Frobenius Algebras: No. II: Tilted and Hochschild Extension Algebras (EMS Textbooks in Mathematics)

This is the second of three volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book is devoted to fundamental results of the representation theory of finite dimensional hereditary algebras and their tilted algebras, which allow the authors to describe the representation theory of prominent classes of Frobenius algebras. The second part is devoted to basic classical and recent results concerning the Hochschild extensions of finite dimensional algebras by duality bimodules and their module categories. Moreover, the shapes of connected components of the stable Auslander-Reiten quivers of Frobenius algebras are described. The only prerequisite for this volume is a basic knowledge of linear algebra and some results of the first volume. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. Introduction Hereditary algebras The quiver of an algebra The tensor algebras of species Exact sequences The Euler forms The Coxeter transformation Postprojective and preinjective components Hereditary algebras of Dynkin type Hereditary algebras of Euclidean type Hereditary algebras of wild type Representations of bimodules Exercises Tilted algebras Torsion pairs Tilting modules The Brenner–Butler theorem Connecting sequences Splitting tilting modules Tilted algebras The criterion of Liu and Skowronski Reflections of hereditary algebras The theorem of Ringel on regular tilting modules Exercises Auslander–Reiten components Functors on module categories The Igusa–Todorov theorem Degrees of irreducible homomorphisms Stable Auslander–Reiten components Generalized standard Auslander–Reiten components Stable equivalence Exercises Selfinjective Hochschild extension algebras Hochschild cohomology spaces Hochschild extension algebras Hochschild extensions by duality modules Non-Frobenius selfinjective Hochschild extensions Hochschild extension algebras of finite field extensions Hochschild extension algebras of path algebras Hochschild extension algebras of hereditary algebras The Auslander–Reiten quivers of Hochschild extension algebras Exercises Bibliography Index This is the first of two volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book serves as a general introduction to basic results and techniques of the modern representation theory of finite dimensional associative algebras over fields, including the Morita theory of equivalences and dualities and the Auslander-Reiten theory of irreducible morphisms and almost split sequences. The second part is devoted to fundamental classical and recent results concerning the Frobenius algebras and their module categories. Moreover, the prominent classes of Frobenius algebras, the Hecke algebras of Coxeter groups, and the finite dimensional Hopf algebras over fields are exhibited. This volume is self contained and the only prerequisite is a basic knowledge of linear algebra. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. The text is primarily addressed to graduate students starting research in the representation theory of algebras as well as mathematicians working in other fields. A publication of the European Mathematical Society (EMS). Distributed within the Americas by the American Mathematical Society. "This is the second of three volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book is devoted to fundamental results of the representation theory of finite dimensional hereditary algebras and their tilted algebras, which allow to describe the representation theory of prominent classes of Frobenius algebras. The second part is devoted to basic classical and recent results concerning the Hochschild extensions of finite dimensional algebras by duality bimodules and their module categories. Moreover, the shapes of connected components of the stable Auslander-Reiten quivers of Frobenius algebras are described. The only prerequisite in this volume is a basic knowledge of linear algebra and some results of the first volume. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. The text is primarily addressed to graduate students starting research in the representation theory of algebras as well mathematicians working in other fields." -- Back cover This is the second of three volumes which will provide a comprehensive introduction to the modern representation theory of Frobenius algebras. The first part of the book is devoted to fundamental results of the representation theory of finite dimensional hereditary algebras and their tilted algebras, which allow to describe the representation theory of prominent classes of Frobenius algebras. The second part is devoted to basic classical and recent results concerning the Hochschild extensions of finite dimensional algebras by duality bimodules and their module categories. Moreover, the shapes of connected components of the stable Auslander-Reiten quivers of Frobenius algebras are described. The only prerequisite in this volume is a basic knowledge of linear algebra and some results of the fi rst volume. It includes complete proofs of all results presented and provides a rich supply of examples and exercises. The text is primarily addressed to graduate students starting research in the representation theory of algebras as well mathematicians working in other fi elds. Includes Bibliographical References (p. [637]-643) And Index.