a_{i} ∈ ℤ}.

Theorem 2.2. For any seed (x, B) of F we have

A(x, B)^{×} = {λx_{n+1}^{a_{n+1}} ⋯ x_{p}^{a_{p}} | λ ∈ K^{×}, a_{i} ∈ ℤ} .

Proof. Let u be an invertible element in A(x, B), and let (y, C) be any seed of A(x, B). By the Laurent phenomenon Theorem 1.1 we know that A(x, B) ⊆ ℒ_{y}. It follows that u is also invertible in ℒ_{y}. Thus by Lemma 2.1 there are a_{1}, …, a_{p} ∈ ℤ and λ ∈ K^{×} such that u = λM, where

M = y_{1}^{a_{1}} ⋯ y_{k}^{a_{k}} ⋯ y_{p}^{a_{p}}.

If all a_{i} with 1 ≤ i ≤ n are zero, we are done. To get a contradiction, assume that there is some 1 ≤ k ≤ n with a_{k} ≠ 0.

Let y_{k}^{∗} := μ_{(y,C)}(y_{k}). Again the Laurent phenomenon yields b_{1}, …, b_{p} ∈ ℤ and ν ∈ K^{×} such that

u = νy_{1}^{b_{1}} ⋯ y_{k−1}^{b_{k−1}}(y_{k}^{∗})^{b_{k}}y_{k+1}^{b_{k+1}} ⋯ y_{p}^{b_{p}}.

Without loss of generality let b_{k} ≥ 0. (Otherwise we can work with u^{−1} instead of u.)

If b_{k} = 0, we get

λy_{1}^{a_{1}} ⋯ y_{k}^{a_{k}} ⋯ y_{p}^{a_{p}} = νy_{1}^{b_{1}} ⋯ y_{k−1}^{b_{k−1}}y_{k+1}^{b_{k+1}} ⋯ y_{p}^{b_{p}},

where λ, ν ∈ K^{×}. This is a contradiction, because a_{k} ≠ 0 and y_{1}, …, y_{m} are algebraically independent, and therefore Laurent monomials in y_{1}, …, y_{m} are linearly independent in F.

Next, assume that b_{k} > 0. By definition we have

y_{k}^{∗} = M_{1} + M_{2}

with

M_{1} = y_{k}^{−1} ∏_{c_{ik}>0} y_{i}^{c_{ik}} and M_{2} = y_{k}^{−1} ∏_{c_{ik}<0} y_{i}^{−c_{ik}},

where the products run over the positive, respectively negative, entries in the kth column of the matrix C.

Thus we get an equality of the form

(2)  u = λM = ν(y_{1}^{b_{1}} ⋯ y_{k−1}^{b_{k−1}})(M_{1} + M_{2})^{b_{k}}(y_{k+1}^{b_{k+1}} ⋯ y_{p}^{b_{p}}).

We know that M_{1} ≠ M_{2}. (Here we use that m > 1 and that exchange matrices are by definition connected. Otherwise, one could get exchange relations of the form x_{k}x′_{k} = 1 + 1.) Thus the right-hand side of Equation (2) is a nontrivial linear combination of b_{k} + 1 ≥ 2 pairwise different Laurent monomials in y_{1}, …, y_{m}. This is again a contradiction, since y_{1}, …, y_{m} are algebraically independent. □

Corollary 2.3. For any seed (x, B) of F the following hold:

(i) Let y and z be non-zero elements in A(x, B). Then y and z are associate if and only if there exist a_{n+1}, …, a_{p} ∈ ℤ and λ ∈ K^{×} with

y = λx_{n+1}^{a_{n+1}} ⋯ x_{p}^{a_{p}}z.

(ii) Let y and z be cluster variables of A(x, B). Then y and z are associate if and only if y = z.

Proof. Part (i) follows directly from Theorem 2.2. To prove (ii), let y and z be clusters of A(x, B). Assume y_{i} and z_{j} are associate for some 1 ≤ i, j ≤ n. By (i) there are a_{n+1}, …, a_{p} ∈ ℤ and λ ∈ K^{×} with y_{i} = λx_{n+1}^{a_{n+1}} ⋯ x_{p}^{a_{p}}z_{j}. By Theorem 1.1 we know that there exist b_{1}, …, b_{n} ∈ ℤ and a polynomial f in ℤ[z_{1}, …, z_{m}] with

y_{i} = \frac{f}{z_{1}^{b_{1}} ⋯ z_{n}^{b_{n}}}

and f is not divisible by any z_{1}, …, z_{n}. The polynomial f and b_{1}, …, b_{n} are uniquely determined by y_{i}. It follows that λ ∈ ℤ and a_{n+1}, …, a_{p} ≥ 0. But we also have z_{j} = λ^{−1}x_{n+1}^{−a_{n+1}} ⋯ x_{p}^{−a_{p}}y_{i}. Reversing the role of y_{i} and z_{j} we get −a_{n+1}, …, −a_{p} ≥ 0 and λ^{−1} ∈ ℤ. This implies y_{i} = z_{j} or −y_{i} = z_{j}. By the remark at the beginning of Section 1.4 we know that z_{j} = f/g for some f, g ∈ ℕ[y_{1}, …, y_{m}]. Assume that −y_{i} = z_{j}. We get z_{j} = −y_{i} = f/g and therefore f + y_{i}g = 0. This is a contradiction to the algebraic independence of y_{1}, …, y_{m}. Thus we proved (ii). □

We thank Giovanni Cerulli Irelli for helping us with the final step of the proof of Corollary 2.3(ii).

Two clusters y and z of a cluster algebra A(x, B) are non-associate if there are no 1 ≤ i, j ≤ n such that y_{i} and z_{j} are associate.

Corollary 2.4. For clusters y and z of A(x, B) the following are equivalent:

(i) The clusters y and z are non-associate.

(ii) The clusters y and z are disjoint.

Proof. Non-associate clusters are obviously disjoint. The converse follows directly from Corollary 2.3(ii). □

3. Irreducibility of cluster variables

In this section we prove Theorem 1.3(ii). The proof is very similar to the proof of Theorem 2.2.

Theorem 3.1. Let (x, B) be a seed of F. Then any cluster variable in A(x, B) is irreducible.

Proof. Let (y, C) be any seed of A(x, B). We know from Theorem 2.2 that the cluster variables of A(x, B) are non-invertible in A(x, B).

Assume that y_{k} is not irreducible for some 1 ≤ k ≤ n. Thus y_{k} = y′_{k}y′′_{k} for some non-invertible elements y′_{k} and y′′_{k} in A(x, B). Since y_{k} is invertible in ℒ_{y}, we know that y′_{k} and y′′_{k} are both invertible in ℒ_{y}. Thus by Lemma 2.1 there are a_{i}, b_{i} ∈ ℤ and λ′, λ′′ ∈ K^{×} with

y′_{k} = λ′y_{1}^{a_{1}} ⋯ y_{s}^{a_{s}} ⋯ y_{p}^{a_{p}} and y′′_{k} = λ′′y_{1}^{b_{1}} ⋯ y_{s}^{b_{s}} ⋯ y_{p}^{b_{p}}.

