# Sympathetic Lie algebras and adjoint cohomology for Lie algebras

## TL;DR

This paper investigates sympathetic Lie algebras, focusing on their structure and adjoint cohomology, and addresses Pirashvili's conjecture relating semisimplicity to cohomology vanishing.

## Contribution

It proves new results on sympathetic Lie algebras and their adjoint cohomology, providing explicit computations for certain semidirect products and advancing understanding of Pirashvili's conjecture.

## Key findings

- Sympathetic Lie algebras are characterized and studied.
- Explicit cohomology results are obtained for specific semidirect products.
- Progress is made towards understanding the conditions for semisimplicity via cohomology.

## Abstract

We study sympathetic Lie algebras, namely perfect and complete Lie algebras. They arise among other things in the study of adjoint Lie algebra cohomology. This is motivated by a conjecture of Pirashvili, which says that a non-trivial finite-dimensional complex perfect Lie algebra is semisimple if and only if its adjoint cohomology vanishes. We prove several results on sympathetic Lie algebras and the adjoint Lie algebra cohomology of Lie algebras in general, using the Hochschild-Serre formula. For certain semidirect products we obtain explicit results for the adjoint cohomology.