(in Polish) Reprezentacje algebr dróg kołczanów

elective monographs

The course will introduce some notions related to abelian categories, which will be developed in the category of representations of quivers. We will define fundamental concepts appearing in all abelian categories. Additionally, for the category of representations of quivers we will introduce the Aulander-Reiten theory and the tilting theory. We will give a classification of hereditary algebras with finitely many isomorphism classes of indecomposable objects.

Abelian categories. Represenations of quivers.

We will introduce the notions of projective and injective objects and define the (higher) extension groups. We will describe projective and injective modules over the path algebra of a quiver and learn to compute the extension groups between objects of these categories. We will characterise basic hereditary algebras as path algebras for quivers with no relations.

Auslander-Reiten theory.

We will describe almost split sequences, define the Aulander-Reiten translation and construct the Auslander-Reiten quiver. We will show exaples of computations of the AR quiver for modules over a quiver and provide a functorial interpretation of the almost split sequences.

Tilting theory.

We will define torsion pairs and tilting objects in the category of the representations of quivers. We will describe the connection between these objects, i.e. the torsion pair assigned to a titling object.

Gabriel's theorem.

We will define the quadratic form of a quiver and prove Gabriel's theorem, which classifies hereditary algebras of finite representation type.

1. Ibrahim Assem, Andrzej Skowroński, Daniel Simson

"Elements of the Representation Theory of Associative Algebras"

2. Matej Brešar, "Introduction to Noncommutative Algebra"

3. Richard Pierce, "Associative Algebras"

4. Nicolae Popescu, "Abelian Categories with Applications to Rings and Modules"

A student has an understanding of

1. the concept of an abelian category,

2. the (higher) extension groups and the Yoneda product between these groups,

3. the connection between modules over the path algebra of a quiver and representations of the quiver,

4. the classical and functorial definitions of almost split sequences,

5. the notion of a torsion pair and of a tilting object,

6. the outline of the proof of Gabriel's theorem.

A student is able to:

1. compute the dimension of an algebra defined by a quiver with relations,

2. compute the extension groups between given modules,

3. write down the projective and injective resolution of a given module,

4. draw the Auslander-Reiten quiver of an algebra of finite representation type.

6) Assessment methods and assessment criteria:

The final mark is given based on the exercises session and the final written exam. The exam may have an oral part if necessary to ascertain the final mark.