Since y_{k} = y′_{k}y′′_{k}, we get a_{s} + b_{s} = 0 for all s ≠ k and a_{k} + b_{k} = 1.

Assume that a_{s} = 0 for all 1 ≤ s ≤ n with s ≠ k. Then y′_{k} = λ′y_{k}^{a_{k}}y_{n+1}^{a_{n+1}} ⋯ y_{p}^{a_{p}} and y′′_{k} = λ′′y_{k}^{b_{k}}y_{n+1}^{b_{n+1}} ⋯ y_{p}^{b_{p}}. If a_{k} ≤ 0, then y′_{k} is invertible in A(x, B), and if a_{k} > 0, then y′′_{k} is invertible in A(x, B). In both cases we get a contradiction.

Next assume a_{s} ≠ 0 for some 1 ≤ s ≤ n with s ≠ k. Let y_{s}^{∗} := μ_{(y,C)}(y_{s}). Thus we have

y_{s}^{∗} = M_{1} + M_{2}

with

M_{1} = y_{s}^{−1} ∏_{c_{is}>0} y_{i}^{c_{is}} and M_{2} = y_{s}^{−1} ∏_{c_{is}<0} y_{i}^{−c_{is}},

where the products run over the positive, respectively negative, entries in the sth column of the matrix C.

Since s ≠ k, we see that y_{k} and therefore also y′_{k} and y′′_{k} are invertible in ℒ_{μ_{s}(y,C)}. Thus by Lemma 2.1 there are c_{i}, d_{i} ∈ ℤ and ν′, ν′′ ∈ K^{×} with

y′_{k} = ν′y_{1}^{c_{1}} ⋯ y_{s−1}^{c_{s−1}}(y_{s}^{∗})^{c_{s}}y_{s+1}^{c_{s+1}} ⋯ y_{p}^{c_{p}}

and

y′′_{k} = ν′′y_{1}^{d_{1}} ⋯ y_{s−1}^{d_{s−1}}(y_{s}^{∗})^{d_{s}}y_{s+1}^{d_{s+1}} ⋯ y_{p}^{d_{p}}.

Note that c_{s} + d_{s} = 0. Without loss of generality we assume that c_{s} ≥ 0. (If c_{s} < 0, we continue to work with y′′_{k} instead of y′_{k}.) If c_{s} = 0, we get

y′_{k} = λ′y_{1}^{a_{1}} ⋯ y_{s}^{a_{s}} ⋯ y_{p}^{a_{p}} = ν′y_{1}^{c_{1}} ⋯ y_{s}^{0} ⋯ y_{p}^{c_{p}}.

This is a contradiction, since a_{s} ≠ 0 and y_{1}, …, y_{m} are algebraically independent. If c_{s} > 0, then

y′_{k} = λ′y_{1}^{a_{1}} ⋯ y_{s−1}^{a_{s−1}}y_{s}^{a_{s}}y_{s+1}^{a_{s+1}} ⋯ y_{p}^{a_{p}}
= ν′y_{1}^{c_{1}} ⋯ y_{s−1}^{c_{s−1}}(y_{s}^{∗})^{c_{s}}y_{s+1}^{c_{s+1}} ⋯ y_{p}^{c_{p}}
= ν′y_{1}^{c_{1}} ⋯ y_{s−1}^{c_{s−1}}(M_{1} + M_{2})^{c_{s}}y_{s+1}^{c_{s+1}} ⋯ y_{p}^{c_{p}}.

We know that M_{1} ≠ M_{2}. Thus the Laurent monomial y′_{k} is a non-trivial linear combination of c_{s} + 1 ≥ 2 pairwise different Laurent monomials in y_{1}, …, y_{m}, a contradiction. □

Note that the coefficients x_{p+1}, …, x_{m} of A(x, B) are obviously irreducible in ℒ_{x}. Since A(x, B) ⊆ ℒ_{x}, they are also irreducible in A(x, B).

4. Factorial cluster algebras

4.1. A factoriality criterion. This section contains the proofs of Theorem 1.4 and Corollary 1.5.

Theorem 4.1. Let y and z be disjoint clusters of A(x, B), and let U be a factorial subalgebra of A(x, B) such that

{y_{1}, …, y_{n}, z_{1}, …, z_{n}, x_{n+1}^{±1}, …, x_{p}^{±1}, x_{p+1}, …, x_{m}} ⊂ U.

Then we have

U = A(x, B) = U(y, z).

Proof. Let u ∈ U(y, z) = ℒ_{y} ∩ ℒ_{z}. Thus we have

u = \frac{f}{y_{1}^{a_{1}}y_{2}^{a_{2}} ⋯ y_{p}^{a_{p}}} = \frac{g}{z_{1}^{b_{1}}z_{2}^{b_{2}} ⋯ z_{p}^{b_{p}}} ,

where f is a polynomial in y_{1}, …, y_{m}, and g is a polynomial in z_{1}, …, z_{m}, and a_{i}, b_{i} ≥ 0 for all 1 ≤ i ≤ p. By the Laurent phenomenon it is enough to show that u ∈ U.

Since y_{i}, z_{i} ∈ U for all 1 ≤ i ≤ m, we get the identity

fz_{1}^{b_{1}}z_{2}^{b_{2}} ⋯ z_{n}^{b_{n}}z_{n+1}^{b_{n+1}} ⋯ z_{p}^{b_{p}} = gy_{1}^{a_{1}}y_{2}^{a_{2}} ⋯ y_{n}^{a_{n}}y_{n+1}^{a_{n+1}} ⋯ y_{p}^{a_{p}}

in U.

By Theorem 3.1 the cluster variables y_{i} and z_{i} with 1 ≤ i ≤ n are irreducible in A(x, B). In particular, they are irreducible in the subalgebra U of A(x, B). The elements y_{n+1}^{a_{n+1}} ⋯ y_{p}^{a_{p}} and z_{n+1}^{b_{n+1}} ⋯ z_{p}^{b_{p}} are units in U. (Recall that x_{i} = y_{i} = z_{i} for all n + 1 ≤ i ≤ m.)

The clusters y and z are disjoint. Now Corollary 2.4 implies that the elements y_{i} and z_{j} are non-associate for all 1 ≤ i, j ≤ n. Thus, by the factoriality of U, the monomial y_{1}^{a_{1}}y_{2}^{a_{2}} ⋯ y_{n}^{a_{n}} divides f in U. In other words there is some h ∈ U with f = hy_{1}^{a_{1}}y_{2}^{a_{2}} ⋯ y_{n}^{a_{n}}. It follows that

u = \frac{f}{y_{1}^{a_{1}}y_{2}^{a_{2}} ⋯ y_{p}^{a_{p}}} = \frac{hy_{1}^{a_{1}}y_{2}^{a_{2}} ⋯ y_{n}^{a_{n}}}{y_{1}^{a_{1}}y_{2}^{a_{2}} ⋯ y_{p}^{a_{p}}} = \frac{h}{y_{n+1}^{a_{n+1}} ⋯ y_{p}^{a_{p}}} = hy_{n+1}^{−a_{n+1}} ⋯ y_{p}^{−a_{p}}.

