习题 17, § 9

(a) 让$c$表示实数可解李代数，其中基$(a, b, c)$使得$[a, b] = c, [a, c] = -b, [b, c] = 0$。让$d$表示实数可解李代数，其中基$(a, b, c, d)$使得$[a, b] = c, [a, c] = -b, [b, c] = d, [a, d] = [b, d] = [c, d] = 0$。令$g$为实数可解李代数。如果$g$包含非零元素$x, y, z$使得$[x, y] = z$和$[x, z] = -y$，则$g$包含子代数同构于$c$或$d$。 （通过$a_i = [a_{i-1}, c_{i-1}], b_i = [b_{i-1}, c_{i-1}], c_i = [a_i, b_i]$归纳写出$a_1 = y, b_1 = z, c_1 = [y, z]$；定义$a_i, b_i, c_i$。考虑最小的整数$k$，使得$c_k = 0$。）

(b) 设$g$和$g^*$为$g$的实可解李代数和$\phi$同态到$g^*$。如果 $g^*$ 包含子代数同构于 $c$ 或 $d$，则 $g$ 具有相同的性质。

(c) 设$h$为复数可解李代数。考虑一个 Jordan–Hölder 级数作为$h$ 的伴随表示； 级数定义一维表示与 $h$ 的商，因此线性形式对 $h$。这些仅依赖于$h$的线性形式，称为$h$的根。如果$h'$是实数可解李代数，则$h'$的根是限制到$h' \otimes_{\mathbf{R}} \mathbf{C}$的根$h'$。

Then let $G$ be a finite-dimensional simply connected solvable real Lie group and $g$ its Lie algebra. Show that the following conditions are equivalent: ($\alpha$) $\exp_G$ is injective; ($\beta$) $\exp_G$ is surjective; ($\gamma$) $\exp_G$ is bijective; ($\delta$) $\exp_G$ is an isomorphism of the analytic manifold $L(G)$ onto the analytic manifold $G$; ($\varepsilon$) $L(G)$ contains no subalgebra isomorphic to $c$ or $d$; ($\zeta$) there exists no quotient algebra of $L(G)$ admitting a subalgebra isomorphic to $c$; ($\eta$) every root of $g$ is of the form $\phi + i\phi'$ where $\phi, \phi'$ are in $g^*$ and $\phi'$ is proportional to $\phi$; ($\theta$) for all $x \in g$, the only pure imaginary eigenvalue (in $\mathbf{C}$) of $\mathrm{ad}\, x$ is 0.\footnote{For more details, cf. M. Saito, Sur certains groupes de Lie résolubles, Sci. Papers of the College of General Education, Univ. of Tokyo, 7 (1957), pp. 1–11 and 157–168.}