PBW property for associative universal enveloping algebras over an operad

TL;DR

This paper investigates the PBW property for associative universal enveloping algebras over operads, providing criteria and conditions for when this property holds, especially in the Koszul case and using Gr"obner bases.

Contribution

It establishes new criteria for the PBW property over operads, including a Koszul case criterion and a tree-structure condition via Gr"obner bases.

Findings

A criterion for PBW property when al P is Koszul.

A necessary condition on the Hilbert series of al P.

A sufficient condition involving Grner basis trees.

Abstract

Given a symmetric operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $V$, the associative universal enveloping algebra ${\mathsf{U}_{\mathcal{P}}}$ is an associative algebra whose category of modules is isomorphic to the abelian category of $V$-modules. We study the notion of PBW property for universal enveloping algebras over an operad. In case $\mathcal{P}$ is Koszul a criterion for the PBW property is found. A necessary condition on the Hilbert series for $\mathcal{P}$ is discovered. Moreover, given any symmetric operad $\mathcal{P}$, together with a Gr\"obner basis $G$, a condition is given in terms of the structure of the underlying trees associated with leading monomials of $G$, sufficient for the PBW property to hold. Examples are provided.