$A_\infty$-deformations of zigzag algebras via Ginzburg dg algebras
Junyang Liu, Zhengfang Wang
TL;DR
This paper provides a concise proof that zigzag algebras of finite trees are intrinsically formal only for ADE types, extending the result to fields of arbitrary characteristic for type E.
Contribution
It offers a simplified proof of the intrinsic formality of zigzag algebras and completes the proof for type E over arbitrary fields.
Findings
Zigzag algebras of ADE trees are intrinsically formal.
Formality depends on the tree being of ADE type.
The result holds over fields of any characteristic for type E.
Abstract
This note aims to give a short proof of the recent result due to Etg\"u-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.
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Taxonomy
TopicsAdvanced Topics in Algebra · Algebraic structures and combinatorial models · Advanced Operator Algebra Research