Cluster Algebras

Introduction

A cluster algebra is a commutative algebra generated inside a field of rational functions by an infinite family of elements obtained from an initial finite set by an operation called mutation. The operation is local: at each step one replaces a single generator by a new one determined by an exchange relation, and the relations are read from a quiver or, more generally, from a skew-symmetrizable integer matrix. The result is a combinatorial structure with three faces. Arithmetically, the Laurent phenomenon says that every generator obtained by mutation is a Laurent polynomial in the initial generators, so that the algebra sits inside the ring of Laurent polynomials despite the exchange relation having a denominator. Combinatorially, the generators and the clusters form a graph on which mutation acts, and the finite type classification of Fomin–Zelevinsky says that mutation produces finitely many generators exactly when the initial exchange matrix is mutation equivalent to a Dynkin diagram of type $A$, $D$, $E$. Representation-theoretically, the acyclic cluster algebras are the coordinate rings of the cluster-tilted algebras, and their clusters correspond to the cluster-tilting objects of the cluster category, so the whole theory is a combinatorial shadow of the tilting theory of the previous article.

This article develops seeds and exchange matrices, the mutation of a seed and of its quiver, the cluster algebra generated by a seed, the Laurent phenomenon, the finite type classification and the relation to the root systems, the cluster complex and the denominator vectors, the representation-theoretic interpretation by cluster-tilting objects, the examples of types $A_1$, $A_2$, $A_3$ and the Markov quiver, and the pointer to the cluster categories. It follows Tilting Theory, Quiver Representations and Representation Type, Auslander–Reiten Theory, Representation Theory and K-Theory of Rings, and it is the last article of the representation-theoretic arc of the category.

Throughout, the ground ring is $\mathbb{Z}$ and the field of rational functions is $\mathbb{Q}(x_1,\dots,x_n)$; the exchange matrix is a skew-symmetrizable integer matrix $B=(b_{ij})$, and the initial variables form a cluster $\mathbf{x}=(x_1,\dots,x_n)$ of algebraically independent transcendentals. The skew-symmetric case corresponds to a quiver without loops and two-cycles; the general skew-symmetrizable case is the general one and is stated where it differs. The theory is combinatorial and algebraic; no topology occurs and the geometric realisations (the cluster varieties, the Newton–Okounkov bodies) belong to Part II. The associated generalised Cartan matrix of an exchange matrix is a matrix of integers; its classification is used through the Dynkin diagrams, and the forms attached to the diagrams are not used.

Seeds and Mutation

Seeds and Exchange Matrices

Definition. A seed is a pair $(\mathbf{x},B)$ where $\mathbf{x}=(x_1,\dots,x_n)$ is a tuple of algebraically independent elements of a field $\mathcal{F}=\mathbb{Q}(x_1,\dots,x_n)$ and $B=(b_{ij})$ is an $n\times n$ integer matrix which is skew-symmetrizable: there are positive integers $d_1,\dots,d_n$ with $d_ib_{ij}=-d_jb_{ji}$ for all $i,j$, and $b_{ii}=0$. In the skew-symmetric case $d_i=1$ for all $i$ and $B$ is the signed adjacency matrix of a quiver $Q$: $b_{ij}=\#\{i\to j\}-\#\{j\to i\}$.

Definition. The mutation of a seed $(\mathbf{x},B)$ in the direction $k$, $1\le k\le n$, is the seed $\mu_k(\mathbf{x},B)=(\mathbf{x}',B')$ with: - $x_k'$ defined by the exchange relation $$ x_kx_k'=\prod_{b_{ik}>0}x_i^{\,b_{ik}}+\prod_{b_{ik}<0}x_i^{\,-b_{ik}}, $$ and $x_i'=x_i$ for $i\neq k$; - the matrix mutation, which reverses the signs in the $k$-th row and the $k$-th column, $b_{ij}'=-b_{ij}$ for $i=k$ or $j=k$, and modifies the remaining entries by the rule $b_{ij}'=b_{ij}+\tfrac12(|b_{ik}|b_{kj}+b_{ik}|b_{kj}|)$; these are the only two cases and together they define the mutated matrix $B'=(b_{ij}')$.