Since h ∈ U and y_{n+1}^{±1}, …, y_{p}^{±1} ∈ U, we get u ∈ U. This finishes the proof. □

Corollary 4.2. Assume that A(x, B) is factorial.

(i) If y and z are disjoint clusters of A(x, B), then A(x, B) = U(y, z). (ii) For any (y, C) ∼ (x, B) we have A(x, B) = U(y, C).

Proof. Part (i) follows directly from Theorem 1.4. To prove part (ii), assume (y, C) ∼ (x, B) and let u ∈ U(y, C). For 1 ≤ k ≤ n let (y_{k}, C_{k}) := μ_{k}(y, C) and y_{k}^{∗} := μ_{(y,C)}(y_{k}). We get

u = \frac{f}{y_{1}^{a_{1}} ⋯ y_{k}^{a_{k}} ⋯ y_{p}^{a_{p}}} = \frac{f_{k}}{y_{1}^{b_{1}} ⋯ (y_{k}^{∗})^{b_{k}} ⋯ y_{p}^{b_{p}}}

for a polynomial f in y_{1}, …, y_{k}, …, y_{m}, a polynomial f_{k} in y_{1}, …, y_{k}^{∗}, …, y_{m}, and a_{i}, b_{i} ≥ 0. This yields an equality

(3)  fy_{1}^{b_{1}} ⋯ (y_{k}^{∗})^{b_{k}} ⋯ y_{p}^{b_{p}} = f_{k}y_{1}^{a_{1}} ⋯ y_{k}^{a_{k}} ⋯ y_{p}^{a_{p}}

in A(x, B). Now we argue similarly as in the proof of Theorem 4.1. The cluster variables y_{1}, …, y_{n}, y_{1}^{∗}, …, y_{n}^{∗} are obviously pairwise different. Now Corollary 2.3(ii) implies that they are pairwise non-associate, and by Theorem 3.1 they are irreducible in A(x, B). Thus by the factoriality of A(x, B), Equation (3) implies that y_{k}^{a_{k}} divides f in A(x, B). Since this holds for all 1 ≤ k ≤ n, we get that y_{1}^{a_{1}} ⋯ y_{k}^{a_{k}} ⋯ y_{n}^{a_{n}} divides f in A(x, B). It follows that u ∈ A(x, B). □

4.2. Existence of disjoint clusters. One assumption of Theorem 4.1 is the existence of disjoint clusters in A(x, B). We can prove this under a mild assumption. But it should be true in general.

Proposition 4.3. Assume that the cluster monomials of A(x, B) are linearly independent. Let (y, C) be a seed of A(x, B), and let

(z, D) := μ_{n} ⋯ μ_{2}μ_{1}(y, C).

Then the clusters y and z are disjoint.

Proof. Set (y[0], C[0]) := (y, C), and for 1 ≤ k ≤ n let (y[k], C[k]) := μ_{k}(y[k − 1], C[k − 1]) and (y_{1}[k], …, y_{m}[k]) := y[k]. We claim that

{y_{1}[k], …, y_{k}[k]} ∩ {y_{1}, …, y_{n}} = ∅.

For k = 1 this is straightforward. Thus let k ≥ 2, and assume that our claim is true for k − 1. To get a contradiction, assume that y_{k}[k] = y_{j} for some 1 ≤ j ≤ n. (By the induction assumption we know that {y_{1}[k], …, y_{k−1}[k]} ∩ {y_{1}, …, y_{n}} = ∅, since y_{i}[k] = y_{i}[k − 1] for all 1 ≤ i ≤ k − 1.)

We have y[k] = (y_{1}[k], …, y_{k}[k], y_{k+1}, …, y_{m}).

Since y_{1}[k], …, y_{k}[k], y_{k+1}, …, y_{m} are algebraically independent and y_{k}[k] ≠ y_{k}, we get 1 ≤ j ≤ k − 1. Since (y[j], C[j]) = μ_{j}(y[j − 1], C[j − 1]), it follows that (y_{j}[j − 1], y_{j}[j]) is an exchange pair of A(x, B). Next, observe that y_{k}[k] = y_{j} = y_{j}[j − 1] and y_{j}[k] = y_{j}[j]. Thus y_{j}[j − 1] and y_{j}[j] are both contained in {y_{1}[k], …, y_{m}[k]}, and therefore y_{j}[j − 1]y_{j}[j] is a cluster monomial. The corresponding exchange relation gives a contradiction to the linear independence of cluster monomials. □

Fomin and Zelevinsky [FZ3, Conjecture 4.16] conjecture that the cluster monomials of A(x, B) are always linearly independent. Under the assumptions that B has maximal rank and that B^{∘} is skew-symmetric, the conjecture follows from [DWZ, Theorem 1.7].

5. The divisibility group of a cluster algebra

Let R be an integral domain, and let Frac(R) be the field of fractions of R. Set Frac(R)^{∗} := Frac(R) \ {0}. The abelian group

G(R) := (Frac(R)^{∗}/R^{×}, ·)

is the divisibility group of R.

For g, h ∈ Frac(R)^{∗} let g ≤ h provided hg^{−1} ∈ R. This relation is reflexive and transitive and it induces a partial ordering on G(R).

Let I be a set. The abelian group (ℤ^{(I)}, +) is equipped with the following partial ordering: We set (x_{i})_{i∈I} ≤ (y_{i})_{i∈I} if x_{i} ≤ y_{i} for all i. (By definition, the elements in ℤ^{(I)} are tuples (x_{i})_{i∈I} of integers x_{i} such that only finitely many x_{i} are non-zero.)

There is the following well-known criterion for the factoriality of R, see for example [C, Section 2].

Proposition 5.1. For an integral domain R the following are equivalent:

(i) R is factorial.

(ii) There is a set I and a group isomorphism

ϕ: G(R) → ℤ^{(I)}

such that for all g, h ∈ G(R) we have g ≤ h if and only if ϕ(g) ≤ ϕ(h).

Not all cluster algebras A(x, B) are factorial, but at least one part of the above factoriality criterion is satisfied:

Proposition 5.2. For any seed (x, B) of F the divisibility group G(A(x, B)) is isomorphic to ℤ^{(I)}, where

I := {f ∈ K[x_{1}, …, x_{m}] | f is irreducible and f ≠ x_{i} for n + 1 ≤ i ≤ p} /K^{×}

is the set of irreducible polynomials unequal to any x_{n+1}, …, x_{p} in K[x_{1}, …, x_{m}] up to non-zero scalar multiples.

Proof. By the Laurent phenomenon and the definition of a seed we get

Frac(A(x, B)) = Frac(ℒ_{x}) = K(x_{1}, …, x_{m}).

Furthermore, by Theorem 2.2 we have

A(x, B)^{×} = {λx_{n+1}^{a_{n+1}} ⋯ x_{p}^{a_{p}} | λ ∈ K^{×}, a_{i} ∈ ℤ}.

