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
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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
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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
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In general, the representation theory of the -subspace quiver
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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) |
Quiver. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quiver&oldid=49385