# PBW-basis for universal enveloping algebras of differential graded Poisson algebras

## TL;DR

This paper develops a PBW-basis for the universal enveloping algebra of differential graded Poisson algebras, providing explicit formulas and applications to DG symplectic ideals and modules.

## Contribution

It introduces a formula for the universal enveloping algebra of DG Poisson algebras and proves a PBW-basis exists for graded commutative polynomial cases.

## Key findings

- Established a PBW-basis for $A^e$ when $A$ is a graded commutative polynomial algebra.
- Provided a formula for computing the universal enveloping algebra of DG Poisson algebras.
- Applied the PBW-basis to characterize DG symplectic ideals as annihilators of simple modules.

## Abstract

For any differential graded (DG for short) Poisson algebra $A$ given by generators and relations, we give a "formula" for computing the universal enveloping algebra $A^e$ of $A$. Moreover, we prove that $A^e$ has a Poincar\'e-Birkhoff-Witt basis provided that $A$ is a graded commutative polynomial algebra. As an application of the PBW-basis, we show that a DG symplectic ideal of a DG Poisson algebra $A$ is the annihilator of a simple DG Poisson $A$-module, where $A$ is the DG Poisson homomorphic image of a DG Poisson algebra $R$ whose underlying algebra structure is a graded commutative polynomial algebra.