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 Poincaré-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. 1 2 XIANGUO HU, JIAFENG LÜ * , AND XINGTING WANG particular, we construct the universal enveloping algebra of any DG Poisson algebra given by generators and relations. Section 3 is devoted to the study of PBW-basis for universal enveloping algebras. To be specific, for a DG Poisson algebra R whose underlying algebra structure is a graded commutative polynomial algebra, by using Gröbner-Shirshov basis theory developed in [8] and [9], we prove that the universal enveloping algebra R e has a PBW-basis, which is analogous to the PBW-basis for universal enveloping algebras of Lie algebras. In the last section, we focus on the simple DG Poisson module. As an application of the PBW-basis theorem for universal enveloping algebras of DG Poisson algebras, we prove 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 R.Throughout the whole paper, Z denotes the set of integers, k denotes a base field and everything is over k unless otherwise stated, all (graded) algebras are assumed to have an identity and all (graded) modules are assumed to be unitary. Universal enveloping algebras of differential graded Poisson algebrasIn this section, first we briefly review some basic definitions and properties of DG Poisson algebras and universal enveloping algebras, then we construct the universal enveloping algebra of any DG Poisson algebra A given by generators and relations.2.1. DG Poisson algebras. By a graded algebra we mean a Z-graded algebra. A DG algebra is a graded algebra with a k-linear homogeneous map d : A → A of degree 1, which is also a graded derivation. Any graded algebra can be viewed as a DG algebra with differential d = 0; in this case it is called a DG algebra with trivial differential. Let A, B be two DG algebras and f : A → B be a graded algebra map of degree zero. Then f is called a DG algebra map if f commutes with the differentials. Definition 2.1. Let A be a graded k-vector space. If there is a k-linear map {·, ·} : A ⊗ A → A of degree 0 such that: (i) (graded antisymmetry): {a, b} = −(−1) |a||b| {b, a}; (ii) (graded Jacobi identity): {a, {b, c}} = {{a, b}, c} + (−1) |a||b| {b, {a, c}}, for any homogeneous elements a, b, c ∈ A, then (A, {·, ·}) is called a graded Lie algebra. Definition 2.2. [12] Let (A, ·) be a graded k-algebra. If there is a k-linear map {·, ·} : A ⊗ A → A of degree 0 such that (i) (A, {·, ·}) is a graded Lie a...

On differential lattices

“…The operad of differential associative algebras was studied in [24]. Furthermore, differential graded Poisson algebras have been studied [18,25]. Under the nilpotent condition d 2 = 0, derivations play an essential role in homological theories [32].…”