A∞-algebra structures associated to K2 algebras

The notion of a K2-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∞-algebra structure. This structure is trivial if the algebra is Koszul. We study the A∞-structure on the Yoneda algebra of a K2-algebra. For each non-negative integer n we prove the existence of a K2-algebra B and a canonical A∞-algebra structure on the Yoneda algebra of B such that the higher multiplications mi are nonzero for all 3⩽i⩽n+3. We also provide examples which show that the K2 property is not detected by any obvious vanishing patterns among higher multiplications.