$A_{\mathrm{\infty }}$-deformations of zigzag algebras via Ginzburg dg algebras

This note aims to give a short proof of the recent result due to Etgü-Lekili (2017) and Lekili-Ueda (2021): the zigzag algebra of any finite tree over a field of characteristic 0 is intrinsically formal if and only if the tree is of type ADE. We also complete the proof of this result by considering a field of arbitrary characteristic for type E, which was still open.