Lorentz Groups and their Lie Algebras

The Lorentz Group $O(1,3)$ (or O(3,1)) is the group of all linear transformations of $\mathbb{R}^4$ (or 4 x 4 real matrices) that preserve the Lorentz inner product

(1)
\begin{align} A \in O(1,3) \leftrightarrow (Ax)_{\mu}(Ay)^{\mu} = x_{\mu}y^{\mu} \mbox{ for all } x, y \leftrightarrow A^{\dagger}gA = g \end{align}

where $g = diag(1,-1,-1,-1)$.

The group $O(1,3)$ has four connected components, determined by the values of 2 homomorphisms from $O(1,3)$ into the 2 element group $\{\pm 1\}$:

(2)
\begin{align} A |\rightarrow det A \end{align}

and

(3)
\begin{align} A |\rightarrow sgn(A_{0}^{0}) \end{align}

Definition: The kernel of a group homomorphism $f: G\rightarrow G^{'}$ is the set of all elements of $G$ which are mapped to the identity element of $G^{'}$.

The kernel of the first of these is the special Lorentz Group $SO(1,3)$ and the kernel of the second group is the orthochronous Lorentz group $O^{\dagger}(1,3)$ — the subgroup of $O(1,3)$ that preserves the direction of time.

The intersection

(4)
\begin{align} SO^{\dagger}(1,3) = SO(1,3) \bigcap O^{\dagger}(1,3) \end{align}

is called the restricted Lorentz group or proper Lorentz group and is the connected component of the identity element in $O(1,3)$.

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