Exercise 8, § 2

Assume that $k$ is of characteristic $p >0$. Denote by $\mathfrak{s}$ a Lie algebra with basis $\{x, y\}$ such that $[x, y] =y$. Let V be a $k$-vector space with basis $\{e_i\}_{i\in\mathbf{Z}/p\mathbf{Z}}$. a) Show that V has a unique $\mathfrak{s}$-module structure such that $xe_i=ie_i$ and $ye_i=e_{i+1}$ for all $i$. This $\mathfrak{s}$-module is simple.

b) Let $\mathfrak{g}=\mathfrak{s}\times V$ be the semi-direct product of $\mathfrak{s}$ by V. Show that $\mathfrak{g}$ is a solvable algebra of rank 1 whose derived algebra is not nilpotent.

c) An element of $\mathfrak{g}$ is regular if and only if its projection on $\mathfrak{s}$ is of the form $ax+by$, with $ab\not= 0$.

d) We have $V^0(x+y) = 0$ and $V^0(x) =ke_0$. Deduce (cf. Exerc. 6) that $\mathfrak{g}$ has Cartan subalgebras of dimension 1 (for example that generated by $x+y$) and Cartan subalgebras of dimension 2 (for example that generated by $x$ and $e_0$).