ELEMENTS OF THE REPRESENTATION THEORY OF ASSOCIATIVE ALGEBRAS

By Y. GAZAE

CONTENTS I ALGEBRAS AND MODULES I 1 ALGEBRAS I 2 MODULES I 3 SEMISIMPLE MODULES AND THE RADICAL OF A MODULE I 4 DIRECT SUM DECOMPOSITIONS I 5 PROJECTIVE AND INJECTIVE MODULES I 6 BASIC ALGEBRAS AND EMBEDDINGS OF MODULE CATEGORIES I 7 EXERCISES II QUIVERS AND ALGEBRAS II 1 QUIVERS AND PATH ALGEBRAS II 2 ADMISSIBLE IDEALS AND QUOTIENTS OF THE PATH ALGEBRA II 3 THE QUIVER OF A ?NITE DIMENSIONAL ALGEBRA II 4 EXERCISES III REPRESENTATIONS AND MODULES III 1 REPRESENTATIONS OF BOUND QUIVERS III 2 THE SIMPLE, PROJECTIVE, AND INJECTIVE MODULES III 3 THE DIMENSION VECTOR OF A MODULE AND THE EULER CHARACTERISTIC III 4 EXERCISES IV AUSLANDER–REITEN THEORY IV 1 IRREDUCIBLE MORPHISMS AND ALMOST SPLIT SEQUENCES IV 2 THE AUSLANDER–REITEN TRANSLATIONS IV 3 THE EXISTENCE OF ALMOST SPLIT SEQUENCES IV 4 THE AUSLANDER–REITEN QUIVER OF AN ALGEBRA IV 5 THE ?RST BRAUER–THRALL CONJECTURE IV 6 FUNCTORIAL APPROACH TO ALMOST SPLIT SEQUENCES IV 7 EXERCISES V NAKAYAMA ALGEBRAS AND REPRESENTATION–?NITE GROUP ALGEBRAS V 1 THE LOEWY SERIES AND THE LOEWY LENGTH OF A MODULE V 2 UNISERIAL MODULES AND RIGHT SERIAL ALGEBRAS V 3 NAKAYAMA ALGEBRAS V 4 ALMOST SPLIT SEQUENCES FOR NAKAYAMA ALGEBRAS V 5 REPRESENTATION–?NITE GROUP ALGEBRAS V 6 EXERCISES VI TILTING THEORY VI 1 TORSION PAIRS VI 2 PARTIAL TILTING MODULES AND TILTING MODULES VI 3 THE TILTING THEOREM OF BRENNER AND BUTLER VI 4 CONSEQUENCES OF THE TILTING THEOREM VI 5 SEPARATING AND SPLITTING TILTING MODULES VI 6 TORSION PAIRS INDUCED BY TILTING MODULES VI 7 EXERCISES VII REPRESENTATION–?NITE HEREDITARY ALGEBRAS VII 1 HEREDITARY ALGEBRAS VII 2 THE DYNKIN AND EUCLIDEAN GRAPHS VII 3 INTEGRAL QUADRATIC FORMS VII 4 THE QUADRATIC FORM OF A QUIVER VII 5 RE?ECTION FUNCTORS AND GABRIEL’S THEOREM VII 6 EXERCISES VIII TILTED ALGEBRAS VIII 1 SECTIONS IN TRANSLATION QUIVERS VIII 2 REPRESENTATION–IN?NITE HEREDITARY ALGEBRAS VIII 3 TILTED ALGEBRAS VIII 4 PROJECTIVES AND INJECTIVES IN THE CONNECTING COMPONENT VIII 5 THE CRITERION OF LIU AND SKOWRO´NSKI VIII 6 EXERCISES IX DIRECTING MODULES AND POSTPROJECTIVE COMPONENTS IX 1 DIRECTING MODULES IX 2 SINCERE DIRECTING MODULES IX 3 REPRESENTATION–DIRECTED ALGEBRAS IX 4 THE SEPARATION CONDITION IX 5 ALGEBRAS SUCH THAT ALL PROJECTIVES ARE POSTPROJECTIVE IX 6 GENTLE ALGEBRAS AND TILTED ALGEBRAS OF TYPE AN IX 7 EXERCISES