Quiver (mathematics)

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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 Γ is a quiver, then the quiver algebra or path algebra KΓ is defined as follows. A path in Γ 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 (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 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 (i.e. Q has no oriented cycles), then KΓ is a finite-dimensional hereditary algebra over K.

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

[edit] Representations of Quivers

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

A morphism, ƒ:V→V', between representations of the quiver Q, is a collection of linear maps such that for every arrow in Q from x to y V'(a)f(x) = f(y)V(a), i.e. the squares that ƒ forms with the arrows of V and V' all commute. A morphism, ƒ, is an isomorphism, if ƒ(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, , is defined by for all vertices x in Q and 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.

For a finite quiver Γ (a quiver with finitely many vertices and edges), let KΓ be its path algebra. Let e_i denote the trivial path at vertex i. Then we can associate to the vertex i, the projective KΓ module KΓe_i consisting of linear combinations of paths which have starting vertex i. This corresponds to the representation of Γ obtained by putting a copy of K at each vertex which lies on a path starting at i and 0 on each other vertex. To each edge joining two copies of K we associate the identity map.

[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:

1. 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, E₆, E₇, E₈.
2. The indecomposable representations are in a one-to-one correspondence with the positive roots of the root system of the Dynkin diagram.

[edit] See also

• ADE classification
• Graph algebra
• Group algebra
• Incidence algebra
• Path algebra
• Quiver diagram

[edit] References

• Quiver Representations, Harm Derksen and Jerzy Weyman, AMS Notices
• Notes on Quiver Representations, William Crawley-Boevey
• 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
• Simson, Daniel; Skowronski, Andrzej; Assem, Ibrahim (2007), Elements of the Representation Theory of Associative Algebras, Cambridge University Press, ISBN 978-0521882187 .