关于特征幂零代数的一个定理

S. Tôgô
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引用次数: 3

摘要

1. 李代数的特征幂零的概念由Dixmier和Lister[3]提出,Leger和Togo[4]对此进行了研究。设L是域Φ上的李代数,®(L)是L的所有派生的集合,设L^ = LcS){L)={Σ1aiDi\aie L, D{e®(Z,)},并定义n>2时的归纳ί ^ =£C»-^©(i)。如果存在一个整数k使得L =(0),则L称为特征幂零,则每一个特征幂零李代数都是幂零的。然而,幂零代数和特征幂零代数的性质之间很少有相似之处,例如,与幂零代数相反,特征幂零李代数的子代数和商代数不一定是特征幂零的。最近,在[_2~],c - y。Chao给出了幂零李代数的一个表征:设Lbea李代数在一个域Φ上,且TV是L的幂零理想,则L是幂零的当且仅当L/N为幂零。这篇笔记的目的是展示一个类似的特征幂零李代数的特征,事实上,更一般的特征幂零非结合代数。我们所说的非结合代数是指不一定是结合的代数,即分配代数[β'J]。非结合代数a的特征幂零的定义是通过用a代替上面提到的李代数中的L而得到的,这是由于T.S. Ravisankar的论文[ΊΓ]的第一版。然而,目前还不知道是否存在一个非李代数的特征性幂零非结合代数。我们将在第3节中证明这种代数的存在性。本笔记中所考虑的所有代数都假定在其基域上是有限维的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
A theorem on characteristically nilpotent algebras
1. The notion of characteristic nilpotency of Lie algebras has been introduced by Dixmier and Lister [3] and studied by Leger and Togo [4]. Let L be a Lie algebra over a field Φ and ® (L) be the set of all derivations of L. Put L^ = LcS){L)={Σ1aiDi\aie L, D{ e ®(Z,)} and define inductively ί ^ = £C»-^©(i) for n>2. L is called characteristically nilpotent provided there exists an integer k such that L = (0). Then every characteristically nilpotent Lie algebra is nilpotent. However, few similarities are known between the properties of nilpotent and characteristically nilpotent algebras, e.g., a subalgebra and a quotient algebra of a characteristically nilpotent Lie algebra are not necessarily characteristically nilpotent, contrary to the case of nilpotent algebras. Recently, in [_2~], C.-Y. Chao has shown a characterization of nilpotent Lie algebras: Let Lbea Lie algebra over a field Φ and TV be a nilpotent ideal of L. Then L is nilpotent if and only if L/N is nilpotent. The purpose of this note is to show a similar characterization of characteristically nilpotent Lie algebras, as a matter of fact, more generally of characteristically nilpotent nonassociative algebras. By a nonassociative algebra we mean an algebra which is not necessarily associative, that is, a distributive algebra [β'J. The definition of characteristic nilpotency of a nonassociative algebra A is obtained by replacing A instead of L in that of a Lie algebra stated above and this is due to the first version of the paper [ΊΓ] of T.S. Ravisankar. However, it has not yet been known that there actually exists a characteristically nilpotent nonassociative algebra which is not a Lie algebra. We shall show the existence of such an algebra in Section 3. All the algebras considered in this note are assumed to be finite dimensional over their base fields.
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