Universal enveloping algebras of weighted differential Poisson algebras

TL;DR

This paper introduces $ ext{lambda}$-differential Poisson algebras as a generalization of differential Poisson algebras, explores their properties, and constructs their universal enveloping algebras using a $ ext{P}$-triple framework.

Contribution

It generalizes differential Poisson algebras to $ ext{lambda}$-differential versions, establishes their algebraic properties, and constructs universal enveloping algebras with new isomorphisms.

Findings

$ ext{lambda}$-DP algebras are closed under tensor product

A $ ext{lambda}$-DP algebra structure exists on cohomology algebra

Universal enveloping algebras are characterized by a $ ext{P}$-triple

Abstract

The $\lambda$-differential operators and modified $\lambda$-differential operators are generalizations of classical differential operators. This paper introduces the notions of $\lambda$-differential Poisson ($\lambda$-DP for short) algebras and modified $\lambda$-differential Poisson ($\lambda$-mDP for short) algebras as generalizations of differential Poisson algebras. The $\lambda$-DP algebra is proved to be closed under tensor product, and a $\lambda$-DP algebra structure is provided on the cohomology algebra of the $\lambda$-DP algebra. These conclusions are also applied to $\lambda$-mDP algebras and their modules. Finally, the universal enveloping algebras of $\lambda$-DP algebras are generalized by constructing a $\mathcal{P}$-triple. Three isomorphisms among opposite algebras, tensor algebras and the universal enveloping algebras of $\lambda$-DP algebras are obtained.