Categorification of acyclic cluster algebras: an introduction

Bernhard Keller

Université Paris 7 – Denis Diderot UFR de Mathématiques Institut de Mathématiques de Jussieu, UMR 7586 du CNRS Case 7012 2, place Jussieu 75251 Paris Cedex 05 France

To Murray Gerstenhaber and Jim Stasheff

Summary. This is a concise introduction to Fomin-Zelevinsky’s cluster algebras and their links with the representation theory of quivers in the acyclic case. We review the definition cluster algebras (geometric, without coefficients), construct the cluster category and present the bijection between cluster variables and rigid indecomposable objects of the cluster category.

1 Introduction

1.1 Context

Cluster algebras were invented by S. Fomin and A. Zelevinsky [28] in the spring of the year 2000 in a project whose aim it was to develop a combinatorial approach to the results obtained by G. Lusztig concerning total positivity in algebraic groups [55] on the one hand and canonical bases in quantum groups [54] on the other hand (let us stress that canonical bases were discovered independently and simultaneously by M. Kashiwara [46]). Despite great progress during the last few years [29] [8] [32], we are still relatively far from these initial aims. Presently, the best results on the link between cluster algebras and canonical bases are probably those of C. Geiss, B. Leclerc and J. Schröer [39] [40] [36] but even they cannot construct canonical bases from cluster variables for the moment. Despite these difficulties, the theory of cluster algebras has witnessed spectacular growth thanks notably to the many links that have been discovered with a wide range of subjects including • Poisson geometry [41] [42] …,

• integrable systems [31] …,

• higher Teichmüller spaces [21] [23] [24] [22] …,

• combinatorics and the study of combinatorial polyhedra like the Stasheff associahedra [18] [17] [53] [26] [59] [27] …,

• commutative and non commutative algebraic geometry, in particular the study of stability conditions in the sense of Bridgeland [10] [9], Calabi-Yau algebras [43] [19], Donaldson-Thomas invariants [65] [50] [51] [52] …,

• and last not least the representation theory of quivers and finite-dimensional algebras, cf. for example the surveys [4] [62] [61].

We refer to the introductory papers [68] [30] [67] [69] [70] and to the cluster algebras portal [25] for more information on cluster algebras and their links with other parts of mathematics.

The link between cluster algebras and quiver representations follows the spirit of categorification: One tries to interpret cluster algebras as combinatorial (perhaps K-theoretic) invariants associated with categories of representations. Thanks to the rich structure of these categories, one can then hope to prove results on cluster algebras which seem beyond the scope of the purely combinatorial methods. It turns out that the link becomes especially beautiful if we use a triangulated category constructed from the category of quiver representations, the so-called cluster category.

In this brief survey, we will review the definition of cluster algebras and Fomin-Zelevinsky’s classification theorem for cluster-finite cluster algebras [29]. We will then recall some basic notions on the representations of a quiver without oriented cycles, introduce the cluster category and describe its link with the cluster algebra.

2 Cluster algebras

The cluster algebras we will be interested in are associated with antisymmetric matrices with integer coefficients. Instead of using matrices, we will use quivers (without loops and 2-cycles), since they are easy to visualize and well-suited to our later purposes.

2.1 Quivers

Let us recall that a quiver Q is an oriented graph. Thus, it is a quadruple given by a set Q_{0} (the set of vertices), a set Q_{1} (the set of arrows) and two maps s : Q_{1} → Q_{0} and t : Q_{1} → Q_{0} which take an arrow to its source respectively its target. Our quivers are ‘abstract graphs’ but in practice we draw them as in this example:

Q : [figure: 3; λ; 5; α; 6; 1; ν; 2; β; μ; 4.; γ]

A loop in a quiver Q is an arrow α whose source coincides with its target; a 2-cycle is a pair of distinct arrows β ≠ γ such that the source of β equals the target of γ and vice versa. It is clear how to define 3-cycles, connected components …. A quiver is finite if both, its set of vertices and its set of arrows, are finite.

