CLUSTER ALGEBRAS AND QUIVER REPRESENTATIONS

BERNHARD KELLER

Abstract. These notes contain an extended abstract and references for a minicourse given at the Summer School ‘Microlocal and Geometric Methods in Representation Theory’ (Schloss Reisensburg, July 2006) of the International Research Training Group Metz–Paderborn.

1. History

Cluster algebras were invented by Fomin-Zelevinsky in 2000, cf. [23], with the aim to find a combinatorial approach to the objects constructed by Lusztig in two series of articles: the first one on canonical bases in quantum groups, cf. [45], …, the second one on total positivity in algebraic groups, cf. [46], …. Despite great progress in the last few years, we are still far from these initial goals. However, in the meantime, cluster algebras themselves have turned into a thriving subject, thanks notably to their many links to other fields. These include Poisson geometry [35] [36], Teichmüller spaces [21] [20], integrable systems [26] [53], combinatorics and combinatorial polyhedra [17] [16] [44], and, last not least, the representation theory of quivers and finite-dimensional algebras [3] [12] [14] [34] [48] [49] [8] … . Here, the idea is that of categorification: One tries to interpret cluster algebras as combinatorial (perhaps K-theoretic) invariants associated with categories of representations. The lift from combinatorial objects to categories provides rich additional structure, which sometimes allows one to prove statements which seem out of reach of the purely combinatorial methods.

We refer to [55] [54] for introductions to cluster algebras and to [25] for a survey of the first three years of their six-year-long history.

2. Reminder on finite root systems

We recall the definition and the classification of finite (reduced, irreducible) root systems from [51]: one proceeds from a choice of positive roots to construct successively the simple roots, the Cartan matrix and the Dynkin diagram.

3. An informal introduction to cluster-finite cluster algebras

We present a brief description of cluster algebras [23] and an informal statement of the classification theorem of cluster-finite cluster algebras [24]. We then construct the cluster algebra (without coefficients) associated to a Dynkin diagram in three examples: A_{2}, A_{3} and G_{2}. On these examples, we observe the main properties proved in [23] and [24]: The Laurent phenomenon, the periodicity and the bijection between non-trivial cluster variables and positive roots given by comparing the exponents of the denominator with the coefficients in the expansion of a positive root as a linear combination of simple roots. We make extensive use of the knitting algorithm, whose invention in representation theory goes back to the late seventies, cf. e.g. [29].

Date: July 31, 2006, last modified on July 31, 2006.

Key words and phrases. Cluster algebra, quiver representations, cluster category.

4. Geometric cluster algebras

4.1. Geometric cluster algebras without coefficients. A quiver is an oriented graph. We define a seed to be a pair (R, u) formed by a quiver R with vertices 1, …, n, without loops and without 2-cycles, and a free generating set u = {u_{1}, …, u_{n}} of the field of fractions of the field of polynomials with integer coefficients in n indeterminates x_{1}, …, x_{n}. We define mutation of quivers and mutation of seeds. Then, for a given quiver Q without loops or 2-cycles, we define the clusters, the cluster variables and the cluster algebra as in [7] and the seed exchange graph as in section 4 of [22]. We point out the link between quiver mutation and matrix mutation. We illustrate these constructions on the examples A_{2}, A_{3} and the Kronecker quiver. We note that the knitting algorithm amounts to performing mutations only at sources and at sinks. Therefore, the knitting algorithm does not yield all clusters (except in rank 2) and does not yield all cluster variables (except for a Dynkin quiver). We illustrate this on a quiver of type A_{2,1}. We state Fomin-Zelevinsky’s criterion [24] for a quiver Q to yield only finitely many cluster variables: This is the case if and only if the quiver is mutation-equivalent to a Dynkin quiver, whose underlying graph Δ is then determined uniquely and called the cluster type of Q. Good criteria for a quiver to be cluster-finite are now known: in terms of quadratic forms [5] or in terms of forbidden minimal cluster-infinite subquivers [52], closely linked to minimal representation-infinite algebras [11]. In general, mutation equivalence is an extremely complex relation. We illustrate this using the quiver mutation application [42]. No normal form for quivers under mutation equivalence is known. For example, it is unknown when a cyclic quiver with three vertices linked respectively by r, s and t arrows is mutation-equivalent to a quiver without oriented cycles.

