Exercise 20, § 4

Let g be a Lie algebra and U its enveloping algebra.

(a) Let $g'$ be an ideal of g, $U' \subset U$ its enveloping algebra, $x \in g$ and $a_1, \ldots, a_n$ in U. Suppose that $a_i = x$ for the indices $j_1, \ldots, j_p$ and $a_i \in U'$ for the other indices (let $k_1, \ldots, k_q$ be these indices, with $k_1 < k_2 < \cdots < k_q$). Then $a_1a_2 \ldots a_n - x^pa_{k_1}a_{k_2} \ldots a_{k_q}$ is a sum of terms of the form $x^{p'}b$, where $b \in U'$ and $p' < p$. (Argue by induction on $p$.)

(b) Suppose henceforth that g is nilpotent and K is of characteristic 0. Let $g'$ be an ideal of codimension 1 in g, $U'$ its enveloping algebra, x an element of g not belonging to $g'$ and $U_0$ (resp. $U'_0$) the subalgebra of U (resp. $U'$) annihilated by a set $\mathfrak{d}$ of derivations of g (which extend to derivations of U) mapping g into $g'$. Suppose that $U_0$ is contained in the centre of U and $U_0 \not\subset U'$. Show that there exist $a_1 \in U'_0,\ a_2 \in U'$ such that $a_1 \neq 0$ and $a = xa_1 + a_2 \in U_0$. (Let $x^mb_m + x^{m-1}b_{m-1} + \cdots + b_0 \in U_0$ with $b_m, \ldots, b_0$ in $U'$, $m > 0,\ b_m \neq 0$. Show, using (a), that, for every derivation $D \in \mathfrak{d}$ of g, $Db_m = 0,\ m(Dx)b_m + Db_{m-1} = 0$ and hence $D(mxb_m + b_{m-1}) = 0$.) Show that $U_0$ is contained in the algebra $K[a, a_1^{-1}, U'_0]$ generated by $a,\ a_1^{-1},\ U'_0$ in the field of fractions of $U_0$ (which can be formed by Corollary 7 to Theorem 1 of § 2). Deduce that the field of fractions of $U_0$ is the field generated by $a$ and $U'_0$. Show that $a$ is transcendental over $K(U'_0)$.

(c) Let $\{0\} = g_0 \subset g_1 \subset \ldots \subset g_n = g$ be a sequence of ideals of $g$ of dimensions $0, 1, \ldots, n$. Let $x_j$ be an element of $g_j$ not belonging to $g_{j-1}$. Let $j_1 < j_2 < \cdots < j_q$ be the indices $j$ such that there exists in $U^j$ (enveloping algebra of $g_j$) an element of the centre $U_0$ of $U$ not belonging to $U^{j-1}$. By (b) there exists $a_{j1} \in U_0 \cap U^j$ and $a_{j2} \in U^{j-1}$ such that $a_{j1} \neq 0$ and $a_j = x_j a_{j1} + a_{j2} \in U_0 \cap U^j$. Show by induction on $n$ that

and that the field of fractions of $U_0$ is generated by the algebraically independent elements $a_{j_1}, \ldots, a_{j_q}$. In particular, the field of fractions of the centre of $U$ is a pure transcendental extension of $K$.