A comment concerning cohomology and invariants of Lie algebras with respect to contractions and deformations

TL;DR

This paper reveals that non-invertible deformations of Lie algebras can produce invariants and alter cohomology, challenging previous expectations about their behavior.

Contribution

It introduces a criterion based on the Poincaré polynomial to determine the invertibility of Lie algebra deformations.

Findings

Non-invertible deformations can generate coadjoint invariants.

Deformations can eliminate cohomology with trivial or adjoint modules.

A criterion for invertibility of deformations is established.

Abstract

Contrary to the expected behavior, we show the existence of non-invertible deformations of Lie algebras which can generate invariants for the coadjoint representation, as well as delete cohomology with values in the trivial or adjoint module. A criterion to decide whether a given deformation is invertible or not is given in dependence of the Poincar\'e polynomial.