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

Junyang Liu, Zhengfang Wang

TL;DR

The paper proves that zigzag algebras $Z(\Gamma)$ of finite trees are intrinsically formal if and only if $\Gamma$ is of type $ADE$ and $\mathrm{char}(\Bbbk)$ is not bad; it also completes the $E$-type case for arbitrary characteristic. It achieves this via an algebraic route using bigraded Hochschild cohomology and a derived Koszul duality between $Z(\Gamma)$ and the $2$-dimensional Ginzburg dg algebra $\Pi_2(Q)$, yielding a key isomorphism $\mathrm{HH}^{2,q}(B,B) \cong (\Lambda_Q/[\Lambda_Q,\Lambda_Q])^{q+2}$. The analysis reduces formality questions to vanishing of $\mathrm{HH}^{2,q}$ for $q>0$ in the ADE/bad-characteristic regime, while nonvanishing in non-ADE cases provides nontrivial $A_\infty$-deformations. This work ties intrinsic formality to the structure of preprojective algebras and Calabi–Yau dualities, with implications for representation theory and symplectic geometry.

Abstract

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.

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

TL;DR

The paper proves that zigzag algebras of finite trees are intrinsically formal if and only if is of type and is not bad; it also completes the -type case for arbitrary characteristic. It achieves this via an algebraic route using bigraded Hochschild cohomology and a derived Koszul duality between and the -dimensional Ginzburg dg algebra , yielding a key isomorphism . The analysis reduces formality questions to vanishing of for in the ADE/bad-characteristic regime, while nonvanishing in non-ADE cases provides nontrivial -deformations. This work ties intrinsic formality to the structure of preprojective algebras and Calabi–Yau dualities, with implications for representation theory and symplectic geometry.

Abstract

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.
Paper Structure (11 sections, 6 theorems, 24 equations)

This paper contains 11 sections, 6 theorems, 24 equations.

Table of Contents

  1. Introduction
  2. Bigraded Hochschild cohomology
  3. $2$-dimensional Ginzburg dg algebras and Hochschild cohomology
  4. $2$-dimensional Ginzburg dg algebras
  5. A small cofibrant resolution
  6. Hochschild cohomology of $2$-dimensional Ginzburg dg algebras
  7. $A_\infty$-deformations of zigzag algebras
  8. $A_\infty$-deformations
  9. Zigzag algebras
  10. Derived Koszul duality between zigzag algebras and $2$-dimensional Ginzburg dg algebras
  11. The proof

Key Result

Theorem 1.1

Let $\Bbbk$ be a field and $\Gamma$ a finite tree. Then the zigzag algebra $Z(\Gamma)$, which is graded by path length, is intrinsically formal if and only if $\Gamma$ is of type ADE and the characteristic of $\Bbbk$ is not $\mathrm{bad}$ ($2$ for type $D$, $2$ or $3$ for $E_6$ and $E_7$, and $2$, $

Theorems & Definitions (12)

  • Theorem 1.1
  • Proposition 3.1: Kel11
  • Proposition 3.2
  • proof
  • Corollary 3.4
  • proof
  • Definition 4.1
  • Theorem 4.2: Kad, ST and RW
  • Definition 4.3: HK
  • Proposition 4.4: EL and Kel03
  • ...and 2 more