I am wondering if somebody could help me with this task. I have some problems with the theory of representations.
Let $\rho \colon \mathbb{R} \rightarrow \mathrm{GL}(2,\mathbb{R})$, $\rho(a) = \left( {\begin{array}{cc} 1 & a \\ 0 & 1 \\ \end{array} } \right) $.
1) Check that this is a representation of the group $\mathbb{R}$.
2) Is it irreducible?
3) Is it indecomposable ?
Thanks in advance!
Since you did the first point, I shall say nothing about that. It is not irreducible, because $V=\left\{(x,0)\,\middle|\,x\in\mathbb{R}\right\}$ is a stable subspace which is neither $\{(0,0)\}$ nor $\mathbb{R}^2$. But it is indecomposable, since there is no stable subspace $W$ such that $\mathbb{R}^2=V\oplus W$.