# External semidirect product of Lie rings

## Definition

### Definition with the left action convention

Suppose $N$ is a Lie ring and $H$ is a Lie ring along with a homomorphism of Lie rings $\varphi$ from $H$ to the Lie ring of derivations $\operatorname{Der}(N)$. The external semidirect product of Lie rings, denoted $N \rtimes H$, is defined as follows:

• As an abelian group, it is the external direct product of $N$ and $H$, denoted $N \times H$ or $N \oplus H$.
• The Lie bracket is as follows:

$[(n_1,h_1),(n_1,h_2)] = [n_1 + \varphi(h_1)n_2, h_1 + h_2]$

where $\varphi(h_1)n_2$ denotes the image of $n_2$ under the derivation $\varphi(h_1)$.

### Why the action convention does not matter

This is basically because every Lie ring is naturally isomorphic to its opposite Lie ring via the negative map.