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

Abstract

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 \(c\).