Any element in K(x_{1}, …, x_{m}) is of the form f_{1} ⋯ f_{s}g_{1}^{−1} ⋯ g_{t}^{−1} with f_{i}, g_{j} irreducible in K[x_{1}, …, x_{m}]. Using that the polynomial ring K[x_{1}, …, x_{m}] is factorial, and working modulo A(x, B)^{×} yields the result. □

6. Examples of non-factorial cluster algebras

6.1. For a matrix A ∈ M_{m,n}(ℤ) and 1 ≤ i ≤ n let c_{i}(A) be the ith column of A.

Proposition 6.1. Let (x, B) be a seed of F. Assume that c_{k}(B) = c_{s}(B) or c_{k}(B) = −c_{s}(B) for some k ≠ s with b_{ks} = 0. Then A(x, B) is not factorial.

Proof. Define (y, C) := μ_{k}(x, B) and (z, D) := μ_{s}(y, C). We get

y_{k} = z_{k} = x_{k}^{−1}(M_{1} + M_{2}),

where

M_{1} := ∏_{b_{ik}>0} x_{i}^{b_{ik}} and M_{2} := ∏_{b_{ik}<0} x_{i}^{−b_{ik}}.

By the mutation rule, we have c_{k}(C) = −c_{k}(B), and since b_{ks} = 0, we get c_{s}(C) = c_{s}(B). Since c_{s}(B) = c_{k}(B) or c_{s}(B) = −c_{k}(B), this implies that

z_{s} = x_{s}^{−1}(M_{1} + M_{2}).

The cluster variables x_{k}, x_{s}, z_{k}, z_{s} are pairwise different. Thus they are pairwise non-associate by Corollary 2.3(ii), and by Theorem 3.1 they are irreducible in A(x, B). Obviously, we have

x_{k}z_{k} = x_{s}z_{s}.

Thus A(x, B) is not factorial. □

To give a concrete example of a cluster algebra, which is not factorial, assume m = n = p = 3, and let B ∈ M_{m,n}(ℤ) be the matrix

B = (\begin{matrix}0 & −1 & 0 \\ 1 & 0 & −1 \\ 0 & 1 & 0 \end{matrix}) .

The matrix B obviously satisfies the assumptions of Proposition 6.1. Note that B = B^{∘} is skew-symmetric, and that Γ(B) is the quiver

\xymatrix{3 \ar[r] & 2 \ar[r] & 1.}

Thus A(x, B) is a cluster algebra of Dynkin type 𝔸_{3}. (Cluster algebras with finitely many cluster variables are classified via Dynkin types, for details see [FZ2].)

Define (z, D) := μ_{3}μ_{1}(x, B). We get z_{1} = x_{1}^{−1}(1 + x_{2}), z_{3} = x_{3}^{−1}(1 + x_{2}) and therefore x_{1}z_{1} = x_{3}z_{3}.

Clearly, the cluster variables x_{1}, x_{3}, z_{1}, z_{3} are pairwise different. Using Corollary 2.3(ii) we get that x_{1}, x_{3}, z_{1}, z_{3} are pairwise non-associate, and by Theorem 3.1 they are irreducible. Thus A(x, B) is not factorial.

6.2. The next example is due to Philipp Lampe. It gives a negative answer to Zelevinsky’s Question 1.6.

Proposition 6.2 ([La]). Let K = ℂ, m = n = 2 and

B = (\begin{matrix}0 & −2 \\ 2 & 0 \end{matrix}) .

Then A(x, B) is not factorial.

The proof of the following result is a straightforward generalization of Lampe’s proof of Proposition 6.2.

Proposition 6.3. Let (x, B) be a seed of F. Assume that there exists some 1 ≤ k ≤ n such that the polynomial X^{d} + Y^{d} is not irreducible in K[X, Y], where d := gcd(b_{1k}, …, b_{mk}) is the greatest common divisor of b_{1k}, …, b_{mk}. Then A(x, B) is not factorial.

Proof. Let X^{d} + Y^{d} = f_{1} ⋯ f_{t}, where the f_{j} are irreducible polynomials in K[X, Y]. Since X^{d} + Y^{d} is not irreducible in K[X, Y], we have t ≥ 2. Let y_{k} := μ_{(x,B)}(x_{k}). The corresponding exchange relation is

x_{k}y_{k} = ∏_{b_{ik}>0} x_{i}^{b_{ik}} + ∏_{b_{ik}<0} x_{i}^{−b_{ik}} = M^{d} + N^{d} = ∏_{j=1}^{t} f_{j}(M, N),

where

M := ∏_{b_{ik}>0} x_{i}^{b_{ik}/d} and N := ∏_{b_{ik}<0} x_{i}^{−b_{ik}/d}.

Clearly, each f_{j}(M, N) is contained in A(x, B). To get a contradiction, assume that A(x, B) is factorial. By Theorem 2.2 none of the elements f_{j}(M, N) is invertible in A(x, B). Since A(x, B) is factorial, each f_{j}(M, N) is equal to a product f_{1j} ⋯ f_{a_{j}j}, where the f_{ij} are irreducible in A(x, B) and a_{j} ≥ 1. By Theorem 3.1 the cluster variables x_{k} and y_{k} are irreducible in A(x, B). It follows that a_{1} + ⋯ + a_{t} = 2, since A(x, B) is factorial. This implies t = 2 and a_{1} = a_{2} = 1. In particular, f_{1}(M, N) and f_{2}(M, N) are irreducible in A(x, B), and we have x_{k}y_{k} = f_{1}(M, N)f_{2}(M, N). For j = 1, 2 the elements x_{k} and f_{j}(M, N) cannot be associate, since f_{j}(M, N) is just a K-linear combination of monomials in {x_{1}, …, x_{m}} \ {x_{k}}. (Here we use Corollary 2.3(i) and the fact that b_{kk} = 0.) This is a contradiction to the factoriality of A(x, B). □

Note that a polynomial of the form X^{d} + Y^{d} is irreducible if and only if X^{d} + 1 is irreducible.

Corollary 6.4. Let K = ℂ, m = n = 2 and

B = (\begin{matrix}0 & −c \\ d & 0 \end{matrix})

with c ≥ 1 and d ≥ 2. Then A(x, B) is not factorial.

Proof. For k = 1 the assumptions of Proposition 6.3 hold. (We have gcd(0, d) = d, and the polynomial X^{d} + 1 is not irreducible in ℂ[X].) □

Corollary 6.5. Let m = n = 2 and

B = (\begin{matrix}0 & −c \\ d & 0 \end{matrix})

with c ≥ 1 and d ≥ 3 an odd number. Then A(x, B) is not factorial.

Proof. For k = 1 the assumptions of Proposition 6.3 hold. (We have gcd(0, d) = d, and for odd d we have

X^{d} + 1 = (X + 1) (∑_{j=0}^{d−1}(−1)^{j}X^{j}) .

