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

Abstract

We consider the algebras ei (Q)ei, where (Q) is the deformed preprojective algebra of weight and i is some vertex of Q, in the case where Q is an extended Dynkin diagram and lies on the hyperplane orthogonal to the minimal positive imaginary root . We prove that the center of ei (Q)ei is isomorphic to O (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 ei (Q)ei. We prove that the algebras AP1,...,Pn;μ = Chx1, . . . , xn|Pi(xi) = 0, Pn i=1 xi = μei make a special case of the algebras ec (Q)ec for star-like quivers Q with the origin