# Cluster algebras I: Foundations

## Abstract

In an attempt to create an algebraic framework for dual canonical bases and total positivity in semisimple groups, we initiate the study of a new class of commutative algebras.

## 1. Introduction

In this paper, we initiate the study of a new class of algebras, which we call *cluster algebras*. Before giving precise definitions, we present some of the main features of these algebras. For any positive integer a cluster algebra , of rank is a commutative ring with unit and no zero divisors, equipped with a distinguished family of generators called *cluster variables*. The set of cluster variables is the (non-disjoint) union of a distinguished collection of called -subsets*clusters*. These clusters have the following *exchange property*: for any cluster and any element there is another cluster obtained from , by replacing with an element related to by a *binomial exchange relation*

where and are two monomials without common divisors in the variables Furthermore, any two clusters can be obtained from each other by a sequence of exchanges of this kind. .

The prototypical example of a cluster algebra of rank 1 is the coordinate ring of the group viewed in the following way. Writing a generic element of , as we consider the entries and as cluster variables, and the entries and as scalars. There are just two clusters and and , is the algebra over the polynomial ring generated by the cluster variables and subject to the binomial exchange relation

Another important incarnation of a cluster algebra of rank 1 is the coordinate ring of the base affine space of the special linear group here ; is the maximal unipotent subgroup of consisting of all unipotent upper triangular matrices. Using the standard notation for the Plücker coordinates on we view , and as cluster variables; then is the algebra over the polynomial ring generated by the two cluster variables and subject to the binomial exchange relation

This form of representing the algebra is closely related to the choice of a linear basis in it consisting of all monomials in the six Plücker coordinates which are not divisible by This basis was introduced and studied in .Reference9 under the name “canonical basis”. As a representation of the space , is the multiplicity-free direct sum of all irreducible finite-dimensional representations, and each of the components is spanned by a part of the above basis. Thus, this construction provides a “canonical” basis in every irreducible finite-dimensional representation of After Lusztig’s work .Reference13, this basis had been recognized as (the classical limit at of) the *dual canonical basis*, i.e., the basis in the algebra -deformed which is dual to Lusztig’s canonical basis in the appropriate universal enveloping algebra (a.k.a. quantum group). The dual canonical basis in the space -deformed was later constructed explicitly for a few other classical groups of small rank: for in Reference16 and for in Reference2. In both cases, can be seen to be a cluster algebra: there are 6 clusters of size 2 for and 14 clusters of size 3 for , .

We conjecture that the above examples can be extensively generalized: for any simply-connected connected semisimple group the coordinate rings , and as well as coordinate rings of many other interesting varieties related to , have a natural structure of a cluster algebra. This structure should serve as an algebraic framework for the study of “dual canonical bases” in these coordinate rings and their , In particular, we conjecture that all monomials in the variables of any given cluster (the -deformations.*cluster monomials*) belong to this dual canonical basis.

A particularly nice and well-understood example of a cluster algebra of an arbitrary rank is the homogeneous coordinate ring of the Grassmannian of subspaces in -dimensional This ring is generated by the Plücker coordinates . for , subject to the relations ,

for all It is convenient to identify the indices . with the vertices of a convex and the Plücker coordinates with its sides and diagonals. We view the sides -gon, as scalars, and the diagonals as cluster variables. The clusters are the maximal families of pairwise noncrossing diagonals; thus, they are in a natural bijection with the triangulations of this polygon. It is known that the cluster monomials form a linear basis in To be more specific, we note that this ring is naturally identified with the ring of polynomial . of an -invariants of points in -tuple Under this isomorphism, the basis of cluster monomials corresponds to the basis considered in .Reference11Reference18. (We are grateful to Bernd Sturmfels for bringing these references to our attention.)

An essential feature of the exchange relations (Equation1.1) is that the right-hand side does not involve subtraction. Recursively applying these relations, one can represent any cluster variable as a *subtraction-free* rational expression in the variables of any given cluster. This positivity property is consistent with a remarkable connection between canonical bases and the theory of total positivity, discovered by G. Lusztig Reference14Reference15. Generalizing the classical concept of totally positive matrices, he defined totally positive elements in any reductive group and proved that all elements of the dual canonical basis in , take positive values at them.