Mutation is an involution: $\mu_k\mu_k$ is the identity on seeds.

Proposition. The matrix mutation is compatible with the quiver mutation: if $B$ is skew-symmetric and corresponds to the quiver $Q$, then $B'$ corresponds to the quiver $\mu_k(Q)$ obtained by (i) reversing every arrow at $k$, and (ii) for each path $i\to k\to j$ inserting an arrow $i\to j$, and then deleting all two-cycles. In this formulation the exchange relation reads $$ x_kx_k'=\prod_{\text{arrows }i\to k}x_i+\prod_{\text{arrows }k\to j}x_j. $$

Proof. The two cases of the matrix formula are exactly the reversal of the arrows at $k$ and the insertion rule for the paths through $k$; the count of the inserted arrows is the number of such paths, and the deletion of the two-cycles corresponds to the cancellation in the integer formula. The exchange relation is the translation of the products over the incoming and the outgoing arrows.

Example (type $A_1$). For $n=1$ and $B=(0)$ the exchange relation is $x_1x_1'=1+1=2$; the mutation is $x_1\mapsto 2/x_1$, of order $2$, and only two cluster variables occur. Over a field of characteristic not $2$ this is the unique rank-one cluster algebra.

Example (type $A_2$). For $n=2$ and the quiver $1\to2$, so $b_{12}=1=-b_{21}$, the exchange relations are $$ x_1x_1'=1+x_2,\qquad x_2x_2'=1+x_1. $$ Mutating at $1$ and then at $2$ produces $x_2''=\dfrac{1+x_1'}{x_2}=\dfrac{x_1+x_2+1}{x_1x_2}$, and mutating once more returns the original variables; there are five cluster variables $x_1,x_2,\frac{1+x_2}{x_1},\frac{1+x_1}{x_2},\frac{x_1+x_2+1}{x_1x_2}$, and the two clusters are $\{x_1,x_2\}$ and $\{\frac{1+x_2}{x_1},\frac{1+x_1}{x_2}\}$. The number five is the number of almost positive roots of the root system $A_2$, the two simple roots and the three positive roots.

Example (Markov quiver). Let $Q$ be the quiver with three vertices and two arrows between each pair, cyclically oriented: $1\rightrightarrows2\rightrightarrows3\rightrightarrows1$. Every mutation at a vertex returns the same quiver up to isomorphism, so the exchange matrix is periodic; the cluster algebra has infinitely many cluster variables and is not of finite type. This is the simplest example of a mutation class closed on one quiver with more than two arrows between some pair.

The Cluster Algebra

Definition. Let $(\mathbf{x},B)$ be a seed with $\mathbf{x}$ in the field $\mathcal{F}$. The cluster variables are the elements obtained from the $x_i$ by all sequences of mutations, and the clusters are the $n$-tuples obtained by the same sequences. The cluster algebra $\mathcal{A}(\mathbf{x},B)$ is the $\mathbb{Z}$-subalgebra of $\mathcal{F}$ generated by all the cluster variables; it is determined up to isomorphism by the mutation class of $B$. The exchange graph has the clusters as vertices and an edge for each mutation between them.

Theorem (Laurent phenomenon, Fomin–Zelevinsky). Every cluster variable of $\mathcal{A}(\mathbf{x},B)$ is a Laurent polynomial in the initial cluster $\mathbf{x}$ with integer coefficients: $$ \mathcal{A}(\mathbf{x},B)\subseteq\mathbb{Z}[x_1^{\pm1},\dots,x_n^{\pm1}]. $$ Moreover every cluster variable is a Laurent polynomial with nonnegative coefficients when the exchange matrix is acyclic (the positivity theorem, proved by the representation-theoretic interpretation below). The cluster algebra is the intersection of the Laurent ring with the field generated by the cluster variables over the clusters, so it is presented as a subalgebra of the Laurent ring and not as a quotient.

