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Quivers are simply diagrams of arrows and points.

This site has been developed by Msc students at Maseno University, who wish to share their ideas in a simple and interesting way. The main source of motivation came from colleagues at the college who were puzzled by our studies. Following the articles on quiver mutation that were published in the iSquared magazine, we felt a need to provide additional information. Hopefully, this site will serve that purpose.

As you might have noticed, this website is about Quiver mutation . Quiver mutation (and cluster mutation) was invented in joint work by S. Fomin and A. Zelevinsky in 2000. It has since been related to a large number of subjects in mathematics and to Seiberg duality in physics.

The operation of mutation transorms one quiver into another. There is a mutation applet which is a tool that does the mutations for us, this is originally available on Kellers website and we thank him for allowing us to use it here.

There is a field of mathematics where quivers and their mutations have been widely studied. That field is Cluster Algebras. It is here that quiver mutation was first defided. Cluster Algebras portal is a web page with a number of links on cluster algebras and other related topics.

Check out the introduction page for a review of the site, or here is a summary of the contents:

Quiver

A generally known meaning is the place for storing arrows in archery. How then do quivers come about in mathematics?

In mathematics, quiver consists of a set of vertices (Nodes), and a set of arrows. Each arrow has a starting and terminating vertex . Multiple arrows are allowe between vertices.

Quivers with relations

In a family, people are related by birth or marriage, in a country by citizenship etc. Amazingly, we have relations in some quivers. These relations are exhibited in paths of a quiver, and they have a very important use in the quiver algebra.

Quivers with potential

I am convinced that most of us like the idea of consolidation "all undr one roof" We can have all relations for a quiver under one roof called a potential. It is the sum of all cycles in a quiver , and we can get all relations from the potential by taking cyclic derivatives.

Quivers with R-charge

Angles and numbers can be assigned to a quiver with potential in such a way that all R-charge conditions are satisfied. We call such a quiver, a quiver with R-charge. These angles are given to arrows and vertices while vertices are assigned numbers.

del-Pezzo quivers

These are a special family of quivers related to geometric surfaces called the del-Pezzo surfaces. Interestingly, they are quivers with potential and R-charge.

Mutation of quivers

Is a well known biological process where a micro-organisms ie viruses changes their genetic forms and gets more resistant to treatment. In mathematcs, we can mutate quivers. Unlike the biologibal mutation, mathematical mitation is a reversible process. We can mutate any quiver, del-Pezzo quivers, quivers with potential as well as quivers with R-charge.

Mutation of del-Pezzo quivers brings a reward, which is pretty patterns.

 

 

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