On τ-tilting modules over trivial extensions of gentle tree 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 τ-tilting modules over the trivial extension T(A) of a gentle tree algebra A depends only on the number of simple A-modules.