It was realized in Reference15Reference5 that the natural geometric framework for total positivity is given by *double Bruhat cells*, the intersections of cells of the Bruhat decompositions with respect to two opposite Borel subgroups. Different aspects of total positivity in double Bruhat cells were explored by the authors of the present paper and their collaborators in Reference1Reference3Reference4Reference5Reference6Reference7Reference12Reference17Reference20. The binomial exchange relations of the form (Equation1.1) played a crucial role in these studies. It was the desire to explain the ubiquity of these relations and to place them in a proper context that led us to the concept of cluster algebras. The crucial step in this direction was made in Reference20, where a family of clusters and exchange relations was explicitly constructed in the coordinate ring of an arbitrary double Bruhat cell. However, this family was not *complete*: in general, some clusters were missing, and not any member of a cluster could be exchanged from it. Thus, we started looking for a natural way to “propagate” exchange relations from one cluster to another. The concept of cluster algebras is the result of this investigation. We conjecture that the coordinate ring of any double Bruhat cell is a cluster algebra.

This article, in which we develop the foundations of the theory, is conceived as the first in a forthcoming series. We attempt to make the exposition elementary and self-contained; in particular, no knowledge of semisimple groups, quantum groups or total positivity is assumed on the part of the reader.

One of the main structural features of cluster algebras established in the present paper is the following *Laurent phenomenon*: any cluster variable viewed as a rational function in the variables of any given cluster is in fact a Laurent polynomial. This property is quite surprising: in most cases, the numerators of these Laurent polynomials contain a huge number of monomials, and the numerator for moves into the denominator when we compute the cluster variable obtained from by an exchange (Equation1.1). The magic of the Laurent phenomenon is that, at every stage of this recursive process, a cancellation will inevitably occur, leaving a single monomial in the denominator.

In view of the positivity property discussed above, it is natural to expect that all Laurent polynomials for cluster variables will have *positive* coefficients. This seems to be a rather deep property; our present methods do not provide a proof of it.

On the bright side, it is possible to establish the Laurent phenomenon in many different situations spreading beyond the cluster algebra framework. One such extension is given in Theorem 3.2. By a modification of the method developed here, a large number of additional interesting instances of the Laurent phenomenon are established in a separate paper Reference8.

The paper is organized as follows. Section 2 contains an axiomatic definition, first examples and the first structural properties of cluster algebras. One of the technical difficulties in setting up the foundations involves the concept of an *exchange graph* whose vertices correspond to clusters, and the edges to exchanges among them. It is convenient to begin by taking the tree -regular as our underlying graph. This tree can be viewed as a universal cover for the actual exchange graph, whose appearance is postponed until Section 7.

The Laurent phenomenon is established in Section 3. In Sections 4 and 5, we scrutinize the main definition, obtain useful reformulations, and introduce some important classes of cluster algebras.

Section 6 contains a detailed analysis of cluster algebras of rank 2. This analysis exhibits deep and somewhat mysterious connections between cluster algebras and Kac-Moody algebras. This is just the tip of an iceberg: these connections will be further explored (for cluster algebras of an arbitrary rank) in the sequel to this paper. The main result of this sequel is a complete classification of cluster algebras of *finite type*, i.e., those with finitely many distinct clusters; cf. Example 7.6. This classification turns out to be yet another instance of the famous Cartan-Killing classification.

## 2. Main definitions

Let be a finite set of size the standard choice will be ; Let . denote the * tree, -regular* whose edges are labeled by the elements of so that the , edges emanating from each vertex receive different labels. By a common abuse of notation, we will sometimes denote by the set of the tree’s vertices. We will write if vertices are joined by an edge labeled by .

