Quotient vector space and representations Supppose $A$ is an $k$-algebra, and $V$ is a finite dimensional representation of $A$ with subrepresentation $W\subset V$. So in the following short exact sequence, everything is a representation of $A$:
$0 \to W \to V \to V/W \to 0$.
Since every short exact sequence of vector spaces splits, we have a homomorphism $V/W \overset{\phi}{\hookrightarrow} V$. But the image of $\phi$ is not necessarily a subrepresentation unless $V$ decomposes into $V/W \oplus W$ as a representation. Can you explain this incompatibility? Thank you.
 A: An example of such situation is the following. Take $G=\mathbb{Z}$ acting on $V=k^2=Span(x,y)$, where $1\in\mathbb{Z}$ acts as the matrix 
$$\begin{pmatrix}
1 & 1\\
0 & 1
\end{pmatrix}$$.
Then obviously $W=Span(x)$ is a subrepresentation. But it is not true that $V\simeq W\oplus U$ as $G$-modules with $U\simeq V/W$. Assume it is true. Then we can pick a vector of the form $y+cx$ from $U$. Then if we apply $1\in\mathbb{Z}$ to $y+cx$ we get $1.(y+cx)=y+(c+1)x$. By the assumption, $U$ is a submodule in $V$, so $y+(c+1)x$ must also be in $U$. But then since both $y+cx$ and $y+(c+1)x$ are in $U$, we must have $x,y\in U$, which is contradiction. So indeed we don't have an isomorphism of $G$-modules $V\simeq W\oplus V/W$.
Any SES of vector spaces splits because every module over a field (i.e. vector space) is free, and hence projective. This exacly means the splitting of any short exact sequence.
But it is not true for general algebras. Hence you might have the situation when there is no such splitting. Here I am talking about algebras since $G$-representation (or $G$-module) is the same as module over the group algebra $k[G]$. 
A representation $V$ of an algebra $A$ is called semisimple if for any submodule $W\subset V$ there exists submodule $U\subset V$ such that $V\simeq U\oplus W$. For example, any module over a group algebra of a finite group is semisimple. The example above shows that for infinite groups there are modules that are not semisimple.
I hope this helps a bit.
