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A quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q0768601.png" /> is given by two sets <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q0768602.png" /> and two mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q0768603.png" />; the elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q0768604.png" /> are called vertices or points, those of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q0768605.png" /> arrows; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q0768606.png" /> is an arrow, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q0768607.png" /> is called its start vertex, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q0768608.png" /> its end vertex, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q0768609.png" /> is said to go from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686010.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686011.png" />, written as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686012.png" />. (Thus, a quiver is nothing else than a directed graph with possibly multiple arrows and loops (cf. [[Graph, oriented|Graph, oriented]]), or a diagram scheme in the sense of A. Grothendieck; the word  "quiver"  is due to P. Gabriel.) Given a quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686013.png" />, there is the opposite quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686014.png" />, with the same set of vertices but with the reversed orientation for all the arrows.
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$#A+1 = 171 n = 0
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$#C+1 = 171 : ~/encyclopedia/old_files/data/Q076/Q.0706860 Quiver
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Given a quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686015.png" />, a path in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686016.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686017.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686018.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686019.png" /> are arrows with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686020.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686021.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686022.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686023.png" />; a path in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686024.png" /> of length 0 is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686025.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686026.png" />. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686027.png" /> is a path, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686028.png" /> is called its start vertex, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686029.png" /> its end vertex; paths <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686030.png" /> of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686031.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686032.png" /> are called cyclic paths.
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Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686033.png" /> be a field. The path algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686034.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686035.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686036.png" /> is the free vector space over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686037.png" /> with as basis the set of paths in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686038.png" />, and with distributive multiplication given on the basis by
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A quiver  $  Q = ( Q _ {0} , Q _ {1} , s, e) $
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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} $
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arrows; if  $  \alpha $
 +
is an arrow, then  $  s ( \alpha ) $
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is called its start vertex,  $  e ( \alpha ) $
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its end vertex, and  $  \alpha $
 +
is said to go from  $  s( \alpha ) $
 +
to  $  e ( \alpha ) $,
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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|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 ) $,
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with the same set of vertices but with the reversed orientation for all the arrows.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686039.png" /></td> </tr></table>
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Given a quiver  $  Q $,
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a path in  $  Q $
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of length  $  l \geq  1 $
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is of the form  $  ( x | \alpha _ {1} \dots \alpha _ {l} | y ) $,
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where  $  \alpha _ {i} $
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are arrows with  $  x = s( \alpha _ {1} ) $,
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$  e ( \alpha _ {i} ) = s( \alpha _ {i+} 1 ) $
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for  $  1 \leq  i < l $,
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and  $  e ( \alpha _ {l} ) = y $;
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a path in  $  Q $
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of length 0 is of the form  $  ( x \mid  x) $
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with  $  x \in Q _ {0} $.
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If  $  \omega = ( x | \alpha _ {1} \dots \alpha _ {l} | y ) $
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is a path, then  $  x = s( \omega ) $
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is called its start vertex,  $  y = e( \omega ) $
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its end vertex; paths  $  \omega $
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of length  $  \geq  1 $
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with  $  s( \omega ) = e( \omega ) $
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are called cyclic paths.
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686040.png" /></td> </tr></table>
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Let  $  k $
 +
be a field. The path algebra  $  kQ $
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of  $  Q $
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over  $  k $
 +
is the free vector space over  $  k $
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with as basis the set of paths in  $  Q $,
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and with distributive multiplication given on the basis by
  
The elements <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686041.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686042.png" /> are primitive and orthogonal idempotents, and in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686043.png" /> is finite, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686044.png" /> is the unit element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686045.png" />. Note that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686046.png" /> is finite-dimensional if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686047.png" /> is finite and has no cyclic path.
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$$
 +
( x | \alpha _ {1} \dots \alpha _ {l} | y) \cdot
 +
( x  ^  \prime  | \alpha _ {1}  ^  \prime  \dots \alpha _ {l  ^  \prime  }  ^  \prime  |
 +
y  ^  \prime  ) =
 +
$$
  
