Perfect and semiperfect restricted enveloping algebras

TL;DR

This paper investigates conditions for the semiperfectness of restricted enveloping algebras of restricted Lie algebras and establishes that such algebras are perfect only when the Lie algebra is finite-dimensional.

Contribution

It provides a characterization of when restricted enveloping algebras are semiperfect and proves that they are perfect precisely when the underlying Lie algebra is finite-dimensional.

Findings

u(L) is semiperfect under specific conditions on L

u(L) is perfect if and only if L is finite-dimensional

The paper establishes a clear link between the algebraic properties of u(L) and the dimension of L

Abstract

For a restricted Lie algebra $L$, the conditions under which its restricted enveloping algebra $u(L)$ is semiperfect are investigated. Moreover, it is proved that $u(L)$ is left (or right) perfect if and only if $L$ is finite-dimensional.