李群中Cartan子群的结构
The Structure of Cartan Subgroups in Lie Groups
摘要：
研究了连通李群中Cartan子群的性质和结构。我们得到了Cartan子群的一个特征，推广了Wüstner的结构定理。我们证明了Cartan子群与根的极大紧子群的中心化子的Cartan子群是相同的。此外，我们描述了在连通可解李群中构造包含某些幂零子群的嘉当子群的方法。我们把商群模闭正规子群中的Cartan子群映射为环境群中Cartan子群的象。我们还研究了连通李群上幂映射的象的稠密性，并证明了如果任何$k$次幂映射的闭正规子群的限制和相应的商群上的映射都有稠密象，则它的象也有稠密象.
英文摘要：
We study properties and the structure of Cartan subgroups in a connected Lie group. We obtain a characterisation of Cartan subgroups which generalises Wüstner's structure theorem for the same. We show that Cartan subgroups are same as those of the centralizers of maximal compact subgroups of the radical. Moreover, we describe a recipe for constructing Cartan subgroups containing certain nilpotent subgroups in a connected solvable Lie group. We characterise the Cartan subgroups in the quotient group modulo a closed normal subgroup as the images of the Cartan subgroups in the ambient group. We also study the density of the images of power maps on a connected Lie group and show that the image of any $k$-th power map has dense image if its restriction to a closed normal subgroup and the corresponding map on the quotient group have dense images.
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