nLab
Lorentz group

Definition

The Lorentz group is the orthogonal group for an invariant bilinear form of signature $(-+++\cdots )$, $O(n,1)$.

In physics the theory of special relativity the Lorentz group acts canonically as the group of linear isometries of Minkowski spacetime preserving a chosen basepoint. This is called the action by Lorentz transformations.

Related concepts

• Lorentzian manifold
• quantum group version: quantum Lorentz group

group	symbol	universal cover	symbol	higher cover	symbol
orthogonal group	$\mathrm{O}(n)$	Pin group	$\mathrm{Pin}(n)$	Tring group	$\mathrm{Tring}(n)$
special orthogonal group	$\mathrm{SO}(n)$	Spin group	$\mathrm{Spin}(n)$	String group	$\mathrm{String}(n)$
Lorentz group	$\mathrm{O}(n,1)$	$$	$\mathrm{Spin}(n,1)$	$$	$$
anti de Sitter group	$\mathrm{O}(n,2)$	$$	$\mathrm{Spin}(n,2)$	$$	$$
Narain group	$O(n,n)$
Poincaré group	$\mathrm{ISO}(n,1)$	$$	$$	$$	$$
super Poincaré group	$\mathrm{sISO}(n,1)$	$$	$$	$$	$$