path algebra of a quiver

Let Q=(Q0,Q1,s,t) be a quiver, i.e. Q0 is a set of vertices, Q1 is a set of arrows, s:Q1Q0 is a source function and t:Q1Q0 is a target function.

Recall that a path of length l1 from x to y in Q is a sequenceMathworldPlanetmath of arrows (a1,,al) such that


for any i=1,2,,l-1,l.

Also we allow paths of length 0, i.e. stationary paths.

If a=(a1,,al) and b=(b1,,bk) are two paths such that t(al)=s(b1) then we say that a and b are compatibile and in this case we can form another path from a and b, namely


Note, that the length of ab is a sum of lengths of a and b. Also a path a=(a1,,al) of positive length is called a cycle if t(al)=s(a1). In this case we can compose a with itself to produce new path.

Also if a is a path from x to y and ex,ey are stationary paths in x and y respectively, then we define aey=a and exa=a.

Let kQ be a vector space with a basis consisting of all paths (including stationary paths). For paths a and b define multiplicationPlanetmathPlanetmath as follows:

If a and b are compatible, then put ab=ab and put ab=0 otherwise. This operationMathworldPlanetmath extendes bilinearly to entire kQ and it can be easily checked that kQ becomes an associative algebra in this manner called the path algebraPlanetmathPlanetmath of Q over k.

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