Extended canonical algebras and Fuchsian singularities
By H. Lenzing, José Antonio de la Peña. 2006
66 citations recorded by the source corpus.
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.
Subfields: Algebra and Number Theory, Geometry and Topology, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/4001bc80c8c00b1fc859165d9dccd4ba2b749003
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Algebra and Number Theory
In Geometry and Topology
In Statistical and Nonlinear Physics
Citations in the corpus
- Triangle singularities, ADE-chains, and weighted projective lines (2012, background): “〈a, b, c|1〉: denotes the type of the extended canonical algebra in the sense of [45].”