Proof (in outline). The proof is an induction on the length of a mutation sequence: the exchange relation of the mutated variable is the quotient of a sum of Laurent monomials, and the key step is that the monomials in the numerator are divisible by $x_k$ in the Laurent ring when the Laurent exponents are read off the matrix; the divisibility is the content of the "caterpillar lemma" and is verified by a bootstrap over the mutation class. The positivity for the acyclic case follows from the categorification of the next section, where the Laurent coefficients count composition factors.

Theorem (Fomin–Zelevinsky, finite type). The cluster algebra $\mathcal{A}(\mathbf{x},B)$ has finitely many cluster variables if and only if the mutation class of $B$ contains the exchange matrix of a quiver whose underlying graph is a Dynkin diagram of type $A_n$, $D_n$, $E_6$, $E_7$ or $E_8$. In that case: - the cluster variables are in bijection with the almost positive roots of the corresponding root system, that is, with the positive roots together with the negatives of the simple roots; - the clusters are the clusters of the root system, that is, the sets of pairwise compatible almost positive roots, and the exchange graph is the graph of the clusters; - the number of cluster variables is $n$ plus the number of positive roots of the root system: it is $\dfrac{n(n+1)}{2}+n$ in type $A_n$, $n(n-1)+n$ in type $D_n$, and $36+6$, $63+7$, $120+8$ in types $E_6,E_7,E_8$.

Proof (in outline). The acyclic case is established by comparing the exchange relations with the reflection functors: the mutation at a vertex corresponds to the reflection at a simple root, the cluster variables correspond to the almost positive roots, and the clusters to the sets of pairwise compatible roots. The general case follows because the mutation class of a Dynkin quiver consists of acyclic quivers, by the classification of the mutation classes of the Dynkin diagrams. The counts are the counts of the almost positive roots of the root systems.

Example. For $A_2$ the almost positive roots are the two simple roots $\alpha_1,\alpha_2$ and the three positive roots, five in all, matching the five cluster variables of the computed example; for $A_3$ there are $3+6=9$ almost positive roots and nine cluster variables, and fourteen clusters.

The Cluster Complex and the Denominator Vectors

Definition. Two cluster variables are compatible if they occur together in some cluster. The cluster complex is the simplicial complex whose faces are the sets of pairwise compatible cluster variables; its maximal faces are the clusters. The denominator vector of a cluster variable $x$ is the vector $\mathbf{d}(x)\in\mathbb{Z}^n$ of the exponents of the denominator of $x$ written as a Laurent polynomial in the initial cluster with the denominator cleared and the numerator not divisible by any $x_i$; the $\mathbf{g}$-vector is the vector of exponents of the numerator with respect to a suitable basis, defined by the tropical rules of Fomin–Zelevinsky.

Proposition. The denominator vector is well defined: writing the cluster variable as an element of $\mathbb{Z}[x_1^{\pm1},\dots,x_n^{\pm1}]$ in lowest terms, the exponents of the denominator are independent of the sequence of mutations producing $x$. The denominator vectors of the cluster variables of a cluster algebra of finite type are in bijection with the positive roots, and the clusters correspond to the sets of almost positive roots with the compatibility relation of the root system.

Proof (in outline). The well-definedness is the Laurent phenomenon together with the uniqueness of the reduced form in the Laurent ring. The finite-type statement is the refinement of the classification theorem: the numerator of the exchange relation identifies the denominator vector shift with the action of the reflection group, and the compatibility of two cluster variables is the vanishing of a suitable bilinear pairing of their denominator vectors; the pairing is the one attached to the root system and its theory of forms is not used here, the statement being taken from the classification theorem.

Example. In type $A_2$ the denominator vectors of the five cluster variables are the five almost positive roots; the compatibility relation pairs the two simple roots with all the positive roots containing them and leaves the two middle variables incompatible, giving exactly the two clusters.

The Representation-Theoretic Interpretation

Remark. Let $Q$ be an acyclic quiver and $A=kQ$ its path algebra. The cluster category $\mathcal{C}_Q$ is the orbit category of the derived category $D^b(A)$ under the action of the functor $\tau^{-1}[1]$, where $\tau$ is the Auslander–Reiten translation and $[1]$ the shift; it is a triangulated category studied by Buan–Marsh–Reineke–Reiten–Todorov. The cluster-tilting objects are the objects $T$ with $\operatorname{Ext}^1_{\mathcal{C}}(T,T)=0$ and $n$ pairwise non-isomorphic indecomposable summands, and the endomorphism algebras $\operatorname{End}_{\mathcal{C}}(T)^{\mathrm{op}}$ are the cluster-tilted algebras, the relatives of the tilted algebras of Tilting Theory.

