Given a symmetric operad $\mathcal{P}$ and a $\mathcal{P}$-algebra $V$, the 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 PBW property is found. Necessary condition on Hilbert series for $\mathcal{P}$ is found. Moreover, given any symmetric operad $\mathcal{P}$ together with a Gr\"obner basis $G$, a condition is given on the structure of the underlying trees associated with leading monomials of $G$ sufficient for the PBW property to hold. Examples are provided. 查看全文>>