On $\tau$-tilting modules over trivial extensions of gentle tree algebras

TL;DR

This paper characterizes trivial extensions of gentle tree algebras as Brauer tree algebras without exceptional vertices and relates their support τ-tilting modules to the number of simple modules.

Contribution

It provides a complete classification of trivial extensions of gentle tree algebras as specific Brauer tree algebras and links their support τ-tilting modules to simple module counts.

Findings

Trivial extensions of gentle tree algebras are exactly Brauer tree algebras without exceptional vertices.

The number of support τ-tilting modules over T(A) depends only on the number of simple A-modules.

Characterizations for when trivial extensions are Brauer line, star, or cycle algebras.

Abstract

We show that trivial extensions of gentle tree algebras are exactly Brauer tree algebras without exceptional vertex. We also give a characterization for the algebras whose trivial extensions are Brauer line/star/cycle algebras. As a consequence, the number of support $\tau$-tilting modules over the trivial extension $T(A)$ of a gentle tree algebra $A$ depends only on the number of simple $A$-modules.