The next lecture in this series is here.
In recent years, Vinberg theory of graded Lie algebras has become relevant in many areas of number theory, from arithmetic statistics (e.g., in the work of Romano–Thorne) to the local Langlands correspondence (e.g., in the work of Reeder-Yu). These lectures will provide the algebraic background for number theory students to engage with research involving graded Lie algebras. We’ll start by discussing some of the relevant aspects of the invariant theory of Lie algebras, including the Chevalley restriction theorem and the pioneering work of Kostant on invariant rings. We’ll then define graded Lie algebras and look at the graded analogues of these theorems, based on work of Vinberg. Time permitting, we’ll look at Slodowy slices and applications to families of algebraic curves. These lectures should give number theory students sufficient background to read, for example, Thorne’s paper Vinberg’s representations and arithmetic invariant theory and other related papers. But the lectures will also be a useful introduction to some beautiful aspects of Lie theory for students in algebra and representation theory. I’ll assume students have some knowledge of Lie algebras, but I will review relevant background and provide examples throughout the lectures.
This video is part of the London Mathematical Society‘s Online Graduate Lecture Series. These are supported by the LMS, and organized jointly by the ROW Diophantine Geometry Seminar, the ERLASS Arithmetic Statistics Seminar, and the Northern Number Theory Seminar.