Quiver

A quiver $ Q = ( Q _ {0} , Q _ {1} , s, e) $ is given by two sets $ Q _ {0} , Q _ {1} $ and two mappings $ s, e: Q _ {1} \rightarrow Q _ {0} $; the elements of $ Q _ {0} $ are called vertices or points, those of $ Q _ {1} $ arrows; if $ \alpha $ is an arrow, then $ s ( \alpha ) $ is called its start vertex, $ e ( \alpha ) $ its end vertex, and $ \alpha $ is said to go from $ s( \alpha ) $ to $ e ( \alpha ) $, written as $ \alpha : s( \alpha ) \rightarrow e ( \alpha ) $. (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 $ Q = ( Q _ {0} , Q _ {1} , s , e ) $, there is the opposite quiver $ Q ^ {*} = ( Q _ {0} , Q _ {1} , e, s ) $, with the same set of vertices but with the reversed orientation for all the arrows.

Given a quiver $ Q $, a path in $ Q $ of length $ l \geq 1 $ is of the form $ ( x | \alpha _ {1} \dots \alpha _ {l} | y ) $, where $ \alpha _ {i} $ are arrows with $ x = s( \alpha _ {1} ) $, $ e ( \alpha _ {i} ) = s( \alpha _ {i+} 1 ) $ for $ 1 \leq i < l $, and $ e ( \alpha _ {l} ) = y $; a path in $ Q $ of length 0 is of the form $ ( x \mid x) $ with $ x \in Q _ {0} $. If $ \omega = ( x | \alpha _ {1} \dots \alpha _ {l} | y ) $ is a path, then $ x = s( \omega ) $ is called its start vertex, $ y = e( \omega ) $ its end vertex; paths $ \omega $ of length $ \geq 1 $ with $ s( \omega ) = e( \omega ) $ are called cyclic paths.

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

$$ ( x | \alpha _ {1} \dots \alpha _ {l} | y) \cdot ( x ^ \prime | \alpha _ {1} ^ \prime \dots \alpha _ {l ^ \prime } ^ \prime | y ^ \prime ) = $$

