On the stability of weight spaces of enveloping algebra in prime characteristic

TL;DR

This paper investigates the stability of weight spaces in the enveloping algebra of Lie algebras over fields of prime characteristic, showing that classical stability results do not always hold and providing conditions for stability.

Contribution

It extends the understanding of weight space stability to prime characteristic fields by identifying conditions under which stability is preserved.

Findings

Classical stability of weight spaces fails in prime characteristic without additional conditions

A specific condition for weight space stability in prime characteristic fields is established

The results contrast with the characteristic zero case, highlighting new complexities

Abstract

By the result of Dixmier, any weight space of enveloping algebra of Lie algebra L over a field of characteristic 0 is adL stable. In this paper we will show that this result need not be true, if F is replaced by a field of prime characteristic. A condition will be given, so a weight space will be adL stable.