Infinitesimal Symmetries and their Representations (Spring 2023)

Lie theory is an area with numerous applications and connections to other branches of mathematics, from harmonic analysis and combinatorics to low-dimensional topology and algebraic geometry. The theory was originally tailored to get a better understanding of certain geometric objects and differential equations by exploiting their families of symmetries. The fundamental idea behind Lie theory is that one can translate problems from geometry to algebra, where some questions are easier to solve, and then export the results back.

Lie algebras are the main character on the algebraic side of this story. They present them- selves as vector spaces of linear operators equipped with an operation which is in general neither associative nor commutative and can be thought of as the commutator [X,Y] = XY − YX of linear maps.

In this project we will learn about the representation theory of some Lie algebras, which essentially boils down to finding matrices satisfying certain commutation relations. We will study first the smallest simple Lie algebra sl2, for which we will get a fairly clear picture of the situation. This will serve as building block and intuition to explore the representation theory of more general Lie algebras.

For more information contact Guillermo Sanmarco (sanmarco@iastate.edu)

People:

Pre-requisites: