Exercise 19, § 13

Let V be a finite dimensional vector space and $E = V\oplus V^*$. Define a non-degenerate bilinear form $\Phi$ on E by

Put $N =\bigwedge(V)$ and $Q(x) =\frac{1}{2}\Phi (x, x)$ for $x\in E$. Denote the spinor representation by $\lambda : C(Q)\rightarrow$ End(N) as in no. 2.IV, define $f:\mathfrak{o}(\Phi )\rightarrow C^+(Q)$ as in Lemma 1 of no. 2, and denote by $\rho$ the linear representation $\lambda \circ f$ of the algebra $\mathfrak{o}(\Phi )$ on N.

a) Associate to any endomorphism $u$ of V the endomorphism $\widetilde{u}$ of E by the formula $\widetilde{u}(x, x^*) = (u(x),-^tu(x^*))$. Show that $u \rightarrow \widetilde{u}$ is a homomorphism of Lie algebras from $\mathfrak{g}\mathfrak{l}(V)$ to $\mathfrak{o}(\Phi )$; moreover, for $u$ in $\mathfrak{g}\mathfrak{l}$(V), $\rho (\widetilde{u})$ is the unique derivation of the algebra $\bigwedge(V) = N$ that coincides with $u$ on V.

b) Let $\Psi$ be a non-degenerate alternating bilinear form on V and $\gamma : V\rightarrow V^*$ the isomorphism defined by $\Psi (x, y) =\langle x, \gamma (y)\rangle$ for $x, y$ in V. Show that the endomorphisms $\widetilde{X}_+$ and $\widetilde{X}_-$ of E defined by

belongs to $\mathfrak{o}(\Phi )$. Put $\widetilde{H}= (-1\widetilde{)}$. Show that $( \widetilde{H},\widetilde{X}_+,\widetilde{X}_-)$ is an $\mathfrak{s}\mathfrak{l}_2$-triplet in the Lie algebra $\mathfrak{o}(\Phi )$.

c) Show that $\rho$ takes the endomorphisms $\widetilde{H},\widetilde{X}_+,\widetilde{X}_-$ of $\mathfrak{o}(\Phi )$ to the endomorphisms of N denoted by $H, X_+$ and $X_-$, respectively, in no. 3.IV. Deduce that $(H, X_+, X_-)$ is an $\mathfrak{s}\mathfrak{l}_2$-triplet in the Lie algebra $\mathfrak{g}\mathfrak{l}(N)$.