To each vertex we will associate a ,*cluster* of generators (“variables”) All these variables will commute with each other and satisfy the following .*exchange relations,* for every edge in :

Here and are two monomials in the variables we think of these monomials as being associated with the two ends of the edge ; .

To be more precise, let be an abelian group without torsion, written multiplicatively. We call the *coefficient group*; a prototypical example is a free abelian group of finite rank. Every monomial in (Equation2.2) will have the form

for some coefficient and some nonnegative integer exponents .

The monomials must satisfy certain conditions (axioms). To state them, we will need a little preparation. Let us write to denote that a polynomial divides a polynomial Accordingly, . means that the monomial contains the variable For a rational function . the notation , will denote the result of substituting for into To illustrate, if . then , .

### Definition 2.1

An *exchange pattern* on with coefficients in is a family of monomials of the form (Equation2.3) satisfying the following four axioms:

We note that in the last axiom, the substitution is effectively monomial, since in the event that neither nor contain condition ( ,Equation2.6) requires that both and do not depend on thus making the whole substitution irrelevant. ,

One easily checks that axiom (Equation2.7) is invariant under the “flip” , so no restrictions are added if we apply it “backwards”. The axioms also imply at once that setting ,

for every edge we obtain another exchange pattern , this gives a natural involution ; on the set of all exchange patterns.

### Remark 2.2

Informally speaking, axiom (Equation2.7) describes the propagation of an exchange pattern along the edges of More precisely, let us fix the . exchange monomials for all edges emanating from a given vertex This choice uniquely determines the ratio . for any vertex adjacent to and any edge (to see this, take and in (Equation2.7), and allow to vary). In view of (Equation2.5), this ratio in turn uniquely determines the exponents of all variables in both monomials and There remains, however, one degree of freedom in determining the coefficients . and because only their ratio is prescribed by (Equation2.7). In Section 5 we shall introduce an important class of *normalized* exchange patterns for which this degree of freedom disappears, and so the whole pattern is uniquely determined by the monomials associated with edges emanating from a given vertex.

Let denote the group ring of with integer coefficients. For an edge we refer to the binomial , as the *exchange polynomial*. We will write or to indicate this fact. Note that, in view of the axiom (Equation2.4), the right-hand side of the exchange relation (Equation2.2) can be written as which is the same as , .

Let be an exchange pattern on with coefficients in Note that since . is torsion-free, the ring has no zero divisors. For every vertex let , denote the field of rational functions in the cluster variables , with coefficients in , For every edge . we define a , field isomorphism -linear by

Note that property (Equation2.4) ensures that The .*transition maps* enable us to identify all the fields with each other. We can then view them as a single field that contains all the elements for all , and Inside . these elements satisfy the exchange relations ( ,Equation2.1)–(Equation2.2).

### Definition 2.3

Let be a subring with unit in containing all coefficients for and The .*cluster algebra* of rank over associated with an exchange pattern is the with unit in -subalgebra generated by the union of all clusters for , .

The smallest possible ground ring is the subring of generated by all the coefficients the largest one is ; itself. An intermediate choice of appears in Proposition 2.6 below.

Since is a subring of a field it is a commutative ring with no zero divisors. We also note that if , is obtained from by the involution (Equation2.8), then the cluster algebra is naturally identified with .

### Example 2.4

Let The tree . has only one edge The corresponding cluster algebra . has two generators and satisfying the exchange relation

where and are arbitrary elements of the coefficient group In the “universal” setting, we take . to be the free abelian group generated by and Then the two natural choices for the ground ring . are the polynomial ring and the Laurent polynomial ring , All other realizations of . can be viewed as specializations of the universal one. Despite the seeming triviality of this example, it covers several important algebras: the coordinate ring of each of the varieties , and (cf. Section 1) is a cluster algebra of rank for an appropriate choice of , , , and .

### Example 2.5

Consider the case The tree . is shown below:

Let us denote the cluster variables as follows:

(the above equalities among the cluster variables follow from (Equation2.1)). Then the clusters look like

We claim that the exchange relations (Equation2.2) can be written in the following form:

where the integers and are either both positive or both equal to and the coefficients , and are elements of satisfying the relations

Furthermore, any such choice of parameters , , , results in a well-defined cluster algebra of rank 2.

