Extended canonical algebras and Fuchsian singularities

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 ?? 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.