# Definition The Lorentz group in $n$-dimensions, $O(n-1, 1)$ is the group of [[Lorentz transformation|Lorentz transformations]]. It can be interpreted as the set of all transformations between [[intertial reference frame|intertial reference frames]]. This group has [[dimension]] $n(n-1)/2$. The more general case $O(p,q)$ with $p+q = n$ is known as the [[indefinite orthogonal group|indefinite orthogonal group]]. The Lorentz group is an [[Isometry Group]] over real [[vector space|vector spaces]] equipped with a [[Minkowski metric]]. See [[Isometry Group#Example 3 O n-1 n|here]] for a discussion. # Lie Algebra $\mathfrak{o}(n-1,1)$ Since $O(n-1,1) \cong \text{Isom}(V)$, element $X$ of its [[Lie Algebra]] $\mathfrak{o}(n-1,1)$ must obey: $ \braket{X \pmb{v} | \pmb{w}} = - \braket{\pmb{v}|X\pmb{w}} \quad \forall \pmb{v}, \pmb{w} \in V $ Now take $V = \mathbb{R}^n$ with the [[Minkowski metric]] and pick an [[orthogonality|orthonormal]] [[basis]]. Then we have: $ \begin{align} (X [\pmb{v}])^T [\eta][\pmb{w}] &= -[\pmb{v}]^T[\eta] X[\pmb{w}]\\ [\pmb{v}]^T X^T[\eta] [\pmb{w}] &= -[\pmb{v}]^T [\eta] X [\pmb{w}]\\ X^T [\eta] &= -[\eta]X \end{align} $ Writing this out in components we get: $ X = \begin{pmatrix} X' & \pmb{a} \\ \pmb{a} & 0 \end{pmatrix} \quad X' \in \mathfrak{o}(n-1), \quad \pmb{a} \in \mathbb{R}^{n-1} $ Thus, $\mathfrak{o}(n-1,1)$ is the set of $n\times n$ matrices of the form $X$ shown above. We can think of $X'$ as generating rotations in the $n-1$ spatial dimensions and $\pmb{a}$ generating a boost along its direction. The Lie algebra has the same dimension as its Lie group, $n(n-1)/2$, as one can easily verify.