4.2. Geometric cluster algebras with coefficients. Starting from a bipartite quiver without loops or 2-cycles (or equivalently, a rectangular integer matrix with antisymmetric principal part), we construct the associated geometric cluster algebra with coefficients as in [7]. Following [23] we show that such algebras occur in geometry as coordinate algebras of varieties: SL(2) (cluster type A_{1}), the cone on the Grassmannian of planes in (n + 3)-space (cluster type A_{n}) and, following [35], the big cell of the Grassmannian of k-dimensional subspaces in n-space (possible cluster types A_{n}, D_{4}, D_{6}, E_{8} or cluster-infinite). Here, the resulting cluster structure is compatible with the Sklyanin Poisson bracket on the Grassmannian, i.e. the variables in each cluster form a log-canonical local coordinate system. More generally, as shown in [35], for each bipartite quiver whose associated matrix is of full rank, the cluster variety (obtained by glueing tori corresponding to the clusters) admits compatible Poisson structures. The field of rational functions central for this bracket is generated by certain monomials in the coefficients. If the matrix associated with the bipartite quiver is not of full rank, there is no compatible Poisson bracket in general, but instead, one can find compatible 2-forms. This is studied in detail in [36], in view of possible applications to Teichmüller spaces.

Finally, we exhibit the cluster algebra structure of the subgroup of unipotent upper triangular matrices in SL(n). On this example, we point out the link with total positivity [46] and with Lusztig’s dual semicanonical basis [47]. More generally, the coordinate algebra of a maximal unipotent subgroup in any simple connected simply connected complex algebraic group has a canonical cluster algebra structure. This cluster algebra and its representation-theoretic pendant, the category of representations of the corresponding preprojective algebra, have been thoroughly studied by Geiss-Leclerc-Schröer in [34] [33] [31] [30] [32].

We point out that many other examples of geometric cluster algebras with coefficients, in particular coordinate algebras of double Bruhat cells, are contained in [6].

5. Quiver representations and Gabriel’s theorem

Several books on the representation theory of quivers and finite-dimensional algebras are now available: The books [2] and [1] are written for the novice, whereas [50] and [28] are research monographs.

We introduce the notion of representation of a (finite) quiver and illustrate it on examples. We introduce morphisms of representations, subrepresentations, direct sums, simple representations and indecomposable representations. We state the decomposition theorem (also known as Azumaya-Krull-Remak-Schmidt theorem). We introduce the Tits form and state Gabriel’s theorem [27] for representationfinite quivers. We indicate on one example how to treat the case of non-simply laced Dynkin diagrams and refer to [18] [19] for the corresponding representation theory of species. We sketch the definition of a tame (resp. wild) algebra and state the extension of Gabriel’s theorem to tame quivers due to Nazarova, Gelfand-Ponomarev, Donovan-Freislich and Dlab-Ringel.

We give a first qualitative link between representation-theory and cluster algebras: For a quiver Q without oriented cycles, let μ be the supremum of the multiplicities of the arrows of the quivers in the mutation class of Q. Then Q is representation-finite iff μ = 1, it is tame iff μ = 2 and it is wild iff μ = ∞. Moreover, as shown in [10], the mutation-class of Q is finite iff Q is representation-finite or tame (if this seems confusing, consider the case of the Kronecker quiver).

Using [42] we illustrate the trichotomy finite–tame–wild on the example of the three quivers whose representations correspond essentially to configurations of 3, 4 or 5 subspaces in a vector space.

6. Cluster variables from quiver representations

A theorem by Fomin-Zelevinsky [24] states that there is a canonical bijection between positive roots of a Dynkin quiver Q and the non trivial cluster variables of the corresponding cluster algebra. By combining this with Gabriel’s theorem, we obtain a bijection V ↦ X_{V} from the set of isoclasses of indecomposable representations of Q to the set of non-trivial cluster variables. Using [42] we illustrate this bijection on the examples A_{2}, A_{3}, D_{4} and point out the link between the number of terms in the numerator of X_{V} and the ‘number’ of subrepresentations of V. This motivates the Caldero-Chapoton formula [12]: If V is a representation of Q of dimension vector d, then

CC(V) = \frac{1}{x_{1}^{d_{1}}x_{2}^{d_{2}} … x_{n}^{d_{n}}} ∑_{0≤e≤d} χ(Gr_{e}(V)) ∏_{i=1}^{n} x_{i}^{∑_{j→i} e_{j}+∑_{i→j}(d_{j}−e_{j})}.

Caldero-Chapoton prove in [12] that if V is an indecomposable representation of a Dynkin quiver, then CC(V) is a cluster variable and equals X_{V}. We illustrate the theorem on examples for A_{2} and D_{4}.

More generally, it is now known that for an arbitrary quiver without oriented cycles, the Caldero-Chapoton map yields a bijection from the set of isoclasses of rigid indecomposable representations (i.e. those with Ext^{1}(V, V) = 0) to the set of non-trivial cluster variables. This was proved in [15] using the Caldero-Chapoton formula as well as results from [9] [4] [3] [48]. A slightly different proof was given in the appendix to [9]. A new and original proof for extended Dynkin quivers is due to A. Hubery [39], who establishes a link with Green’s theorem on the comultiplication in the Hall algebra.

