Poisson-like cohomologies associated with some Lie superalgebras

TL;DR

This paper generalizes Poisson cohomology to Lie superalgebras, linking it with de Rham cohomology and introducing a new perspective on Euler numbers in superalgebra contexts.

Contribution

It introduces Poisson-like cohomology groups for Lie superalgebras, extending classical Poisson cohomology concepts and relating them to de Rham cohomology.

Findings

Poisson-like cohomology groups generalize classical Poisson cohomology.

De Rham cohomology matches Poisson-like cohomology for differential forms.

New insights into Euler numbers within Lie superalgebra frameworks.

Abstract

The concept of Poisson cohomology groups associated with Poisson manifolds is a part of the theory of Lie superalgebras of vector fields. Therefore, we abstracted them as Poisson-like cohomology groups for general Lie superalgebras. In doing so, we obtained a generalization of the concept of the Euler number. For the Lie superalgebras of differential forms on a manifold, we found the de Rham cohomology groups match with the Poisson-like cohomology groups in the special case. In order to understand the development of the discussion, we presented some simple examples and show ideas of how our discussion unfolds.