Superconformal algebra

#### 1` `General definition and generators

It is important to remember that the classical SUGRA background has a large algebra of
symmetries, the superconformal algebra a.k.a. . We will denote it :

It has the subalgebra which consists of those generators which preserve a fixed point:

Indeed, is a coset space:

Let us use a calligraphic index to parametrize the generators of :

The rest of the generators of are denoted:

The superindex corresponds to some -grading which we do not need now, we just have
to remember that the generators generate ,
and generate the rest of .

#### 2` `Structure of commutation relations

The commutation relations are parametrized by the structure constants.

For our present purpose, we have to know the following facts:

- First of all, acts by the adjoint representation:
- We need to know that the commutator of and falls into :
- The commutator is proportional to , and moreover there is a basis where the structure constants are proportional to the gamma-matrices:

We see that the superconformal algebra is somewhat similar to the algebra (11),
or maybe we should say “to a sum of two copies of it”.