Exercise 5, § 7

Let K be a field of characteristic $p > 0$, $g$ a Lie algebra of finite dimension $n$ over K and U the enveloping algebra of $g$.

(a) A p-polynomial (in one indeterminate) over K is any polynomial in $K[X]$ whose only terms $\neq 0$ are of degree a power of $p$. Show that for every polynomial $f(X) \neq 0$ in $K[X]$ there exists $g(X) \in K[X]$ such that $f(X)g(X)$ is a $p$-polynomial (consider the remainders in Euclidean division of the $X^{p^k}$ by $f(X)$).

(b) Show that for every element $z \in g$ there exists a non-zero polynomial $f(X) \in K[X]$ such that $f(z)$ belongs to the centre C of U. (Consider the minimal polynomial of the endomorphism $\operatorname{ad} z$ and apply (a) to this polynomial, and also formula (1) of Exercise 19 of § 1.)

(c) Let $(e_i)_{1 \leq i \leq n}$ be a basis of $g$. For each $i$ let $f_i \neq 0$ be a polynomial in $K[X]$ such that $f_i(e_i) \in C$; let $d_i$ be its degree and L the two-sided ideal of U generated by the $y_i = f_i(e_i)$. Show that the classes mod. L of the elements $e_1^{\alpha_1} \ldots e_n^{\alpha_n}$, where $0 \leq \alpha_i < d_i$ form a basis of $U/L$.

(d) Show that $g$ admits a finite-dimensional faithful linear representation (choose the $f_i$ so that $d_i > 1$ for all $i$).

(e) Show that if $g \neq \{0\}$ there exists a non-semi-simple finite-dimensional linear representation of $g$. (Assuming that the $f_i$ are of degrees $d_i > 1$, replace the $f_i$ by $f_i^2$ in the definition of L and note that there then exist nilpotent elements $\neq 0$ in the centre of $U/L$ and hence that $U/L$ is not semi-simple.)