We will now extend the superconformal algebra to some infinite-dimensional algebra, just like was extended to SYM.
We will take two copies and of the SYM algebra (7), add a copy of , and glue them all together:
In other words:
The consistency of this definition can be verified using the general theory of quadratic-linear algebras, the PBW theorem by Polishchuk-Positselski-Braverman-Gaitsgory.
The analogue of (17) is:
It involves relative Lie algebra cohomology. The cochain complex consists of -invariant linear functions:
Therefore an element of is a function of Lie-algebra valued variables with taking values in :
The actual map between and goes as follows. Given a representative cocycle , the representative of the corresponding element in is obtained by the substitution of in place of :