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This book offers an original introduction to the representation theory of algebras, suitable for beginning researchers in algebra. It includes many results and techniques not usually covered in introductory books, some of which appear here for the first time in book form. The exposition employs methods from linear algebra (spectral methods and quadratic forms), as well as categorical and homological methods (module categories, Galois coverings, Hochschild cohomology) to present classical aspects of ring theory under new light. This includes topics such as rings with several objects, the Harada-Sai lemma, chain conditions, and Auslander-Reiten theory. Noteworthy and significant results covered in the book include the Brauer-Thrall conjectures, Drozd's theorem, and criteria to distinguish tame from wild algebras. This text may serve as the basis for a second graduate course in algebra or as an introduction to research in the field of representation theory of algebras. The originality of the exposition and the wealth of topics covered also make it a valuable resource for more established researchers.

Representations of Algebras: Tame and Wild Behavior

1 Introduction and First Examples.- 2 A Categorical Approach.- 3 Constructive Methods.- 4 Spectral Methods in Representation Theory.- 5 Group Actions on Algebras and Module Categories.- 6 Reflections and Weyl Groups.- 7 Simply Connected Algebras.- 8 Degenerations of Algebras.- 9 Further Comments.