Thus X^{d} + 1 is not irreducible in K[X].) □

7. Examples of factorial cluster algebras

7.1. Cluster algebras of Dynkin type 𝔸 as polynomial rings. Assume m = n + 1 = p + 1, and let B ∈ M_{m,n}(ℤ) be the matrix

B = (\begin{matrix}0 & −1 \\ 1 & 0 & −1 \\ & 1 & 0 & ... \\ & & 1 & ... & −1 \\ & & & ... & 0 & −1 \\ & & & & 1 & 0 \\ & & & & & 1 \end{matrix}) .

Obviously, B^{∘} is skew-symmetric, Γ(B) is the quiver

\xymatrix{m \ar[r] & ⋯ \ar[r] & 2 \ar[r] & 1,}

and A(x, B) is a cluster algebra of Dynkin type 𝔸_{n}. Note that A(x, B) has exactly one coefficient, and that this coefficient is non-invertible.

Let (x[0], B[0]) := (x, B). For each 1 ≤ i ≤ m − 1 we define inductively a seed by

(x[i], B[i]) := μ_{m−i} ⋯ μ_{2}μ_{1}(x[i − 1], B[i − 1]).

For 0 ≤ i ≤ m − 1 set (x_{1}[i], …, x_{m}[i]) := x[i].

For simplicity we define x_{0}[i] := 1 and x_{−1}[i] := 0 for all i.

Lemma 7.1. For 0 ≤ i ≤ m − 2, 1 ≤ k ≤ m − 1 − i and 0 ≤ j ≤ i we have

(4) μ_{(x[i],B[i])}(x_{k}[i]) = \frac{x_{k−1}[i] + x_{k+1}[i]}{x_{k}[i]} = \frac{x_{k−1+j}[i − j] + x_{k+1+j}[i − j]}{x_{k+j}[i − j]}.

Proof. The first equality follows from the definition of (x[i], B[i]) and the mutation rule. The second equality is proved by induction on i. □

Corollary 7.2. For 0 ≤ i ≤ m − 2 we have

(5)  x_{i+2} = x_{1}[i + 1]x_{i+1} − x_{i},
(6)  x_{i+1}[1] = x_{1}[i + 1]x_{i}[1] − x_{i−1}[1].

Proof. Equation (5) follows from (4) for k = 1 and j = i. The case k = 1 and j = i − 1 yields Equation (6). □

Proposition 7.3. The elements x_{1}[0], x_{1}[1], …, x_{1}[m − 1] are algebraically independent and

K[x_{1}[0], x_{1}[1], …, x_{1}[m − 1]] = A(x, B).

In particular, the cluster algebra A(x, B) is a polynomial ring in m variables.

Proof. It follows from Equation (5) that

x_{1}[i] ∈ K(x_{1}, …, x_{i+1}) \ K(x_{1}, …, x_{i})

for all 1 ≤ i ≤ m − 1. Since x_{1}, …, x_{m} are algebraically independent, this implies that x_{1}[0], x_{1}[1], …, x_{1}[m − 1] are algebraically independent as well. Thus

U := K[x_{1}[0], x_{1}[1], …, x_{1}[m − 1]]

is a polynomial ring in m variables. In particular, U is factorial. Equation (5) implies that x_{1}, …, x_{m} ∈ U, and Equation (6) yields that x_{1}[1], …, x_{m}[1] ∈ U. Clearly, the clusters x and x[1] are disjoint. Thus the assumptions of Theorem 4.1 are satisfied, and we get U = A(x, B). □

The cluster algebra A(x, B) as defined above has been studied by several people. It is related to a T -system of Dynkin type 𝔸_{1} with a certain boundary condition, see [DK]. Furthermore, for K = ℂ the cluster algebra A(x, B) is naturally isomorphic to the complexified Grothendieck ring of the category C_{n} of finitedimensional modules of level n over the quantum loop algebra of Dynkin type 𝔸_{1}, see [HL, N2]. It is well known, that A(x, B) is a polynomial ring. We just wanted to demonstrate how to use Theorem 4.1 in practise.

7.2. Acyclic cluster algebras as polynomial rings. Let C = (c_{ij}) ∈ M_{n,n}(ℤ) be a generalized Cartan matrix, i.e. C is symmetrizable, c_{ii} = 2 for all i and c_{ij} ≤ 0 for all i ≠ j.

Assume that m = 2n = 2p, and let (x, B) be a seed of F, where B = (b_{ij}) ∈ M_{2n,n}(ℤ) is defined as follows: For 1 ≤ i ≤ 2n and 1 ≤ j ≤ n let

b_{ij} := \begin{cases}0 & if i = j, \\ −c_{ij} & if 1 ≤ i < j ≤ n, \\ c_{ij} & if 1 ≤ j < i ≤ n, \\ 1 & if i = n + j, \\ c_{i−n,j} & if n + 1 ≤ i ≤ 2n and i − n < j, \\ 0 & if n + 1 ≤ i ≤ 2n and i − n > j. \end{cases}

Thus we have

B = (\begin{matrix}0 & b_{12} & b_{13} & ⋯ & b_{1n} \\ b_{21} & 0 & b_{23} & ⋯ & b_{2n} \\ b_{31} & b_{32} & 0 & ... & ⋮ \\ ⋮ & ⋮ & ... & ... & b_{n−1,n} \\ b_{n1} & b_{n2} & ⋯ & b_{n,n−1} & 0 \\ 1 & −b_{12} & −b_{13} & ⋯ & −b_{1n} \\ 0 & 1 & −b_{23} & ⋯ & −b_{2n} \\ 0 & 0 & 1 & ... & ⋮ \\ ⋮ & ⋮ & ... & ... & −b_{n−1,n} \\ 0 & 0 & 0 & 0 & 1 \end{matrix}) .

Clearly, (x, B) is an acyclic seed. Namely, if i → j is an arrow in Σ(B), then i < j. Up to simultaneous reordering of columns and rows, each acyclic skewsymmetrizable matrix in M_{n,n}(ℤ) is of the form B^{∘} with B defined as above. Note that A(x, B) has exactly n coefficients, and that all these coefficients are non-invertible.

For 1 ≤ i ≤ n let

(x[1], B[1]) := μ_{n} ⋯ μ_{2}μ_{1}(x, B)

and (x_{1}[1], …, x_{2n}[1]) := x[1]. Let B_{0} := B, and for 1 ≤ i ≤ n let B_{i} := μ_{i}(B_{i−1}). Thus we have B_{n} = B[1]. It is easy to work out the matrices B_{i} explicitly: The matrix B_{i} is obtained from B_{i−1} by changing the sign in the ith row and the ith column of the principal part B_{i−1}^{∘}. Furthermore, the (n + i)th row

(0, …, 0, 1, −b_{i,i+1}, −b_{i,i+2}, …, −b_{in})

of B_{i−1} gets replaced by

(−b_{i1}, −b_{i2}, …, −b_{i,i−1}, −1, 0, …, 0).

