Symmetries of AdS$_3$, $SO(2,2)$ and $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$

Question

Basically, I want to know how one can see the $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ symmetry of AdS$_3$ explicitly.

AdS$_3$ can be defined as hyperboloid in $\mathbb{R}^{2,2}$ as

$$X_{-1}^2+X_0^2-X_1^2-X_2^2=L^2$$ where $L$ is the AdS radius. Since the metric of $\mathbb{R}^{2,2}$, $$ds^2=-dX_{-1}^2-dX_0^2+dX_1^2+dX_2^2,$$ is invariant under $SO(2,2)$ transformations and also the hyperboloid defined above is invariant we can conclude that AdS$_3$ has an $SO(2,2)$ symmetry.

One can probably show with pure group theoretical arguments that the $SO(2,2)$ symmetry is isomorphic to an $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ symmetry. I would like to know however, if one can see this symmetry more explicitly in some representation of AdS$_3$?

I suppose a starting point might be, that one can write the hyperboloid constraint equation as

$$\frac{1}{L^2}\text{det}\;\begin{pmatrix} X_{-1}-X_1 & -X_0+X_2 \\ X_0+X_2 & X_{-1}+X_1\end{pmatrix}=1$$ i.e. there is some identification of the hyperboloid with the group manifold of $SL(2,\mathbb{R})$ itself. However, that does not tell us anything about the symmetries.

The only explanation that I have found (on page 12 of this pdf Master thesis) was that the group manifold of $SL(2,\mathbb{R})$ carries the Killing-Cartan metric

$$g=\frac{1}{2}\text{tr}\,\left(g^{-1}dg\right)^2$$ which is invariant under the actions $$g\rightarrow k_L\, g \qquad\text{and}\qquad g\rightarrow g\, k_R$$ with $k_L,k_R\in SL(2,\mathbb{R})$. But how does one get from the metric on $\mathbb{R}^{2,2}$ to this Killing-Cartan metric? Also, I don't find this very explicit and was wondering if there is a more direct way.

This post imported from StackExchange Physics at 2014-10-27 19:55 (UTC), posted by SE-user physicus

Comments

Isn't $SO(2,2)$ non-compact as well. It's algebra should be isomorphic to $so(4)$, but onlike $SO(4)$ it is non-compact, or am I wrong? Anyways, the isomorphism between $SO(2,2)$ and $SL(2,\mathbb{R})\times SL(2,\mathbb{R})$ might be only on the level of the Lie algebras. In the literature, both are refered to being the isometry group of AdS$_3$, for example in the MAGOO AdS/CFT review page 150.

This post imported from StackExchange Physics at 2014-10-27 19:55 (UTC), posted by SE-user physicus

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Related result for other signature: physics.stackexchange.com/q/28505/2451

This post imported from StackExchange Physics at 2014-10-27 19:55 (UTC), posted by SE-user Qmechanic

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@physicus Do you have a good reference about the stuff you already know? Thanks!