Exercise 3, § 1

Tag 00YG, permanent. Cite it as [Bourbaki, Tag 00YG].

Lie Groups and Lie Algebras, VIII, 219

Let E be a vector space having a basis $(e_n)_{n\in\mathbf{Z}}$ . Let

a(n) =a_0+a_1n, b(n) =b_0+b_1n, c(n) =c_0+c_1n

be three affine functions with coefficients in $k$ . Define endomorphisms $X_+, X_-$ , $H$ of E by the formulas

X_+e_n=a(n)e_{n-1}, X_-e_n=b(n)e_{n+1}, He_n=c(n)e_n

This gives an $\mathfrak{s}$ -module structure on E if and only if:

a_1b_1= 1, c_1=-2, c_0=-a_0b_1-a_1b_0

Under what condition is this $\mathfrak{s}$ -module simple?

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