Nakayama algebras and Fuchsian singularities

TL;DR

This paper studies Nakayama algebras, showing their realization as endomorphism algebras of tilting objects in derived categories, classifies those of Fuchsian type, and offers a new proof for piecewise hereditary cases.

Contribution

It classifies Nakayama algebras of Fuchsian type and provides a novel proof for Nakayama algebras of piecewise hereditary type.

Findings

Certain Nakayama algebras are realized as endomorphism algebras of tilting objects.

Classified all Nakayama algebras of Fuchsian type.

Provided a new proof for Nakayama algebras of piecewise hereditary type.

Abstract

This present paper is devoted to the study of a class of Nakayama algebras $N_n(r)$ given by the path algebra of the equioriented quiver $\mathbb{A}_n$ subject to the nilpotency degree $r$ for each sequence of $r$ consecutive arrows. We show that the Nakayama algebras $N_n(r)$ for certain pairs $(n,r)$ can be realized as endomorphism algebras of tilting objects in the bounded derived category of coherent sheaves over a weighted projective line, or in its stable category of vector bundles. Moreover, we classify all the Nakayama algebras $N_n(r)$ of Fuchsian type, that is, derived equivalent to the bounded derived categories of extended canonical algebras. We also provide a new way to prove the classification result on Nakayama algebras of piecewise hereditary type, which have been done by Happel--Seidel before.