1. Context
In one of my exercise classes I am asked to do the following:
"Verify that Lie algebras (resp. associative algebras) over a field F form an F-linear category LieAlgF (resp. AssocAlgF) with algebra morphisms (resp. Lie algebra morphisms) as morphisms in the category."
2. My thoughts
If you define the addition (scalar multiplication) of morphisms via pointwise addition (scalar multiplication) the respective hom-spaces are not vector spaces, however.
That is, these operations are not defined, i.e. the sum of (Lie) algebra morphisms is not necessarily a (Lie) algebra morphism. Similarly, a (Lie) algebra morphism mutiplied by a scalar is not necessarily a (Lie) algebra morphism.
For instance: Let $\mathfrak{g}$ be a non-abelian Lie algebra over a field $\mathbb{F}$ with more than two elements. Consider the Lie algebra morphism $id: \mathfrak{g}\rightarrow \mathfrak{g}$. Let $x,y \in \mathfrak{g}$ such that $[x,y] \neq 0$ and $\lambda \in \mathbb{F} \setminus \{0, 1\}$. Then $[\lambda id(x), \lambda id(y)] =\lambda ^2 \cdot[x,y] \neq \lambda \cdot[x,y] $.
3. Questions
Am I overlooking something here? Is there another way to define addition and scalar multiplication on the hom-spaces so that the respective categories becomes $k$-linear?