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

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

$$ = \ \left \{ \begin{array}{ll} ( x | \alpha _ {1} \dots \alpha _ {l} , \alpha _ {1} ^ \prime \dots \alpha _ {l ^ \prime } ^ \prime | y ^ \prime ) & \textrm{ if } \ y = x ^ \prime , \\ 0 &{ \textrm{ if } y \neq x ^ \prime . } \\ \end{array} \right .$$

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 $ \mathop{\rm dim} V $ has, by definition, integral coordinates: $ ( \mathop{\rm dim} V) _ {x} = \mathop{\rm dim} _ {k} V _ {x} $ for $ x \in Q _ {0} $; and $ \sum _ {x \in Q _ {0} } ( \mathop{\rm dim} V ) _ {x} $ is called the dimension of $ V $. In case $ Q $ has no cyclic path, $ ( \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 of the form $ A _ {n} $, $ D _ {n} $, $ E _ {6} $, $ E _ {7} $, $ E _ {8} $, see [a4], [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 [a3], [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 [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 $ \mathop{\rm dim} V $ is a positive root for $ \Delta $, and all positive roots are obtained in this way; the number of isomorphism classes of indecomposable representations $ V $ with fixed $ \mathop{\rm dim} V $ depends on whether $ \mathop{\rm dim} V $ is a real root (then there is just one class) or an imaginary root [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, [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 $k$ as they arise in algebra, geometry and analysis. Typical tame quivers are the Kronecker quiver

$$ \circ \Rightarrow \circ , $$

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

$$ \begin{array}{ll} \circ \ &{} \\ \circ &{} \\ {} &\circ. \\ \circ &{} \\ \circ &{} \\ \end{array} $$

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

$$ \begin{array}{ll} \circ \ &{} \\ . &{} \\ . &\circ . \\ . &{} \\ \circ &{} \\ \end{array} $$

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].

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