To prove this, we notice that, in view of (Equation2.4)–(Equation2.5), both monomials and do not contain the variable and at most one of them contains , If . enters neither nor then these two are simply elements of , But then ( .Equation2.6) forces *all* monomials to be elements of while ( ,Equation2.7) implies that it is possible to give the names and to the two monomials corresponding to each edge so that (Equation2.11)–(Equation2.12) hold with .

Next, consider the case when precisely one of the monomials and contains Applying if necessary the involution ( .Equation2.8) to our exchange pattern, we may assume that and for some positive integer and some Thus, the exchange relation associated to the edge . takes the form By ( .Equation2.6), we have and for some positive integer and some Then the exchange relation for the edge . takes the form At this point, we invoke ( .Equation2.7):

By (Equation2.5), we have and for some satisfying Continuing in the same way, we obtain all relations ( .Equation2.11)–(Equation2.12).

For fixed and the “universal” coefficient group , is the multiplicative abelian group generated by the elements and for all subject to the defining relations (Equation2.12). It is easy to see that this is a free abelian group of infinite rank. As a set of its free generators, one can choose any subset of that contains four generators and precisely one generator from each pair for .

A nice specialization of this setup is provided by the homogeneous coordinate ring of the Grassmannian Recall (cf. Section .1) that this ring is generated by the Plücker coordinates where , and are distinct elements of the cyclic group We shall write . for and adopt the convention , see Figure ;1.

The ideal of relations among the Plücker coordinates is generated by the relations

for A direct check shows that these relations are a specialization of the relations ( .Equation2.11), if we set , , and , for all The coefficient group . is the multiplicative free abelian group with generators It is also immediate that the elements . and defined in this way satisfy the relations (Equation2.12).

We conclude this section by introducing two important operations on exchange patterns: *restriction* and *direct product*. Let us start with restriction. Let be an exchange pattern of rank with an index set and coefficient group Let . be a subset of size in Let us remove from . all edges labeled by indices in and choose any connected component , of the resulting graph. This component is naturally identified with Let . denote the restriction of to i.e., the collection of monomials , for all and Then . is an exchange pattern on whose coefficient group is the direct product of with the multiplicative free abelian group with generators , We shall say that . is obtained from by restriction from to Note that . depends on the choice of a connected component so there can be several different patterns obtained from , by restriction from to (We thank the anonymous referee for pointing this out.) .

### Proposition 2.6

Let be a cluster algebra of rank associated with an exchange pattern and let , be obtained from by restriction from to using a connected component The . of -subalgebra generated by is naturally identified with the cluster algebra where , is the polynomial ring .

### Proof.

If then ( ,Equation2.1) implies that stays constant as varies over Therefore, we can identify this variable with the corresponding generator . of the coefficient group and the statement follows. ,

Let us now consider two exchange patterns and of ranks and respectively, with index sets , and and coefficient groups , and We will construct the exchange pattern . (the direct product of and of rank ) with the index set , and coefficient group , Consider the tree . whose edges are colored by and, for , let , be a map with the following property: if in and (resp., then ), in (resp., Clearly, such a map ). exists and is essentially unique: it is determined by specifying the image of any vertex of We now introduce the exchange pattern . on by setting, for every and the monomial , to be equal to the latter monomial coming from the exchange pattern , The axioms ( .Equation2.4)–(Equation2.7) for are checked directly.

### Proposition 2.7

Let and be cluster algebras. Let and Then the cluster algebra . is canonically isomorphic to the tensor product of algebras tensor products are taken over all .

### Proof.

Let us identify each cluster variable for , and (resp., with ), (resp., Under this identification, the exchange relations for the exchange pattern . become identical to the exchange relations for and .

## 3. The Laurent phenomenon

In this section we prove the following important property of cluster algebras.

