习题 14, § 1

设$a$为李代数，$b$子代数为$a$。 A composition series joining $a$ to $b$ is a decreasing sequence $(a_i)_{0 \leqslant i \leqslant n}$ of subalgebras of $a$ such that $a_0 = a, a_n = b, a_{i+1}$ is an ideal of $a_i$ for $0 \leqslant i < n$. $b$ is called a subinvariant subalgebra of $a$ if there exists a composition series joining $a$ to $b$. (a) Let $b$ be a subinvariant subalgebra of $a$ and $(a_i)_{0 \leqslant i \leqslant n}$ a composition series joining $a$ to $b$. 从习题 13 推导出$[\mathscr{C}^{n+p}b, a] \subset \mathscr{C}^p b$ 通过注意到 $[a_i, b] \subset a_{i+1}$ 得到$1 \leqslant i < n$。 (b) 由(a)推导出$\mathscr{C}^p b$($p = 0, 1, 2, \ldots$)的交$\mathscr{C}^\infty b$是$a$的理想。 (c) 从 (b) 推导出 $a$ 的次不变量子代数 $c$，使得$c = \mathscr{D}c$ 是 $a$ 的理想。 (d) 当$K$为域和$\dim_K a < +\infty$时，证明$\mathscr{D}^p b$（$p = 0, 1, 2, \ldots$）的交$\mathscr{D}^\infty b$是$a$的理想。 （证明这个交是 $a$ 的次不变量子代数并应用 (c)。）