Nonsemisimple translation representation of the infinite cyclic group
= Nonsemisimple translation representation of the infinite cyclic group
On polynomials of degree at most one, let the generator of $\mathbb Z$ act by $p(x)\mapsto p(x+1)$. In the basis $(1,x)$ its matrix is
$$
\begin{pmatrix}1&1\\0&1\end{pmatrix}.
$$
This nontrivial <Jordan block> is not diagonalizable, so the representation is not a direct sum of one-dimensional subrepresentations.