Let $k$ be a field of characteristic $0$. For which smooth algebraic groups $G$ over $k$ does the abelian category of linear representations $\mathsf{Rep}_k(G)$ (not assumed to be finite-dimensional) have the property that every monomorphism splits? Is it true when $G$ is reductive? I am looking for references.
PS: There seem to be two definitions of a semisimple abelian category. One says that every object is semisimple, i.e. a direct sum of simple objects. The other says that monomorphisms split. Are these conditions equivalent?
I couldn't find a reference stating explicitly what you want for arbitrary representations, but I think it's true that the category of all representations of a reductive group over a field of characteristic zero is semisimple. The references below are to Milne's coursenotes on algebraic groups.
First, it's enough to show that every representation is a direct sum of simples, since this implies that monomorphisms split by a Zorn's Lemma argument: If $N$ is a subrepresentation of $M$, where $M=\bigoplus_{i\in I}S_i$ is a direct sum of simples, then by Zorn there is a maximal $J\subseteq I$ such that $N\cap\bigoplus_{j\in J}S_j=0$, and then $\bigoplus_{j\in J}S_j$ is a complement to $N$.
Let $G$ be an affine algebraic group over a field of characteristic zero.
By Theorem 14.41 in Milne, every finite-dimensional representation of $G$ is a direct sum of simples if and only if the connected component of the identity in $G$ is reductive.
By Corollary 5.7 in Milne, every representation $M$ of $G$ is a filtered union of finite-dimensional subrepresentations, and is therefore a sum $\sum_{i\in I}S_i$ of simple subrepresentations. By Zorn, there is a maximal $J\subseteq I$ so that the sum $\sum_{j\in J}S_j$ is direct, and then $M=\bigoplus_{j\in J}S_j$.