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Extremal elements in Lie algebras, buildings and structurable algebras
Journal of Algebra ( IF 0.8 ) Pub Date : 2021-03-26 , DOI: 10.1016/j.jalgebra.2021.03.014
Hans Cuypers , Jeroen Meulewaeter

An extremal element in a Lie algebra g over a field of characteristic not 2 is an element xg such that [x,[x,g]] is contained in the linear span of x. The linear span of an extremal element, called an extremal point, is an inner ideal of g, i.e. a subspace I satisfying [I,[I,g]]I. We show that in characteristic different from 2,3 the geometry with point set the set of extremal points and as lines the minimal inner ideals containing at least two extremal points is a Moufang spherical building, or in case there are no lines a Moufang set.

This last result on the Moufang sets is obtained by connecting Lie algebras to structurable algebras, a class of non-associative algebras with involution generalizing Jordan algebras. It is shown that in characteristic different from 2,3 each finite-dimensional simple Lie algebra generated by extremal elements is either a symplectic Lie algebra or can be obtained by applying the Tits-Kantor-Koecher construction to a skew-dimension one structurable algebra. Various relations between the Lie algebra g and its extremal geometry on the one hand and the associated structurable algebra on the other hand are investigated.



中文翻译:

李代数,建筑物和可构造代数中的极值元素

李代数中的极值元素 G 在特征不是2的字段上是一个元素 XG 这样 [X[XG]]包含在x的线性范围内。极值元素的线性跨度(称为极值点)是G,即满足的子空间[一世[一世G]]一世。我们显示出与2个3 具有点的几何设置了一组极点,而包含至少两个极点的最小内部理想作为线是Moufang球形建筑物,或者在没有线的情况下设置了Moufang集。

在Moufang集上的最后结果是通过将Lie代数连接到可构造代数而获得的,可构造代数是一类具有对合广义Jordan代数的非缔合代数。结果表明,在特性上与2个3由极值元素生成的每个有限维简单Lie代数要么是辛Lie代数,要么可以通过将Tits-Kantor-Koecher构造应用于偏维一可构造代数来获得。李代数之间的各种关系G 一方面研究了其极值几何,另一方面研究了相关的可构造代数。

更新日期:2021-04-09
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