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

arXiv comment13 pages, accepted for publication in Kodai Mathematical Journal

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