Titlebook: Introduction to the Representation Theory of Algebras; Michael Barot Textbook 2015 Springer International Publishing Switzerland 2015 Ausl

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Module Categories,nique. Several categorical notions for modules are developed: projective and injective, simple and semisimple modules are defined. In each case a full characterization of indecomposable modules with that particular property is achieved. This is a first step towards understanding the structure of module categories.

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Matrix Problems, with a given subdivision is given and a “normal form” is sought: that is a family of matrices such that of each equivalence class one and only matrix, the representative, is chosen..There is little theory within this chapter as the main goal is to establish the normal form of certain examples: the

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Representations of Quivers,ly to a more sophisticated language known as categories and functors and large part of the chapter is devoted to the development of this new language. The benefit of it will be that the list of “normal forms” will be enhanced by some internal structure. At the end a the important example of a linear

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Algebras,up the whole chapter. With this three different languages are developed for the same thing, each of which with a distinctive flavour. All three languages have their own advantages and it is convenient to be able to switch freely between them as in the literature all three of them are used..As we wil

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The Auslander-Reiten Theory,s of notions and results named after Auslander and Reiten is presented, all of them constitute the Auslander-Reiten theory: The “Auslander-Reiten translate”, which associates to every indecomposable non-projective module an indecomposable non-injective module and the the “Auslander Reiten sequences”

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Knitting,the Auslander-Reiten quiver including the structure of the contained indecomposable modules. The development of the knitting technique takes up the main part of this chapter and will exhibited at concrete examples. It underlines the importance of combinatorial invariants, which will be studied with

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Combinatorial Invariants,l useful tools are presented for working with dimension vectors: The first of them, called Grothendieck group, is based on homological algebra. The second linearizes the Auslander-Reiten translation as an action on the Grothendieck group and is called Coxeter transformation. The third is a quadratic