Isomorphism between path algebra and direct sum of path algebras of it connected components

Question

I search for a reference that point the isomorphism between path algebra and direct sum of path algebras of it connected component if exists. In other words, I search for a reference that proves:

Let $F$ denote a field.

Let $E$ be a finite acyclic quiver and let $C_1, \dots, C_n$ be connected components of $E$ where $n$ is a positive integer. Then, $FE \cong \bigoplus_i FC_i$.

Thanks.

Answer 1

You mentioned "bound quiver algebra" in your comment, but I am not familiar with these. Here are some more details on your initial question.

Let $E$ be any quiver/(directed multi-)graph. Let $E^0$ and $E^1$ be the sets of vertices and arrows of $E$, and $s,r\colon E^1\to E^0$ the source and range maps.

By a path of length $n$ in $E$ I will mean a finite string $e_1\ldots e_n$ of edges such that $s(e_{i+1})=r(e_i)$. Source and range extend to paths as $$s(e_1\cdots e_n)=s(e_1)\text{ and }r(e_1\cdots e_n)=r(e_n).$$ We also add vertices as possible paths of "length zero", and the source and range of these paths is that same vertex

The path algebra $FE$ is defined as the free $F$-algebra generated by paths in $E$ subject to the relations that state that the product $\mu\cdot\nu$ of two paths is just the concatenated path $\mu\nu$ if $r(\mu)=s(\nu)$, and $0$ otherwise.

Now for the isomorphism: Let $\left\{C_i\right\}_i$ be the connected components of $E$.

Given a path $\mu$ in $E$, the source of $\mu$ is a vertex in precisely one of the $C_{i(\mu)}$. Define $\phi(\mu)=\left(\phi(\mu)_i\right)\in\bigoplus_j FC_j$ as

• $\phi(\mu)_{i(\mu)}$ being the element of $C_{i(\mu)}$ corresponding to $\mu$;
• $\phi(\mu)_j=0$ in $C_j$ for $j\neq i(\mu)$;

Prove that $\phi(\mu)\cdot\phi(\nu)=\phi(\mu\nu)$ (where the dot means product in the algebra) for paths $\mu\nu$ which can be concatenated, and $0$ for paths which cannot be concatenated.

By the universal property of algebras generated by sets and relations we e "extend" $\phi$ to an $F$-algebra morphism $\phi\colon FE\to\bigoplus_j FC_j$.

The inverse of $\phi$ is constructed in a similar manner: First find algebra morphisms $\psi_j\colon FC_j\to FE$, by "reinterpreting paths in $C_j$ as paths in $E$"; use the universal property of direct sums to obtain a specific morphism $\psi\colon\bigoplus_j FC_j\to FE$; and then prove that $\phi$ and $\psi$ are inverse to each other by proving that $\phi\circ\psi$ and $\psi\circ\psi$ take paths back to themselves.