Classification silted algebras for a quiver of Dynkin type $\mathbb{A}_{n}$ via geometric models

TL;DR

This paper classifies silted algebras of Dynkin type A_n using geometric models, showing they are either tilted of type A_n or of a product type, and provides a formula for their count.

Contribution

It offers a complete classification of silted algebras of type A_n via geometric models and characterizes them as tilted or product-tilted algebras, including a counting formula.

Findings

Classification of silted algebras as tilted or product-tilted

Complete geometric model description for type A_n

Explicit formula for counting silted algebras

Abstract

Let $\Q$ be the quiver of Dynkin type $\mathbb{A}_n$ with linear orientation and $A_{n}=k\Q$. In this paper, we give a complete classification of the silted algebras of type $A_{n}$ by using the geometric models of gentle algebras. We show that any finite-dimensional algebra is a silted of type $A_{n}$ if and only if it is a tilted of type $A_{n}$ or a tilted algebra of type $A_{m}\times A_{n-m}$ for any positive integer $1\leq m\leq n-1$. Based on the classification, we obtain a formula for computing the number of silted algebras of type $A_{n}$.