Universal Enveloping Algebras of Poisson Superalgebras

TL;DR

This paper develops the theory of universal enveloping algebras for Poisson superalgebras, establishing new PBW theorems and exploring their properties in the context of Hopf superalgebras and specific classes like Poisson symplectic superalgebras.

Contribution

It introduces a new PBW Theorem for Lie-Rinehart superalgebras and extends it to Poisson superalgebras, also analyzing their universal enveloping algebras in various algebraic contexts.

Findings

Proved a PBW Theorem for Lie-Rinehart superalgebras.

Established that universal enveloping algebras of Poisson Hopf superalgebras are Hopf superalgebras.

Studied universal enveloping algebras for Poisson symplectic superalgebras.

Abstract

In this paper, we define and study the universal enveloping algebra of a Poisson superalgebra. In particular, a new PBW Theorem for Lie-Rinehart superalgebras is proved leading to a PBW Theorem for Poisson superalgebras, we show the universal enveloping algebra of a Poisson Hopf superalgebra (resp. Poisson-Ore extension) is a Hopf superalgebra (resp. iterated Ore extension), and we study the universal enveloping algebra for interesting classes of Poisson superalgebras such as Poisson symplectic superalgebras.