On $τ$-tilting modules over trivial extensions of gentle tree algebras
Qi Wang, Yingying Zhang
TL;DR
This work addresses when trivial extensions $T(A)$ of gentle algebras yield Brauer graph algebras and how many support $\tau$-tilting modules arise. It leverages the correspondence $T(A)\cong B_{\Gamma_A}$ from Schroll to transfer questions to Brauer graph algebras and applies known results for Brauer trees, lines, stars, and cycles. The main contributions are (i) a characterization that $A$ is a gentle tree algebra if and only if $T(A)$ is a Brauer tree algebra without an exceptional vertex, (ii) a precise classification of when $T(A)$ is a Brauer line or Brauer star and the corresponding structural conditions on $A$, and (iii) a classification for Brauer cycle outcomes with exact counts, such as $\#\mathsf{s\tau-tilt}\,T(A)=\binom{2n}{n}$ for $n$ simples and $2^{2n-1}$ in the odd-cycle case when rad$^2 A=0$. These results illuminate the deep links between gentle and Brauer graph algebras within $\tau$-tilting theory and yield orientation-independent counts in key families.
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 $τ$-tilting modules over the trivial extension $T(A)$ of a gentle tree algebra $A$ depends only on the number of simple $A$-modules.