unital path algebras

Let Q be a quiver and k an arbitrary field.

PropositionPlanetmathPlanetmath. The path algebra kQ is unitary if and only if Q has a finite number of vertices.

Proof. ,,” Assume, that Q has an infiniteMathworldPlanetmath number of vertices and let 1kQ be an identityPlanetmathPlanetmathPlanetmathPlanetmath. Then we can express 1 as


where λnk and wn are paths (they form a basis of kQ as a vector space). Since Q has an infinite number of vertices, then we can take a stationary path ex for some vertex x such that there is no path among w1,,wn ending in x. By definition of kQ and by the fact that 1 is an identity we have:



,,” If the set Q0 of vertices of Q is finite, then put


where eq 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 exw=w and wey=w. All other combinationsMathworldPlanetmathPlanetmath of w with eq yield 0 and thus we obtain that


This completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

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