Quiver

A quiver is given by two sets and two mappings ; the elements of are called vertices or points, those of arrows; if is an arrow, then is called its start vertex, its end vertex, and is said to go from to , written as . (Thus, a quiver is nothing else than a directed graph with possibly multiple arrows and loops (cf. Graph, oriented), or a diagram scheme in the sense of A. Grothendieck; the word "quiver" is due to P. Gabriel.) Given a quiver , there is the opposite quiver , with the same set of vertices but with the reversed orientation for all the arrows.

Given a quiver , a path in of length is of the form , where are arrows with , for , and ; a path in of length 0 is of the form with . If is a path, then is called its start vertex, its end vertex; paths of length with are called cyclic paths.

Let be a field. The path algebra of over is the free vector space over with as basis the set of paths in , and with distributive multiplication given on the basis by

The elements with are primitive and orthogonal idempotents, and in case is finite, is the unit element of . Note that is finite-dimensional if and only if is finite and has no cyclic path.

Recall that a ring of global dimension is said to be hereditary, and a finite-dimensional -algebra with radical is said to be split basic provided is a product of copies of . The path algebras with a finite quiver without a cyclic path are precisely the finite-dimensional -algebras which are hereditary and split basic.

Let be a quiver and a field. A representation of over is given by a family of vector spaces () and a family of linear mappings (). Given two representations , a mapping is given by linear mappings such that for any one has . Let be finite. The category of right -modules is equivalent to the category of representations of (provided one applies all the vector space mappings , as well as the module homomorphisms in , on the right), and usually one identifies these categories. For any vertex , there is the one-dimensional representation of defined by , for and for . Then is equal to the number of arrows with and . Given a finite-dimensional representation , its dimension vector has, by definition, integral coordinates: for ; and is called the dimension of . In case has no cyclic path, is just the Jordan–Hölder multiplicity of in .

A finite quiver is called representation-finite, tame or wild if the path algebra has this property. A connected quiver is representation-finite if and only if the underlying graph of (obtained from by deleting the orientation of the edges) is a Dynkin diagram of the form , , , , , see [a4], [a1]; and is tame if and only if is of the form , , , , , see [a3], [a8]. More precisely, recall that an -matrix with and for all is called a symmetric generalized Cartan matrix [a6]. To a symmetric generalized Cartan -matrix one associates the following quiver : its set of vertices is , and for one draws arrows from to . Note that the quivers of the form with a symmetric generalized Cartan matrix are precisely the quivers without a cyclic path.

Let be a symmetric generalized Cartan matrix. If is an indecomposable representation of , then is a positive root for , and all positive roots are obtained in this way; the number of isomorphism classes of indecomposable representations with fixed depends on whether is a real root (then there is just one class) or an imaginary root [a7].

Let be a quiver. A non-zero -linear combination of paths of length with the same start vertex and the same end vertex is called a relation on . Given a set of relations, let be the ideal in generated . Then is said to be an algebra defined by a quiver with relations. A finite-dimensional -algebra is isomorphic to one defined by a quiver with relations if and only if is split basic. Thus, if is algebraically closed, then any finite-dimensional -algebra is Morita equivalent to one defined by a quiver with relations. All representation-finite and certain minimal representation-infinite -algebras over an algebraically closed field are defined by quivers with relations of the form , and , where are paths (the multiplicative basis theorem, [a2]); this shows that the study of representation-finite algebras is a purely combinatorial problem; it was a decisive step for the proof of the second Brauer–Thrall conjecture (see Representation of an associative algebra).

The representation theory of quivers has been developed in order to deal effectively with certain types of matrix problems over a fixed field as they arise in algebra, geometry and analysis. Typical tame quivers are the Kronecker quiver

its representations are just the matrix pencils (pairs of matrices of the same size, considered with respect to the equivalence relation: if and only if there are invertible matrices with , ), and the four-subspace quiver

In general, the representation theory of the -subspace quiver

deals with the mutual position of -subspaces in a vector space.

Using the language of quivers, these problems are transformed to problems dealing with finite-dimensional split basic -algebras.

In order to deal with an arbitrary finite-dimensional -algebra one needs the notion of a species (instead of a quiver), see [a5]. In this way, one deals with vector space problems which involve different fields. The representation-finite species are those corresponding to arbitrary Dynkin diagrams , the tame ones correspond to the Euclidean diagrams [a9].

References

[a1]	I.N. Bernstein, I.M. Gel'fand, V.A. Ponomarev, "Coxeter functors and Gabriel's theorem" Russian Math. Surveys , 28 : 2 (1973) pp. 17–32 Uspekhi Mat. Nauk , 28 : 2 (1973) pp. 19–34
[a2]	R. Bautista, P. Gabriel, A. Rojter, L. Salmeron, "Representation-finite algebras and multiplicative basis" Invent. Math. , 81 (1985) pp. 217–285
[a3]	P. Donovan, M.R. Freislich, "The representation of finite graphs and associated algebras" Carleton Lecture Notes , 5 (1973)
[a4]	P. Gabriel, "Unzerlegbare Darstellungen I" Manuscripta Math. , 6 (1972) pp. 71–103
[a5]	P. Gabriel, "Indecomposable representations II" , Symp. Math. INDAM (Rome, 1971) , XI , Acad. Press (1973) pp. 81–104
[a6]	V.G. Kac, "Infinite dimensional Lie algebras" , Cambridge Univ. Press (1985)
[a7]	V.G. Kac, "Infinite root systems, representations of graphs and invariant theory" Invent. Math. , 56 (1980) pp. 57–92
[a8]	L.A. Nazarova, "Representations of quivers of infinite type" Math. USSR Izv. , 7 (1973) pp. 749–792 Izv. Akad. Nauk SSSR Ser. Mat. , 37 (1973) pp. 752–791
[a9]	V. Dlab, C.M. Ringel, "Indecomposable representations of graphs and algebras" Memoirs Amer. Math. Soc. , 173 (1976)