Math Club
Wednesday October 28th, 2015
Time: 1:00 PM
Title: Quivers and their representations
Speaker: Alexander Kirillov, Stony Brook
Location: Math Tower P-131
Abstract:
$\hspace{5mm}$It is well known that a subspace in a vector space is completely determined by its dimension: if V is a k-dimensional subspace in n-dimensional space W, we can choose a basis in W to make the pair V, W look like the standard subspace $C^k$ in $C^n$. What about pairs of subspace? is there a standard model for pairs? or for triples? or...
$\hspace{5mm}$All these questions are easiest answered using the notion of a quiver. A quiver is just a directed graph - a collection of points and arrows, and a representation of a quiver is a collection of vector spaces and linear maps between them. Even though this looks like a very trivial thing, it turns out that quiver representations have a number of unexpected connections with other branches of mathematics -- from classification problems like the subspace problem above to the theory of root systems and semisimple Lie algebras.
$\hspace{5mm}$In this talk, we will give an introduction to this theory and discuss its relation with the ADE classification of Dynkin diagrams, Platonic solids, and finite subgroups in $SU(2)$.