Algebra formed by covariant derivatives
We will start by relating the cohomology of (6) with some Lie algebra cohomology.

So, we have to first define that infinite-dimensional Lie superalgebra. The definition is motivated by the super-Yang-Mills theory, and the resulting algebra is called SYM.

Let us proceed to its definition.

Classical equations of motion of the ten-dimensional super-Yang-Mills algebra are encoded in the constraints:


As a consequence of these constraints, there exist operators such that:


The actual definition of the operators and in the super-Yang-Mill theory is:



I am sorry for the abuse of notations; notice that and are completely different fields; but in my notations the only difference between them is greek subindex vs. latin subindex .

Now let us forget about the ``internal structure'' of and given by (9) and (10) and think of Eq. (7) as defining relations in an abstract algebra formed by the letters .

This is an infinite-dimensional algebra. Moreover, this is actually a quadratic algebra, because the constraint (7) is quadratic in generators.