Lie algebra of $O(1, n)$

Question

I would like to know the Lie algebra of the Lorentz group $SO(1, n)$. Can you tell me, what the answer is? Thank you in advance!

Answer 1

Consider the $n\times n$-matrix $J$ such that $a_{ij}=0$ if $i\neq j$, $a_{ii}=1, i<n, a_{nn}=-1$. $A\in SO(n,1)$ if and only if $A^tJA=I$. Thus the Lie algebra of $SO(n,1)=\{M\in M(n,R), M^tJ+JM=0\}$.

Answer 2

Lie algebra of general $SO(N)$ group consists of skew-symmetric matrices with zero trace with ordinary matrix commutator as Lie bracket. You can easily deduce it by differentiating group conditions $A^TA = E$ and $\mbox{det}A = 1$ at zero: $d(A^TA) = dA^T(0) + dA(0) = 0$.

If it is necessary, look at more details about Lorentz group's Lie algebra on Wikipedia.