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

• Published in: Algebra and Discrete Mathematics
• Date: 2004
• Main Author: Mellit, A.
• Format: Article
• Language: English
• Published: Інститут прикладної математики і механіки НАН України 2004
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/156459
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Kleinian singularities and algebras generated by elements that have given spectra and satisfy a scalar sum relation / A. Mellit // Algebra and Discrete Mathematics. — 2004. — Vol. 3, № 3. — С. 89–110. — Бібліогр.: 5 назв. — англ.

We consider the algebras eiΠλ (Q)ei , where Πλ (Q) is the deformed preprojective algebra of weight λ and i is some vertex of Q, in the case where 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 eiΠλ (Q)ei is isomorphic to Oλ (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 eiΠλ (Q)ei . We prove that the algebras AP₁,...,Pn;µ = Chx₁, . . . , xn|Pi(xi) = 0, Pn i=1 x₁ = µei make a special case of the algebras ecΠλ (Q)ec for star-like quivers Q with the origin c.