[S,S]的李结构

Lie structures of [S,S]

设 R是带对合的单结合环 ,Z是 R的中心 ,S是 R的对称元的全体组成的集合 ,K是 R的斜对称元的全体组成的集合。作者证明了下列结论 :(1 )若 R作为 Z上的向量空间的维数大于 4,则 [S,S]=[K,K]且 [S,S]=R;(2 )若 R带第一类对合且 R作为 Z上的向量空间的维数大于 1 6 ,则 [S,S]是单李环且[[S,S],[S,S]]=[S,S];(3)若 R带第二类对合 ,R的特征不为 2 ,R作为 Z上的向量空间的维数不为 4,则对于 [S,S]的任意李理想 U,有 U Z或 U=[S,S]

Let R be a simple associative ring with an involution and Z the center of R.Let S be the set of all symmetric elements in R and K the set of all skew symmetric elements in R.The following results:(1) If the dimension of R as a vector space over Z is larger than 4,then [S,S]=[K,K] and [S,S] =R;(2) If R is with the first involution,and the dimension of R as a vector space over Z is larger than 16,then [S,S]is a simple Lie ring,and [[S,S],[S,S]]=[S,S];(3) If R is with the second involution,and the dimension of R as a vector space over Z is not equal to 4,and the character of R is not equal to 2,then UZ or U=[S,S] for any Lie ideal U of [S,S] are proved.