"Monoidal categorifications of cluster algebras"의 두 판 사이의 차이
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| 15번째 줄: | 15번째 줄: | ||
** given a quiver Q, a path p is a sequence of arrows with some conditions | ** 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) | ** 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) | + | ** p_1p_2 will correspond to a composition <math>p_2\circ p_1</math> of two maps (<math>U\overset{P_2}{\rightarrow }V\overset{P_1}{\rightarrow }W</math>) |
* quiver representation is in fact, a representaion of path algebra of a quiver | * quiver representation is in fact, a representaion of path algebra of a quiver | ||
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| + | <h5>finite type quiver</h5> | ||
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* quiver has finite type of there are finitely many indecomposables | * quiver has finite type of there are finitely many indecomposables | ||
| 32번째 줄: | 39번째 줄: | ||
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| + | <h5>periodicity conjecture</h5> | ||
outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams | outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams | ||
| 51번째 줄: | 60번째 줄: | ||
<h5>related items</h5> | <h5>related items</h5> | ||
| + | * [[quiver representations|Quiver representations]] | ||
* [[categorification of quantum groups]] | * [[categorification of quantum groups]] | ||
| 84번째 줄: | 94번째 줄: | ||
* Keller, Bernhard. 2008. “Cluster algebras, quiver representations and triangulated categories”. <em>0807.1960</em> (7월 12). http://arxiv.org/abs/0807.1960. | * Keller, Bernhard. 2008. “Cluster algebras, quiver representations and triangulated categories”. <em>0807.1960</em> (7월 12). http://arxiv.org/abs/0807.1960. | ||
| + | * [http://www.mpim-bonn.mpg.de/node/365 Total positivity, cluster algebras and categorification] | ||
2011년 4월 13일 (수) 04:50 판
introduction
- replace cluster variables by modules
notions
- 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
finite type 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)
periodicity conjecture
outline of a proof of the periodicity conjecture for pairs of Dynkin diagrams
history
encyclopedia
- http://en.wikipedia.org/wiki/
- http://www.scholarpedia.org/
- http://eom.springer.de
- http://www.proofwiki.org/wiki/
- Princeton companion to mathematics(Companion_to_Mathematics.pdf)
books
expositions
- Keller, Bernhard. 2008. “Cluster algebras, quiver representations and triangulated categories”. 0807.1960 (7월 12). http://arxiv.org/abs/0807.1960.
- Total positivity, cluster algebras and categorification
articles
-
- http://www.ams.org/mathscinet
- http://www.zentralblatt-math.org/zmath/en/
- http://arxiv.org/
- http://www.pdf-search.org/
- http://pythagoras0.springnote.com/
- http://math.berkeley.edu/~reb/papers/index.html
- http://dx.doi.org/
question and answers(Math Overflow)
- http://mathoverflow.net/search?q=
- http://mathoverflow.net/search?q=
- http://math.stackexchange.com/search?q=
- http://math.stackexchange.com/search?q=
blogs
- 구글 블로그 검색
- http://ncatlab.org/nlab/show/HomePage
experts on the field