path algebra of a disconnected quiver

Let Q be a disconnected quiver, i.e. Q can be written as a disjoint unionMathworldPlanetmathPlanetmath of two quivers Q and Q′′ (which means that there is no path starting in Q and ending in Q′′ and vice versa) and let k be an arbitrary field.

PropositionPlanetmathPlanetmath. The path algebraMathworldPlanetmathPlanetmath kQ is isomorphicPlanetmathPlanetmathPlanetmath to the productMathworldPlanetmathPlanetmathPlanetmath of path algebras kQ×kQ′′.

Proof. If w is a path in Q, then w belongs either to Q or Q′′. Define linear map


by T(w)=(w,0) if wQ or T(w)=(0,w) if wQ′′ and extend it linearly to entire kQ. We will show that T is an isomorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmath of algebras.

If w,w are paths in Q, then since Q and Q′′ are disjoint, then each of them entirely lies in Q or Q′′. Now since Q and Q′′ don’t have common vertices it follows that ww=ww=0. Without loss of generality we may assume, that w is in Q and w is in Q′′. Then we have


If both lie in the same component, for example in Q, then


Since T preservers multiplication on paths, then T preserves multiplication and thus T is an algebra homomorphism.

Obviously by definition T is 1-1.

It remains to show, that T is onto. Assume that (a,b)kQkQ′′. Then we can write


where vi are paths in Q and wj are paths in Q′′. It can be easily checked, that


Here we consider all vi and wj as paths in Q.

Thus T is an isomorphism, which completesPlanetmathPlanetmathPlanetmathPlanetmathPlanetmath the proof.

Title path algebra of a disconnected quiver
Canonical name PathAlgebraOfADisconnectedQuiver
Date of creation 2013-03-22 19:16:25
Last modified on 2013-03-22 19:16:25
Owner joking (16130)
Last modified by joking (16130)
Numerical id 4
Author joking (16130)
Entry type Theorem
Classification msc 14L24