An important fact used in physics is that \({SU(2)}\) is the universal covering group of \({SO(3)}\). For example, the four complex numbers associated with an element of \({SU(2)}\) are called **Cayley-Klein parameters**, and via this homomorphism can be used to specify a proper rotation in \({\mathbb{R}^{3}}\), i.e. an element of \({SO(3)}\). \({SU(2)}\) is a double cover, so there are two Cayley-Klein parameters corresponding to every proper rotation.

The connected components of matrix groups are usually related to determinant signs; we can see this by studying \({O\left(3,1\right)}\), which in physics is called the (homogeneous) **Lorentz group**. It is of dimension 6, and consists of rotations and reflections in **Minkowski space** (AKA spacetime), \({\mathbb{R}^{4}}\) with the Minkowski metric, where the positive signatures correspond to space and the negative signature to time. \({O\left(3,1\right)}\) has 4 connected components, corresponding to whether the orientation of time and/or space is reversed. The **proper Lorentz group** \({SO\left(3,1\right)}\) consists of the identity component and the connected component of transformations that reverse both space and time, while the **orthochronous Lorentz group** consists of the two components that preserve the orientation of time. The identity component \({SO\left(3,1\right)^{e}}\) is then called the **proper orthochronous Lorentz group** (AKA restricted Lorentz group). A “rotation” by an angle \({\phi}\) in a “time-like plane” which includes vectors with negative lengths is called a **Lorentz boost** of **rapidity** \({\phi}\).

The **Poincaré group** (AKA inhomogeneous Lorentz group) is the semidirect product of the translations in 4 possible directions with \({O\left(3,1\right)}\); the product is semidirect since the translations are a normal subgroup and every element can be written in exactly one way as a translation followed by a rotation. It has dimension 10, or \({n\left(n+1\right)/2}\) in general. The above adjectives can also be applied to the Poincaré group. All of the above analysis was for the “mostly pluses” signature \({\left(3,1\right)}\), but the results also hold for the “mostly minuses” signature \({\left(1,3\right)}\).

Similarly, the **Euclidean group** \({E(n)\equiv\mathbb{R}^{n}\rtimes O(n)}\) is the semidirect product of translations in \({\mathbb{R}^{n}}\) with \({O\left(n\right)}\), while the **special Euclidean group** \({SE(n)}\) takes the semidirect product with \({SO\left(n\right)}\). These are subgroups of the **affine group** \({\mathit{Aff}(n,\mathbb{R})\equiv\mathbb{R}^{n}\rtimes GL(n,\mathbb{R})}\) and its identity component the **special affine group**, respectively. All of these “inhomogeneous” groups, i.e. groups formed by semidirect products whose elements are a translation \({v}\) followed by a linear transformation \({A}\), can be viewed as matrix groups of the form

\(\displaystyle \begin{pmatrix}A & v\\ 0 & 1 \end{pmatrix},\)

where \({v}\) is a column vector and the matrices can be considered as acting on vectors in **homogeneous coordinates** (AKA projective coordinates), in which a component 1 is appended:

\(\displaystyle \begin{pmatrix}A & v\\ 0 & 1 \end{pmatrix}\begin{pmatrix}w\\ 1 \end{pmatrix}=\begin{pmatrix}Aw+v\\ 1 \end{pmatrix}\)

Δ Note that in homogeneous coordinates scalar multiples of vectors are identified, e.g. \({\left(1,2,3,1\right)=\left(2,4,6,2\right)}\). A possible source of confusion is that inhomogeneous groups act on vectors in homogeneous coordinates. |