Lie Theory Day 2023

• Crystal Hoyt (Bar-Ilan University), Weight representations of finite-dimensional Lie algebras

Abstract: This talk will serve as a quick introduction to the theory of weight modules for simple Lie algebras. We will recall the classification of finite-dimensional simple modules by their highest weight, and then we will consider infinite-dimensional weight modules, including a special class of weight modules that are not highest weight.

• Ivan Penkov (Constructor University Bremen, Batsheva de Rothschild visitor), Weight representations of infinite-dimensional matrix Lie algebras 

• Jeremy Schiff (Bar-Ilan University), Lie Algebras and Differential Equations

Abstract: In a first course on differential equations, typically a lot of time is devoted to learning tricks for solving various kinds of differential equations. Sophus Lie realized that a single principle is at work in the background - roughly speaking, an equation can be solved when it has a sufficiently large continuous symmetry group. I will explain in this lecture how infinitesimal symmetries of differential equations  form a Lie algebra, how the Lie algebra of symmetries can be computed and how the symmetries can be used in the solution process. For quite a few nonlinear equations of interest in various physical situations, infinite dimensional algebras of symmetries have been identified, and I will give at least one example and try to describe some ongoing research in this area.