If we write N_{+} (resp. N_{−}) for the upper (resp. lower) triangular part of B^{∘}, we get

B = (\begin{matrix}1 & & & & N_{+} \\ & . & . & . \\ N_{−} & & & & 1 \\ 1 & & & & −N_{+} \\ & . & . & . \\ 0 & & & & 1 \end{matrix}) and B[1] = (\begin{matrix}1 & & & & N_{+} \\ & . & . & . \\ N_{−} & & & & 1 \\ −1 & & & & 0 \\ & . & . & . \\ −N_{−} & & & & −1 \end{matrix}) .

In particular, the principal part B^{∘} of B is equal to the principal part B[1]^{∘} of B[1].

Documenta Mathematica 18 (2013) 249–274

Now the definition of seed mutation yields

(7)  x_{k}[1] = x_{k}^{−1} (x_{n+k} + ∏_{i=1}^{k−1} x_{i}[1]^{b_{ik}} ∏_{i=k+1}^{n} x_{i}^{−b_{ik}})

for 1 ≤ k ≤ n.

Proposition 7.4. The elements x_{1}, …, x_{n}, x_{1}[1], …, x_{n}[1] are algebraically independent and

K[x_{1}, …, x_{n}, x_{1}[1], …, x_{n}[1]] = A(x, B).

In particular, the cluster algebra A(x, B) is a polynomial ring in 2n variables.

Proof. By Equation (7) and induction we have

x_{k}[1] ∈ K(x_{1}, …, x_{n+k}) \ K(x_{1}, …, x_{n+k−1})

for all 1 ≤ k ≤ n. It follows that x_{1}, …, x_{n}, x_{1}[1], …, x_{n}[1] are algebraically independent, and that the clusters x and x[1] are disjoint. Let

U := K[x_{1}, …, x_{n}, x_{1}[1], …, x_{n}[1]].

Thus U is a polynomial ring in 2n variables. In particular, U is factorial. It follows from Equation (7) that

(8)  x_{n+k} = x_{k}[1]x_{k} − ∏_{i=1}^{k−1} x_{i}[1]^{b_{ik}} ∏_{i=k+1}^{n} x_{i}^{−b_{ik}}.

This implies x_{n+k} ∈ U for all 1 ≤ k ≤ n. Thus the assumptions of Theorem 4.1 are satisfied, and we can conclude that U = A(x, B). □

Proposition 7.4 is a special case of a much more general result proved in [GLS2]. But the proof presented here is new and more elementary.

Next, we compare the basis

P_{GLS} := {x[a] := x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}}x_{1}[1]^{a_{n+1}} ⋯ x_{n}[1]^{a_{2n}} | a = (a_{1}, …, a_{2n}) ∈ ℕ^{2n}}

of A(x, B) resulting from Proposition 7.4 with a basis constructed by Berenstein, Fomin and Zelevinsky [BFZ]. For 1 ≤ k ≤ n let

(9)  x′_{k} := μ_{(x,B)}(x_{k}) = x_{k}^{−1} (x_{n+k} ∏_{i=1}^{k−1} x_{i}^{b_{ik}} + ∏_{i=k+1}^{n} x_{i}^{−b_{ik}} ∏_{i=1}^{k−1} x_{n+i}^{b_{ik}})

and set

P_{BFZ} := {x′[a] := x_{1}^{a_{1}} ⋯ x_{n}^{a_{n}}x_{n+1}^{a_{n+1}} ⋯ x_{2n}^{a_{2n}}(x′_{1})^{a_{2n+1}} ⋯ (x′_{n})^{a_{3n}} |

a = (a_{1}, …, a_{3n}) ∈ ℕ^{3n}, a_{k}a_{2n+k} = 0 for 1 ≤ k ≤ n}.

Proposition 7.5 ([BFZ, Corollary 1.21]). The set P_{BFZ} is a basis of A(x, B).

Note that the basis P_{GLS} is constructed by using cluster variables from two seeds, namely (x, B) and μ_{n} ⋯ μ_{1}(x, B), whereas P_{BFZ} uses cluster variables from n + 1 seeds, namely (x, B) and μ_{k}(x, B), where 1 ≤ k ≤ n.

Now we insert Equation (8) into Equation (9) and obtain

(10) x_{k}x′_{k} = (x_{k}[1]x_{k} − ∏_{i=1}^{k−1} x_{i}[1]^{b_{ik}} ∏_{i=k+1}^{n} x_{i}^{−b_{ik}}) ∏_{i=1}^{k−1} x_{i}^{b_{ik}}+

∏_{i=k+1}^{n} x_{i}^{−b_{ik}} ∏_{i=1}^{k−1} (x_{i}[1]x_{i} − ∏_{j=1}^{i−1} x_{j}[1]^{b_{ji}} ∏_{j=i+1}^{n} x_{j}^{−b_{ji}} )^{b_{ik}} .

Then we observe that the right-hand side of Equation (10) is divisible by x_{k} and that x′_{k} is a polynomial in x_{1}, …, x_{n}, x_{1}[1], …, x_{n}[1]. Thus we can express every element of the basis P_{BFZ} explicitely as a linear combination of vectors from the basis P_{GLS}.

One could use Equation (10) to get an alternative proof of Proposition 7.4 as pointed out by Zelevinsky [Z]. Vice versa, using Propostion 7.4 yields another proof that P_{BFZ} is a basis.

As an illustration, for n = 3 the matrices B_{i} look as follows:

B_{0} = (\begin{matrix}0 & b_{12} & b_{13} \\ b_{21} & 0 & b_{23} \\ b_{31} & b_{32} & 0 \\ 1 & −b_{12} & −b_{13} \\ 0 & 1 & −b_{23} \\ 0 & 0 & 1 \end{matrix} ) ,  B_{1} = (\begin{matrix}0 & −b_{12} & −b_{13} \\ −b_{21} & 0 & b_{23} \\ −b_{31} & b_{32} & 0 \\ −1 & 0 & 0 \\ 0 & 1 & −b_{23} \\ 0 & 0 & 1 \end{matrix}) ,

B_{2} = (\begin{matrix}0 & b_{12} & −b_{13} \\ b_{21} & 0 & −b_{23} \\ −b_{31} & −b_{32} & 0 \\ −1 & 0 & 0 \\ −b_{21} & −1 & 0 \\ 0 & 0 & 1 \end{matrix}) ,  B_{3} = (\begin{matrix}0 & b_{12} & b_{13} \\ b_{21} & 0 & b_{23} \\ b_{31} & b_{32} & 0 \\ −1 & 0 & 0 \\ −b_{21} & −1 & 0 \\ −b_{31} & −b_{32} & −1 \end{matrix}) .

