unital path algebras

Let $Q$ be a quiver and $k$ an arbitrary field.

Proposition. The path algebra $k Q$ is unitary if and only if $Q$ has a finite number of vertices.

Proof. ,, $\Rightarrow$ ” Assume, that $Q$ has an infinite number of vertices and let $1\in kQ$ be an identity. Then we can express $1$ as

1=\sum_{i=1}^{n}\lambda_{n}\cdot w_{n}

where $\lambda_{n}\in k$ and $w_{n}$ are paths (they form a basis of $k Q$ as a vector space). Since $Q$ has an infinite number of vertices, then we can take a stationary path $e_{x}$ for some vertex $x$ such that there is no path among ${w_{1},\ldots,w_{n}}$ ending in $x$ . By definition of $k Q$ and by the fact that $1$ is an identity we have:

e_{x}=1\cdot e_{x}=\left(\sum_{i=1}^{n}\lambda_{n}\cdot w_{n}\right)\cdot e_{x% }=\sum_{i=1}^{n}\lambda_{n}\cdot(w_{n}\cdot e_{x})=\sum_{i=1}^{n}\lambda_{n}% \cdot 0=0.

Contradiction. $\square$

,, $\Leftarrow$ ” If the set $Q_{0}$ of vertices of $Q$ is finite, then put

1=\sum_{q\in Q_{0}}e_{q}

where $e_{q}$ denotes the stationary path (note that $1$ is well-defined, since the sum is finite). If $w$ is a path in $Q$ from $x$ to $y$ , then $e_{x}\cdot w=w$ and $w\cdot e_{y}=w$ . All other combinations of $w$ with $e_{q}$ yield $0$ and thus we obtain that

1\cdot w=w=w\cdot 1.

This completes the proof. $\square$

Title	unital path algebras
Canonical name	UnitalPathAlgebras
Date of creation	2013-03-22 19:16:23
Last modified on	2013-03-22 19:16:23
Owner	joking (16130)
Last modified by	joking (16130)
Numerical id	4
Author	joking (16130)
Entry type	Theorem
Classification	msc 14L24