The superconformal algebra 
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*How does one derive the superconformal algebra? 

*Especialy how to argue the existence of the operator $S$ which doesn't exist either in either the supersymmetric algebra or the conformal algebra? 

*What is the explanation of the convention of choosing If $J _ \alpha ^\beta$ as a generator of $SU(2) \times SU(2)$ and $R^A _B$ and $r$ are the generators of the $U(m)$ R-symmetry ( $m \neq 4$)? How does one get these $R$ and $r$? 

*One further defines $R_1, R_2$ and $R_3$ as the generators of the Cartan subalgebra of $SU(4)$ (the $N=4$ R-symmetry group) where these are defined such that $R_k$ has $1$ on the $k^{th}$ diagonal entry and $-1$ on the $(k+1)^{th}$ diagonal entry. 
Using the above notation apparently one of the $Q-S$ commutation looks like,
$\{ S^A_\alpha , Q^\beta _B\} = \delta ^A_B J^\beta _\alpha + \delta ^\beta _\alpha R^A_B + \delta ^A_B \delta ^\beta _\alpha (\frac{H}{2} + r \frac{4-m}{4m})$
I would like to know how the above comes about and why the above equation implies the following equation (..each of whose sides is defined as $\Delta$) 
$2\{ Q_4 ^- , S^4_ - \} = H - 3J_3 + 2\sum _{k=1} ^ 3 \frac{k}{4} R_k$ 
I can't understand how to get the second of the above equations from the first of the above and why is this $\Delta$ an important quantity. 
{Is there any open source online reference available which explains the above issues? 
 A: I'm not sure what you mean by "derving" the algebra, but one way of obtaining a superconformal algebra (let me consider $N=4$) is to start with the four-dimensional conformal algebra and the $N=4$ supersymmetry algebra and consider the closure of the combined algebra. The conformal algebra consists of the Lorentz generators $J^\alpha{}_\beta$ and $\dot{J}^{\dot{\alpha}}{}_{\dot{\beta}}$ (I write the Lorentz group as $SL(2)\times SL(2)$, which is what I think you do in your question), the translations $P^{\alpha\dot{\beta}}$, the special conformal transformations $K_{\alpha\dot{\beta}}$ and the dilatation $H$. The $N=4$ supersymmetry group is generated by the $SU(4)$ R-symmetry generators $R^A{}_B$, the translations $P^{\alpha\dot{\beta}}$ and the supercharges $Q^A{}_\alpha$ and $\dot{Q}_{\dot{\alpha}A}$. These two algebras are both closed, but if we want to consider the combined symmetry the above generators are not enough to close the algebra. In particular the commutator between $K^{\alpha\dot{\beta}}$ and $\dot{Q}_{\dot{\gamma}A}$ gives rise to a new generator:
$[K^{\alpha\dot{\beta}},\dot{Q}_{\dot{\gamma}C}] = \delta^{\dot{\beta}}_{\dot{\gamma}} S^{\alpha}{}_C$
By also including the superconformal charges $S^{\alpha}{}_A$ (and a correpsonding charge $\dot{S}^{A\dot{\alpha}}$) we ge a closed algebra. The other (anti-)commutation relations follow if we require the the Jacobi identity is satisfied. In particular we get
$\{S^{\alpha}{}_a,Q^b{}_\beta\} = \delta^b_a J^{\alpha}{}_\beta + \delta^{\alpha}_{\beta} R^B{}_A + \frac{1}{2} \delta^b_a \delta^{\alpha}_{\beta} H$
which is your next-to-last equation (in somewhat different conventions when it comes to indices). To obtain your last equation you just have to take the particular case of $A=B=4$ and $\alpha=\beta=-$ and express the result in terms of your Cartan generators $R_1$, $R_2$, $R_3$ and $J_3$. The importance of this last expression really depends on in which context you have encountered the algebra. If you give some more details maybe someone can answer.