### Theorem 3.1

In a cluster algebra, any cluster variable is expressed in terms of any given cluster as a Laurent polynomial with coefficients in the group ring .

We conjecture that each of the coefficients in these Laurent polynomials is actually a *nonnegative* integer linear combination of elements in .

We will obtain Theorem 3.1 as a corollary of a more general result, which applies to more general underlying graphs and more general (not necessarily binomial) exchange polynomials.

Since Theorem 3.1 is trivial for we shall assume that , For every . let , be a tree of the form shown in Figure 2. The tree has vertices of degree in its “spine” and vertices of degree 1. We label every edge of the tree by an element of an index set -element so that the , edges incident to each vertex on the spine receive different labels. (The reader may wish to think of the tree as being part of the tree -regular of the cluster-algebra setup.)

We fix two vertices and of that do not belong to the spine and are connected to its opposite ends. This gives rise to the orientation on the spine: away from and towards (see Figure 2).

As before, let be an abelian group without torsion, written multiplicatively. Let denote the additive semigroup generated by in the integer group ring Assume that a nonzero polynomial . in the variables with coefficients in , is associated with every edge , of We call . an *exchange polynomial,* and write to describe this situation. Suppose that the exchange polynomials associated with the edges of satisfy the following conditions:

(Note the orientation of the edge in (Equation3.2).)

For every vertex on the spine, let denote the family of exchange polynomials associated with the edges emanating from Also, let . denote the collection of all Laurent polynomials that appear in condition (Equation3.2), for all possible choices of and let , denote the subring with unit generated by all coefficients of the Laurent polynomials from where , is the vertex on the spine connected with .

As before, we associate a *cluster* to each vertex and consider the field , of rational functions in these variables with coefficients in All these fields are identified with each other by the transition isomorphisms . defined as in (Equation2.9). We then view the fields as a single field that contains all the elements for , and These elements satisfy the exchange relations ( .Equation2.1) and the following version of (Equation2.2):

for any edge in .

### Theorem 3.2

If conditions (Equation3.1)–(Equation3.2) are satisfied, then each element of the cluster is a Laurent polynomial in the cluster with coefficients in the ring , .

We note that Theorem 3.2 is indeed a generalization of Theorem 3.1, for the following reasons:

- •
is naturally embedded into ;

### Proof.

We start with some preparations. We shall write any Laurent polynomial in the variables in the form

where all coefficients are nonzero, is a finite subset of the lattice (i.e., the lattice of rank with coordinates labeled by and ), is the usual shorthand for The set . is called the *support* of and denoted by .

Notice that once we fix the collection condition ( ,Equation3.2) can be used as a recursive rule for computing from for any edge , on the spine. It follows that the whole pattern of exchange polynomials is determined by the families of polynomials and Moreover, since these polynomials have coefficients in . and the expression for , in (Equation3.2) does not involve subtraction, it follows that the support of any exchange polynomial is uniquely determined by the supports of the polynomials from and Note that condition ( .Equation3.1) can be formulated as a set of restrictions on these supports. In particular, it requires that in the situation of (Equation3.2), the Laurent polynomial does not depend on and is a *polynomial* in in other words, every ; should have and .

We now fix a family of supports for all , and assume that this family complies with ( ,Equation3.1). As is common in algebra, we shall view the coefficients for all , and as ,*indeterminates*. Then all the coefficients in all exchange polynomials become “canonical” (i.e., independent of the choice of polynomials in these indeterminates, with positive integer coefficients. )

The above discussion shows that it suffices to prove our theorem in the following “universal coefficients” setup: let be the free abelian group (written multiplicatively) with generators for all , and Under this assumption, . is simply the integer polynomial ring in the indeterminates .

Recall that we can view all cluster variables as elements of the field of rational functions in the cluster with coefficients in For . let , denote the ring of Laurent polynomials in the cluster with coefficients in , We view each . as a subring of the ambient field .