7. Sketch of proof of Caldero-Chapoton’s theorem

The aim is to show that CC(V) is a cluster variable (showing that it is X_{V} is then not hard). The idea for this is to show that the map V ↦ X_{V} satisfies the initial conditions and the recurrence relation for a suitable version of the knitting algorithm. One first shows that CC(V) = X_{V} holds if V is an indecomposable projective module. Then, one shows that CC is multiplicative on split short exact sequences and ‘almost multiplicative’ on almost split short exact sequences [2]. Finally, one uses the knitting algorithm for representations to conclude that CC(V) is a cluster variable for each indecomposable representation V.

8. Cluster algebras from cluster categories

8.1. The repetition and the derived category. We formalize the definition of the repetition ℤQ of a quiver Q without oriented cycles. We define the mesh category associated to the repetition and state Happel’s theorem [37] [38] according to which the mesh category is isomorphic to the category of indecomposables of the bounded derived category of Q if Q is a Dynkin quiver. We extend the map V ↦ X_{V} to the derived category so that it becomes multiplicative on split exact sequences and ‘almost multiplicative’ on almost split triangles. The resulting map takes the shifted indecomposable projectives to the trivial cluster variables. It is periodic with respect to a certain periodicity automorphism, namely F = τ^{−1}S, where S is the shift functor for complexes. This motivates the definition of the cluster category.

8.2. Clusters from the cluster category. Following [3] we define the cluster category as the orbit category of the bounded derived category under the action of the automorphism F = τ^{−1}S, where S is the shift functor. In type A_{n}, an isomorphic category was independently defined in [13] using the diagonals of an (n + 3)-gon as objects. A theorem of [3] states that for a Dynkin quiver, the isoclasses of indecomposables of the cluster category are in bijection with all the cluster variables (trivial and non trivial) and that under this bijection, the clusters correspond exactly to the cluster-tilting subsets of indecomposables, i.e. the sets of n indecomposables T_{1}, …, T_{n} with Ext^{1}(T_{i}, T_{j}) = 0 for all i, j. We illustrate this on the example A_{3} and then state how to obtain the quiver R in a seed (R, u) whose cluster u corresponds to T_{1}, …, T_{n}. It follows that the cluster determines the seed. This statement was extended from Dynkin quivers to arbitrary quivers without oriented cycles by Buan-Marsh-Reiten-Todorov [9], who thus proved a conjecture by Fomin-Zelevinsky.

8.3. Relations from triangles. The derived category is triangulated and the cluster category admits a canonical triangulation for which the projection functor becomes a triangle functor [43]. For objects L, M, N of the cluster category of a Dynkin quiver, denote by Ext^{1}(L, M) the set of morphisms from L to SM and by Ext^{1}(L, M)_{N} the subset formed by the morphisms ε : L → SM such that there is a triangle

M → N → L → SM

whose third arrow is ε. It is shown in [14] that if Ext^{1}(L, M) does not vanish, we have the relation

X_{L}X_{M} = ∑_{N} \frac{χ(ℙExt^{1}(L, M)_{N}) + χ(ℙExt^{1}(M, L)_{N})}{dim Ext^{1}(L, M)} .

It yields a sufficient set of relations (including those of the form xx′ = 1+y), a basis and structure constants if Q is Dynkin. The key ingredients of the proof are the Caldero-Chapoton formula and the fact that the cluster category is 2-Calabi-Yau, i.e. we have the functorial isomorphisms

DExt^{1}(L, M) →^{∼} Ext^{1}(M, L) ,

where D is the duality over the ground field. The formula has been adapted to the setting of modules over preprojective algebras in [32]. It can be extended to the case where Q does not have oriented cycles if, inspired by [32], we slightly modify our viewpoint and the notations:

X_{L}X_{M} = \frac{1}{dim Ext^{1}(L, M)} (∫_{ε∈P (L,M)} X_{mt(ε)}dχ + ∫_{ε∈P (M,L)} X_{mt(ε)}dχ)

where mt(ε) denotes the middle term of the extension associated with ε and P (L, M) the projectivization of Ext^{1}(L, M). Here, as in [47], the integral ∫_{Y} fdχ is defined if the integrand is a function on the complex algebraic variety Y all of whose fibers are constructible and such that only finitely many fibers have non zero Euler characteristic. In this case, the value of the integral is the sum of the Euler characteristics of the fibers of f.

8.4. Quivers with oriented cycles. Quivers which are not mutation-equivalent to quivers without oriented cycles are very frequent. The problem of obtaining a representation-theoretic interpretation of the cluster-algebra in this case is being studied in [40] [41]. In very rough terms, the strategy consists in replacing path algebras by ‘Calabi-Yau algebras’.

Acknowledgments

I thank Joachim Hilgert, Henning Krause and Tilmann Würzbacher for the invitation to the summer school and for their excellent work in organizing it. I am grateful to all the participants for their interest and feedback. I thank Sven-Ake Wegener for providing me with an excellent set of handwritten notes of the lectures.

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

   E-mail address: keller@math.jussieu.fr