Given a Path Algebra $\mathbb{C}Q$ of an acyclic directed graph(also called a Quiver in this theory) $Q$. I am interested in finding a corresponding matrix algebra for it if possible.
I am able to do this for all Quivers $Q$ for which for any two vertices $i$ and $j$ there exists at most one directed path from $i$ to $j$. Say there are $n$ vertices.
The result I got is for such quivers: $$\mathbb{C}Q \cong M$$ where $M$ is the algebra of $n \times n$ complex matrices for which the entries $(l, m)$(lth row and mth column) is $0$ if there is no directed path from vertex $l$ to $m$. All other entries can range freely. The operations are as usual.
Note that my quiver could have undirected cycles. An example is: For the following quiver
The algebra is:
\begin{bmatrix} K & K & K & K & 0 & 0 & 0 & 0 & 0 \\ 0 & K & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & K & 0 & 0 & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & K & K & K & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & K & 0 & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & K & 0 & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & K & 0 & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & K & 0 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & K \\ \end{bmatrix}
But I dont know how to deal with quivers which have no directed cycles (else the algebra is infinite dimensional) but may have mulitple directed paths from some vertex to another.
**Note:**The path algebra splits into direct sums of path algebras for the connected components and the issue im facing about multiple paths between two vertices is also relevant inside component so it is not a loss of generality to assume that the direct graph is connected.
Update 2: I think I now have a an answer using tensor products
Let $Q$ be the underlying quiver. Let $V_{i, j}$ be the vector space with basis given by paths from $i$th vertex to $j$th vertex.
Then the path algebra of $Q$ is isomorphic to the algebra of all matrices with $i, j$th entry coming from $V_{i, j}$ and the product in the matrix multiplication is by concatenation of paths.
Everything can be said in terms of tensor products when the path algebra is identified with the Tensor algebra. But then I see that all this is just bookeeping.
Thanks.