= Solution
The <path algebra> $kQ$ is the $k$-vector space with basis all directed paths, including the length-zero path $e_i$ at each vertex. Extend path composition bilinearly, declaring the product zero when the paths cannot be composed. We use the left-module convention: $pq$ means first traverse $q$, then $p$; for an arrow $\rho:i\to j$, $e_j\rho e_i=\rho$.
The constant paths are mutually orthogonal <idempotents>. For any path $p:i\to j$,
$$
e_jp=p=pe_i,\qquad e_\ell p=0\ (\ell\ne j),\qquad pe_\ell=0\ (\ell\ne i).
$$
Consequently, for the finite vertex set used here,
$$
\boxed{1_{kQ}=\sum_{i\in Q_0}e_i.}
$$
It acts as an identity on every basis path, hence on the entire <path algebra>. Finiteness of the vertex set is what makes this a genuine element of the algebra; an infinite-vertex path algebra instead has finite vertex sums as local units.
Back to article page
