PBW-deformations and deformations \`a la Gerstenhaber of N-Koszul algebras

TL;DR

This paper explores the connection between Gerstenhaber deformations, Hochschild cohomology, and PBW-deformations of N-Koszul algebras, providing a cohomological perspective and recovering key theorems in the field.

Contribution

It establishes an explicit cohomological link between classical Gerstenhaber deformations and PBW-deformations of homogeneous algebras, enhancing understanding of their structure.

Findings

Revealed a cohomological framework linking Gerstenhaber deformations and PBW-deformations.

Recovered a theorem by Berger and Ginzburg on PBW property conditions.

Provided explicit criteria for filtered algebras to satisfy PBW conditions.

Abstract

In this article we establish an explicit link between the classical theory of deformations \`a la Gerstenhaber -- and a fortiori with the Hochschild cohomology-- and (weak) PBW-deformations of homogeneous algebras. Our point of view is of cohomological nature. As a consequence, we recover a theorem by R. Berger and V. Ginzburg, which gives a precise condition for a filtered algebra to satisfy the so-called PBW property, under certain assumptions.