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

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 $3\le i\le n+3$. We also provide examples which show that the ${\mathcal{K}}_2$ property is not detected by any obvious vanishing patterns among higher multiplications.

Fixed statement of Lemma 2.4 with correct annihilator of b_ic_i. Corrected proof of Lemma 3.1; minor change in last paragraph

References in corpus (2)

• Generalized Koszul properties for augmented algebras
• $d$-Koszul algebras, 2-$d$ determined algebras and 2-$d$-Koszul algebras