Super homologies associated with low dimensional Lie algebras

TL;DR

This paper explores the construction of Lie superalgebras from low-dimensional Lie algebras using Schouten-like brackets, analyzing their homology groups and Betti numbers to understand their structure.

Contribution

It introduces a method to construct non-trivial Lie superalgebras from abstract Lie algebras via Schouten-like brackets and investigates their homology properties.

Findings

Constructed Lie superalgebras from low-dimensional Lie algebras.

Analyzed super homology groups and Betti numbers for these structures.

Provided insights into the control of superalgebras by core Lie algebras.

Abstract

A Poisson structure on a manifold is characterized by the Schouten bracket. The graded algebra of the tangent bundle with the Schouten bracket is a prototype of Lie superalgebra. The Poisson condition means that a cycle in the 2-chain space. Given a graded Lie superalgebra, the 0-graded subspace is a Lie algebra. In this note, using the DGA of tangent bundle with the Schouten bracket as a model, we start from an abstract Lie algebra, construct non-trivial Lie superalgebra by Schouten-like bracket. Then it is natural to ask how the core Lie algebra control the Lie superalgebra. One trial here is to investigate the Betti numbers of the super homology groups. For abelian Lie algebras, the boundary operator is trivial, so we study super homology groups for low dimensional non-abelian Lie algebras of dimension smaller than 4.