Quiver (mathematics)

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 "K"Γ is defined as follows. A path in "Q" is a sequence of arrows a₁ a₂ a₃ ... a_n such that the head of a_i+1 = tail of a_i, using the convention of concatenating paths from right to left. Then the path algebra is a vector space having all the paths 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"Γ 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 (i.e. Q has no oriented cycles), then "K"Γ is a finite-dimensional hereditary algebra over "K".

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 $f(x):V(x)\; ightarrow\; V\text{'}(x)$ such that for every arrow in Q from x to y $V\text{'}(a)f(x)=f(y)V(a)$, i.e. the squares that f forms with the arrows of V and V' all commute. 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, $Voplus\; W$, is defined by $(Voplus\; W)(x)=V(x)oplus\; W(x)$ for all vertices x in Q and $(Voplus\; W)(a)$ is the direct sum of the linear mappings V(a) and W(a).

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

A categorical definition of a quiver representation can also be given. The quiver itself can be considered a category, where the vertices are objects and paths are morphisms. Then a representation of Q is just a covariant functor from this category to the category of finite dimensional vector spaces.

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 (connected) 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: $A\_n$, $D\_n$, $E\_6$, $E\_7$, $E\_8$.
# The indecomposable representations are in a one-to-one correspondence with the positive roots of the root system of the Dynkin diagram.

See also

* Quiver diagram
* ADE classification

References

* [http://www.ams.org/notices/200502/fea-weyman.pdf Quiver Representations] , Harm Derksen and Jerzy Weyman, AMS Notices
* [http://www.amsta.leeds.ac.uk/~pmtwc/quivlecs.pdf Notes on Quiver Representations]
* [http://www.arxiv.org/pdf/math/0505082 Finite-dimensional algebras and quivers] , Alistair Savage, Encyclopedia of Mathematical Physics, eds. J.-P. Françoise, G.L. Naber and Tsou S.T. Oxford: Elsevier, 2006, volume 2, pages 313-320
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