Let be a Lie group. It is possible to introduce the structure of a Lie group on . In fact:

and the structure of the group is introduced by pointwise multiplication. Consider the corresponding Lie superalgebra:

We can interpret as the algebra of maps from to ; therefore the elements of are -valued functions of an odd parameter . It is possible to extend by an extra odd element with the following commutation relations:

We will call this extended algebra .

Let us consider the universal enveloping and look at it as a quadratic-linear algebra. The dual algebra is a differential algebra; it has a nonzero differential because the original algebra was quadratic-linear; the differential corresponds to the commutator. We will call it :

This is generated by the linear space: .

Let be the basis of as a linear space. Then is generated by odd elements and even elements :

As a dual to a non-homogeneous quadratic-linear algebra, the supercommutative algebra comes with the differential:

But since was actually a DLA, the differential being , the dual map defines the second differential on :

These two differentials commute:

The total differential is called :

Schematically:

Notice that:

The cohomology of is ; this can be proven using the homotopy operator: