有限维结合代数上表示的可约性

Reducibility of Representation on a Finite Dimensional Associative Algebra

给出有限维结合代数上表示可约性的两个判别法。它们是，（Ⅰ）若是有限维结合代数Ａ上的表示，其表示矩阵为ａ，且存在非零元ａ∈Ｚ（Ａ），使得Ｔ（ａ）≠０，而ｄｅｔＴ（ａ）≠０，则是可约的；（Ⅱ）若是有限维结合代数Ａ上的正则表示，其反表示矩阵为Ｓ（ａ），则是既约的充要条件为：ａ∈Ａ，ａ≠０，有ｄｅｔＳ（ａ）≠０。

Two criteria for the reducibility of presentations of finite dimensional associatie algebras are presented. One is that let φ be a representation on a finite dimensional associative algebra A , its representative matrix be T(a),and there exist an element a∈Z(A), a≠0,where Z(A) is the centralizer of A, such that T(a)≠0 and det T(a)=0,then φ is reducible. The other is that let φ be a regular representation on a finite dimensional associative algebra A, and its anti represetative matrix is S(a), then φ is irreducible if and only if det S(a)≠0 for any non zero element a in A.