Exercise 7, § 1

An involutive Lie algebra is a (real) Lie algebra $\mathfrak{g}$ equipped with an automorphism $s$ such that $s\circ s= 1_{\mathfrak{g}}$. Denote by $\mathfrak{g}^+$ (resp. $\mathfrak{g}^-$) the eigenspace of $\mathfrak{g}$ relative to the eigenvalue +1 (resp. $-1$) of $s$.

a) Show that $\mathfrak{g}^+$ is a subalgebra of $\mathfrak{g}$ and that $\mathfrak{g}^-$ is a $\mathfrak{g}^+$-module; we have $[\mathfrak{g}^-,\mathfrak{g}^-]\subset \mathfrak{g}^+$, and $\mathfrak{g}^+$ and $\mathfrak{g}^-$ are orthogonal with respect to the Killing form.

b) Show that the following conditions are equivalent:

(i) The $\mathfrak{g}^+$-module $\mathfrak{g}^-$ is simple.

(ii) The algebra $\mathfrak{g}^+$ is maximal among the subalgebras of $\mathfrak{g}$ distinct from $\mathfrak{g}$. If these conditions are satisfied, and if $\mathfrak{g}^+$ contains no non-zero ideal of $\mathfrak{g}$, the involutive Lie algebra $(\mathfrak{g}, s)$ is said to be irreducible.

c) Assume that $\mathfrak{g}$ is semi-simple. Show that the following conditions are equivalent:

(i) The only ideals of $\mathfrak{g}$ stable under $s$ are $\{0\}$ and $\mathfrak{g}$;

(ii) $\mathfrak{g}$ is either simple or the sum of two simple ideals interchanged by $s$.

Show that these conditions are satisfied when $(\mathfrak{g}, s)$ is irreducible (observe that $\mathfrak{g}$ is the direct sum of ideals stable under $s$ and satisfying (ii)).

d) Assume from now on that $\mathfrak{g}$ is compact semi-simple. Show that the involutive Lie algebra $(\mathfrak{g}, s)$ is irreducible if and only if $s$ is different from the identity and $(\mathfrak{g}, s)$ satisfies the equivalent conditions in c) (let $\mathfrak{p}$ be a $\mathfrak{g}^+$-submodule of $\mathfrak{g}^-$ and $\mathfrak{q}$ its orthogonal complement with respect to the Killing form; observe that $[\mathfrak{p},\mathfrak{q}] = 0$ and deduce that $\mathfrak{p}+ [\mathfrak{p},\mathfrak{p}]$ is an ideal of $\mathfrak{g}$).

e) Prove that $\mathfrak{g}$ is the direct sum of a family $(\mathfrak{g}_i)_{0\leq i\leq n}$ of ideals stable under $s$, such that $\mathfrak{g}_0$ is fixed by $s$ and such that the involutive algebras $(\mathfrak{g}_i, s|\mathfrak{g}_i)$ are irreducible for $1\leq i\leq n$.