# Compatible actions of Lie algebras

## TL;DR

This paper explores compatible actions in Lie algebras, introducing a diagrammatic approach to describe the Peiffer product and establishing its universal property as a coproduct in the category of crossed modules over Lie algebras.

## Contribution

It introduces a new diagrammatic method for the Peiffer product in Lie algebras and extends known group results to the Lie algebra context.

## Key findings

- Peiffer product described via a new diagrammatic approach.
- Transfer of results linking compatible actions and crossed modules from groups to Lie algebras.
- Peiffer product has the universal property of a coproduct in the category of crossed modules.

## Abstract

We study compatible actions (introduced by Brown and Loday in their work on the non-abelian tensor product of groups) in the category of Lie algebras over a fixed ring. We describe the Peiffer product via a new diagrammatic approach, which specializes to the known definitions both in the case of groups and of Lie algebras. We then use this approach to transfer a result linking compatible actions and pairs of crossed modules over a common base object $L$ from groups to Lie algebras. Finally, we show that the Peiffer product, naturally endowed with a crossed module structure, has the universal property of the coproduct in $\mathbf{XMod}_L(\mathbf{Lie}_R)$.