Commutation relations of the generators of the conformal group My question is from P.98 of the book by Di Francesco on Conformal Field theory.  He gives the six non-vanishing commutation relations between the elements $P_{\mu}, D, L_{\mu \nu}$ and $K_{\mu}$ comprising the four dimensional space of generators of the conformal group.  These are then rewritten by defining a set of four more generators: $$J_{\mu \nu} = L_{\mu \nu}\,\,\,,\,\,\,J_{-1, \mu} = \frac{1}{2}(P_{\mu} - K_{\mu})\,\,\,,\,\,\,J_{-1,0} = D\,\,\,,\,\,\,J_{0,\mu} = \frac{1}{2}(P_{\mu} + K_{\mu}),$$ and I believe the motivation for doing so is so that the six commutation relations mentioned above can be eloquently recast in a single one line commutation relation $$[J_{ab}, J_{cd}] = i(\eta_{ad}J_{bc} + \eta_{bc}J_{ad} - \eta_{ac}J_{bd} - \eta_{bd}J_{ac})$$ It also says that $a,b \in \left\{-1,0,1,\dots,d\right\}$ and that the new generators obey the $SO(d+1,1)$ commutation relations (which I think is the one-line equation above).
My question is: What do the indices $a,b$ represent and what does the notation $SO(d+1,1)$ mean? I think there are $d$ spatial dimensions but I can't see what the significance of the $-1$ and $0$ elements are.
Many thanks.
 A: The indices $a$ and $b$ are chosen in such a way that inserting different combinations of values from $-1$ to $d$ gives just the original six commutation relations for the generators of conformal symmetry transformations. As you have correctly suggested, the point of this is to rewrite the six relations in compact form.
The latter shows that the conformal group is actually given by $\text{SO}(d+1,1)$, which is the group of special orthogonal transformations in $d+1$ spacelike and $1$ timelike dimensions, where $d$ is the number of spatial space-time dimensions. $0$ represents the time-component, while $-1$ is spacelike and appears because we have dilatations and special conformal transformations. Note that that $-1$ is not a spatial dimension with respect to spacetime but rather an additional dimension with respect to the group action. 
A: Just following up on Frederic's answer.  I wouldn't get too hung up on thinking about a metric that includes  $\eta_{-1-1}$.  The metric really refers to spacetime and there is no new spacetime dimension we've introduced.  It's just a way of labeling the generators - their indices don't necessarily refer to spacetime dimensions, though they do for most of the $J$ here.  
You should separate the idea of the symmetry group from the spacetime somewhat.  What I mean is, you seem to be hung up on thinking of SO(3) as the group of isometries of 3-space and maybe SO(3,1) as the isometries of Minkowski space (maybe you didn't know that last part but it's true), and therefore thinking you need a larger spacetime to have SO(d+1,1).
However, it is not necessarily the case that the symmetry group of a theory in d spatial dimensions and n timelike dimensions is SO(d,n).  It happens to be true for Euclidean space and Minkowski space without conformal invariance, so that prejudices us.  But really, it's just a prejudice to assume that the (maximal) symmetry group is SO(d,n).  I mean, it is true in most of the QFTs we first learn.  And furthermore, in theories with conformal invariance, SO(d,n) is contained as a subgroup, and we didn't need to worry about whether it was part of a larger group when we were learning QFT.
I think there is one mistake in Frederic's answer.  The conformal group in d spatial and n timelike dimensions is SO(d+1,n+1).  So a CFT in 4-D Euclidean space is a rep of SO(5,1) and in 4-D Minkowski space is a rep of SO(4,2).
One very interesting thing about all this: you might ask, what is a spacetime where SO(4,2) really is just generalized rotations (as opposed to rotations + SCTs + dilatations)?  Well, $AdS_5$ is one!  This might be your first clue towards the existence of the AdS-CFT correspondence!  A CFT in 3+1-dimensional spacetime obeys the same algebra as the isometries of $AdS_5$.  See "ANTI-DE SITTER SPACE" by Ingemar Bengtsson - pages 1-5 give a nice concise introduction to AdS spacetime and it's isometries.
