A non-restricted counterexample to the first Kac-Weisfeiler conjecture

TL;DR

This paper constructs the first known example of a non-restricted Lie algebra that contradicts the first Kac-Weisfeiler conjecture, challenging assumptions about module dimensions in Lie algebra representation theory.

Contribution

It provides the first counterexample to the first Kac-Weisfeiler conjecture for non-restricted Lie algebras, using pairs of Lie algebras with isomorphic enveloping algebras but different indexes.

Findings

Counterexample to the conjecture for non-restricted Lie algebras

Pairs of Lie algebras with isomorphic enveloping algebras but different indexes

Challenges previous assumptions in Lie algebra representation theory

Abstract

In 1971 Kac and Weisfeiler made two important conjectures regarding the representation theory of restricted Lie algebras over fields of positive characteristic. The first of these predicts the maximal dimension of the simple modules, and can be stated without the hypothesis that the Lie algebra is restricted. In this short article we construct the first example of a non-restricted Lie algebra for which the prediction of the first Kac--Weisfeiler conjecture fails. Our method is to present pairs of Lie algebras which have isomorphic enveloping algebras but distinct indexes.