Extremal properties of concealed-canonical algebras

TL;DR

This paper investigates the extremal properties of canonical algebras, revealing their unique characteristics among endomorphism rings of tilting bundles on weighted projective lines, and explores related antipodal classes.

Contribution

It characterizes canonical algebras by extremal properties and provides new insights into the nature of concealed-canonical algebras, addressing why canonical algebras are considered canonical.

Findings

Canonical algebras are uniquely characterized by extremal properties.

The study identifies properties of antipodal classes of these algebras.

Insights into the fundamental nature of concealed-canonical algebras.

Abstract

Canonical algebras, introduced by C.M. Ringel in 1984, play an important role in the representation theory of finite dimensional algebras. They are equipped with a large contact surface to many further mathematical subjects like function theory, 3-manifolds, singularity theory, commutative algebra and algebraic geometry. We show in this paper that canonical algebras are characterized by a number of interesting extremal properties (among the class of endomorphism rings of tilting bundles on a weighted projective line). We also study the corresponding class of algebras antipodal to canonical ones. Our study sheds new insight in the nature of concealed-canonical algebras, and sheds light on an old question: Why are the canonical algebras canonical?