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Extended canonical algebras and Fuchsian singularities

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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.

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Authors and Affiliations

  1. Institut für Mathematik, Universität Paderborn, 33095, Paderborn, Germany

    Helmut Lenzing

  2. Instituto de Matemáticas, Universidad Nacional Autónoma de México, Ciudad Universitaria, Mexico, 04510, DF, Mexico

    José A. de la Peña

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  1. Helmut Lenzing
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  2. José A. de la Peña
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Correspondence to José A. de la Peña.

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Lenzing, H., de la Peña, J.A. Extended canonical algebras and Fuchsian singularities. Math. Z. 268, 143–167 (2011). https://doi.org/10.1007/s00209-010-0663-z

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