In this terminology, our goal is to show that the cluster is contained in We proceed by induction on . the size of the spine. The claim is trivial for , so let us assume that , and furthermore assume that our statement is true for all “caterpillars” with smaller spine. ,

Let us abbreviate and and suppose that the path from , to starts with the following two edges: Let . be the vertex such that .

The following lemma plays a crucial role in our proof.

### Lemma 3.3

The clusters , and , are contained in Furthermore, . elements of as .

Note that is a unique factorization domain, so any two elements have a well-defined greatest common divisor which is an element of , defined up to a multiple from the group of units (that is, invertible elements) of In our “universal” situation, . consists of Laurent monomials in the cluster with coefficients .

### Proof.

The only element from the clusters , and , whose inclusion in is not immediately obvious is To simplify the notation, let us denote . , , , and , so that these variables appear in the clusters at , as shown below: ,

Note that the variables for , do not change as we move among the four clusters under consideration. The lemma is then restated as saying that ,

Another notational convention will be based on the fact that each of the polynomials has a distinguished variable on which it depends, namely for and and , for (In view of ( .Equation3.1), and do not depend on while , does not depend on With this in mind, we will routinely write .) , and , as polynomials in one (distinguished) variable. In the same spirit, the notation , etc., will refer to the partial derivative with respect to the distinguished variable. ,

We will prove the statements (Equation3.3), (Equation3.4), and (Equation3.5) one by one, in this order.

By (Equation3.2), the polynomial is given by

where is an “honest” polynomial in and a Laurent polynomial in the “mute” variables , (Recall that . does not depend on We then have: .)

Since

and

(Equation3.3) follows.

We next prove (Equation3.4). We have

Since and are invertible in we conclude that , Now the trouble that we took in passing to universal coefficients finally pays off: since . and are nonzero polynomials in the cluster whose coefficients are distinct generators of the polynomial ring it follows that , proving ( ,Equation3.4).

It remains to prove (Equation3.5). Let

Then

Our goal is to show that to this end, we are going to compute ; as “explicitly” as possible. We have, ,

Hence

Note that the right-hand side is a *linear* polynomial in whose coefficients are Laurent polynomials in the rest of the variables of the cluster , Thus our claim will follow if we show that . This, again, is a consequence of our “universal coefficients” setup since the coefficients of . , and are distinct generators of the polynomial ring .

We can now complete the proof of Theorem 3.2. We need to show that any variable belongs to Since both . and are closer to than we can use the inductive assumption to conclude that , belongs to both and Since . it follows from ( ,Equation2.1) that can be written as for some and On the other hand, since . it follows from ( ,Equation2.1) and from the inclusion guaranteed by Lemma 3.3 that has the form for some and some The inclusion . now follows from the fact that, by the last statement in Lemma 3.3, the denominators in the two obtained expressions for are coprime in .

Several examples that can be viewed as applications of Theorem 3.2 are given in Reference8.

## 4. Exchange relations: The exponents

Let be an exchange pattern (see Definition Equation2.1). In this section we will ignore the coefficients in the monomials and take a closer look at the dynamics of their exponents. (An alternative point of view that the reader may find helpful is to assume that all exchange patterns considered in this section will have all their coefficients equal to For every edge .) in let us write the ratio , of the corresponding monomials as

where (cf. (Equation2.3)); we note that ratios of this kind have already appeared in (Equation2.7). Let us denote by the integer matrix whose entries are the exponents in (Equation4.1). In view of (Equation2.5), the exponents in and are recovered from :

Thus, the family of matrices encodes all the exponents in all monomials of an exchange pattern.

We shall describe the conditions on the family of matrices imposed by the axioms of an exchange pattern. To do this, we need some preparation.

### Definition 4.1

A square integer matrix is called *sign-skew-symmetric* if, for any and either , or else , and are of opposite sign; in particular, for all .

### Definition 4.2

Let and be square integer matrices of the same size. We say that is obtained from by the *matrix mutation* in direction and write if

An immediate check shows that is *involutive*, i.e., its square is the identity transformation.

