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.

Read more about this topic:  Anti De Sitter Space

Famous quotes containing the words homogeneous and/or space:

    “O my Brothers! love your Country. Our Country is our home, the home which God has given us, placing therein a numerous family which we love and are loved by, and with which we have a more intimate and quicker communion of feeling and thought than with others; a family which by its concentration upon a given spot, and by the homogeneous nature of its elements, is destined for a special kind of activity.”
    —Giuseppe Mazzini (1805–1872)

    “Art and power will go on as they have done,—will make day out of night, time out of space, and space out of time.”
    —Ralph Waldo Emerson (1803–1882)