The Lorentz group is the orthogonal group for an invariant bilinear form of signature $(-+++\cdots)$, $O(d-1,1)$.
In physics, in the theory of 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.
The elements in the Lorentz group in the image of the special orthogonal group $SO(d-1) \hookrightarrow O(d-1,1)$ are rotations in space. The further elements in the special Lorentz group $SO(d-1,1)$, which mathematically are “hyperbolic rotations” in a space-time plane, are called boosts in physics.
One distinguishes the following further subgroups of the Lorentz group $O(d-1,1)$:
the proper Lorentz group $SO(d-1,1)$ is the subgroup of elements which have determinant +1 (as elements $SO(d-1,1)\hookrightarrow GL(d)$ of the general linear group);
the proper orthochronous (or restricted) Lorentz group $SO^+(d-1,1) \hookrightarrow SO(d-1,1)$ is the further subgroup of elements which do not act by reflection along the timelike axis.
As a smooth manifold, the Lorentz group $O(d-1,1)$ has four connected components. The connected component of the identity is the the proper orthochronous group $SO^+(3,1)$. The other three components are
$SO^+(d-1,1)\cdot P$
$SO^+(d-1,1)\cdot T$
$SO^+(d-1,1)\cdot P T$,
where, as matrices
is the operation of point reflection at the origin in space, where
is the operation of reflection in time and hence where
is point reflection in spacetime.
While the proper orthochronous Lorentz group $SO^+(d-1,1)$ is connected, it is not simply connected. Its universal double cover is the Lorentzian spin group $Spin(d-1,1)$
quantum group version: quantum Lorentz group
group | symbol | universal cover | symbol | higher cover | symbol |
---|---|---|---|---|---|
orthogonal group | $\mathrm{O}(n)$ | Pin group | $Pin(n)$ | Tring group | $Tring(n)$ |
special orthogonal group | $SO(n)$ | Spin group | $Spin(n)$ | String group | $String(n)$ |
Lorentz group | $\mathrm{O}(n,1)$ | $\,$ | $Spin(n,1)$ | $\,$ | $\,$ |
anti de Sitter group | $\mathrm{O}(n,2)$ | $\,$ | $Spin(n,2)$ | $\,$ | $\,$ |
conformal group | $\mathrm{O}(n+1,t+1)$ | $\,$ | |||
Narain group | $O(n,n)$ | ||||
Poincaré group | $ISO(n,1)$ | Poincaré spin group | $\widehat {ISO}(n,1)$ | $\,$ | $\,$ |
super Poincaré group | $sISO(n,1)$ | $\,$ | $\,$ | $\,$ | $\,$ |
superconformal group |