For example, for

B = B_{0} = (\begin{matrix}0 & 2 & 0 \\ −2 & 0 & 1 \\ 0 & −1 & 0 \\ 1 & −2 & 0 \\ 0 & 1 & −1 \\ 0 & 0 & 1 \end{matrix})

Documenta Mathematica 18 (2013) 249–274

the quivers Γ(B_{0}) and Γ(B_{3}) look as follows:

\xymatrix{Γ(B_{0}): &  & 4 \ar[rr] &  & 1 \ar[dl] \ar[dl] & Γ(B_{3}): &  & 4 &  & 1 \ar[ll] \ar[dl] \ar[dl] \\  & 5 \ar[rr] &  & 2 \ar[ul] \ar[ul] \ar[dl] &  &  & 5 \ar[urrr] \ar[urrr] &  & 2 \ar[ll] \ar[dl] \\ 6 \ar[rr] &  & 3 \ar[ul] &  &  & 6 \ar[urrr] &  & 3 \ar[ll]}

The cluster algebra A(x, B) is a polynomial ring in the 6 variables x_{1}, x_{2}, x_{3} and

x_{1}[1] = \frac{x_{2}^{2} + x_{4}}{x_{1}} ,
x_{2}[1] = \frac{x_{2}^{4}x_{3} + 2x_{2}^{2}x_{3}x_{4} + x_{3}x_{4}^{2} + x_{1}^{2}x_{5}}{x_{1}^{2}x_{2}} ,
x_{3}[1] = \frac{x_{2}^{4}x_{3} + 2x_{2}^{2}x_{3}x_{4} + x_{3}x_{4}^{2} + x_{1}^{2}x_{5} + x_{1}^{2}x_{2}x_{6}}{x_{1}^{2}x_{2}x_{3}} .

7.3. Cluster algebras arising in Lie theory as polynomial rings. The next class of examples can be seen as a fusion of the examples discussed in Sections 7.1 and 7.2. In the following we use the same notation as in [GLS2].

Let C ∈ M_{n,n}(ℤ) be a symmetric generalized Cartan matrix, and let 𝔤 be the associated Kac-Moody Lie algebra over K = ℂ with triangular decomposition 𝔤 = 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫, see [K].

Let U(𝔫)_{gr}^{∗} be the graded dual of the enveloping algebra U(𝔫) of 𝔫. To each element w in the Weyl group W of 𝔤 one can associate a subalgebra R(C_{w}) of U(𝔫)_{gr}^{∗} and a cluster algebra A(C_{w}), see [GLS2]. Here C_{w} denotes a Frobenius category associated to w, see [BIRS, GLS2].

In [GLS2] we constructed a natural algebra isomorphism

A(C_{w}) → R(C_{w}).

This yields a cluster algebra structure on R(C_{w}).

Let i = (i_{r}, …, i_{1}) be a reduced expression of w. In [GLS2] we studied two cluster-tilting modules V_{i} = V_{1} ⊕ ⋯ ⊕ V_{r} and T_{i} = T_{1} ⊕ ⋯ ⊕ T_{r} in C_{w}, which are associated to i. These modules yield two disjoint clusters (δ_{V_{1}}, …, δ_{V_{r}}) and (δ_{T_{1}}, …, δ_{T_{r}}) of R(C_{w}). The exchanges matrices are of size r × (r − n). In contrast to our conventions in this article, the n coefficients are δ_{V_{k}} = δ_{T_{k}} with k^{+} = r + 1, where k^{+} is defined as in [GLS2], and none of these coefficients is invertible. Furthermore, we studied a module M_{i} = M_{1} ⊕⋯⊕M_{r} in C_{w}, which yields cluster variables δ_{M_{1}}, …, δ_{M_{r}} of R(C_{w}). (These do not form a cluster.) Using methods from Lie theory we obtained the following result.

Theorem 7.6 ([GLS2, Theorem 3.2]). The cluster algebra R(C_{w}) is a polynomial ring in the variables δ_{M_{1}}, …, δ_{M_{r}}.

To obtain an alternative proof of Theorem 7.6, one can proceed as follows:

(i) Show that the cluster variables δ_{M_{1}}, …, δ_{M_{r}} are algebraically independent.

(ii) Show that for 1 ≤ k ≤ r the cluster variables δ_{V_{k}} and δ_{T_{k}} are polyno-

mials in δ_{M_{1}}, …, δ_{M_{r}}. (iii) Apply Theorem 4.1.

Part (i) can be done easily using induction and the mutation sequence in [GLS2, Section 13]. Part (ii) is not at all straightforward.

Let us give a concrete example illustrating Theorem 7.6. Let 𝔤 be the Kac-Moody Lie algebra associated to the generalized Cartan matrix

C = (\begin{matrix}2 & −2 \\ −2 & 2 \end{matrix}) ,

and let i = (2, 1, 2, 1, 2, 1, 2, 1). Then A(C_{w}) = A(x, B_{i}), where r = n + 2 = 8, x_{7} and x_{8} are the (non-invertible) coefficients, and

B_{i} = (\begin{matrix}0 & 2 & −1 \\ −2 & 0 & 2 & −1 \\ 1 & −2 & 0 & 2 & −1 \\ & 1 & −2 & 0 & 2 & −1 \\ & & 1 & −2 & 0 & 2 \\ & & & 1 & −2 & 0 \\ & & & & 1 & −2 \\ & & & & & 1 \end{matrix}) .

The principal part B_{i}^{∘} of B_{i} is skew-symmetric, and the quiver Γ(B_{i}) looks as follows:

\xymatrix{8 \ar[rr] &  & 6 \ar[dl] \ar[dl] \ar[rr] &  & 4 \ar[dl] \ar[dl] \ar[rr] &  & 2 \ar[dl] \ar[dl] \\  & 7 \ar[rr] &  & 5 \ar[ul] \ar[ul] \ar[rr] &  & 3 \ar[ul] \ar[ul] \ar[rr] &  & 1 \ar[ul] \ar[ul]}

Define

(x[0], B[0]) := (x, B_{i}),
(x[1], B[1]) := μ_{5}μ_{3}μ_{1}(x[0], B[0]), (x[2], B[2]) := μ_{6}μ_{4}μ_{2}(x[1], B[1]),
(x[3], B[3]) := μ_{3}μ_{1}(x[2], B[2]),  (x[4], B[4]) := μ_{4}μ_{2}(x[3], B[3]),
(x[5], B[5]) := μ_{1}(x[4], B[4]),  (x[6], B[6]) := μ_{2}(x[5], B[5]),

and for 0 ≤ k ≤ 6 let (x_{1}[k], …, x_{8}[k]) := x[k].

Under the isomorphism A(C_{w}) → R(C_{w}) the cluster x[0] of A(C_{w}) = A(x, B_{i}) corresponds to the cluster (δ_{V_{1}}, …, δ_{V_{8}}) of R(C_{w}), the cluster x[6] corresponds to (δ_{T_{1}}, …, δ_{T_{8}}), and we have

x_{1}[0] ↦ δ_{M_{1}}, x_{2}[0] ↦ δ_{M_{2}}, x_{1}[2] ↦ δ_{M_{3}}, x_{2}[2] ↦ δ_{M_{4}},
x_{1}[4] ↦ δ_{M_{5}}, x_{2}[4] ↦ δ_{M_{6}}, x_{1}[6] ↦ δ_{M_{7}}, x_{2}[6] ↦ δ_{M_{8}}.

By Theorem 7.6 we know that the cluster algebra A(x, B_{i}) is a polynomial ring in the variables x_{1}[0], x_{2}[0], x_{1}[2], x_{2}[2], x_{1}[4], x_{2}[4], x_{1}[6], x_{2}[6].

