无穷列支泡上的不变度量

R. García-Delgado
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引用次数: 0

摘要

我们阐述了一个无熵的李代数 $\g$ 允许一个无变量的标准。我们利用 $\g$ 拥有两个规范无边理想 $\ide(\g)\subset \mathfrak{J}(\g)$ 来分解 $\g$ 的底层向量空间,然后我们说明了 $\g$ 允许一个不变度量的充分条件。根据理想$mathfrak{J}(\g)$的性质,我们可以证明,如果一个现李代数$\g \times \Sa$允许一个不变度量,那么从$\Sa \times \Sa$到$\g/\mathfrak{J}(\g)$的中心空间一定有一个不变且非退化的双线性映射。我们还证明了在任何无穷烈代数 $\g$ 中都存在一个非零的、对称的和不变的双线性形式。这个双线性形式允许通过一个带单位的代数来重构 $\g$ 。我们证明了这个代数是简单的,当且仅当双线性形式是 $\g$ 上的一个不变度量。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Invariant Metrics on Nilpotent Lie algebras
We state criteria for a nilpotent Lie algebra $\g$ to admit an invariant metric. We use that $\g$ possesses two canonical abelian ideals $\ide(\g) \subset \mathfrak{J}(\g)$ to decompose the underlying vector space of $\g$ and then we state sufficient conditions for $\g$ to admit an invariant metric. The properties of the ideal $\mathfrak{J}(\g)$ allows to prove that if a current Lie algebra $\g \otimes \Sa$ admits an invariant metric, then there must be an invariant and non-degenerate bilinear map from $\Sa \times \Sa$ into the space of centroids of $\g/\mathfrak{J}(\g)$. We also prove that in any nilpotent Lie algebra $\g$ there exists a non-zero, symmetric and invariant bilinear form. This bilinear form allows to reconstruct $\g$ by means of an algebra with unit. We prove that this algebra is simple if and only if the bilinear form is an invariant metric on $\g$.
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