Lie identities on symmetric elements of restricted enveloping algebras

TL;DR

This paper investigates the algebraic properties of symmetric elements in restricted enveloping algebras of restricted Lie algebras over fields of characteristic p > 2, focusing on conditions for Lie solvability, nilpotency, and bounded Engel properties.

Contribution

It characterizes when the symmetric elements of u(L) exhibit Lie solvability, nilpotency, or bounded Engel properties based on the structure of L.

Findings

Identifies conditions for symmetric elements to be Lie solvable.

Determines when symmetric elements are Lie nilpotent.

Establishes criteria for bounded Lie Engel properties.

Abstract

Let L be a restricted Lie algebra over a field of characteristic p > 2 and denote by u(L) its restricted enveloping algebra. We determine the conditions under which the set of symmetric elements of u(L) with respect to the principal involution is Lie solvable, Lie nilpotent, or bounded Lie Engel.