0
$\begingroup$

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!

$\endgroup$
  • $\begingroup$ 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
1
$\begingroup$

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

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.