"Monoidal categorifications of cluster algebras"의 두 판 사이의 차이

2011년 4월 13일 (수) 04:39 판

quiver : oriented graph
represetation of a quiver : collection of vector space and linear maps between them
homomorphism of 2 quiver representations
path algebra of a quiver
- given a quiver Q, a path p is a sequence of arrows with some conditions
- path algebra : set of all k-linear combinations of all paths (including e_i's)
- p_1p_2 will correspond to a composition \(p_2\circ p_1\) of two maps (U\overset{P_2}{\rightarrow }V\overset{P_1}{\rightarrow }W)
quiver representation is in fact, a representaion of path algebra of a quiver
quiver has finite type of there are finitely many indecomposables

\thm (Gabriel)

A connected quiver Q has finite type iff corresponding graph is Dynking diagram (A,D,E)

outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams

Keller, Bernhard. 2008. “Cluster algebras, quiver representations and triangulated categories”. 0807.1960 (7월 12). http://arxiv.org/abs/0807.1960.

@@ 13번째 줄: / 13번째 줄: @@
 * homomorphism of 2 quiver representations
 *  path algebra of a quiver<br>
-** given a quiver Q, a path p is a sequence of arrows with soem
+** given a quiver Q, a path p is a sequence of arrows with some conditions
-** set of all k-linear combinations of all paths (including e_i's)
+** path algebra : set of all k-linear combinations of all paths (including e_i's)
-**
+** p_1p_2 will correspond to a composition <math>p_2\circ p_1</math> of two maps (U\overset{P_2}{\rightarrow }V\overset{P_1}{\rightarrow }W)
 * quiver representation is in fact, a representaion of path algebra of a quiver
+* quiver has finite type of there are finitely many indecomposables
+\thm (Gabriel)
+A connected quiver Q has finite type iff corresponding graph is Dynking diagram (A,D,E)