Bourbaki

Proposition 4

Tag 09VZ, permanent. Cite it as [Bourbaki, Tag 09VZ].

Lie Groups and Lie Algebras, chapter I, § 3. Representations

Let M1,M2M_1, M_2 be two gg-modules. The canonical KK-linear mappings (Algebra, Chapter II, § 4, no. 2, Proposition 2 and no. 1, Proposition 1):

M1∗⊗KM2→ϕLK(M1,M2),LK(M1,M2∗)→ψ(M1⊗KM2)∗ M_1^* \otimes_K M_2 \xrightarrow{\phi} \mathscr{L}_K(M_1, M_2), \quad \mathscr{L}_K(M_1, M_2^*) \xrightarrow{\psi} (M_1 \otimes_K M_2)^*

(where the second is bijective) are gg-module homomorphisms.

We write

N=M1∗⊗M2,P=L(M1,M2),Q=L(M1,M2∗),R=(M1⊗M2)∗. N = M_1^* \otimes M_2, \quad P = \mathscr{L}(M_1, M_2), \quad Q = \mathscr{L}(M_1, M_2^*), \quad R = (M_1 \otimes M_2)^*.

Then, for x∈g,f∈M1∗,m1∈M1,m2∈M2x \in g, f \in M_1^*, m_1 \in M_1, m_2 \in M_2,

((ϕxN)(f⊗m2))⋅m1=(ϕ(xM1∗f⊗m2+f⊗xM2m2))⋅m1=⟨xM1∗f,m1⟩m2+⟨f,m1⟩xM2m2 ((\phi x_N)(f \otimes m_2)) \cdot m_1 = (\phi(x_{M_1^*} f \otimes m_2 + f \otimes x_{M_2} m_2)) \cdot m_1 \\ = \langle x_{M_1^*} f, m_1 \rangle m_2 + \langle f, m_1 \rangle x_{M_2} m_2
((xPϕ)(f⊗m2))⋅m1=xM2(ϕ(f⊗m2)⋅m1)−ϕ(f⊗m2)(xM1m1)=⟨f,m1⟩xM2m2−⟨f,xM1m1⟩m2 ((x_P \phi)(f \otimes m_2)) \cdot m_1 = x_{M_2}(\phi(f \otimes m_2) \cdot m_1) - \phi(f \otimes m_2)(x_{M_1} m_1) \\ = \langle f, m_1 \rangle x_{M_2} m_2 - \langle f, x_{M_1} m_1 \rangle m_2

and hence ϕxN=xPϕ\phi x_N = x_P \phi. On the other hand, for x∈g,u∈L(M1,M2∗),m1∈M1,m2∈M2x \in g, u \in \mathscr{L}(M_1, M_2^*), m_1 \in M_1, m_2 \in M_2:

(ψxQu)(m1⊗m2)=⟨(xQu)⋅m1,m2⟩=⟨xM2∗um1−uxM1m1,m2⟩(xRψu)(m1⊗m2)=−⟨ψu,xM1m1⊗m2+m1⊗xM2m2⟩=−⟨uxM1m1,m2⟩−⟨um1,xM2m2⟩ (\psi x_Q u)(m_1 \otimes m_2) = \langle (x_Q u) \cdot m_1, m_2 \rangle = \langle x_{M_2^*} u m_1 - u x_{M_1} m_1, m_2 \rangle \\ (x_R \psi u)(m_1 \otimes m_2) = -\langle \psi u, x_{M_1} m_1 \otimes m_2 + m_1 \otimes x_{M_2} m_2 \rangle \\ = -\langle u x_{M_1} m_1, m_2 \rangle - \langle u m_1, x_{M_2} m_2 \rangle

and hence ψxQ=xRψ\psi x_Q = x_R \psi, which completes the proof.

The gg-modules L(M1,M2∗)\mathscr{L}(M_1, M_2^*) and (M1⊗M2)∗(M_1 \otimes M_2)^* are identified under the isomorphism ψ\psi. If M1M_1 and M2M_2 have finite bases, ϕ\phi is an isomorphism (Algebra, Chapter II, § 4, no. 2, Proposition 2), which allows us to identify the gg-modules M1∗⊗M2M_1^* \otimes M_2 and L(M1,M2)\mathscr{L}(M_1, M_2); in that case, we can therefore identify the gg-modules M1∗⊗M2∗,L(M1,M2∗)M_1^* \otimes M_2^*, \mathscr{L}(M_1, M_2^*) and (M1⊗M2)∗(M_1 \otimes M_2)^*.

Bourbaki puts the proof directly after the statement, so what is above is the statement and its proof together.

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Cites

Transcribed from the printed volume. Not translated.