Exercise 25, § 6

Let K be a field of arbitrary characteristic $p$. Let $sp(2n, K)$ denote the Lie algebra of the formal symplectic group in $2n$ variables over K (§ 1, Exercise 26 (c)). Show that this Lie algebra, which is identified with a sub-algebra of $gl(2n, K)$, has basis consisting of the elements

(a) Show that, if $p \neq 2$ and $n \geq 1$, the algebra $sp(2n, K)$ is simple (for $n = 1, sp(2, K) = sl(2, K)$). (Same method as in Exercise 24.)

(b) If $p = 2$ and $n \geq 3$, show that the $H_i, F_{ij}, G'_{ij}$ and $G''_{ij}$ form an ideal $\mathfrak{a}$ of dimension $n(2n-1)$; the elements of $\mathfrak{a}$ of the form

such that $\sum_{i=1}^{n} \lambda_i = 0$ form an ideal $\mathfrak{b} \subset \mathfrak{a}$ of dimension $n(2n-1) - 1$ and the multiples of $\sum_{i=1}^{n} H_i$ an ideal $\mathfrak{c}$, the centre of $sp(2n, K)$; $\mathfrak{b}, \mathfrak{c}$ are the only ideals of $a, b = \mathscr{D}(\mathfrak{sp}(2n, K))$ and $b/c$ is simple. (Same method.) What are the ideals of $\mathfrak{sp}(4, K)$ when $K$ is of characteristic 2?