The Dense Orbits of the Quiver

Student: Adara Andonie

Mentor: Danny Lara

Abstract

A quiver is a quadruple consisting of sets of vertices and sets of arrows with two maps
which associates each arrow’s source and target respectively. Quivers provide a way to visualize finite dimensional algebras. To study algebras, one may study its modules, which are generalized versions of vector spaces. Modules are difficult to study so we look at special types of modules called indecomposable modules; they are modules that cannot be broken down and are used to build other modules. If modules were legos figures, indecomposable modules would be the base pieces. There are three types of finite dimensional algebras: finite algebras, tame algebras, and wild algebras. A finite representation type algebra only has finitely many indecomposable modules, metaphorically, there are finitely many lego pieces to work with. A tame algebra has infinitely many indecomposable modules but can be described by a single parameter. We will be working with wild algebras, where a wild algebra has infinitely many indecomposables with infinitely many parameters and there is no possible way to describe the indecomposable modules. We look at specific algebras and show that it is a dense orbit algebra, so if we were to pick a module and the orbit gives us a set of all modules that are “the same”, this would be a dense orbit. We are doing this because we have a conjecture that tells us that dense orbit algebras only have finitely many indecomposables that are dense, including wild algebras. Thus we’re building evidence towards that conjecture.

Presentation

1 thought on “The Dense Orbits of the Quiver”

  1. This is a HARD math topic and you presented an introduction to it in an understandable engaging manner. Nicely done!

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