### Proposition 4.3

A family of integer matrices corresponds to an exchange pattern if and only if the following conditions hold:

(1) is sign-skew-symmetric for any .

(2) If then , .

### Proof.

We start with the “only if” part, i.e., we assume that the matrices are determined by an exchange pattern via (Equation4.1) and check the conditions (1)–(2). The condition follows from (Equation2.4). The remaining part of (1) (dealing with follows at once from ( ),Equation2.6). Turning to part (2), the equality is immediate from the definition (Equation4.1). Now suppose that In this case, we apply the axiom ( .Equation2.7) to the edge taken together with the two adjacent edges emanating from and and labeled by Taking ( .Equation4.2) into account, we obtain:

where Comparing the exponents of . on both sides of (Equation4.4) yields Finally, if . then comparing the exponents of , on both sides of (Equation4.4) gives

To complete the proof of (2), it remains to notice that, in view of the already proven part (1), the condition is equivalent to which makes the last formula equivalent to ( ,Equation4.3).

To prove the “if” part, it suffices to show that if the matrices satisfy (1)–(2), then the monomials given by the first equality in (Equation4.2) (with satisfy the axioms of an exchange pattern. This is done by a direct check. )

Since all matrix mutations are involutive, any choice of an initial vertex and an arbitrary integer matrix gives rise to a unique family of integer matrices satisfying condition (2) in Proposition 4.3 and such that Thus, the exponents in all monomials . are uniquely determined by a single matrix By Proposition .4.3, in order to determine an exchange pattern, must be such that all matrices obtained from it by a sequence of matrix mutations are sign-skew-symmetric. Verifying that a given matrix has this property seems to be quite nontrivial in general. Fortunately, there is another restriction on that is much easier to check, which implies the desired property, and still leaves us with a large class of matrices sufficient for most applications.

### Definition 4.4

A square integer matrix is called *skew-symmetrizable* if there exists a diagonal *skew-symmetrizing* matrix with positive integer diagonal entries such that is skew-symmetric, i.e., for all and .

### Proposition 4.5

For every choice of a vertex and a skew-symmetrizable matrix there exists a unique family of matrices , associated with an exchange pattern on and such that Furthermore, all the matrices . are skew-symmetrizable, sharing the same skew-symmetrizing matrix.

### Proof.

The proof follows at once from the following two observations:

1. Every skew-symmetrizable matrix is sign-skew-symmetric.

2. If is skew-symmetrizable and then , is also skew-symmetrizable, with the same skew-symmetrizing matrix.

We call an exchange pattern—and the corresponding cluster algebra—*skew-symmetrizable* if all the matrices given by (Equation4.1) (equivalently, one of them) are skew-symmetrizable. In particular, all cluster algebras of rank are skew-symmetrizable: for we have while for , the calculations in Example ,Equation2.5 show that one can take

for all in the notation of ( ,Equation2.10)–(Equation2.11).

### Remark 4.6

Skew-symmetrizable matrices are closely related to symmetrizable (generalized) Cartan matrices appearing in the theory of Kac-Moody algebras. More generally, to every sign-skew-symmetric matrix we can associate a generalized Cartan matrix of the same size by setting

There seem to be deep connections between the cluster algebra corresponding to and the Kac-Moody algebra associated with We exhibit such a connection for the rank . case in Section 6 below. This is however just the tip of an iceberg: a much more detailed analysis will be presented in the sequel to this paper.

In order to show that non-skew-symmetrizable exchange patterns do exist, we conclude this section by exhibiting a 3-parameter family of such patterns of rank .

### Proposition 4.7

Let , and , be three positive integers such that There exists a unique family of matrices . associated with a non-skew-symmetrizable exchange pattern on and such that the matrix at a given vertex is equal to

### Proof.

First of all, the matrix is sign-skew-symmetric but not skew-symmetrizable. Indeed, any skew-symmetrizable matrix satisfies the equation However, this equation for . holds only when is equal to or .

For the purpose of this proof only, we refer to a matrix as *cyclical* if its entries follow one of the two sign patterns

In particular,