Algebras Generated by Elements with given Spectrum and Scalar Sum and Kleinian Singularities

We consider algebras eiΠ (Q)ei where Π (Q) is the deformed preprojective algebra of weight λ and i is some vertex of Q in the case when Q is an extended Dynkin diagram and λ lies on the hyperplane orthogonal to the minimal positive imaginary root δ. We prove that the center of eiΠ (Q)ei is isomorphic to O (Q) which is a deformation of coordinate ring of Kleinian singularity which corresponds to Q. Also we find the minimal k for which the standard identity of degree k holds in eiΠ (Q)ei. We prove that algebras AP1,...,Pn;μ = C〈x1, . . . , xn|Pi(xi) = 0, ∑n i=1 xi = μe〉 are the special case of algebras ecΠ (Q)ec for star-like quivers Q with origin c. Introduction Consider the problem of description of n-tuples of hermitian operators {Ai} in a Hilbert space satisfying given restrictions on spectra σ(Ai) ⊂ Mi with Mi ⊂ R finite and relation ∑n i=1 Ai = μI, with I the identity and μ ∈ R. Study of such n-tuples is equivalent to study of *-representations of certain *-algebra. Forgetting the *-structure we arrive to the following class of algebras. Definition 1. Let P1, . . . , Pn be complex polynomials in one variable and μ ∈ C. We put inessential restriction Pi(0) = 0. Define algebra AP1,...,Pn;μ = C〈x1, . . . , xn|Pi(xi) = 0 (i = 1, . . . , n), n ∑