Piecewise Hereditary algebras of Dynkin and extended Dynkin type

TL;DR

This paper characterizes piecewise hereditary incidence algebras of Dynkin and extended Dynkin types, introduces a new family called ANS, and develops a computational method to identify relevant cutting sets.

Contribution

It provides a detailed description of Phia algebras of Dynkin type and introduces the ANS family for extended Dynkin types, along with a computational tool for analysis.

Findings

Explicit quiver with relations for Dynkin type Phia algebras

Introduction of the ANS family of extended Dynkin type

A computer program to identify cutting sets on trivial extensions

Abstract

We present a study on the description of incidence algebras that are piecewise hereditary, which we denominate Phia algebras. We describe the quiver with relations of the Phia algebras of Dynkin type and introduce a new family of Phia algebras of extended Dynkin type, which we call ANS family, in reference to Assem, Nehring, and Skowro\'nski. In this description, the important method was the one of cutting sets on trivial extensions, inspired by this we made of a computer program which shows exactly the cutting sets on the given trivial extension that result on incidence algebras.