Restricted enveloping algebras whose skew and symmetric elements are Lie metabelian

TL;DR

This paper investigates conditions under which the symmetric or skew elements of the restricted enveloping algebra of a restricted Lie algebra are Lie metabelian, focusing on algebraic structures in characteristic p>2.

Contribution

It characterizes when symmetric or skew elements in restricted enveloping algebras are Lie metabelian, providing new insights into their algebraic properties.

Findings

Identifies conditions for symmetric elements to be Lie metabelian

Determines when skew elements are Lie metabelian

Advances understanding of algebraic structures in characteristic p>2

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 establish when the symmetric or skew elements of $u(L)$ under the principal involution are Lie metabelian.