Anti de Sitter Space - Anti de Sitter As Homogeneous and Symmetric Space

Anti De Sitter As Homogeneous and Symmetric Space

In the same way that the sphere, anti de Sitter with parity aka reflectional symmetry and time reversal symmetry can be seen as a quotient of two groups whereas AdS without P or C can be seen as

This quotient formulation gives to a homogeneous space structure. The Lie algebra of is given by matrices

\mathcal{H}= \begin{pmatrix} \begin{matrix} 0&0\\ 0&0 \end{matrix} & \begin{pmatrix} \cdots 0\cdots\\ \leftarrow v^t\rightarrow \end{pmatrix}\\ \begin{pmatrix} \vdots & \uparrow\\ 0 & v \\ \vdots & \downarrow \end{pmatrix} & B \end{pmatrix} ,

where is a skew-symmetric matrix. A complementary in the Lie algebra of is

\mathcal{Q}= \begin{pmatrix} \begin{matrix} 0&a\\ -a&0 \end{matrix} & \begin{pmatrix} \leftarrow w^t\rightarrow \\ \cdots 0\cdots\\ \end{pmatrix}\\ \begin{pmatrix} \uparrow & \vdots\\ w & 0\\ \downarrow & \vdots \end{pmatrix} & 0 \end{pmatrix}.

These two fulfil . Then explicit matrix computation shows that . So anti de Sitter is a reductive homogeneous space, and a non-Riemannian symmetric space.

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