2.2 Seeds and mutations

Fix an integer n ≥ 1. A seed is a pair (R, u), where

a) R is a finite quiver without loops or 2-cycles with vertex set {1, …, n};

b) u is a free generating set {u_{1}, …, u_{n}} of the field ℚ(x_{1}, …, x_{n}) of fractions of the polynomial ring ℚ[x_{1}, …, x_{n}] in n indeterminates.

Notice that in the quiver R of a seed, all arrows between any two given vertices point in the same direction (since R does not have 2-cycles). Let (R, u) be a seed and k a vertex of R. The mutation μ_{k}(R, u) of (R, u) at k is the seed (R′, u′), where

a) R′ is obtained from R as follows:

1) reverse all arrows incident with k;

2) for all vertices i ≠ j distinct from k, modify the number of arrows between i and j as follows:

[figure: R R′; i; r; s; j; k; t; i; r+st; j; t; k; s; i; r; j; t; k; s; i; r−st; s; j; k; t]

where r, s, t are non negative integers, an arrow \xymatrix{i \ar[r]^{l} & j} with l ≥ 0

means that l arrows go from i to j and an arrow \xymatrix{i \ar[r]^{l} & j} with l ≤ 0 means that −l arrows go from j to i.

b) u′ is obtained from u by replacing the element u_{k} with

u′_{k} = \frac{1}{u_{k}} ( ∏_{arrows i → k} u_{i} + ∏_{arrows k → j} u_{j}) .  (1)

In the exchange relation (1), if there are no arrows from i with target k, the product is taken over the empty set and equals 1. It is not hard to see that μ_{k}(R, u) is indeed a seed and that μ_{k} is an involution: we have μ_{k}(μ_{k}(R, u)) = (R, u).

2.3 Examples of mutations

Let R be the cyclic quiver

[figure: 1; 2; 3]

and u = {x_{1}, x_{2}, x_{3}}. If we mutate at k = 1, we obtain the quiver

[figure: 1; 2; 3]

and the set of fractions given by u′_{1} = (x_{2} + x_{3})/x_{1}, u′_{2} = u_{2} = x_{2} and u′_{3} = u_{3} = x_{3}. Now, if we mutate again at 1, we obtain the original seed. This is a general fact: Mutation at k is an involution. If, on the other hand, we mutate (R′, u′) at 2, we obtain the quiver

[figure: 1; 2; 3]

and the set u′′ given by u′′_{1} = u′_{1} = (x_{2} + x_{3})/x_{1}, u′_{2} = \frac{x_{1}+x_{2}+x_{3}}{x_{1}x_{2}} and u′′_{3} = u′_{3} = x_{3}.

Let us consider the following, more complicated quiver glued together from four 3-cycles:

[figure: 1; 2; 3; 4; 5; 6.]

If we successively perform mutations at the vertices 5, 3, 1 and 6, we obtain the sequence of quivers (we use [47])

[figure: 1; 2; 3; 4; 5; 6; 1; 2; 3; 4; 5; 6; 1; 2; 3; 4; 5; 6; 1; 2; 3; 4; 5; 6.]

Notice that the last quiver no longer has any oriented cycles and is in fact an orientation of the Dynkin diagram of type D_{6}. The sequence of new fractions appearing in these steps is

u′_{5} = \frac{x_{3}x_{4} + x_{2}x_{6}}{x_{5}} , u′_{3} = \frac{x_{3}x_{4} + x_{1}x_{5} + x_{2}x_{6}}{x_{3}x_{5}} ,
u′_{1} = \frac{x_{2}x_{3}x_{4} + x_{3}^{2}x_{4} + x_{1}x_{2}x_{5} + x_{2}^{2}x_{6} + x_{2}x_{3}x_{6}}{x_{1}x_{3}x_{5}} , u′_{6} = \frac{x_{3}x_{4} + x_{4}x_{5} + x_{2}x_{6}}{x_{5}x_{6}} .

