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 
:
has a large algebra of
symmetries, the superconformal algebra a.k.a. . We will denote it :
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It has the subalgebra 
which consists of those generators which preserve a fixed point:
which consists of those generators which preserve a fixed point:
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Indeed, is a coset space:
is a coset space:
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Let us use a calligraphic index to parametrize the generators of 
:
:
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The rest of the generators of are denoted:
are denoted:
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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 .
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”.
algebra (11),
or maybe we should say “to a sum of two copies of it”.