First, such an algebra can be presented by generators and relations using homogeneous relators of degree N, and it is natural to consider those algebras which are obtained by changing the relators by adding to them terms of lower degree. Among these, there is a subclass of algebras characterized by a certain Poincaré-Birkhoff-Witt propery, which is particularly interesting in view of their applications. These deformations have studied by R. Berger and V. Ginzburg [BG] under the name of PBW-deformations, generalizing previous work of A. Braverman and D. Gaitsgory for the quadratic case. In particular, they were able to exhibit a sequence of conditions on non-homogeneous deformations, generalizing the Jacobi condition of Lie algebras, which are equivalent to the PBW property in the general situation.

On the other hand, we have the general theory of formal deformations of algebras as initialted by M. Gerstenhaber [G] in terms of Hochschild cohomology.

The purpose of the talk is to present a result showing the equivalence of these two notions in the case of N-Koszul algebras.

I will recall the general definition of N-Koszul algebras, introduce with some detail the two notions of deformations which are involved in the result, discuss examples and explain some of the details of the proof.

This is joint work with A. Solotar and E. Herscovich.

[BG] Berger, R.; Ginzburg, V. Higher symplectic reflection algebras and non-homogeneous N-Koszul property. J. of Algebra 304 (2006), 577-601.

[G] Gerstenhaber, M. On the deformation of rings and algebras, Ann. of Math. (2) 79 (1964), 59-103.

[BG] Braverman, A.; Gaitsgory, D. The Poincare-Birkhoff-Witt theorem for quadratic algebras of Koszul type. J. of Algebra 181 (1996), 315-328.