nLab
Lorentz Lie algebra

Contents

Context

Riemannian geometry

Lie theory

∞-Lie theory (higher geometry)

Background

Smooth structure

Higher groupoids

Lie theory

∞-Lie groupoids

∞-Lie algebroids

Formal Lie groupoids

Cohomology

Homotopy

Examples

\infty-Lie groupoids

\infty-Lie groups

\infty-Lie algebroids

\infty-Lie algebras

Contents

Idea

The Lorentz Lie algebra 𝔬(d1,1)\mathfrak{o}(d-1,1) is the Lie algebra of the Lorentz group O(d1,1)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 𝔰𝔬(d1,1)\mathfrak{so}(d-1,1) and 𝔰𝔬 +(d1,1)\mathfrak{so}^+(d-1,1) corresponding to the special orthogonal group SO(d1,1)SO(d-1,1) and the proper orthochronous Lorentz group SO +(d1,1)SO^+(d-1,1), respectively. Indeed, the Lie algebra of any Lie group remains unchanged after passing to a subgroup of finite index.

References

See also

Last revised on July 4, 2020 at 15:42:01. See the history of this page for a list of all contributions to it.