The notation $SO(n,m)$ refers to the special orthogonal group of matrices that leave invariant a non-degenerate quadratic form of signature $(n,m)$ on a real vector space. Here's a more detailed breakdown:
1. Orthogonal Group: The orthogonal group, denoted as $O(n,m)$, consists of all $(n+m) \times (n+m)$ real matrices $A$ such that
$$ A^T g A = g $$where $A^T$ is the transpose of $A$ and $g$ is a diagonal matrix with $n$ entries of +1 and $m$ entries of -1. These matrices preserve the quadratic form given by $g$.
2. Special Orthogonal Group: The special orthogonal group $SO(n,m)$ is the subgroup of $O(n,m)$ that consists of matrices with determinant +1. In other words, these are the "volume-preserving" orthogonal transformations.
For $SO(n)$ (or $SO(n,0)$), which is a common special case, the quadratic form is the standard dot product in $\mathbb{R}^n$, and the group consists of $n \times n$ orthogonal matrixs with determinant +1.
The case $O(1,3)$ is Lorentz group.
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Author of the notes: Antonio J. Pan-Collantes
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