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!

  • $\begingroup$ What did you do? $\endgroup$ – José Carlos Santos Apr 28 '18 at 21:33
  • $\begingroup$ The first point I did. But how to check 2 and 3 I don't understand. $\endgroup$ – gdmkr Apr 28 '18 at 21:36
  • $\begingroup$ Do you mean a representation of $\mathbb{R}$? $\endgroup$ – leibnewtz Apr 28 '18 at 21:42
  • $\begingroup$ Yes, sorry, it was a typo $\endgroup$ – gdmkr Apr 28 '18 at 21:49

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 $\mathbb{R}^2=V\oplus W$.


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