Idempotent operators over the exterior algebra

Question

I am wondering if there exists a (reasonably) well-known set of operators $A_i$ over the exterior algebra such that

$\{A_i,A_j\} = \frac{1}{2}(A_i +A_j)$,

where $\{X,Y\}=(XY+YX)/2$.

Answer 1

I'm not familiar with differential forms, as such. But in geometric algebra (which supposedly encompasses differential forms), where $A$ is a k-blade and $A\wedge A= \langle AA \rangle_{2k}$, $1$ and $0$ are the only elements that satisfy the 1st property. But neither $1$ nor $0$ hold for the 2nd. Thus I think that there is no such element.

Answer 2

I think the solution is not possible, though with you may have a solution to similar systems:

By 1), $A$ must be a function, i.e., a $0$-form, and , since the wedge of functions is their product, you must have $A==\pm 1$, But then, substituting in 2), you will have $\pm\eta+\pm\eta=\eta$, which is not possible.