具有二维导理想的实可解李代数的表示及相应李群的余伴轨道几何
Representation of Real Solvable Lie Algebras Having 2-dimensional Derived Ideal and Geometry of Coadjoint Orbits of Corresponding Lie Groups
摘要：
设{ \Lnk }是所有具有$k$维导理想的$n$维实可解李代数的类. 2020年，作者等人给出了{ \Li }的所有非2步幂零李代数的分类.本文主要研究这类李代数的表示以及相应的连通和单连通李群。也就是说，对于每一个代数，我们给出了一个忠实表示的最小度的上界。然后，我们给出了相应群的余伴随轨道的几何描述。此外，我们还证明了K。P. Shum和第二作者等人的结论对于这里考虑的李群仍然成立。也就是说，我们证明了，对于每一个考虑组，家庭的最大维coadjoint轨道形成一个可测的叶理意义上的Connes。文中还给出了这些叶理的拓扑分类。
英文摘要：
Let {\Lnk} be the class of all $n$-dimensional real solvable Lie algebras having $k$-dimensional derived ideals. In 2020 the authors et al. gave a classification of all non 2-step nilpotent Lie algebras of {\Li}. We propose in this paper to study representations of these Lie algebras as well as their corresponding connected and simply connected Lie groups. That is, for each algebra, we give an upper bound of the minimal degree of a faithful representation. Then, we give a geometrical description of coadjoint orbits of corresponding groups. Moreover, we show that the characteristic property of the family of maximal dimensional coadjoint orbits of a MD-group studied by K. P. Shum and the second author et al. is still true for the Lie groups considered here. Namely, we prove that, for each considered group, the family of the maximal dimensional coadjoint orbits forms a measurable foliation in the sense of Connes. The topological classification of these foliations is also provided.
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