Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 3 / 2 / a

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 $p:i\to j$,Consequently, for the finite vertex set used here,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.