# Cluster algebras I: Foundations

To the memory of Sergei Kerov

## 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

The prototypical example of a cluster algebra of rank 1 is the coordinate ring

Another important incarnation of a cluster algebra of rank 1 is the coordinate ring

This form of representing the algebra *dual canonical basis*, i.e., the basis in the

We conjecture that the above examples can be extensively generalized: for any simply-connected connected semisimple group *cluster monomials*) belong to this dual canonical basis.

A particularly nice and well-understood example of a cluster algebra of an arbitrary rank

for all

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

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

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

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 whose edges are labeled by the elements of

To each vertex *cluster* of *exchange relations,* for every edge

Here

To be more precise, let *coefficient group*; a prototypical example is a free abelian group of finite rank. Every monomial

for some coefficient

The monomials

We note that in the last axiom, the substitution

One easily checks that axiom (Equation 2.7) is invariant under the “flip”

for every edge

Let *exchange polynomial*. We will write

Let

Note that property (Equation 2.4) ensures that *transition maps*

The smallest possible ground ring

Since