Recall that a ring of global dimension <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686048.png" /> is said to be hereditary, and a finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686049.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686050.png" /> with radical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686051.png" /> is said to be split basic provided <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686053.png" /> is a product of copies of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686054.png" />. The path algebras <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686055.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686056.png" /> a finite quiver without a cyclic path are precisely the finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686057.png" />-algebras which are hereditary and split basic.
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$$
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= \
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\left \{
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686058.png" /> be a quiver and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686059.png" /> a field. A representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686060.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686061.png" /> over <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686062.png" /> is given by a family of vector spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686063.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686064.png" />) and a family of linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686065.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686066.png" />). Given two representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686067.png" />, a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686068.png" /> is given by linear mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686069.png" /> such that for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686070.png" /> one has <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686071.png" />. Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686072.png" /> be finite. The category <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686073.png" /> of right <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686074.png" />-modules is equivalent to the category of representations of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686075.png" /> (provided one applies all the vector space mappings <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686076.png" />, as well as the module homomorphisms in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686077.png" />, on the right), and usually one identifies these categories. For any vertex <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686078.png" />, there is the one-dimensional representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686079.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686080.png" /> defined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686081.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686082.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686083.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686084.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686085.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686086.png" /> is equal to the number of arrows <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686087.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686088.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686089.png" />. Given a finite-dimensional representation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686090.png" />, its dimension vector <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686091.png" /> has, by definition, integral coordinates: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686092.png" /> for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686093.png" />; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686094.png" /> is called the dimension of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686095.png" />. In case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686096.png" /> has no cyclic path, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686097.png" /> is just the Jordan–Hölder multiplicity of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686098.png" /> in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q07686099.png" />.
+
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.
  
A finite quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860100.png" /> is called representation-finite, tame or wild if the path algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860101.png" /> has this property. A connected quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860102.png" /> is representation-finite if and only if the underlying graph <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860103.png" /> of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860104.png" /> (obtained from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860105.png" /> by deleting the orientation of the edges) is a [[Dynkin diagram|Dynkin diagram]] of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860106.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860107.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860108.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860109.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860110.png" />, see [[#References|[a4]]], [[#References|[a1]]]; and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860111.png" /> is tame if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860112.png" /> is of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860113.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860114.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860115.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860116.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860117.png" />, see [[#References|[a3]]], [[#References|[a8]]]. More precisely, recall that an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860118.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860119.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860120.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860121.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860122.png" /> is called a symmetric generalized Cartan matrix [[#References|[a6]]]. To a symmetric generalized Cartan <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860123.png" />-matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860124.png" /> one associates the following quiver <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860125.png" />: its set of vertices is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860126.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860127.png" /> one draws <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860128.png" /> arrows from <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860129.png" /> to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860130.png" />. Note that the quivers of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860131.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860132.png" /> a symmetric generalized Cartan matrix are precisely the quivers without a 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860133.png" /> be a symmetric generalized Cartan matrix. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860134.png" /> is an indecomposable representation of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860135.png" />, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860136.png" /> is a positive [[Root|root]] for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860137.png" />, and all positive roots are obtained in this way; the number of isomorphism classes of indecomposable representations <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860138.png" /> with fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860139.png" /> depends on whether <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860140.png" /> is a real root (then there is just one class) or an imaginary root [[#References|[a7]]].
+
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 $.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860141.png" /> be a quiver. A non-zero <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860142.png" />-linear combination of paths of length <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860143.png" /> with the same start vertex and the same end vertex is called a relation on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860144.png" />. Given a set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860145.png" /> of relations, let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860146.png" /> be the ideal in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860147.png" /> generated <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860148.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860149.png" /> is said to be an algebra defined by a quiver with relations. A finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860150.png" />-algebra <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860151.png" /> is isomorphic to one defined by a quiver with relations if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860152.png" /> is split basic. Thus, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860153.png" /> is algebraically closed, then any finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860154.png" />-algebra is Morita equivalent to one defined by a quiver with relations. All representation-finite and certain minimal representation-infinite <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860155.png" />-algebras over an algebraically closed field are defined by quivers with relations of the form <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860156.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860157.png" />, where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860158.png" /> 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]]).
+
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.
  
