Tag - Auslander-Reiten theory

Sondre Kvamme: Higher torsion classes and silting complexes

20th June, 2024

Higher Auslander-Reiten theory was introduced by Iyama in 2007 as a generalization of classical Auslander-Reiten theory. The main objects of study in the theory are d-cluster tilting subcategories of module categories. It turns out that many notions in algebra and representation theory have generalizations to higher Auslander-Reiten theory. In particular, in 2016 Jørgensen introduced a generalization of torsion classes, called higher torsion classes.

In this talk, I will recall the definition of higher torsion classes. I will then explain how functorially finite d-torsion classes give rise to (d+1)-term silting complexes, and hence to derived equivalences. The construction is analogous to the construction of 2-term silting complexes due to Adachi-Iyama-Reiten in 2014. I will illustrate the constructions and results on higher Nakayama algebras of type An.

Francesca Fedele: Ext-projectives in subcategories of triangulated categories

21st February, 2023

Let T be a suitable triangulated category and C a full subcategory of T closed under summands and extensions. An indecomposable object c in C is called Ext-projective if Ext1(c,C)=0. Such an object cannot appear as the endterm of an Auslander-Reiten triangle in C. However, if there exists a minimal right almost split morphism b→c in C, then the triangle x→b→c→ extending it is a so called left-weak Auslander-Reiten triangle in C. We show how in some cases removing the indecomposable c from the subcategory C and replacing it with the indecomposable x gives a new extension closed subcategory C' of T and see how this operation is related to Iyama-Yoshino mutation of C with respect to a rigid subcategory. Time permitting, we will see the application of the result to cluster categories of type A.

Jan Trlifaj: Tree modules, and limits of the approximation theory

19th August, 2016

Classes of modules closed under transfinite extensions often provide for precovers, and hence fit in the machinery of relative homological algebra. However, there are important exceptions: the Whitehead groups, and flat Mittag-Leffler modules over non-perfect rings. The latter class is just the zero dimensional instance (for T = R and n = 0) of non-precovering of the class of all locally T-free modules, where T is any n-tilting module which is not Σ-pure split. The phenomenon occurs even for finite dimensional algebras, when R is hereditary of infinite representation type, and T is the Lukas tilting module. The key tools here are the tree modules, which have recently been generalized in order to solve Auslander's problem on the existence of almost split sequences.

Julian Külshammer: Higher Nakayama algebras

19th August, 2016

Nakayama algebras are among the best understood representation-finite algebras. They are defined as those algebras such that each indecomposable projective and each indecomposable injective module admits a unique composition series. An equivalent characterisation is that τjS is simple (or zero) for all j ∈ ℤ and every simple module S. Here, τ denotes the Auslander–Reiten translation. Nakayama algebras can be classified by the sequence of lengths of their indecomposable projective modules, called the Kupisch series.

In this talk, we introduce a higher analogue of a Nakayama algebra for each Kupisch series 𝓁 in the sense of Iyama's higher Auslander–Reiten theory. More precisely, (in type A) the higher Nakayama algebra A𝓁(d) is a quotient of the higher Auslander algebra An(d) of type A, constructed by Iyama and studied extensively by Oppermann and Thomas. In type ̃A, one has to use an infinite version of An(d). The higher Nakayama algebra has a d-cluster-tilting module, i.e. a module M with

add(M) = {N | Exti(M,N) = 0 ∀i = 1, . . . , d−1 } = {N | Exti(N,M) = 0 ∀i = 1, . . . , d−1 }.

There are n simple modules in add(M) and they satisfy that τdjS is simple for all j ∈ ℤ and every simple module S in add(M), where τd = τΩd−1 is Iyama's higher Auslander–Reiten translation.

Jiarui Fei: Tensor multiplicity via upper cluster algebras

19th August, 2016

By tensor multiplicity we mean the multiplicities in the tensor product of any two finite-dimensional irreducible representations of a simply connected Lie group. Finding their polyhedral models is a long-standing problem. The problem asks to express the multiplicity as the number of lattice points in some convex polytope.

Accumulating from the works of Gelfand, Berenstein and Zelevinsky since 1970’s, around 1999 Knutson and Tao invented their hive model for the type A cases, which led to the solution of the saturation conjecture. Outside type A, Berenstein and Zelevinsky’s models are still the only known polyhedral models up to now. Those models lose a few nice features of the hive model.

In this talk, I will explain how to use upper cluster algebras, an interesting class of commutative algebras introduced by Berenstein-Fomin-Zelevinsky, to discover new polyhedral models for all Dynkin types. Those new models improve the ones of Berenstein-Zelevinsky's, or in some sense generalize the hive model.

It turns out that the quivers of relevant upper cluster algebras are related to the Auslander-Reiten theory of presentations, which can be viewed as a categorification of these quivers. The upper cluster algebras are graded by triple dominant weights, and the dimension of each graded component counts the corresponding tensor multiplicity.

The proof also invokes another categorification – Derksen-Weyman-Zelevinsky’s quiver-with-potential model for the cluster algebra. The bases of these upper cluster algebras are parametrized by µ-supported g-vectors. The polytopes will be described via stability conditions.

Steffen Oppermann: d-tilting bundles for Geigle-Lenzing weighted projective spaces

16th August, 2016

This talk is based on joint work with Martin Herschend, Osamu Iyama, and Hiroyuki Minamoto. Classically, the classes of tame (representation infinite, connected) hereditary algebras and Fano Geigle-Lenzing weighted projective lines coincide up to derived equivalence. With the development of Iyama's higher AR-theory, and our work on Geigle-Lenzing projective spaces, it has become natural to ask if there is a higher dimensional analogue of this fact. Here dimension refers to, on the one side the global dimension of the algebra, and on the other side the dimension of the space. Unfortunately, so far a general answer (or general strategy) is elusive. In my talk I will focus on the hypersurface case, and more specifically certain weight sequences within the hypersurface case. For these, I will explain how one may find suitable tilting bundles on the Geigle-Lenzing weighted projective space.