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- Title
Extended canonical algebras and Fuchsian singularities.
- Authors
Lenzing, Helmut; de la Peña, José A.
- Abstract
The authors introduce a new class of finite dimensional algebras called extended canonical, and determine the shape of their derived categories. Extended canonical algebras arise from a canonical algebra Λ by onepoint extension or coextension by an indecomposable projective module. Our main results concern the case of negative Euler characteristic of the corresponding weighted projective line $${\mathbb{X}}$$; more specifically we establish, for a base field of arbitrary characteristic, a link to the Fuchsian singularity R of $${\mathbb{X}}$$ which for the base field of complex numbers is isomorphic to an algebra of automorphic forms. By means of a recent result of Orlov we show that the triangulated category of the graded singularities of R (in the sense of Buchweitz and Orlov) admits a tilting object whose endomorphism ring is the corresponding extended canonical algebra. Of particular interest are those cases where the attached Coxeter transformation has spectral radius one. A K-theoretic analysis then shows that this happens exactly for 38 cases including Arnold's 14 exceptional unimodal singularities. The paper is related to recent independent work by Kajiura, Saito and Takahashi.
- Subjects
ALGEBRA; AUTOMORPHIC functions; MATHEMATICAL singularities; COMPLEX numbers; ENDOMORPHISMS; COXETER groups
- Publication
Mathematische Zeitschrift, 2011, Vol 268, Issue 1/2, p143
- ISSN
0025-5874
- Publication type
Article
- DOI
10.1007/s00209-010-0663-z