PBW-Basis for Universal Enveloping Algebras of Differential Graded Poisson Algebras

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 Poincare–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.