Quiver (mathematics)

Bol

Please note that the content of this book primarily consists of articles available from Wikipedia or other free sources online. In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed, i.e. a multidigraph. They are commonly used in representation theory: a representation, V, of a quiver assigns a vector space V(x) to each vertex x of the quiver and a linear map V(a) to each arrow a. If K is a field and ¿ is a quiver, then the quiver algebra or path algebra K¿ is defined as follows. A path in ¿ is a sequence of arrows a1 a2 a3 ... an such that the head of ai+1 = tail of ai, using the convention of concatenating paths from right to left. Then the path algebra is a vector space having all the paths (of length ¿ 0) in the quiver as basis, and multiplication given by concatenation of paths. If two paths cannot be concatenated because the end vertex of the first is not equal to the starting vertex of the second, their product is defined to be zero. This defines an associative algebra over K. This algebra has a unit element if and only if the quiver has only finitely many vertices. In this case, the modules over K¿ are naturally identified with the representations of ¿.

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Pagina's: 88, Paperback, Betascript Publishers