$A_{\infty}$-algebra Structures Associated to $\mathcal{K}_2$-algebras

TL;DR

This paper investigates the $A_{ abla}$-algebra structures on the Yoneda algebra of $ abla$-algebras, revealing that the higher multiplications can be nonzero up to any specified degree, and that the $ abla$-property isn't detectable by simple vanishing patterns.

Contribution

It constructs explicit examples of $ abla$-algebras with nontrivial higher multiplications, demonstrating the complexity of the $ abla$-property beyond obvious vanishing patterns.

Findings

Existence of $ abla$-algebras with nonzero higher multiplications up to degree n+3.

Higher multiplications do not vanish in patterns detectable by simple criteria.

The $ abla$-property cannot be characterized solely by vanishing of certain higher multiplications.

Abstract

The notion of a $\mathcal{K}_2$-algebra was recently introduced by Cassidy and Shelton as a generalization of the notion of a Koszul algebra. The Yoneda algebra of any connected graded algebra admits a canonical $A_{\infty}$-algebra structure. This structure is trivial if the algebra is Koszul. We study the $A_{\infty}$-structure on the Yoneda algebra of a $\mathcal{K}_2$-algebra. For each non-negative integer $n$ we prove the existence of a $\mathcal{K}_2$-algebra $B$ and a canonical $A_{\infty}$-algebra structure on the Yoneda algebra of $B$ such that the higher multiplications $m_i$ are nonzero for all $3 \leq i \leq n+3$. We also provide examples which show that the $\mathcal{K}_2$ property is not detected by any obvious vanishing patterns among higher multiplications.