$$ = \ \left \{ The elements $ ( x \mid x ) $ with $ x \in Q _ {0} $ are primitive and orthogonal idempotents, and in case $ Q _ {0} $ is finite, $ 1 = \sum _ {x \in Q _ {0} } ( x \mid x) $ is the unit element of $ kQ $. Note that $ k Q $ is finite-dimensional if and only if $ Q $ is finite and has no cyclic path. Recall that a ring of global dimension $ \leq 1 $ is said to be hereditary, and a finite-dimensional $ k $- algebra $ A $ with radical $ N $ is said to be split basic provided $ A/N $ is a product of copies of $ k $. The path algebras $ kQ $ with $ Q $ a finite quiver without a cyclic path are precisely the finite-dimensional $ k $- algebras which are hereditary and split basic. Let $ Q $ be a quiver and $ k $ a field. A representation $ V = ( V _ {x} , V _ \alpha ) $ of $ Q $ over $ k $ is given by a family of vector spaces $ V _ {x} $( $ x \in Q _ {0} $) and a family of linear mappings $ V _ \alpha : V _ {s( \alpha ) } \rightarrow V _ {e( \alpha ) } $( $ \alpha \in Q _ {1} $). Given two representations $ V, V ^ \prime $, a mapping $ f = ( f _ {x} ): V \rightarrow V ^ \prime $ is given by linear mappings $ f _ {x} : V _ {x} \rightarrow V _ {x} ^ \prime $ such that for any $ \alpha \in Q _ {1} $ one has $ f _ {s ( \alpha ) } V _ \alpha ^ \prime = V _ \alpha f _ {e( \alpha ) } $. Let $ Q $ be finite. The category $ \mathop{\rm mod} kQ $ of right $ kQ $- modules is equivalent to the category of representations of $ Q $( provided one applies all the vector space mappings $ V _ \alpha , f _ {x} $, as well as the module homomorphisms in $ \mathop{\rm mod} kQ $, on the right), and usually one identifies these categories. For any vertex $ x \in Q _ {0} $, there is the one-dimensional representation $ S( x) $ of $ Q $ defined by $ S( x) _ {x} = k $, $ S ( x) _ {y} = 0 $ for $ y \neq x \in Q _ {0} $ and $ S( x) _ \alpha = 0 $ for $ \alpha \in Q _ {1} $. Then $ \mathop{\rm dim} _ {k} \mathop{\rm Ext} ^ {1} ( S( i), S( j)) $ is equal to the number of arrows $ \alpha $ with $ s( \alpha ) = i $ and $ e ( \alpha ) = j $. Given a finite-dimensional representation $ V $, its dimension vector $ bold \mathop{\rm dim} V $ has, by definition, integral coordinates: $ ( bold \mathop{\rm dim} V) _ {x} = \mathop{\rm dim} _ {k} V _ {x} $ for $ x \in Q _ {0} $; and $ \sum _ {x \in Q _ {0} } ( bold \mathop{\rm dim} V ) _ {x} $ is called the dimension of $ V $. In case $ Q $ has no cyclic path, $ ( bold \mathop{\rm dim} V ) _ {x} $ is just the Jordan–Hölder multiplicity of $ S( x) $ in $ V $. A finite quiver $ Q $ is called representation-finite, tame or wild if the path algebra $ kQ $ has this property. A connected quiver $ Q $ is representation-finite if and only if the underlying graph $ \overline{Q}\; $ of $ Q $( obtained from $ Q $ by deleting the orientation of the edges) is a [[Dynkin diagram|Dynkin diagram]] of the form $ A _ {n} $, $ D _ {n} $, $ E _ {6} $, $ E _ {7} $, $ E _ {8} $, see [[#References|[a4]]], [[#References|[a1]]]; and $ Q $ is tame if and only if $ \overline{Q}\; $ is of the form $ {\widetilde{A} } _ {n} $, $ {\widetilde{D} } _ {n} $, $ {\widetilde{E} } _ {6} $, $ {\widetilde{E} } _ {7} $, $ {\widetilde{E} } _ {8} $, see [[#References|[a3]]], [[#References|[a8]]]. More precisely, recall that an $ ( n \times n ) $- matrix $ ( a _ {ij} ) _ {ij} $ with $ a _ {ii} = 2 $ and $ a _ {ij} = a _ {ji} \leq 0 $ for all $ i \neq j $ is called a symmetric generalized Cartan matrix [[#References|[a6]]]. To a symmetric generalized Cartan $ ( n \times n ) $- matrix $ \Delta = ( a _ {ij} ) _ {ij} $ one associates the following quiver $ Q ( \Delta ) $: its set of vertices is $ Q( \Delta ) _ {0} = \{ 1 \dots n \} $, and for $ 1 \leq i < j \leq n $ one draws $ - a _ {ij} $ arrows from $ i $ to $ j $. Note that the quivers of the form $ Q( \Delta ) $ with $ \Delta $ a symmetric generalized Cartan matrix are precisely the quivers without a cyclic path. Let $ \Delta $ be a symmetric generalized Cartan matrix. If $ V $ is an indecomposable representation of $ Q ( \Delta ) $, then $ bold \mathop{\rm dim} V $ is a positive [[Root|root]] for $ \Delta $, and all positive roots are obtained in this way; the number of isomorphism classes of indecomposable representations $ V $ with fixed $ bold \mathop{\rm dim} V $ depends on whether $ bold \mathop{\rm dim} V $ is a real root (then there is just one class) or an imaginary root [[#References|[a7]]]. Let $ Q $ be a quiver. A non-zero $ k $- linear combination of paths of length $ \geq 2 $ with the same start vertex and the same end vertex is called a relation on $ Q $. Given a set $ \{ \rho _ {i} \} _ {i} $ of relations, let $ \langle \rho _ {i} \mid i \rangle $ be the ideal in $ kQ $ generated $ \{ \rho _ {i} \} _ {i} $. Then $ A = kQ / \langle \rho _ {i} \mid i \rangle $ is said to be an algebra defined by a quiver with relations. A finite-dimensional $ k $- algebra $ A $ is isomorphic to one defined by a quiver with relations if and only if $ A $ is split basic. Thus, if $ k $ is algebraically closed, then any finite-dimensional $ k $- algebra is Morita equivalent to one defined by a quiver with relations. All representation-finite and certain minimal representation-infinite $ k $- algebras over an algebraically closed field are defined by quivers with relations of the form $ \omega $, and $ \omega _ {1} - \omega _ {2} $, where $ \omega , \omega _ {1} , \omega _ {2} $ are paths (the multiplicative basis theorem, [[#References|[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|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 $ k $ as they arise in algebra, geometry and analysis. Typical tame quivers are the Kronecker quiver $$ \circ \ \ \circ , $$ its representations are just the matrix pencils (pairs of matrices $ A , B $ of the same size, considered with respect to the equivalence relation: $ ( A, B) \sim ( A ^ \prime , B ^ \prime ) $ if and only if there are invertible matrices $ P , Q $ with $ A ^ \prime = PAQ $, $ B ^ \prime = PBQ $), and the four-subspace quiver $$

In general, the representation theory of the $ n $- subspace quiver

$$

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

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

In order to deal with an arbitrary finite-dimensional $ k $- 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 $ ( A _ {n} , B _ {n} , C _ {n} \dots G _ {2} ) $, 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)