Algebras sharing the same support $\tau$-tilting poset with tree quiver algebras

TL;DR

This paper characterizes finite-dimensional algebras sharing the same support τ-tilting poset as tree quiver algebras, extending classical results on hereditary algebras to a broader class.

Contribution

It provides a full characterization of algebras with support τ-tilting posets identical to those of tree quiver algebras, generalizing previous hereditary algebra results.

Findings

Identified conditions for algebras to share support τ-tilting posets with tree quiver algebras

Extended Happel-Unger's reconstruction results beyond hereditary algebras

Characterized a class of finite-dimensional algebras with identical support τ-tilting posets

Abstract

Happel and Unger reconstructed hereditary algebras from their posets of tilting modules. Inspired by this result, we try removing the assumption to be hereditary. However, it would be unfortunately fail in general: e.g. every selfinjective algebra has the poset consisting of only one point. Therefore, we should consider a generalization of the Happel-Unger's result for posets of support $\tau$-tilting modules, which contains those of tilting modules. In this paper, we spotlight finite dimensional algebras whose support $\tau$-tilting posets coincide with those of tree quiver algebras and give a full characterization of such algebras.