= Matrix model of anti-de Sitter three-space
{title2=$\operatorname{AdS}_3\cong SL(2,\mathbb R),\quad g=B/8$}
For ambient <quadratic form> $Q=-x_0^2-x_4^2+x_1^2+x_2^2$, associate $M=\left(\begin{smallmatrix}x_4+x_1&x_2+x_0\\x_2-x_0&x_4-x_1\end{smallmatrix}\right)$. Its <determinant> is $-Q$, so the unit <Anti-de Sitter spacetime> hyperboloid is the <special linear group> of real two-by-two matrices. On its tangent space at the identity, the induced <bilinear form> is $\tfrac12\operatorname{tr}(UV)$. The <Killing form> is $4\operatorname{tr}(UV)$, and left and right matrix multiplication preserve the ambient form. Consequently the induced <bi-invariant pseudo-Riemannian metric> is $B/8$. This model has periodic time; its causally unwrapped version is the <universal covering Lie group>.
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