The previous iteration of the PITA-seminar was organized by Louis-Phillipe Thibault. It was sadly cut short due to the COVID pandemic.
Peder Thompson, 12. Mar 2020
Title: Trace modules and enveloping classes
Abstract: The idea of trace modules appears in a number of settings, often related to endomorphism invariance. For example, the classic notion of a quasi-injective module is simply a module that is trace in its injective envelope. I will introduce some general properties of trace modules and sketch some ongoing joint work with Haydee Lindo on relating trace modules and enveloping classes of modules over any ring. In particular, we will give some characterizations of rings (such as semi-simple, regular, and Gorenstein rings) in terms of their ideals being trace in certain envelopes. I will also discuss a number of open questions related to the theory of trace modules, such as their relationship to the Auslander-Reiten conjecture.
Didrik Fosse, 27. Feb 2020
Title: A combinatorial rule for tilting mutation
Abstract: Tilting objects are important in the study of the representation theory of algebras. For example, Rickard’s derived Morita theorem tells us that two rings are derived equivalent precisely when there exists a certain tilting object over one of them. Tilting mutation is a way to modify a known tilting object in such a way that you get a new tilting object, which thus allows us to create an algebra that is derived equivalent to a given algebra. In this talk we will develop a set of combinatorial rules for tilting mutation of algebras which are given as path algebras of a certain class of quivers with relations. We will also see an explicit example of how we can use these rules to calculate a sequence of derived equivalent algebras, which in turn can be used as a tool for identifying when two given algebras are derived equivalent. If we manage to find such a sequence which starts with one of the algebras and ends with the other, then they must be derived equivalent.
Mads Hustad Sandøy, 13. Feb 2020
Title: Generalized T-Koszul algebras
Abstract: Koszul algebras are in some sense the graded algebras that are easiest to understand while not being semi-simple. Moreover, they abound in nature. We will review key properties of Koszul algebras before introducing a generalization of them, namely generalized T-Koszul algebras, and sketch connections to $n$-hereditary algebras. The novel parts of this talk are based on joint work with Johanne Haugland.
Louis-Phillipe Thibault, 30. Jan 2020
Title: The quiver of $n$-hereditary algebras
Abstract: Auslander-Reiten theory is fundamental in the study of modules over Artin algebras. In this setting, the number ‘$2$’ appears often. For example, the Auslander correspondence gives a bijection between Morita-equivalence classes of Artin algebras of finite representation type and algebras $A$ satisfying $gl.dim A \leq 2 \leq dom.dim A.$ It is thus natural to generalize some of the central ideas to a “higher dimensional” Auslander-Reiten theory. This was introduced by Iyama in 2004 and has generated many interesting ideas over the years.
One important concept is that of $n$-hereditary algebras, which enjoy some of the key properties of hereditary algebras in the context of higher AR-theory. They are divided into two classes: the $n$-representation-finite and the $n$-representation-infinite algebras. A lot of research has been done on these objects but, in contrast to classical hereditary algebras, very little is known as to what their quivers actually look like. In this talk, I will introduce (higher) AR-theory, give some examples of $n$-hereditary algebras and ask whether we can deduce some of their quiver properties. These questions are part of a project with Mads.
Johanne Haugland, 16. Jan 2020
Title: Extriangulated categories – functors and subcategories
Abstract: In this talk we give a basic introduction to extriangulated categories. Extriangulated categories were first presented in a paper by Hiroyuki Nakaoka and Yann Palu in 2016, and has turned out to be a useful framework to unify and extend known concepts and results. We discuss the relevant definitions and explain how extriangulated categories is a simultaneous generalization of exact and triangulated categories. After this, we move on to the question of what the correct definitions of extriangulated functors and extriangulated subcategories should be.