. I will go back to this example in Part III.

Abstract. This is a joint work with Victor Ginzburg [4] in which we study a class of associative algebras associated to finite groups acting on a vector space. These algebras are non-homogeneous N-Koszul algebra generalizations of sym-plectic reflection algebras. We realize the extension of the N-Koszul property to non-homogeneous algebras through a Poincaré-Birkhoff-Witt property. PART I- HOMOGENEOUS N-KOSZUL ALGEBRAS I introduced these algebras in [2]. These algebras extend classic Koszul algebras (Priddy, 1970) corresponding to N = 2. A natural question is: why higher N ’s? I list below four answers. 1. There are some relevant examples coming from- noncommutative projective algebraic geometry: cubic Artin-Schelter regular algebras [1] of global dimension 3, as A = C〈x, y〉 (ay2x+ byxy + axy2 + cx3, x ↔ y) where the second relation is obtained from the first one by exchanging x and y. The two generators x and y have degree one, and the two relations are cubic. Artin-Schelter regular algebras are noncommutative analogues of polynomial rings which are used to make noncommutative projective algebraic geometry in sense of M. Artin and J. Zhang.- representation theory: skew-symmetrizer killing algebras (introduced in [2])