# nLab Lorentz Lie algebra

Contents

### Context

#### Riemannian geometry

Riemannian geometry

# Contents

## Idea

The Lorentz Lie algebra $\mathfrak{o}(d-1,1)$ is the Lie algebra of the Lorentz group $O(d-1,1)$ (the group of linear isometries of Minkowski spacetime). In particular, it is an orthogonal Lie algebra.

It coincides with the Lie algebras $\mathfrak{so}(d-1,1)$ and $\mathfrak{so}^+(d-1,1)$ corresponding to the special orthogonal group $SO(d-1,1)$ and the proper orthochronous Lorentz group $SO^+(d-1,1)$, respectively. Indeed, the Lie algebra of any Lie group remains unchanged after passing to a subgroup of finite index.