# Cohomologies of a Lie algebra with a derivation and applications

## TL;DR

This paper studies the cohomology theory of Lie algebras with derivations, called LieDer pairs, and explores their deformations, extensions, and classifications using cohomology groups.

## Contribution

It introduces the concept of representations and cohomologies for LieDer pairs, and classifies their extensions and higher structures via cohomology.

## Key findings

- A LieDer pair is rigid if its second cohomology is trivial.
- Deformations are extensible when the obstruction class in third cohomology is trivial.
- Central extensions are classified by the second cohomology group.

## Abstract

The main object of study of this paper is the notion of a LieDer pair, i.e. a Lie algebra with a derivation. We introduce the concept of a representation of a LieDer pair and study the corresponding cohomologies. We show that a LieDer pair is rigid if the second cohomology group is trivial, and a deformation of order n is extensible if its obstruction class, which is defined to be an element is the third cohomology group, is trivial. We classify central extensions of LieDer pairs using the second cohomology group with the coefficient in the trivial representation. For a pair of derivations, we define its obstruction class and show that it is extensible if and only if the obstruction class is trivial. Finally, we classify Lie2Der pairs using the third cohomology group of a LieDer pair.