unital path algebras
Let be a quiver and an arbitrary field.
where and are paths (they form a basis of as a vector space). Since has an infinite number of vertices, then we can take a stationary path for some vertex such that there is no path among ending in . By definition of and by the fact that is an identity we have:
,,” If the set of vertices of is finite, then put
where denotes the stationary path (note that is well-defined, since the sum is finite). If is a path in from to , then and . All other combinations of with yield and thus we obtain that
This completes the proof.
|Title||unital path algebras|
|Date of creation||2013-03-22 19:16:23|
|Last modified on||2013-03-22 19:16:23|
|Last modified by||joking (16130)|