Proposition 5

Let E, F be finite dimensional $\mathfrak{s}\mathfrak{l}(2, k)$-modules, and let $f\in$ Hom$_k(E,F)$. The following conditions are equivalent:

(i) $f$ is a homomorphism of $\mathfrak{s}\mathfrak{l}(2, k)$-modules;

(ii) $f$ is a homomorphism of $\mathbf{S}\mathbf{L}(2, k)$-modules.

Condition (i) means that $f$ is an invariant element of the $\mathfrak{s}\mathfrak{l}(2, k$)-module Hom$_k(E,F)$, and condition (ii) means that $f$ is an invariant element of the $\mathbf{S}\mathbf{L}(2, k$)-module Hom$_k(E,F)$. Since these module structures are associated by Chap. VII, §5, no. 4, Lemma 1 (iii), the proposition follows from Th. 2 (iii).