Definition
Consider
The subgroup orthogonal group ) such that
generalized orthogonal group .
The subgroup special generalized orthogonal group .
Another way to identify the generalized orthogonal group is with the matrices
Misplaced & & \ddots &&&& \\ &&1&&&\\ &&&-1&&\\ &&&&\ddots&\\ &&&&&-1 \end{bmatrix}$$ Where the first $n$ entries along the diagonal are 1 and the remaining $k$ are $-1$. # Lie group structure We can show that $\text{O}(n;k)$ is a closed subgroup since if $A \in \text{O}(n;k)$ then it must satisfy the following equations: \begin{align*}
[A_{-,j}, A_{-,l}]{n,k} &= 0 \qquad j\neq l\
[A {-,j}, A_{-,l}]{n,k} &= 1 \qquad 1 \leq j \leq n\
[A {-,j}, A_{-,l}]_{n,k} &= -1 \qquad n+1 \leq j \leq n + k
\end{align*}
You can't use 'macro parameter character #' in math mode These equations are polynomials in the entries of $A$, so looking at the inverse image of the closed set defined above give the (closed) subgroup $\text{O}(n;k)$. Clearly, $\text{SO}(n;k)$ will also be closed. Therefore, by the [[202406170727|closed subgroup theorem]], both are [[202405061454|Lie groups]]. ## Lie algebra $\mathfrak{so}(n;k)$ Let $\mathfrak{so}(n;k)$ be the [[202405062209|Lie algebra]] of $\text{O}(n;k)$ and $\text{SO}(n;k)$. $$ \mathfrak{so}(n;k) = \{X \in \text{Mat}_{n + k \times n+k}(\mathbb{R}) \ | \ gX^{T}g = -X\}$$