带有对偶基的代数

Algebras with dual basis.

特征为零域上的中心单代数具有很好的性质是因为它们都有对偶基 ;而当域的特征不为零时 ,同样是中心单代数 ,情况就不一样了 ,主要是它们不具有对偶基 .因而给出了在域 F的特征 p为不零情况下 ,单分离代数存在对偶基的充分必要条件是很有趣的 .设 A是域 F上的 n维单分离代数 ,且其特征 p不整除 n,则对 A的任何一组基 { a1,a2 ,… ,an} ,都存在惟一的对偶基 .如果 C是 A的中心 ,A在 C上的维数不能被 p整除 ,则其任何基都存在对偶基 .我们还利用对偶基来刻画这样的分离代数的中心 ,进而推广了特征为零时的 Whitehead引理 .最后 ,利用迹函数的方法刻画了非半单代数的幂零根和相应的直和分解 .

Central simple algebras over a field with characteristic zero have good propertie since they have a dual basis. However, central simple algebras over a field with characteristic p≠0 have no such good properties, since they have no dual basis. So it is very interesting to characterize algebras with dual basis. A necessary and sufficient condition for a simple algebra with a dual basis is given. Using the dual basis the center of these algebras is characterized and generalize some results are generalized. In addition, The nilpotent radicals of some algebras over a field with nonzero characteristic p by using trace function are described.