= Action map of a Lie algebra representation
{title2=$a:\mathfrak g\otimes V\to V,\quad x\otimes v\mapsto xv$}
Give $\mathfrak g$ its <Adjoint representation>. The representation identity gives $a(y\cdot(x\otimes v))=[y,x]v+xyv=y(xv)$, so the action map is a <Lie algebra representation homomorphism>. For a nontrivial simple module it is surjective. If the acting algebra is semisimple and the module is finite dimensional, <Weyl complete reducibility theorem> splits the surjection and makes $V$ a direct summand.
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