Path algebra

In mathematics Wegealgebren provide a way to be interpreted as representations of tubulars modules and thus to transfer results, which are known for the modules, and on representations of tubulars. This follows, for example, that every finite- dimensional representation of a finite quiver without oriented loops is isomorphic to a direct sum of indecomposable representations ( by simple application of the theorem of Krull - Remak - Schmidt).

A path ( or path ) in a boiler ( = directed graph ) is a sequence of arrows, so that the tip of the arrow -th is the start of th arrow, wherein paths are successively hung from the right to the left.

The paths algebra ( or path algebra) to be an algebra over a field and defined as follows: As a vector space is the vector space as the base has all the way in, with the multiplication of two paths is given as a series connection of the way if they are stuck together can. Is the end of a path does not match the beginning of the path, so is the product is equal to zero. (Note that the " stopping" at a point defines a way, the trivial way to the point. )

Thus we obtain an associative algebra over the field. The algebra then has exactly one identity element, if the quiver has only a finite number of points, namely the sum of all trivial paths to all points. In this case, one can identify the moduli in a natural way with representations of quivers.

If the quiver only a finite number of points and arrows and there are no oriented circles in him, is a finite-dimensional hereditary algebra.