PBW Property for Associative Universal Enveloping Algebras over an Operad

Given a symmetric operad P and a P-Algebra V, the associative universal enveloping algebra UP 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 P is Koszul a criterion for the PBW property is found. A necessary condition on the Hilbert series for P is discovered. Moreover, given any symmetric operad P, together with a Gröbner 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.