Theorem (Buan–Marsh–Reineke–Reiten–Todorov; Caldero–Chapoton). For an acyclic initial quiver $Q$: - the cluster-tilting objects of the cluster category are in bijection with the clusters of the cluster algebra $\mathcal{A}(\mathbf{x},B(Q))$, and the indecomposable summands of a cluster-tilting object correspond to the cluster variables of the cluster; - the mutation of a cluster-tilting object corresponds to the mutation of the seed, and the cluster-tilted algebra of a mutation is the endomorphism algebra of the mutated object; - each cluster variable is the image of a rigid indecomposable object of the cluster category under a map to the cluster algebra, and for a Dynkin quiver the indecomposables of the cluster category are in bijection with the almost positive roots, so the two classifications agree.

Proof (in outline). The cluster category has finitely many indecomposables when $Q$ is Dynkin, namely the indecomposables of the derived category modulo the translation and the shift, and their number is that of the almost positive roots by a direct computation on the AR quiver. The mutations of the cluster-tilting objects are realised by the exchange triangles, which are the triangles with two cluster-tilting objects at the ends and are the categorical form of the exchange relations; the endomorphism algebras mutate as the cluster-tilted algebras, and this gives the bijection.

Example. For $Q$ of type $A_2$ the cluster category has five indecomposables, three of them in the orbit of the projective indecomposables of the two-vertex quiver and two in the orbit of the simples; the five cluster variables of the cluster algebra correspond to them, and the two clusters to the two cluster-tilting objects, matching the list computed above.

Remark. The cluster-tilted algebras are the endomorphism algebras of the cluster-tilting objects, and they are the algebraic relatives of the tilted algebras; their quivers are obtained from the initial quiver by the mutation rule, and their representation theory inherits the description by the cluster category. The full categorical treatment, including the $\infty$-categorical and model-categorical refinements, belongs to Part II, wheretreats the homotopy-coherent version, and the present article cites that treatment rather than developing it.

The Cartan Counterpart and the Classification

Definition. Let $B$ be a skew-symmetrizable exchange matrix. Its Cartan counterpart $A(B)$ is the integer matrix with $a_{ii}=2$ and $a_{ij}=-|b_{ij}|$ for $i\neq j$; an exchange matrix is of finite type when its mutation class is finite.

Theorem (Fomin–Zelevinsky). The exchange matrix $B$ is of finite type if and only if its Cartan counterpart $A(B)$ is a Cartan matrix of finite type, that is, a Cartan matrix of a Dynkin diagram. Consequently the finite types are $A_n$, $B_n=C_n$, $D_n$, $E_6$, $E_7$, $E_8$, $F_4$ and $G_2$, and the Dynkin diagram is determined up to isomorphism by the mutation class of $B$.

Proof (in outline). The Cartan counterpart is not invariant under mutation, but the mutation class of $B$ determines a mutation class of Cartan matrices, obtained as follows: a matrix of the mutation class is a quasi-Cartan companion of the exchange matrix, and mutation of $B$ corresponds to the standard reflection of that companion, as Fomin and Zelevinsky show by comparing the exchange rule entry by entry and using the sign coherence of the companion. The mutation classes of Cartan matrices that are finite are exactly the Dynkin diagrams, which is the classical classification of the finite Coxeter groups; the positive definiteness of the symmetric bilinear form attached to the diagram is what rules out the mutations that would produce a matrix of indefinite type.

Remark. The theorem reduces the classification of the cluster algebras of finite type to a statement about Cartan matrices, and it explains why the affine diagrams appear in the mutation-finite but infinite-type cases, where the analysis of the rank-two example below is the first instance.

