Definition and elementary properties. The module category of a cluster
tilted algebra. Homological properties. Quiver mutation and the
ordinary quiver of a cluster tilted algebra. Relation extensions and
the bound quiver of a cluster tilted algebra. The cluster repetitive
algebra. The Auslander-Reiten quiver of a cluster tilted algebra:
local slices and reflexions. Gentle cluster tilted algebras. Modules
determined by their composition factors. Hochschild cohomology of
cluster tilted algebras.
Lecture notes.
This course's aim is to present cluster characters for 2-Calabi-Yau
triangulated categories and their main properties. The original
motivation for the study of these notions resides in their application
to Fomin-Zelevinsky's cluster algebras; however, we will focus on the
theory itself. First, we shall define triangulated 2-Calabi-Yau
categories; our main examples will be orbit categories of derived
categories of Dynkin quivers. Then, we will define cluster
characters, which are kinds of generating series for "counting"
submodules of a given modules according to their dimension. Finally,
we will see how these cluster characters interact, giving rise to
various multiplication formulas.
Exercise sheet of the first lecture.
Exercise sheet of the second
lecture.
The course will be introductory and treat of the following topics.
I. The K-theory of exact categories.
I.1. Quillen's Q-construction.
I.2. Quillen's fundamental theorems: Localization, Dévissage, Additivity and Resolution.
I.3. Negative K-theory.
II. Algebraic K-theory and triangulated categories.
II.1 The bounded derived category of an exact category.
II.2 Exact sequences of triangulated categories and the Thomason-Waldhausen localization theorem.
II.3 Quillen's fundamental theorems revisited.
Brauer graph algebras are symmetric algebras of tame representation
type. They first appeared, as Brauer tree algebras, in the modular
representation theory of finite groups. Since then they have been
studied in their own right and much of their representation theory is
well understood. It has recently been shown that the class of Brauer
graph algebras coincides with the class of symmetric special biserial
algebras, which is another class of tame algebras, that has been
well-studied. These two presentations, as Brauer graph algebras and as
symmetric special biserial algebras, provide two different approaches
to the subject.
In this course we will define Brauer graph algebras as well as
symmetric special biserial algebras and give an overview of their
representation theory based on the two approaches. In particular, we
will show how Brauer graph algebras have recently been connected to
Jacobian algebras associated to triangulations of surfaces.