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

• Veröffentlicht in: Algebra and Discrete Mathematics
• Datum: 2004
• 1. Verfasser: Mellit, A.
• Format: Artikel
• Sprache: Englisch
• Veröffentlicht: Інститут прикладної математики і механіки НАН України 2004
• Online Zugang: https://nasplib.isofts.kiev.ua/handle/123456789/156459
• Назва журналу: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Zitieren: 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.