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

There is one constant path and $r$ loop generators; every other path is a word in those loops. Distinct words are distinct basis paths, so there are no relations among the generators. Thereforethe free associative algebra on $r$ generators, with each loop sent to its corresponding generator. For $r=1$ this is $k[x_1]$; for $r\ge 2$ it is noncommutative. The quiver with one loop is the $r=1$ case of the quiver with r loops.

Number the source vertices $1,\dots ,r$ and the sink $c=r+1$, and write $a_i:i\to c$ for the arrows. The only basis paths are $e_1,\dots ,e_r,e_c,a_1,\dots ,a_r$. Using standard matrix units, defineThe relations $e_c{a}_i=a_i=a_i{e}_i$, orthogonality of the vertex idempotents, and $a_i{a}_j=0$ agree exactly with multiplication of these matrix units. The map is therefore an algebra homomorphism, and its images form a basis of the algebraThis is an explicit isomorphism of the path algebra of the subspace quiver with a $(2r+1)$-dimensional triangular matrix algebra.