Lie Properties of Restricted Enveloping Algebras

TL;DR

This paper surveys known results on the Lie structure of restricted enveloping algebras of restricted Lie algebras over fields of positive characteristic, and introduces a new upper bound for their Lie nilpotency class.

Contribution

It provides a comprehensive survey of the Lie properties of u(L) and presents a new theorem establishing an upper bound for its Lie nilpotency class.

Findings

Survey of known Lie structure results of u(L)

Discussion on units and symmetric elements in u(L)

New theorem on upper bound for Lie nilpotency class

Abstract

Let L be a restricted Lie algebra over a field of positive characteristic. We survey the known results about the Lie structure of the restricted enveloping algebra u(L) of L. Related results about the structure of the group of units and the symmetric and skew-symmetric elements of u(L) are also discussed. Moreover, a new theorem about an upper bound for the Lie nilpotency class of u(L) is proved.