It is remarkable that all the denominators appearing here are monomials and that all the coefficients in the numerators are positive.

Finally, let us consider the quiver

[figure: 1; 2; 3; 4; 5; 6; 7; 8; 9; 10.]

One can show [49] that it is impossible to transform it into a quiver without oriented cycles by a finite sequence of mutations. However, its mutation class (the set of all quivers obtained from it by iterated mutations) contains many quivers with just one oriented cycle, for example

[figure: 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 1; 2.; 3; 4; 5; 6; 7; 8; 9; 10]

In fact, in this example, the mutation class is finite and it can be completely computed using, for example, [47]: It consists of 5739 quivers up to isomorphism. The above quivers are members of the mutation class containing relatively few arrows. The initial quiver is the unique member of its mutation class with the largest number of arrows. Here are some other quivers in the mutation class with a relatively large number of arrows:

[figure: ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘; ∘]

Only 84 among the 5739 quivers in the mutation class contain double arrows (and none contain arrows of multiplicity ≥ 3). Here is a typical example

[figure: 1; 2; 3; 4; 5; 6; 7; 8; 9; 10; 2]

2.4 Definition of cluster algebras

Let Q be a finite quiver without loops or 2-cycles with vertex set {1, …, n}. Consider the seed (Q, x) consisting of Q and the set x formed by the variables x_{1}, …, x_{n}. Following [28] we define

• the clusters with respect to Q to be the sets u appearing in seeds (R, u) obtained from (Q, x) by iterated mutation,

• the cluster variables for Q to be the elements of all clusters,

• the cluster algebra A_{Q} to be the ℚ-subalgebra of the field ℚ(x_{1}, …, x_{n}) generated by all the cluster variables.

Thus the cluster algebra consists of all ℚ-linear combinations of monomials in the cluster variables. It is useful to define another combinatorial object associated with this recursive construction: The exchange graph associated with Q is the graph whose vertices are the seeds modulo simultaneous renumbering of the vertices and the associated cluster variables and whose edges correspond to mutations.

2.5 The example A_{3}

Let us consider the quiver

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

obtained by endowing the Dynkin diagram A_{3} with a linear orientation. By applying the recursive construction to the initial seed (Q, x) one finds exactly fourteen seeds (modulo simultaneous renumbering of vertices and cluster variables). These are the vertices of the exchange graph, which is isomorphic to the third Stasheff associahedron [64] [18]:

[figure: 2; ∘; ∘; ∘; 3; ∘; ∘; ∘; 1; ∘; ∘; ∘; ∘; ∘]

The vertex labeled 1 corresponds to (Q, x), the vertex 2 to μ_{2}(Q, x), which is given by

\xymatrix{1 \ar[rr] & 2 \ar[l] & 3 \ar[l]} , \{x_{1}, \frac{x_{1} + x_{3}}{x_{2}} , x_{3}\} ,

and the vertex 3 to μ_{1}(Q, x), which is given by

\xymatrix{1 & 2 \ar[l] \ar[r] & 3} , \{\frac{1 + x_{3}}{x_{1}} , x_{2}, x_{3}\}.

We find a total of 9 cluster variables, namely

x_{1} , x_{2} , x_{3}, \frac{1 + x_{2}}{x_{1}} , \frac{x_{1} + x_{3} + x_{2}x_{3}}{x_{1}x_{2}} , \frac{x_{1} + x_{1}x_{2} + x_{3} + x_{2}x_{3}}{x_{1}x_{2}x_{3}} ,

\frac{x_{1} + x_{3}}{x_{2}} , \frac{x_{1} + x_{1}x_{2} + x_{3}}{x_{2}x_{3}} , \frac{1 + x_{2}}{x_{3}} .

