  For a quotient algebra $U$ of the tensor algebra we give explicit conditions
on its relations for $U$ being a PBW-deformation of an $N$-Koszul algebra $A$.
We show there is a one-one correspondence between such deformations and a class
of $A_\infty$-structures on the Yoneda algebra $Ext_A^*(k,k)$ of $A$. We
compute the PBW-deformations of the algebra whose relations are the
anti-symmetrizers of degree $N$ and also of cubic Artin-Schelter algebras.
