Cluster algebras I: Foundations
To the memory of Sergei Kerov
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.
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 clusters. These clusters have the following exchange property: for any cluster -subsets 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 .Reference 9 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 .Reference 13, 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 Reference 16 and for in Reference 2. 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 cluster monomials) belong to this dual canonical basis. -deformations.
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 .Reference 11Reference 18. (We are grateful to Bernd Sturmfels for bringing these references to our attention.)
An essential feature of the exchange relations (Equation 1.1) 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 Reference 14Reference 15. 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 Reference 15Reference 5 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 Reference 1Reference 3Reference 4Reference 5Reference 6Reference 7Reference 12Reference 17Reference 20. The binomial exchange relations of the form (Equation 1.1) 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 Reference 20, 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 (Equation 1.1). 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 Reference 8.
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, whose edges are labeled by the elements of -regular 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 (Equation 2.2) 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 ,.
We note that in the last axiom, the substitution is effectively monomial, since in the event that neither nor contain condition ( ,Equation 2.6) requires that both and do not depend on thus making the whole substitution irrelevant. ,
One easily checks that axiom (Equation 2.7) 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.
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 (Equation 2.4), the right-hand side of the exchange relation (Equation 2.2) 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 (Equation 2.4) 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 ( ,Equation 2.1)–(Equation 2.2).
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 (Equation 2.8), then the cluster algebra is naturally identified with .