By Andrzej Skowronski, Kunio Yamagata
This can be the 1st of 2 volumes in an effort to supply a finished advent to the fashionable illustration idea of Frobenius algebras. the 1st a part of the booklet serves as a common creation to uncomplicated effects and methods of the fashionable illustration thought of finite dimensional associative algebras over fields, together with the Morita concept of equivalences and dualities and the Auslander-Reiten conception of irreducible morphisms and virtually break up sequences. the second one half is dedicated to primary classical and up to date effects about the Frobenius algebras and their module different types. furthermore, the admired periods of Frobenius algebras, the Hecke algebras of Coxeter teams, and the finite dimensional Hopf algebras over fields are exhibited. This quantity is self contained and the single prerequisite is a uncomplicated wisdom of linear algebra. It contains whole proofs of all effects awarded and gives a wealthy provide of examples and routines. The textual content is essentially addressed to graduate scholars beginning examine within the illustration concept of algebras in addition to mathematicians operating in different fields. A ebook of the eu Mathematical Society (EMS). disbursed in the Americas through the yankee Mathematical Society.
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Extra info for Frobenius Algebras I: Basic Representation Theory
Example text
Q0 ; Q1 ; s; t / be a finite quiver and I an admissible ideal of the path algebra KQ of Q over a field K. Q/. ˛1 / ! ˛l / called the evaluation map of M on the path w. Then for a K-linear combination %D r X i wi iD1 of paths in Q with a common source a and a common target b, we define the K-linear map r X '% D i 'wi W Ma ! Q/ is said to be bound by I , or satisfying the relations of I , if we have '% D 0 for all relations % 2 I . 6 that the ideal I is generated, as a two-sided ideal, by a finite set f%1 ; : : : ; %r g of relations.
Let A be a K-algebra, M be a right A-module and M1 ; : : : ; Mr , ˚ Mr if and only if the r 1, right A-submodules of M . Then M D M1 ˚ following conditions are satisfied: (1) M D M1 C C Mr , P (2) Mi \ j 2f1;:::;rgnfig Mj D 0 for each i 2 f1; : : : ; rg. 20 Chapter I. Algebras and modules Proof. Clearly, M D M1 C C Mr is just the fact that every element m 2 M has an expression of the form m D m1 C C mr with m1 2 M1 ; : : : ; mr 2 Mr . Moreover, if m1 C C mr D m D m01 C C m0r are two expressions of m 2 M with m1 ; m01 2 M1 ; : : : ; mr ; m0r 2 Mr , then, for each i 2 f1; : : : ; rg, we have Á X X Mj : mj0 mj 2 Mi \ mi m0i D j 2f1;:::;rgnfig j 2f1;:::;rgnfig Hence the required equivalence follows.
Ei / D fi for i 2 f1; : : : ; ng. Proof. Let I m D 0A for a positive integer m, and for elements x1 ; : : : ; xn in A we denote by hx1 ; : : : ; xn i the set of K-linear combinations of elements of the form x1r1 : : : xnrn for nonnegative integers r1 ; : : : ; rn . ei / D fi and ei 2 hx1 ; : : : ; xn i for i 2 f1; : : : ; ng. First consider the case n D 1. x/ D x C I for some x 2 A. x x 2 /m D 0A . x x 2 /m D x m x mC1 y, where y D Pm i 1 m i 1 x . Then we obtain x m D x mC1 y and xy D yx. We claim iD1 .