Again we observe that all denominators are monomials. Notice also that 9 = 3 + 6 and that 3 is the rank of the root system associated with A_{3} and 6 its number of positive roots. Moreover, if we look at the denominators of the non trivial cluster variables (those other than x_{1}, x_{2}, x_{3}), we see a natural bijection with the positive roots

α_{1}, α_{1} + α_{2}, α_{1} + α_{2} + α_{3}, α_{2}, α_{2} + α_{3}, α_{3}

of the root system of A_{3}, where α_{1}, α_{2}, α_{3} denote the three simple roots.

2.6 Cluster algebras with finitely many cluster variables

The phenomena observed in the above example are explained by the following key theorem:

Theorem 1 (Fomin-Zelevinsky [29]). Let Q be a finite connected quiver without loops or 2-cycles with vertex set {1, …, n}. Let A_{Q} be the associated cluster algebra.

a) All cluster variables are Laurent polynomials, i.e. their denominators are monomials.

b) The number of cluster variables is finite iff Q is mutation equivalent to an orientation of a simply laced Dynkin diagram Δ. In this case, Δ is unique and the non trivial cluster variables are in bijection with the positive roots of Δ; namely, if we denote the simple roots by α_{1}, …, α_{n}, then for each positive root ∑ d_{i}α_{i}, there is a unique non trivial cluster variable whose denominator is ∏ x_{i}^{d_{i}}.

3 Categorification

We refer to the books [63] [34] [2] and [1] for a wealth of information on the representation theory of quivers and finite-dimensional algebras. Here, we will only need very basic notions.

Let Q be a finite quiver without oriented cycles. For example, Q can be an orientation of a simply laced Dynkin diagram or the quiver

\xymatrix{ & 2 \ar[dr]^{β} \\ 1 \ar[rr]_{γ} \ar[ur]^{α} &  & 3.}

Let k be an algebraically closed field. Recall that a representation of Q is a diagram of finite-dimensional vector spaces of the shape given by Q. Thus, in the above example, a representation of Q is a (not necessarily commutative) diagram

\xymatrix{ & V_{2} \ar[dr]^{V_{β}} \\ V_{1} \ar[rr]_{V_{γ}} \ar[ur]^{V_{α}} &  & V_{3}}

formed by three finite-dimensional vector spaces and three linear maps. A morphism of representations is a morphism of diagrams. We thus obtain the category of representations rep(Q). Notice that it is an abelian category (since it is a category of diagrams in an abelian category, that of finite-dimensional vector spaces): Sums, kernels and cokernels in the category rep(Q) are computed componentwise. We denote by D_{Q} its bounded derived category. Thus, the objects of D_{Q} are the bounded complexes

\xymatrix{… \ar[r] & V^{p} \ar[r]^{d} & V^{p+1} \ar[r] & …}

of representations and its morphisms are obtained from morphisms of complexes by formally inverting all quasi-isomorphisms (=morphisms inducing isomorphisms in homology). The category D_{Q} is still an additive category (direct sums are given by direct sums of complexes) but it is almost never abelian. In fact, it is abelian if and only if Q does not have any arrows. But it is always triangulated. This means that D_{Q} is additive and endowed with

a) a suspension functor Σ : D_{Q} →^{∼} D_{Q}, namely the functor taking a complex V to V [1], where V [1]^{p} = V^{p+1} for all p ∈ ℤ and d_{V [1]} = −d_{V};

b) a class of triangles, namely the sequences

\xymatrix{U \ar[r] & V \ar[r] & W \ar[r] & ΣU}

which are ‘induced’ from short exact sequences of complexes.

The triangulated category D_{Q} admits a Serre functor, i.e. an autoequivalence S : D_{Q} →^{∼} D_{Q} which makes the Serre duality formula true: We have

D Hom(X, Y) →^{∼} Hom(Y, SX)

bifunctorially in X, Y belonging to D_{Q}, where D denotes the duality functor Hom_{k}(?, k) over the ground field k. The cluster category is defined as the orbit category

C_{Q} = D_{Q}/(S^{−1} ∘ Σ^{2})^{ℤ}

of D_{Q} under the action of the cyclic group generated by the automorphism S^{−1} ∘ Σ^{2}. Thus, its objects are the same as those of D_{Q} and its morphisms are defined by

Hom_{C_{Q}}(X, Y) = ⨁_{p∈ℤ} Hom_{D_{Q}}(X, (S^{−1} ∘ Σ^{2})^{p}Y).

