Asymptotic triangulations can be viewed as limits of triangulations under the action of the mapping class group. In the case of the annulus, such triangulations have been introduced in K. Baur and G. Dupont (Compactifying exchange graphs: Annuli and tubes, Ann. Comb.3(18) (2014), 797–839). We construct an alternative method of obtaining these asymptotic triangulations using Coxeter transformations. This provides us with an algebraic and combinatorial framework for studying these limits via the associated quivers.

Let $A$ be a truncated polynomial ring over a complete discrete valuation ring ${\mathcal{O}}$ , and we consider the additive category consisting of $A$ -lattices $M$ with the property that $M\otimes {\mathcal{K}}$ is projective as an $A\otimes {\mathcal{K}}$ -module, where ${\mathcal{K}}$ is the fraction field of ${\mathcal{O}}$ . Then, we may define the stable Auslander–Reiten quiver of the category. We determine the shape of the components of the stable Auslander–Reiten quiver that contain Heller lattices.

With applications in mind to the representations and cohomology of block algebras, we examine elements of the graded center of a triangulated category when the category has a Serre functor. These are natural transformations from the identity functor to powers of the shift functor that commute with the shift functor. We show that such natural transformations that have support in a single shift orbit of indecomposable objects are necessarily of a kind previously constructed by Linckelmann. Under further conditions, when the support is contained in only finitely many shift orbits, sums of transformations of this special kind account for all possibilities. Allowing infinitely many shift orbits in the support, we construct elements of the graded center of the stable module category of a tame group algebra of a kind that cannot occur with wild block algebras. We use functorial methods extensively in the proof, developing some of this theory in the context of triangulated categories.

Assume that $A$ is a finite-dimensional algebra over some field, and assume that $A$ is weakly symmetric and indecomposable, with radical cube zero and radical square nonzero. We show that such an algebra of wild representation type does not have a nonprojective module $M$ whose ext-algebra is finite dimensional. This gives a complete classification of weakly symmetric indecomposable algebras which have a nonprojective module whose ext-algebra is finite dimensional. This shows in particular that existence of ext-finite nonprojective modules is not equivalent with the failure of the finite generation condition (Fg), which ensures that modules have support varieties.

If $T$ and $T^{\prime }$ are two cluster-tilting objects of an acyclic cluster category related by a mutation, their endomorphism algebras are nearly Morita equivalent (Buan et al., Cluster-tilted algebras, Trans. Amer. Math. Soc. 359(1) (2007), 323–332 (electronic)); that is, their module categories are equivalent “up to a simple module”. This result has been generalized by Yang, using a result of Plamondon, to any simple mutation of maximal rigid objects in a 2-Calabi–Yau triangulated category. In this paper, we investigate the more general case of any mutation of a (non-necessarily maximal) rigid object in a triangulated category with a Serre functor. In that setup, the endomorphism algebras might not be nearly Morita equivalent, and we obtain a weaker property that we call pseudo-Morita equivalence. Inspired by Buan and Marsh (From triangulated categories to module categories via localization II: calculus of fractions, J. Lond. Math. Soc. (2) 86(1) (2012), 152–170; From triangulated categories to module categories via localisation, Trans. Amer. Math. Soc. 365(6) (2013), 2845–2861), we also describe our result in terms of localizations.

We study certain integer valued length functions on triangulated categories, and establish a correspondence between such functions and cohomological functors taking values in the category of finite length modules over some ring. The irreducible cohomological functions form a topological space. We discuss its basic properties, and include explicit calculations for the category of perfect complexes over some specific rings.

In the context of varieties of representations of arbitrary quivers, possibly carrying loops, we define a generalization of Lusztig Lagrangian subvarieties. From the combinatorial study of their irreducible components arises a structure richer than the usual Kashiwara crystals. Along with the geometric study of Nakajima quiver varieties, in the same context, this yields a notion of generalized crystals, coming with a tensor product. As an application, we define the semicanonical basis of the Hopf algebra generalizing quantum groups, which was already equipped with a canonical basis. The irreducible components of the Nakajima varieties provide the family of highest weight crystals associated to dominant weights, as in the classical case.

We describe how Mirković–Vilonen (MV) polytopes arise naturally from the categorification of Lie algebras using Khovanov–Lauda–Rouquier (KLR) algebras. This gives an explicit description of the unique crystal isomorphism between simple representations of KLR algebras and MV polytopes. MV polytopes, as defined from the geometry of the affine Grassmannian, only make sense in finite type. Our construction on the other hand gives a map from the infinity crystal to polytopes for all symmetrizable Kac–Moody algebras. However, to make the map injective and have well-defined crystal operators on the image, we must in general decorate the polytopes with some extra information. We suggest that the resulting ‘KLR polytopes’ are the general-type analogues of MV polytopes. We give a combinatorial description of the resulting decorated polytopes in all affine cases, and show that this recovers the affine MV polytopes recently defined by Baumann, Kamnitzer, and the first author in symmetric affine types. We also briefly discuss the situation beyond affine type.

In this paper, we study the poset of basic tilting kQ-modules when Q is a Dynkin quiver, and the poset of basic support τ-tilting kQ-modules when Q is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if Q is of types $\mathbb{A}_{1}$ , $\mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a non-linear orientation and the second poset is a distributive lattice if and only if Q is of type $\mathbb{A}_{1}$ .

Let $R$ be a commutative Gorenstein ring. A result of Araya reduces the Auslander–Reiten conjecture on the vanishing of self-extensions to the case where $R$ has Krull dimension at most one. In this paper we extend Araya’s result to certain $R$-algebras. As a consequence of our argument, we obtain examples of bound quiver algebras that satisfy the Auslander–Reiten conjecture.

