generalized orthogonal group

Definition

Consider ${\mathbb{R}}^{n+k}$ for $n,k\in \mathbb{N}$. Let $[-,-{]}_{n,k}$ be a bilinear form defined as $[x,y{]}_{n,k}=x_1{y}_1+\cdots +x_n{y}_n-x_{n+1}{y}_{n+1}-\cdots -x_{n+k}{y}_{n+k}.$

The subgroup $\text{O}(n;k)\subset \text{GL}(n+k,\mathbb{R})$ that preserves this bilinear form (as opposed to the dot product that is preserved by the orthogonal group) such that $[Ax,Ay{]}_{n,k}=[x,y{]}_{n,k}$ is called the generalized orthogonal group.

The subgroup $\text{SO}(n;k)\subset \text{O}(n;k)$ with determinant 1 is called the special generalized orthogonal group.

Another way to identify the generalized orthogonal group is with the matrices $A\in \text{GL}(n+k,\mathbb{R})$ such that $A^{T}gA=g$ for the matrix

& \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*}

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 $\text{SO}(n;k)$. $$ \mathfrak{so}(n;k) = \{X \in \text{Mat}_{n + k \times n+k}(\mathbb{R}) \ | \ gX^{T}g = -X\}$$