The representation theory of quivers has been developed in order to deal effectively with certain types of matrix problems over a fixed field <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860159.png" /> as they arise in algebra, geometry and analysis. Typical tame quivers are the Kronecker quiver
+
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]]].
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860160.png" /></td> </tr></table>
+
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]]).
  
its representations are just the matrix pencils (pairs of matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860161.png" /> of the same size, considered with respect to the equivalence relation: <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860162.png" /> if and only if there are invertible matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860163.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860164.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860165.png" />), and the four-subspace quiver
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860166.png" /></td> </tr></table>
+
$$
 +
\circ \  \  \circ ,
 +
$$
  
In general, the representation theory of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860168.png" />-subspace quiver
+
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
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860169.png" /></td> </tr></table>
+
$$
  
deals with the mutual position of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860170.png" />-subspaces in a vector space.
+
In general, the representation theory of the  $  n $-
 +
subspace quiver
  
Using the language of quivers, these problems are transformed to problems dealing with finite-dimensional split basic <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860171.png" />-algebras.
+
$$
  
In order to deal with an arbitrary finite-dimensional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860172.png" />-algebra one needs the notion of a species (instead of a quiver), see [[#References|[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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/q/q076/q076860/q076860173.png" />, the tame ones correspond to the Euclidean diagrams [[#References|[a9]]].
+
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 [[#References|[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 [[#References|[a9]]].
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Bautista,  P. Gabriel,  A. Rojter,  L. Salmeron,  "Representation-finite algebras and multiplicative basis"  ''Invent. Math.'' , '''81'''  (1985)  pp. 217–285</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Donovan,  M.R. Freislich,  "The representation of finite graphs and associated algebras"  ''Carleton Lecture Notes'' , '''5'''  (1973)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Gabriel,  "Unzerlegbare Darstellungen I"  ''Manuscripta Math.'' , '''6'''  (1972)  pp. 71–103</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Gabriel,  "Indecomposable representations II" , ''Symp. Math. INDAM (Rome, 1971)'' , '''XI''' , Acad. Press  (1973)  pp. 81–104</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  V.G. Kac,  "Infinite dimensional Lie algebras" , Cambridge Univ. Press  (1985)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  V.G. Kac,  "Infinite root systems, representations of graphs and invariant theory"  ''Invent. Math.'' , '''56'''  (1980)  pp. 57–92</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  V. Dlab,  C.M. Ringel,  "Indecomposable representations of graphs and algebras"  ''Memoirs Amer. Math. Soc.'' , '''173'''  (1976)</TD></TR></table>
 
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  R. Bautista,  P. Gabriel,  A. Rojter,  L. Salmeron,  "Representation-finite algebras and multiplicative basis"  ''Invent. Math.'' , '''81'''  (1985)  pp. 217–285</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  P. Donovan,  M.R. Freislich,  "The representation of finite graphs and associated algebras"  ''Carleton Lecture Notes'' , '''5'''  (1973)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  P. Gabriel,  "Unzerlegbare Darstellungen I"  ''Manuscripta Math.'' , '''6'''  (1972)  pp. 71–103</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  P. Gabriel,  "Indecomposable representations II" , ''Symp. Math. INDAM (Rome, 1971)'' , '''XI''' , Acad. Press  (1973)  pp. 81–104</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  V.G. Kac,  "Infinite dimensional Lie algebras" , Cambridge Univ. Press  (1985)</TD></TR><TR><TD valign="top">[a7]</TD> <TD valign="top">  V.G. Kac,  "Infinite root systems, representations of graphs and invariant theory"  ''Invent. Math.'' , '''56'''  (1980)  pp. 57–92</TD></TR><TR><TD valign="top">[a8]</TD> <TD valign="top">  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</TD></TR><TR><TD valign="top">[a9]</TD> <TD valign="top">  V. Dlab,  C.M. Ringel,  "Indecomposable representations of graphs and algebras"  ''Memoirs Amer. Math. Soc.'' , '''173'''  (1976)</TD></TR></table>

Revision as of 08:09, 6 June 2020


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)
How to Cite This Entry:
Quiver. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Quiver&oldid=48406
This article was adapted from an original article by C.M. Ringel (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article