Toric quiver varieties (moduli spaces of quiver representations) are studied. Given a quiver and a weight, there is an associated quasi-projective toric variety together with a canonical embedding into projective space. It is shown that for a quiver with no oriented cycles the homogeneous ideal of this embedded projective variety is generated by elements of degree at most 3. In each fixed dimension d up to isomorphism there are only finitely many d-dimensional toric quiver varieties. A procedure for their classification is outlined.

Let $R$ be a commutative noetherian local ring. As an analog of the notion of the dimension of a triangulated category defined by Rouquier, the notion of the dimension of a subcategory of finitely generated $R$-modules is introduced in this paper. We found evidence that certain categories over nice singularities have small dimensions. When $R$ is Cohen–Macaulay, under a mild assumption it is proved that finiteness of the dimension of the full subcategory consisting of maximal Cohen–Macaulay modules which are locally free on the punctured spectrum is equivalent to saying that $R$ is an isolated singularity. As an application, the celebrated theorem of Auslander, Huneke, Leuschke, and Wiegand is not only recovered but also improved. The dimensions of stable categories of maximal Cohen–Macaulay modules as triangulated categories are also investigated in the case where $R$ is Gorenstein, and special cases of the recent results of Aihara and Takahashi, and Oppermann and Št́ovíček are recovered and improved. Our key technique involves a careful study of annihilators and supports of $\mathsf{Tor}$, $\mathsf{Ext}$, and $\underline{\mathsf{Hom}}$ between two subcategories.

We construct the quantized enveloping algebra of any simple Lie algebra of type $\mathbb{A}\mathbb{D}\mathbb{E}$ as the quotient of a Grothendieck ring arising from certain cyclic quiver varieties. In particular, the dual canonical basis of a one-half quantum group with respect to Lusztig’s bilinear form is contained in the natural basis of the Grothendieck ring up to rescaling. This paper expands the categorification established by Hernandez and Leclerc to the whole quantum groups. It can be viewed as a geometric counterpart of Bridgeland’s recent work for type $\mathbb{A}\mathbb{D}\mathbb{E}$.

We consider an artin algebra A and its crossed product algebra A α#σ G, where G is a finite group with its order invertible in A. Then, we prove that A is a tilted algebra if and only if so is A α#σ G.

The notions of central endolength and semigeneric tameness are introduced, and their behaviour under base field extension for finite-dimensional algebras over perfect fields are analysed. For k a perfect field, K an algebraic closure and Λ a finite-dimensional k-algebra, here there is a proof that Λ is semigenerically tame if and only if Λ ⊗kK is tame.

Let be a field and let Q be a minimal Hopf quiver, i.e. a cyclic quiver or the infinite linear quiver, and let repln(Q) denote the category of locally nilpotent finite-dimensional -representations of Q. The category repln(Q) has natural tensor structures induced from graded Hopf structures on the path coalgebra . Tensor categories of the form repln(Q) are an interesting class of tame hereditary pointed tensor categories that are not finite. The aim of this paper is to compute the Clebsch–Gordan formulae and Green rings of such tensor categories.

The (usual) Caldero–Chapoton map is a map from the set of objects of a category to a Laurent polynomial ring over the integers. In the case of a cluster category, it maps reachable indecomposable objects to the corresponding cluster variables in a cluster algebra. This formalizes the idea that the cluster category is a categorification of the cluster algebra. The definition of the Caldero–Chapoton map requires the category to be 2-Calabi-Yau, and the map depends on a cluster-tilting object in the category. We study a modified version of the Caldero–Chapoton map which requires only that the category have a Serre functor and depends only on a rigid object in the category. It is well known that the usual Caldero–Chapoton map gives rise to so-called friezes, for instance, Conway–Coxeter friezes. We show that the modified Caldero–Chapoton map gives rise to what we call generalized friezes and that, for cluster categories of Dynkin type A, it recovers the generalized friezes introduced by combinatorial means in recent work by the authors and Bessenrodt.

We present a computer algebra package based onMagma for performing computations in rational Cherednik algebras with arbitrary parameters and in Verma modules for restricted rational Cherednik algebras. Part of this package is a new general Las Vegas algorithm for computing the head and the constituents of a module with simple head in characteristic zero, which we develop here theoretically. This algorithm is very successful when applied to Verma modules for restricted rational Cherednik algebras and it allows us to answer several questions posed by Gordon in some specific cases. We can determine the decomposition matrices of the Verma modules, the graded $G$ -module structure of the simple modules, and the Calogero–Moser families of the generic restricted rational Cherednik algebra for around half of the exceptional complex reflection groups. In this way we can also confirm Martino’s conjecture for several exceptional complex reflection groups.

Supplementary materials are available with this article.

We show how some of the refined tropical counts of Block and Göttsche emerge from the wall-crossing formalism. This leads naturally to a definition of a class of putative $q$-deformed Gromov–Witten invariants. We prove that this coincides with another natural $q$-deformation, provided by a result of Reineke and Weist in the context of quiver representations, when the latter is well defined.

Let Λ be a finite-dimensional algebra and G be a finite group whose elements act on Λ as algebra automorphisms. Assume that Λ has a complete set E of primitive orthogonal idempotents, closed under the action of a Sylow p-subgroup S ≤ G. If the action of S on E is free, we show that the skew group algebra Λ G and Λ have the same finitistic dimension, and have the same strong global dimension if the fixed algebra ΛS is a direct summand of the ΛS-bimodule Λ. Using a homological characterization of piecewise hereditary algebras proved by Happel and Zacharia, we deduce a criterion for Λ G to be piecewise hereditary.