Some Problems in the Representation Theory of Simple Modular Lie Algebras

TL;DR

This paper surveys the progress and open problems in the representation theory of finite-dimensional restricted simple Lie algebras over fields of characteristic p > 5, focusing on classical and Cartan-type algebras.

Contribution

It reviews existing results, compares classical and Cartan-type cases, and formulates conjectures and open problems in the representation theory of these algebras.

Findings

Classical algebras have well-developed representation theory.

Progress has been made in understanding Cartan-type algebras.

Several conjectures and open problems remain in the field.

Abstract

The finite-dimensional restricted simple Lie algebras of characteristic p > 5 are classical or of Cartan type. The classical algebras are analogues of the simple complex Lie algebras and have a well-advanced representation theory with important connections to Kazhdan-Lusztig theory, quantum groups at roots of unity, and the representation theory of algebraic groups. We survey progress that has been made towards developing a representation theory for the restricted simple Cartan-type Lie algebras, discuss comparable results in the classical case, formulate a couple of conjectures, and pose a dozen open problems for further study.