Background and motivation: I'm wondering about the existence of an algebra which is in some ways similar to the exterior algebra, but is generated by idempotents rather than nilpotents.
Let $V$ be a finite-dimensional real vector space and let $\{e_i\}$ be a basis of $V$. Given that the exterior algebra $\Lambda(V)$ associated to a vector space is the algebra $T(V)/I$, where $T(V)$ is the tensor algebra associated with $V$ and $I$ is the ideal generated by elements of the form $x\otimes x$, where $x\in V$, it then follows that the inclusion of $V$ in $\Lambda(V)$ is an injection. Let $n_i$ denote the image of $e_i$ under this inclusion. It then follows that the elements $n_i$ satisfy the anticommutation relations $\{n_i, n_j\} = 0$.
I am curious about the existence or non-existence of an algebra $A(V)$ associated with $V$ which satisfies the following properties: first, as with the exterior algebra, it is possible to inject $V$ into $A(V)$. Second, if $p_i$ is the image of $e_i$ under this injection, then the elements $p_i$ satisfy the anticommutation relations $\{p_i, p_j\} = \delta_{ij} p_i$. If such an algebra exists, by universality of the tensor algebra $T(V)$ it should be isomorphic to $T(V)/J$ for some ideal $J$. What would be this ideal J?
$J$ would be the ideal of $T(V)$ generated by elements of the following set $\{p_i\otimes p_j+p_j\otimes p_i\mid i\neq j, i,j\in I\}\cup\{(p_i\otimes p_i)-p_i \mid i\in I \}$
This effectively forces the relations you seek to hold for the $p_i$.
(I haven't checked the details that $p_i\mapsto p_i$ is an injection, but my gut says it is so, since none of the relations look like they can introduce linear dependence among the $p_i$.)