Exercise 3, § 10

Assume that $k$ is $\mathbf{R},\mathbf{C}$, or a non-discrete complete ultrametric field. Give the grassmannian $\mathbf{G}(\mathfrak{g})$ of vector subspaces of $\mathfrak{g}$ its natural structure of analytic manifold over $k($Differentiable and Analytic Manifolds, Results, 5.2.6). Consider the subsets of $\mathbf{G}(\mathfrak{g})$ formed by:

(i) the subalgebras

(ii) the solvable subalgebras

(iii) the nilpotent subalgebras

(iv) the subalgebras consisting of nilpotent elements

(v) the Borel subalgebras.

Show that these subsets are closed (for (v), use Exerc. 2). Deduce that these subsets are compact when $k$ is locally compact.

Show by examples that the subsets of $\mathbf{G}(\mathfrak{g})$ formed by:

(vi) the Cartan subalgebras

(vii) the subalgebras reductive in $\mathfrak{g}$

(viii) the semi-simple subalgebras

(ix) the decomposable subalgebras

are not necessarily closed, even when $k=\mathbf{C}$.