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 Grobner 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.

中文翻译：

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操作数上关联通用包络代数的 PBW 属性

给定对称操作数 $\mathcal{P}$ 和 $\mathcal{P}$-代数 $V$，结合泛包络代数 ${\mathsf{U}_{\mathcal{P}}}$ 是其模范畴同构于$V$-模的阿贝尔范畴的结合代数。我们研究了操作数上通用包络代数的 PBW 属性的概念。如果 $\mathcal{P}$ 是 Koszul，则可以找到 PBW 属性的标准。发现了 $\mathcal{P}$ 的希尔伯特级数的一个必要条件。此外，给定任何对称操作数 $\mathcal{P}$，连同 Grobner 基 $G$，根据与 $G$ 的前导单项式相关联的底层树的结构给出条件，足以满足 PBW 属性举行。提供了示例。