Kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation

We consider the algebras $e_i\Pi^\lambda(Q)e_i$, where $\Pi^\lambda(Q)$ is the deformed preprojective algebra of weight $\lambda$ and $i$ is some vertex of $Q$, in the case where $Q$ is an extended Dynkin diagram and $\lambda$ lies on the hyperplane orthogonal to the minimal positive imaginary root $\delta$. We prove that the center of $e_i\Pi^\lambda(Q) e_i$ is isomorphic to $\mathcal{O}^\lambda(Q)$, a deformation of the coordinate ring of the Kleinian singularity that corresponds to $Q$. We also find a minimal $k$ for which a standard identity of degree $k$ holds in $e_i\Pi^\lambda(Q) e_i$. We prove that the algebras $A_{P_1,\dots,P_n;\mu}=\mathbb{C}\langle x_1,\dots, x_n | P_i(x_i)=0,\sum_{i=1}^n x_i=\mu e\rangle$ make a special case of the algebras $e_c \Pi^\lambda(Q) e_c$ for star-like quivers $Q$ with the origin $c$.