# $U(\mathfrak g)$ module

Let $$\mathfrak g$$ be a Lie algebra and $$U(\mathfrak g)$$ is the universal enveloping algebra of $$\mathfrak g.$$ Let $$V$$ be a vector space is a unital $$U(\mathfrak g)$$ module. I want to show that the map $$V\to V$$, $$v\mapsto Xv$$ is linear where $$X\in\mathfrak g$$. I can easily show that $$X(v+w)=Xv+Xw$$ which follows from definition. But now how to show that $$X(cv)=cXv$$ for any scalar $$c\in\mathbb C.$$

• What exactly is your definition of what it means for a vector space to be a $U(\mathfrak{g})$-module? By some definitions, $X(cv)=cXv$ would simply be part of the definition. In any case, the definition must include some compatibility between the scalar multiplication of $V$ and the modulle structure, and you should be able to use that compatibility to deduce that $X(cv)=cXv$. – Eric Wofsey May 19 at 16:05