Path algebra

In mathematics, path algebras provide a way of interpreting representations of quivers as modules and thus transferring results that are known for modules to representations of quivers. It follows, for example, that every finite-dimensional representation of a finite quiver without oriented circles is isomorphic to a direct sum of indecomposable representations (by simply applying the Krull-Remak-Schmidt theorem ).

A path (or path ) in a quiver (= directed graph) is a sequence of arrows , so that the tip of the -th arrow forms the beginning of the -th arrow, with paths being placed one behind the other from right to left. ${\displaystyle Q}$ ${\displaystyle a_{1}a_{2}a_{3}\ldots a_{n}}$ ${\displaystyle a_{i}}$ ${\displaystyle (i+1)}$ ${\displaystyle i}$

The path algebra (or path algebra ) zu is an algebra over a field and is defined as follows: As a vector space, it is the vector space that has all paths in as a basis , where the multiplication of two paths is given as a series of paths if they are connected to one another can. If the end of a path does not coincide with the beginning of the path , the product is set equal to zero. (Note that "stopping" at a point also defines a path, the trivial path to the point .) ${\displaystyle KQ}$ ${\displaystyle Q}$ ${\displaystyle K}$ ${\displaystyle K}$ ${\displaystyle Q}$ ${\displaystyle \beta }$ ${\displaystyle \alpha }$ ${\displaystyle \alpha \beta }$ ${\displaystyle i}$ ${\displaystyle i}$

This gives an associative algebra over the body . Algebra has a unit element if and only if the quiver has finitely many points, namely the sum of all trivial paths to all points. In this case one can identify the modules naturally with representations of quivers. ${\displaystyle K}$ ${\displaystyle Q}$ ${\displaystyle KQ}$

If the quiver has only a finite number of points and arrows and there are no oriented circles in it, then a finite- dimensional hereditary algebra is over . ${\displaystyle Q}$ ${\displaystyle KQ}$ ${\displaystyle K}$

literature

• M. Auslander, I. Riding, S. Smalø: Representation theory of Artin algebras. Cambridge Studies in Advanced Mathematics, 36. Cambridge University Press, Cambridge, 1997. xiv + 425 pp. ISBN 0-521-41134-3 ; ISBN 0-521-59923-7

Web links

• W. Crawley-Boevey, Lectures on Representations of Quivers