Consider vector space with coordinates x1, ... xn. Consider polynomial deRham complex (also known as Koszul complex) which is generated by xi and dx_i. As an algebra it is just $C[x_i]\otimes \Lambda [dx_i]$.

From the point of view of super-mathematics this algebra is just polynomial algebra on the super-vector space $C^{n|n}$.

**Question** I wonder if there any application of super-math to usual math in this particular context ? For example does Lie supergroup GL(n|n), its the Lie algebra and its universal enveloping U(gl(n|n)) which acts on $C^{n|n}$ have meaning/interpretation/application to differential forms ?

Well there might be some relation in opposite direction - if so it is also welcome.