# A class of nilpotent Lie algebras whose center acts nontrivially in cohomology

## TL;DR

This paper demonstrates that for a specific class of nilpotent Lie algebras with a codimension one abelian ideal, the central representation acts nontrivially in cohomology, revealing new insights into their structure.

## Contribution

It establishes the nontriviality of the central representation in cohomology for all one-dimensional central extensions of certain nilpotent Lie algebras.

## Key findings

- Central representation is nontrivial in these cases.
- Applicable to all one-dimensional central extensions with a codimension one abelian ideal.
- Provides a new understanding of cohomological actions in nilpotent Lie algebras.

## Abstract

We show that the central representation is nontrivial for all one-dimensional central extensions of nilpotent Lie algebras possessing a codimension one abelian ideal.