path algebra of a quiver
Let be a quiver, i.e. is a set of vertices, is a set of arrows, is a source function and is a target function.
Recall that a path of length from to in is a sequence of arrows such that
for any .
Also we allow paths of length , i.e. stationary paths.
If and are two paths such that then we say that and are compatibile and in this case we can form another path from and , namely
Note, that the length of is a sum of lengths of and . Also a path of positive length is called a cycle if . In this case we can compose with itself to produce new path.
Also if is a path from to and are stationary paths in and respectively, then we define and .
If and are compatible, then put and put otherwise. This operation extendes bilinearly to entire and it can be easily checked that becomes an associative algebra in this manner called the path algebra of over .
|Title||path algebra of a quiver|
|Date of creation||2013-03-22 19:16:19|
|Last modified on||2013-03-22 19:16:19|
|Last modified by||joking (16130)|