Symmetry transformation in AdS space In AdS/CFT papers the action of the SO(D,2) symmetry is usually given at the boundary where the transformations are just the conformal transformations (Poincare, scaling and special) for D+1 Minkowskian space.
I would like to know how the SO(1,2) transformations acts for arbitary point in AdS, lets say in coordinate system  
$ds^2 = 1/z^2(dz^2+dx^2+dy^2)$.
Is is possible to write $z'=z(z,x,y)$, $x'=x(z,x,y)$ and $y'=y(z,x,y)$ so that the above metric stays invariant?  
 A: To see the whole symmetry, it is better to see $AdS4$ as a 4-surface in a 5-dimensional space.
We suppose that the 5-metrics is $ds^2 = dx^2 + dy^2 + dz^2 - dt^2 - du^2$
The 4-surface $AdS4$ is defined as : 
$x^2 + y^2 + z^2 - t^2 - u^2  = -1 $
With this definition, the whole $SO(3,2)$ symmetry, for transformations $(x,y,z,t,u) \rightarrow (x',y',z',t',u') $ is obvious.
Now, from the intrinsic 4-metric $ds^2=1/(z^2)(+dx^2+dy^2+dz^2-dt^2)$ 
some invariant-metrics transformations are easy to see:


*

*Three translations for the coordinates $t,x,y$

*Scale transformation : multiplying $t,x,y,z$ by a constant term $\lambda$

A: Just to follow up on Trimok's answer, you might find the following references useful:


*

*Correlation functions in the CFT(d)/AdS(d+1) Correspondence  (http://arxiv.org/abs/hep-th/9804058)

*Supersymmetric Gauge Theories and the AdS/CFT (Correspondence
http://arxiv.org/abs/hep-th/0201253)
The first paper contains a discussion of the discrete inversion isometry of AdS, which is crucial for evaluating the bulk integrals associated with correlation functions in AdS/CFT. Since it is a discrete isometry there is no infinitesimal form. The second paper (especially section 8.1) discusses the continuous isometries of the metric, in addition to the inversion isometry. 
