Higher Koszul algebras and A-infinity algebras

We study a class of A∞-algebras, named (2,p)-algebras, which is related to the class of p-homogeneous algebras, especially to the class of p-Koszul algebras. A general method to construct (2,p)-algebras is given. Koszul dual of a connected graded algebra is defined in terms of A∞-algebra. It is proved that a p-homogeneous algebra A is p-Koszul if and only if the Koszul dual E(A) is a reduced (2,p)-algebra and generated by E1(A). The (2,p)-algebra structure of the Koszul dual E(A) of a p-Koszul algebra A is described explicitly. A necessary and sufficient condition for a p-homogeneous algebra to be a p-Koszul algebra is also given when the higher multiplications on the Koszul dual are ignored.