Quivers with Relations and Cluster Tilted Algebras

Cluster algebras are commutative algebras that were introduced by Fomin and Zelevinsky in order to model the dual canonical basis of a quantum group and total positivity in algebraic groups. Cluster categories were introduced as a representation-theoretic model for cluster algebras. In this article we use this representation-theoretic approach to prove a conjecture of Fomin and Zelevinsky, that for cluster algebras with no coefficients associated to quivers with no oriented cycles, a seed is determined by its cluster. We also obtain an interpretation of the monomial in the denominator of a non-polynomial cluster variable in terms of the composition factors of an indecomposable exceptional module over an associated hereditary algebra.

Methods and results from the representation theory of finite di- mensional algebras have led to many interactions with other areas of mathe- matics. Such areas include the theory of Lie algebras and quantum groups, commutative algebra, algebraic geometry and topology, and in particular the new theory of cluster algebras. The aim of this workshop was to further de- velop such interactions and to stimulate progress in the representation theory of algebras.

Cluster algebras were conceived by Fomin and Zelevinsky (1) in the spring of 2000 as a tool for studying dual canonical bases and total positivity in semisimple Lie groups. However, the theory of cluster algebras has since taken on a life of its own, as connections and applications have been discovered in diverse areas of mathematics, including representation theory of quivers and finite dimensional algebras, cf., for example, refs. 2⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓⇓–15; Poisson geometry (16⇓⇓–19); Teichmuller theory (20⇓⇓⇓–24); string theory (25⇓⇓⇓⇓⇓–31); discrete dynamical systems and integrability (6, 32⇓⇓⇓⇓⇓–38); and combinatorics (39⇓⇓⇓⇓⇓⇓⇓–47). Quite remarkably, cluster algebras provide a unifying algebraic and combinatorial framework for a wide variety of phenomena in these and other settings. We refer the reader to the survey papers (36, 48⇓⇓⇓⇓–53) and to the cluster algebras portal (www.math.lsa.umich.edu/~fomin/cluster.html) for various introductions to cluster algebras and their links with other subjects in mathematics (and physics). In brief, a cluster algebra A of rank k is a subring of an ambient field ℱ of rational functions in k variables, say x 1, …, x k . Unlike most commutative rings, a cluster algebra is not presented at the outset via a complete set of generators and relations. Instead, from the data of the initial seed — which includes the k initial cluster variables x 1, …, x k , plus an exchange matrix — one uses an iterative procedure called “mutation” to produce the rest of the cluster variables. In particular, each new cluster … [↵][1]1To whom correspondence should be addressed. Email: williams{at}math.berkeley.edu. [1]: #xref-corresp-1-1

Cluster algebras were introduced by S. Fomin and A. Zelevinsky in connection with dual canonical bases. Let U U be a cluster algebra of type A n A_n . We associate to each cluster C C of U U an abelian category C C \mathcal {C}_C such that the indecomposable objects of C C \mathcal {C}_C are in natural correspondence with the cluster variables of U U which are not in C C . We give an algebraic realization and a geometric realization of C C \mathcal {C}_C . Then, we generalize the “denominator theorem” of Fomin and Zelevinsky to any cluster.

Cluster categories were introduced in the paper “Tilting theory and cluster combinatorics” [8] in order to better understand the combinatorics of cluster algebras, by giving new, module theoretic and categorical meanings to the combinatorics of the well known cluster algebras [20]. Subsequently, correspondences between the many notions in these two areas were given, e.g. [5, 6, 11, 9] and others.

An approach to non-simply laced cluster algebras

In this paper, we introduce a notion of unistructural cluster algebras, for which the set of cluster variables uniquely determines the clusters, as well as the notion of weak unistructural cluster algebras, for which the set of cluster variables determines the clusters, provided that the type of the cluster algebra is fixed. We prove that, for cluster algebras of the Dynkin type, the two notions of unistructural and weakly unistructural coincide, and that cluster algebras of rank 2 are always unistructural. We then prove that a cluster algebra $\mathcal A$ is weakly unistructural if and only if any automorphism of the ambient field, which restricts to a permutation of cluster variables of $\mathcal A$, is a cluster automorphism. We also investigate the Fomin-Zelevinsky conjecture that two cluster variables are compatible if and only if one does not appear in the denominator of the Laurent expansions of the other.

A strong connection between cluster algebras and representation theory was established by the cluster category. Cluster characters, like the original Caldero-Chapoton map, are maps from certain triangulated categories to cluster algebras and they have generated much interest. Holm and Jørgensen constructed a modified Caldero-Chapoton map from a sufficiently nice triangulated category to a commutative ring, which is a generalised frieze under some conditions. In their construction, a quotient K0sp(T)/M of a Grothendieck group of a cluster tilting subcategory T is used. In this article, we show that this quotient is the Grothendieck group of a certain extriangulated category, thereby exposing the significance of it and the relevance of extriangulated structures. We use this to define another modified Caldero-Chapoton map that recovers the one of Holm–Jørgensen.We prove our results in a higher homological context. Suppose S is a (d+2)-angulated category with subcategories X⊆T⊆S, where X is functorially finite and T is 2d-cluster tilting, satisfying some mild conditions. We show there is an isomorphism between the Grothendieck group K0(S,EX,sX) of the category S, equipped with the d-exangulated structure induced by X, and the quotient K0sp(T)/N, where N is the higher analogue of M above. When X=T the isomorphism is induced by the higher index with respect to T introduced recently by Jørgensen. Thus, in the general case, we can understand the map taking an object in S to its K0-class in K0(S,EX,sX) as a higher index with respect to the rigid subcategory X.

Equivalences between cluster categories

Cluster algebras arising from cluster tubes II: The Caldero-Chapoton map

The unrestricted T-system is a family of relations in the Grothendieck ring of the category of the finite-dimensional modules of Yangian or quantum affine algebra associated with a complex simple Lie algebra. The unrestricted T-system admits a reduction called the restricted T-system. In this paper we formulate the periodicity conjecture for the restricted T-systems, which is the counterpart of the known and partially proved periodicity conjecture for the restricted Y-systems. Then, we partially prove the conjecture by various methods: the cluster algebra and cluster category method for the simply laced case, the determinant method for typesAandC, and the direct method for typesA, D, andB(level 2).

On cluster-tilting graphs for hereditary categories

Following the historical track in pursuing $T$-equivariant flat toric degenerations of flag varieties and spherical varieties, we explain how powerful tools in algebraic geometry and representation theory, such as canonical bases, Newton–Okounkov bodies, PBW- filtrations and cluster algebras come to push the subject forward.

A new construction of the associahedron was recently given by Arkani‐Hamed, Bai, He, and Yan in connection with the physics of scattering amplitudes. We show that their construction (suitably understood) can be applied to construct generalized associahedra of any simply laced Dynkin type. Unexpectedly, we also show that this same construction produces Newton polytopes for all the ‐polynomials of the corresponding cluster algebras. In addition, we show that the toric variety associated to the ‐vector fan has the property that its nef cone is simplicial.

The greedy basis equals the theta basis: A rank two haiku