Quiver (mathematics)

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In mathematics, a quiver is a directed graph where loops and multiple arrows between two vertices are allowed. 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 Γ is a quiver, then the quiver algebra or path algebra KΓ is defined as follows: it is the vector space having all the paths in the quiver as basis; multiplication is 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 KΓ are naturally identified with the representations of Γ.

If the quiver has finitely many vertices and arrows and the end vertex and starting vertex of any path are always distinct, then KΓ is a finite-dimensional hereditary algebra over K, i.e. submodules of projective modules over KΓ are projective.

1 Representations of Quivers
2 Gabriel's Theorem
3 See also
4 References

[edit] Representations of Quivers

A representation of a quiver, Q, is said to be trivial if V(x)=0 for all vertices x in Q.

A morphism, f:V->V', between representations of the quiver Q, is a collection of linear maps $f(x):V(x)\rightarrow V'(x)$ such that for every arrow in Q from x to y $V'(a) f (x) = f (y) V (a)$ . A morphism, f, is an isomorphism, if f(x) is invertible for all vertices x in the quiver. With these definitions the representations of a quiver form a category.

If V and W are representations of a quiver Q, then the direct sum of these representations, $V\oplus W$ , is defined by $(V\oplus W)(x)=V(x)\oplus W(x)$ for all vertices x in Q and $(V\oplus W)(a)$ is the direct sum of the linear mappings V(a) and W(a).

A representation is said to be decomposable if its isomorphic to the direct sum of non-zero representations.

[edit] Gabriel's Theorem

A quiver is of finite type if it has finitely many non-isomorphic indecomposable representations. Gabriel's theorem classifies all quiver representations of finite type. More precisely, it states that:

A quiver is of finite type if and only if its underlying graph (when the directions of the arrows are ignored) is one of the following Dynkin diagrams: $A n$ , $D n$ , $E 6$ , $E 7$ , $E 8$ .
These indecomposable representations are in a one-to-one correspondence with the positive roots of the root system of the Dynkin diagram.