一类欧氏李代数的$Z$ -分解
On $Z$-decomposition of some euclidian lie algebras
摘要：
本文利用$Z-$分解作为工具，寻找某些李群上的局部对称左不变黎曼度量.为此，我们需要计算曲率算子的谱。由于这类谱的研究是非常困难的，我们对黎曼李群的类和维数作了限制。研究了关联李代数为$\mathbb{A}_2\oplus 2\mathbb{A}_1,\,\mathbb{A}_{3,1}\oplus \mathbb{A}_1,\, \mathbb{A}_{3,3}\oplus \mathbb{A}_1$的$4-$维连通黎曼李群以及$3$和$4-$维黎曼李群的子类为$\mathcal{C}-$空间.
英文摘要：
In this paper we use the $Z-$decomposition as a tool to find locally symmetric left invariant Riemannian metrics on some Lie groups. For this purpose, we need to compute the spectrum of the curvature operator. Since the study of this spectrum is very difficult, we impose restriction on the class of Riemannian Lie groups and their dimension. We investigate the $4-$dimensional connected Riemannian Lie groups whose associated Lie algebras are $\mathbb{A}_2\oplus 2\mathbb{A}_1,\,\mathbb{A}_{3,1}\oplus \mathbb{A}_1,\, \mathbb{A}_{3,3}\oplus \mathbb{A}_1$ and the subclasses of $3$ and $4-$dimensional Riemannian Lie groups which are $\mathcal{C}-$spaces.
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