8. Applications

8.1. Prime elements in the dual semicanonical basis. As in Section 7.3 let C ∈ M_{n,n}(ℤ) be a symmetric generalized Cartan matrix, and let 𝔤 = 𝔫_{−} ⊕ 𝔥 ⊕ 𝔫 be the associated Lie algebra.

As before let W be the Weyl group of 𝔤. To C one can also associate a preprojective algebra Λ over ℂ, see for example [GLS2, R]

Lusztig [Lu] realized the universal enveloping algebra U(𝔫) of 𝔫 as an algebra of constructible functions on the varieties Λ_{d} of nilpotent Λ-modules with dimension vector d ∈ ℕ^{n}. He also constructed the semicanonical basis S of U(𝔫). The elements of S are naturally parametrized by the irreducible components of the varieties Λ_{d}.

An irreducible component Z of Λ_{d} is called indecomposable if it contains a Zariski dense subset of indecomposable Λ-modules, and Z is rigid if it contains a rigid Λ-module M, i.e. M is a module with Ext_{Λ}^{1}(M, M) = 0.

Let S^{∗} be the dual semicanonical basis of the graded dual U(𝔫)_{gr}^{∗} of U(𝔫). The elements ρ_{Z} in S^{∗} are also parametrized by irreducible components Z of the varieties Λ_{d}. We call ρ_{Z} indecomposable (resp. rigid) if Z is indecomposable (resp. rigid). An element b ∈ S^{∗} is called primitive if it cannot be written as a product b = b_{1}b_{2} with b_{1}, b_{2} ∈ S^{∗} \ {1}.

Theorem 8.1 ([GLS1, Theorem 1.1]). If ρ_{Z} is primitive, then Z is indecomposable.

Theorem 8.2 ([GLS2, Theorem 3.1]). For w ∈ W all cluster monomials of the cluster algebra R(C_{w}) belong to the dual semicanonical basis S^{∗} of U(𝔫)_{gr}^{∗}. More precisely, we have

{cluster variables of R(C_{w})} ⊆ {ρ_{Z} ∈ S^{∗} | Z is indecomposable and rigid},
{cluster monomials of R(C_{w})} ⊆ {ρ_{Z} ∈ S^{∗} | Z is rigid}.

Combining Theorems 3.1, 7.6 and 8.2 we obtain a partial converse of Theorem 8.1.

Theorem 8.3. The cluster variables in R(C_{w}) are prime, and they are primitive elements of S^{∗}.

Conjecture 8.4. If ρ_{Z} ∈ S^{∗} is indecomposable and rigid, then ρ_{Z} is prime in U(𝔫)_{gr}^{∗}.

8.2. Monoidal categorifications of cluster algebras. Let C be an abelian tensor category with unit object I_{C}. We assume that C is a Krull-Schmidt category, and that all objects in C are of finite length. Let M(C) := K_{0}(C) be the Grothendieck ring of C. The class of an object M ∈ C is denoted by [M]. The addition in M(C) is given by [M] + [N] := [M ⊕ N] and the multiplication is defined by [M][N] := [M ⊗ N]. We assume that [M ⊗ N] = [N ⊗ M]. (In general this does not imply M ⊗ N ≅ N ⊗ M.) Thus M(C) is a commutative ring.

Tensoring with K over ℤ yields a K-algebra M_{K}(C) := K ⊗_{ℤ} K_{0}(C) with Kbasis the classes of simple objects in C. Note that the unit object I_{C} is simple.

A monoidal categorification of a cluster algebra A(x, B) is an algebra isomorphism

Φ: A(x, B) → M_{K}(C),

where C is a tensor category as above, such that each cluster monomial y = y_{1}^{a_{1}} ⋯ y_{m}^{a_{m}} of A(x, B) is mapped to a class [S_{y}] of some simple object S_{y} ∈ C. In particular, we have

[S_{y}] = [S_{y_{1}}]^{a_{1}} ⋯ [S_{y_{m}}]^{a_{m}} = [S_{y_{1}}^{⊗a_{1}} ⊗ ⋯ ⊗ S_{y_{m}}^{⊗a_{m}}].

For an object M ∈ C let x_{M} be the element in A(x, B) with Φ(x_{M}) = [M].

The concept of a monoidal categorification of a cluster algebra was introduced in [HL, Definition 2.1]. But note that our definition uses weaker conditions than in [HL].

An object M ∈ C is called invertible if [M] is invertible in M_{K}(C). An object M ∈ C is primitive if there are no non-invertible objects M_{1} and M_{2} in C with M ≅ M_{1} ⊗ M_{2}.

Proposition 8.5. Let Φ: A(x, B) → M_{K}(C) be a monoidal categorification of a cluster algebra A(x, B). Then the following hold:

(i) The invertible elements in M_{K}(C) are

M_{K}(C)^{×} = {λ[I_{C}][S_{x_{n+1}}]^{a_{n+1}} ⋯ [S_{x_{p}}]^{a_{p}} | λ ∈ K^{×}, a_{i} ∈ ℤ} .

(ii) Let M be an object in C such that the element x_{M} is irreducible in A(x, B). Then M is primitive.

Proof. Part (i) follows directly from Theorem 2.2. To prove (ii), assume that M is not primitive. Thus there are non-invertible objects M_{1} and M_{2} in C with M ≅ M_{1} ⊗ M_{2}. Thus in M_{K}(C) we have [M] = [M_{1}][M_{2}]. Since Φ is an algebra isomorphism, we get x_{M} = x_{M_{1}}x_{M_{2}} with x_{M_{1}} and x_{M_{2}} non-invertible in A(x, B). Since x_{M} is irreducible, we have a contradiction. □

Combining Proposition 8.5 with Theorem 3.1 we get the following result.

Corollary 8.6. Let Φ: A(x, B) → M_{K}(C) be a monoidal categorification of a cluster algebra A(x, B). For each cluster variable y of A(x, B), the simple object S_{y} is primitive.

Examples of monoidal categorifications of cluster algebras can be found in [HL, N1], see also [Le].

Acknowledgements. We thank Giovanni Cerulli Irelli, Sergey Fomin, Daniel Labardini Fragoso, Philipp Lampe and Andrei Zelevinsky for helpful discussions. We thank Giovanni Cerulli Irelli for carefully reading several preliminary versions of this article. The first author likes to thank the Max-Planck Institute for Mathematics in Bonn for a one year research stay in 2010/2011. The third author thanks the Sonderforschungsbereich/Transregio SFB 45 for financial support, and all three authors thank the Hausdorff Institute for Mathematics in Bonn for support and hospitality.

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Christof Geiß

Instituto de Matemáticas Universidad Nacional Autónoma de México Ciudad Universitaria 04510 México D.F. México christof@math.unam.mx Bernard Leclerc Université de Caen LMNO CNRS UMR 6139 Institut Universitaire de France 14032 Caen cedex France bernard.leclerc@unicaen.fr

Jan Schröer

Mathematisches Institut Universität Bonn Endenicher Allee 60 53115 Bonn Germany schroer@math.uni-bonn.de