Modular Lie algebras and the Gelfand-Kirillov conjecture

TL;DR

This paper investigates the types of simple Lie algebras for which the Gelfand-Kirillov conjecture holds, concluding it is limited to types A_n, C_n, and G_2, thus narrowing the conjecture's scope.

Contribution

It establishes a necessary condition linking the Gelfand-Kirillov conjecture to specific Lie algebra types, advancing understanding of the conjecture's applicability.

Findings

Gelfand-Kirillov conjecture holds only for Lie algebras of type A_n, C_n, or G_2.

The result restricts the class of Lie algebras satisfying the conjecture.

Provides a classification constraint for future research on the conjecture.

Abstract

Let g be a finite dimensional simple Lie algebra over an algebraically closed field of characteristic zero. We show that if the Gelfand-Kirillov conjecture holds for g, then g has type A_n, C_n or G_2.