Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$. 20a. Show that there exists a unique algebra morphism $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ such that $$\forall \phi \in \mathcal{U}_q, \quad \Psi_a(\phi) \circ P_a = P_a \circ \phi$$ 20b. Show that $\phi \in \mathcal{U}_q$ is contained in the kernel of $\Psi_a$ if and only if the image of $\phi$ is in the subspace of $V$ spanned by the vectors $v_i - a^p v_r, i \in \mathbf{Z}$, where $i = p\ell + r$ is the Euclidean division of $i$ by $\ell$.
Let $\mathcal{U}_q$ be the set of endomorphisms $\phi \in \mathcal{L}(V)$ that are compatible with $P_a$.
20a. Show that there exists a unique algebra morphism $\Psi_a : \mathcal{U}_q \rightarrow \mathcal{L}(W_{\ell})$ such that
$$\forall \phi \in \mathcal{U}_q, \quad \Psi_a(\phi) \circ P_a = P_a \circ \phi$$
20b. Show that $\phi \in \mathcal{U}_q$ is contained in the kernel of $\Psi_a$ if and only if the image of $\phi$ is in the subspace of $V$ spanned by the vectors $v_i - a^p v_r, i \in \mathbf{Z}$, where $i = p\ell + r$ is the Euclidean division of $i$ by $\ell$.