One can show [48] that C_{Q} admits a structure of triangulated category such that the projection functor D_{Q} → C_{Q} becomes a triangle functor (in general, the orbit category of a triangulated category under the action of an automorphism group is no longer triangulated). It is not hard to see that the cluster category has finite-dimensional morphism spaces, and that it admits a Serre functor induced by that of the derived category. The definition of the cluster category then immediately yields an isomorphism

S →^{∼} Σ^{2}

and this means that C_{Q} is 2-Calabi-Yau. The definition of the cluster category is due to Buan-Marsh-Reineke-Reiten-Todorov [5] (for arbitrary Q without oriented cycles) and, independently and with a very different, more geometric description, to Caldero-Chapoton-Schiffler [13] (for Q of type A_{n}).

To state the close relationship between the cluster category C_{Q} and the cluster algebra A_{Q}, we need some notation: For two objects L and M of C_{Q}, we write

Ext^{1}(L, M) = Hom_{C_{Q}}(L, ΣM).

Notice that it follows from the Calabi-Yau property that we have a canonical isomorphism

Ext^{1}(L, M) →^{∼} D Ext^{1}(M, L).

An object L of C_{Q} is rigid if we have Ext^{1}(L, L) = 0. It is indecomposable if it is non zero and in each decomposition L = L_{1} ⊕ L_{2}, we have L_{1} = 0 or L_{2} = 0.

Theorem 2 ([15]). Let Q be a finite quiver without oriented cycles with vertex set {1, …, n}.

a) There is an explicit bijection L ↦ X_{L} from the set of isomorphism classes of rigid indecomposables of the cluster category C_{Q} onto the set of cluster variables of the cluster algebra A_{Q}.

b) Under this bijection, the clusters correspond exactly to the cluster-tilting subsets, i.e. the sets T_{1}, …, T_{n} of rigid indecomposables such that

Ext^{1}(T_{i}, T_{j}) = 0

for all i, j.

c) If L and M are rigid indecomposables such that the space Ext^{1}(L, M) is one-dimensional, then we have the generalized exchange relation

X_{L} = \frac{X_{B} + X_{B′}}{X_{M}}  (2)

where B and B′ are the middle terms of ‘the’ non split triangles

\xymatrix{L \ar[r] & B \ar[r] & M \ar[r] & ΣL & and & M \ar[r] & B′ \ar[r] & L \ar[r] & ΣM}

and we define

X_{B} = ∏_{i=1}^{s} X_{B_{i}} ,

where B = B_{1} ⊕ ⋯ ⊕ B_{s} is a decomposition into indecomposables.

The relation (2) in part c) of the theorem can be generalized to the case where the extension group is of higher dimension, cf. [14] [44] [66]. One can show using [7] that relation (2) generalizes the exchange relation (1) which appeared in the definition of the mutation.

The proof of the theorem builds on work by many authors notably Buan-Marsh-Reiten-Todorov [11], Buan-Marsh-Reiten [6], Buan-Marsh-Reineke-Reiten-Todorov [5], Marsh-Reineke-Zelevinsky [58], … and especially on Caldero-Chapoton’s explicit formula for X_{L} proved in [12] for orientations of simply laced Dynkin diagrams. We include the formula below. Another crucial ingredient of the proof is the Calabi-Yau property of the cluster category. An alternative proof of part c) was given by A. Hubery [44] for quivers whose underlying graph is an extended simply laced Dynkin diagram.

