Exercise 18, § 13

Let V be a finite dimensional vector space, Q a non-degenerate quadratic form on V, and $\Psi$ the symmetric bilinear form associated to Q. Denote the Lie algebra $\mathfrak{o}(\Psi )$ by $\mathfrak{g}$ and the extension of $\Psi$ to $\bigwedge^2(V)$ by $\Psi_2$.

a) Show that there exists an isomorphism of vector spaces $\theta :\bigwedge^2(V)\rightarrow \mathfrak{g}$ characterized by the following equivalent properties:

$^{22}$For more details on Exercises 16 and 17, see: N. JACOBSON, Lie Algebras, Interscience Publ. (1962), Chap. X and G. B. SELIGMAN, Modular Lie Algebras,

Springer-Verlag (1967), Chap. IV.

(i) For $a, b$ and $x$ in V, we have $\theta (a\wedge b).x=a.\Psi (x, b)-b.\Psi (x, a)$.

(ii) For $x, y$ in V and $u$ in $\bigwedge^2(V)$ we have $\Psi_2(x\wedge y, u) =\Psi (x, \theta (u).y)$.

Let $\sigma$ be the identity representation of $\mathfrak{g}$ on V; then $\theta$ is an isomorphism of $\bigwedge^2(V)$ with the adjoint representation of $\mathfrak{g}$.

b) Define the linear map $f:\mathfrak{g}\rightarrow C^+(Q)$ as in Lemma 1 of no. 2. Show that $f \theta (a\wedge b) =\frac{1}{2}(ab-ba)$ for $a, b$ in V, and deduce a new proof of assertions (iii), (iv) and (v) of Lemma 1 of no. 2.

c) The notations $l,F,F', e_0,N, \lambda$ and $\rho$ are those of no. 2. Choose $e\not= 0$ in $\bigwedge(F')$ and define the bilinear form $\Phi$ on N by

Show that $\Phi (\lambda (a).x, y) +\Phi (x, \lambda (a).y) = 0$ for $x, y$ in N and $a$ in $F\oplus F'$. Deduce that $\Phi$ is invariant under the representation $\rho$ of $\mathfrak{g}$ (remark that the Lie algebra $\rho (\mathfrak{g})$ is generated by $\lambda (F\oplus F')$ by $a$) and b) above).