Tag - Tilting theory

Martin Frankland: Toda brackets in n-angulated categories

17th October, 2024

Geiss, Keller, and Oppermann introduced n-angulated categories to capture the structure found in certain cluster tilting subcategories in quiver representation theory. Jasso and Muro investigated Toda brackets and Massey products in such cluster tilting subcategories by using the ambient triangulated category. In joint work with Sebastian Martensen and Marius Thaule, we introduce Toda brackets in n-angulated categories, generalizing Toda brackets in triangulated categories (the case n = 3). We will look at different constructions of the brackets, their properties, some examples, and some applications.

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.

Aslak Bakke Buan: From exceptional to τ-exceptional sequences in module categories

30th May, 2024

Exceptional sequences and their mutations were first considered in triangulated categories by the Moscow school of algebraic geometers. In the early nineties, Crawley-Boevey and Ringel studied exceptional sequences for module categories of hereditary algebras. We first recall their definitions and their main results, and then proceed to discuss a natural generalization to all (not necessarily hereditary) finite-dimensional algebras. This is the theory of τ-exceptional sequences, which was developed in joint work with Marsh, motivated by τ-tilting theory, by Adachi-Iyama-Reiten, by Jasso's reduction techniques for such modules and corresponding torsion pairs, and by the introduction of signed exceptional sequences by Igusa-Todorov.

The interplay between theories for τ-rigid modules, torsion pairs, and wide subcategories is central to our discussions.

Kaveh Mousavand: Some applications of bricks in classical and modern problems in representation theory

12th December, 2023

Bricks (also known as Schur representations) form a special subfamily of indecomposable modules, and they are used in the algebraic and geometric study of representation theory of algebras. We start by looking at some classical results on bricks, including a characterization of locally representation-directed algebras (due to Dräxler). Then, we consider some new directions of research in which bricks have played crucial roles. More specifically, we briefly recall an elegant correspondence between bricks and indecomposable τ-rigid-modules (due to Demonet-Iyama-Jasso), which has many applications in τ-tilting theory. We use the notion of τ-rigidity to give a new characterization of locally representation-directed algebras, and to further generalize this family. If time permits, we also report on some new results on an open conjecture (so-called the 2nd brick-Brauer-Thrall conjecture) which I posed in 2019.

Gustavo Jasso: (Non-)uniqueness of strong enhancements

30th May, 2023

The Derived Auslander-Iyama Correspondence, a recent theorem of Muro and myself, guarantees the existence of unique (DG) enhancements for algebraic triangulated categories that satisfy mild finiteness conditions as well as a dZ-cluster tilting object. I will explain a by-product of our work that permits us to construct, to the best of our knowledge, the first examples of algebraic triangulated categories that admit a unique enhancement but not a unique strong enhancement in the sense of Lunts and Orlov. This talk is based on joint work with Fernando Muro (Sevilla)..

Bernhard Keller: On Amiot’s conjecture

17th January, 2023

In 2010, Claire Amiot conjectured that algebraic 2-Calabi-Yau categories with a cluster-tilting object must come from quivers with potential. This would extend a structure theorem obtained with Idun Reiten in the case where the endomorphism algebra of the cluster-tilting object is hereditary. Many other classes of examples are also known. We will report on recent progress in the general case obtained in joint work with Junyang Liu and based on Van den Bergh's structure theorem for complete Calabi-Yau algebras.

Fernando Muro: Uniqueness of enhancements for Hom-finite triangulated categories with an n-cluster tilting object

1st March, 2022

In this talk, we will report on ongoing joint work with Gustavo Jasso. The goal is to show that algebraic triangulated categories satisfying the assumptions in the title have a unique DG-enhancement over a ground perfect field, up to Morita equivalence. This extends previous work on finite triangulated categories. The key step is the connection with Geiss-Keller-Oppermann's notion of n-angulated categories, which are like triangulated categories but with longer ‘triangles'. 

Bethany Marsh: 𝜏-exceptional sequences

5th October, 2021

We introduce the notion of a (signed) 𝜏-exceptional sequence for a finite dimensional algebra. We show that there is a bijection between the set of complete signed exceptional sequences and ordered basic support 𝜏-tilting objects.

Qi Wang: On τ-tilting finiteness of Schur algebras

17th November, 2020

Support τ-tilting modules are introduced by Adachi, Iyama and Reiten in 2012 as a generalization of classical tilting modules. One of the importance of these modules is that they are bijectively corresponding to many other objects, such as two-term silting complexes and left finite semibricks. Let V be an n-dimensional vector space over an algebraically closed field 𝔽 of characteristic p. Then, the Schur algebra S(n,r) is defined as the endomorphism ring End𝔽Gr(V⊗r) over the group algebra 𝔽Gr of the symmetric group Gr. In this talk, we discuss when the Schur algebra S(n,r) has only finitely many pairwise non-isomorphic basic support τ-tilting modules.