Distributive lattices of tilting modules and support $\tau$-tilting modules over path algebras

TL;DR

This paper characterizes when the posets of basic tilting and support $ au$-tilting modules over path algebras of Dynkin quivers form distributive lattices, revealing specific conditions on quiver types and orientations.

Contribution

It provides a complete classification of when these posets are distributive lattices based on the quiver type and orientation, extending understanding of module categories over path algebras.

Findings

Poset of basic tilting modules is a distributive lattice iff Q is of type A with a nonlinear orientation.

Poset of support $ au$-tilting modules is a distributive lattice iff Q is of type A.

Characterizes the structure of tilting modules over Dynkin quivers.

Abstract

In this paper we study the poset of basic tilting $kQ$-modules when $Q$ is a Dynkin quiver, and the poset of basic support $\tau$-tilting $kQ$-modules when $Q$ is a connected acyclic quiver respectively. It is shown that the first poset is a distributive lattice if and only if $Q$ is of types $\mathbb{A}_{1}, \mathbb{A}_{2}$ or $\mathbb{A}_{3}$ with a nonlinear orientation and the second poset is a distributive lattice if and only if $Q$ is of type $\mathbb{A}_{1}$.