The theorem does shed new light on cluster algebras. In particular, we have the following

Corollary 1 (Caldero-Reineke [16]). Suppose that Q does not have oriented cycles. Then all cluster variables of A_{Q} belong to ℕ[x_{1}^{±}, …, x_{n}^{±}].

This settles a conjecture of Fomin-Zelevinsky [28] in the case of cluster algebras associated with acyclic quivers. The proof is based on Lusztig’s [57] and in this sense it does not quite live up to the hopes that cluster theory ought to explain Lusztig’s results. However, it does show that the conjecture is true for this important class of cluster algebras.

4 Caldero-Chapoton’s formula

We describe the bijection of part a) of theorem 2. Let k be an algebraically closed field and Q a finite quiver without oriented cycles with vertex set {1, …, n}. Let L be an object of the cluster category C_{Q}. With L, we will associate an element X_{L} of the field ℚ(x_{1}, …, x_{n}). According to [5], the object L decomposes into a sum of indecomposables L_{i}, 1 ≤ i ≤ s, unique up to isomorphism and permutation. By defining

X_{L} = ∏_{i=1}^{s} X_{L_{i}}

we reduce to the case where L is indecomposable. Now again by [5], if L is indecomposable, it is either isomorphic to an object π(V), or an object Σπ(P_{i}), where π : D_{Q} → C_{Q} is the canonical projection functor, Σ is the suspension functor of C_{Q}, V is a representation of Q (identified with a complex of representations concentrated in degree 0) and P_{i} is the projective representation associated with a vertex i (P_{i} is characterized by the existence of a functorial isomorphism

Hom(P_{i}, W) = W_{i}

for each representation W). If L is isomorphic to Σπ(P_{i}), we put X_{L} = x_{i}. If L is isomorphic to π(V), we define

X_{L} = X_{V} = \frac{1}{∏_{i=1}^{n} x_{i}^{d_{i}}} ∑_{0≤e≤d} χ(Gr_{e}(V)) ∏_{i=1}^{n} x_{i}^{∑_{j→i} e_{j}+∑_{i→j}(d_{j}−e_{j})} ,

where d_{i} = dim V_{i}, 1 ≤ i ≤ n, the sum is taken over all elements e ∈ ℕ^{n} such that 0 ≤ e_{i} ≤ d_{i} for all i, the quiver Grassmannian Gr_{e}(V) is the variety of n-tuples of subspaces U_{i} ⊂ V_{i} such that dim U_{i} = e_{i} and the U_{i} form a subrepresentation of V, the Euler characteristic χ is taken with respect to étale cohomology (or with respect to singular cohomology with coefficients in a field if k = ℂ) and the sums in the exponent of x_{i} are taken over all arrows j → i respectively all arrows i → j. This formula was invented by P. Caldero and F. Chapoton in [12] for the case of a quiver whose underlying graph is a simply laced Dynkin diagram. It is still valid for arbitrary quivers without oriented cycles [15] and further generalizes to arbitrary triangulated 2-Calabi-Yau categories containing a cluster-tilting object [60].

5 Some further developments

The extension of the results presented here to quivers containing oriented cycles is the subject of ongoing research. In a series of papers [39] [35] [40]

[36] [37], Geiss-Leclerc-Schröer have obtained remarkable results for a class of quivers which are important in the study of (dual semi-)canonical bases. They use an analogue [38] of the Caldero-Chapoton map due ultimately to Lusztig [56]. The class they consider has been further enlarged by Buan-Iyama-Reiten-Scott [3]. Thanks to their results, an analogue of Caldero-Chapoton’s formula and a version of theorem 2 was proved in [33] for an even larger class.

Building on [58] Derksen-Weyman-Zelevinsky are developing a representation-theoretic model for mutation of general quivers in [20]. Their approach is related to Kontsevich-Soibelman’s [52] and Chuang-Rouquier’s [19] forthcoming work, where 3-Calabi-Yau categories play an important rôle, as was already the case in [45].

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