有限维结合代数上表示可约性的两个判别法则

Two Criterions for Representing Reducibility on Finite Dimentional Associative Algebras

本文给出了有限维结合代数上表示可约性的两个判别法则。它们是:若φ是有限维结合代数上的表示,其表示矩阵为aφ=T(a)∈F_n,n>1,并且存在a∈Z(A),a≠0,使得T(a)≠0 而det T(a)=0,则φ是可约的;若φ是有限维结合代数A上的正则表示,其反表示矩阵为S(a)∈F_n,n>1,则φ是既约的充要条件为:(?)a∈A,a≠0,有det S(a)≠0。

In this paper the author gives two criterions for representing reducibility on finite dimensional associative algebras. They are: (1) Let φ be a representation on a finite dimensional associative algebra A, its representative matrix is T_(a) (=aφ∈ F_n, n>1) for any a∈A, and there exists an element a∈Z(A), a≠0, so that T_(a)≠0 and det T_(a) =0, then φ is reducible. (2) Let φ be a regular representation on a finite dimensional associative algebra A, its inverse representative matrix is S_(a) ∈F_(n) (n>1) for any a∈A, then φ is irreducible if and only if det S_a≠0 for arbitrary nonzero etement a in A.