Example. For a rank-two exchange matrix the mutation class is finite exactly for $|b_{12}b_{21}|\le3$: $|b_{12}b_{21}|=0,1,2,3$ give the types $A_1\times A_1$, $A_2$, $B_2$ and $G_2$, and for $|b_{12}b_{21}|\ge4$ the cluster algebra has infinitely many cluster variables. The skew-symmetric case with $b_{12}=b$, $b_{21}=-b$ has $-b_{12}b_{21}=b^2$, so the finite rank-two skew-symmetric types are $|b|\le1$, that is, $A_1\times A_1$ and $A_2$; the types $B_2$ and $G_2$ require the skew-symmetrizable matrices with $(b_{12},b_{21})=(-2,1)$ and $(-3,1)$.

g-Vectors and Cluster Monomials

Definition. A cluster monomial is a product $x_1^{a_1}\cdots x_n^{a_n}$ of the elements of a single cluster; the $\mathbf{g}$-vectors of a cluster are the vectors of exponents of the tropical evaluation of the cluster variables, as in the definition of the denominator vectors above.

Theorem (Derksen–Weyman–Zelevinsky; Gross–Hacking–Keel–Kontsevich). The $\mathbf{g}$-vectors of the elements of a cluster are sign-coherent: for every $i$ the $i$-th coordinates of the $\mathbf{g}$-vectors of the cluster variables of a fixed cluster are all non-negative or all non-positive. Moreover the $\mathbf{g}$-vectors of the cluster variables of a cluster form a $\mathbb{Z}$-basis of $\mathbb{Z}^n$, and the cluster monomials of the cluster algebra are linearly independent; the two statements together are the $\mathbf{g}$-vector conjecture of Fomin and Zelevinsky.

Remark. In finite type the denominator vectors of the cluster variables are the almost positive roots, the indexing set of the clusters of the previous section, and the $\mathbf{g}$-vectors are the complementary invariants of the numerators of the Laurent expansions, transforming by the tropical rule under mutation; the Cartan counterpart of the previous subsection relates the two descriptions to the root system of the exchange matrix.

Summary

A seed is a cluster of algebraically independent variables together with a skew-symmetrizable exchange matrix, and the mutation of a seed replaces one variable by the quotient of a sum of two monomials and updates the matrix by integer rules. For a skew-symmetric matrix the rules are those of the mutation of a quiver: reverse the arrows at the mutated vertex, insert the arrows for the paths through it, and delete the two-cycles. The cluster algebra is the subalgebra generated by all the variables obtained by mutation, and the Laurent phenomenon says that it sits inside the ring of Laurent polynomials in the initial cluster, with nonnegative coefficients in the acyclic case. The finite type classification says that only the Dynkin diagrams $A$, $D$, $E$ give finitely many cluster variables, and then the variables are the almost positive roots and the clusters the clusters of the root system.

The cluster complex records compatibility, the denominator vectors organise the variables, and the categorification identifies the clusters with the cluster-tilting objects of the cluster category of the acyclic quiver, the mutations with the exchange triangles, and the cluster-tilted algebras with the endomorphism algebras of the cluster-tilting objects. In this way the cluster algebra is the combinatorial shadow of the tilting theory and the Auslander–Reiten theory of the preceding articles, and its higher-categorical and geometric refinements belong to Part II.

Summary of Notation

Symbol	Meaning
$(\mathbf{x},B)$	seed, cluster and exchange matrix
$n$	rank of the seed, number of cluster variables in a cluster
$x_i$	cluster variables
$\mu_k$	mutation in the direction $k$
$b_{ij}$	entries of the skew-symmetrizable exchange matrix
$\mathbf{d}(x)$	denominator vector of a cluster variable
$\mathcal{A}(\mathbf{x},B)$	cluster algebra
$\mathcal{C}_Q$	cluster category of an acyclic quiver $Q$
$\tau$	Auslander–Reiten translation
$\alpha_i$	simple root of the associated root system
$A(B)$, $a_{ij}$	Cartan counterpart of the exchange matrix
$x_1^{a_1}\cdots x_n^{a_n}$	a cluster monomial
$\mathbf{g}(x)$	$\mathbf{g}$-vector of a cluster variable