Isomorphism invariants of restricted enveloping algebras

TL;DR

This paper investigates how the structure of restricted enveloping algebras determines the underlying restricted Lie algebras, establishing isomorphism results under specific conditions such as abelianness and p-nilpotency.

Contribution

It proves that under certain conditions, isomorphism of restricted enveloping algebras implies isomorphism of the underlying restricted Lie algebras, providing new invariants for classification.

Findings

If L is p-nilpotent and abelian, then L is isomorphic to H.

If L is abelian over an algebraically closed field, then L is isomorphic to H.

L/p-L'^p + γ_3(L) is isomorphic to H/p-L'^p + γ_3(H).

Abstract

Let $L$ and $H$ be finite-dimensional restricted Lie algebras over a perfect field $\F$ such that $u(L)\cong u(H)$, where $u(L)$ is the restricted enveloping algebra of $L$. We prove that if $L$ is $p$-nilpotent and abelian, then $L\cong H$. We deduce that if $L$ is abelian and $\F$ is algebraically closed, then $L\cong H$. We use these results to prove the main result of this paper stating that if $L$ is $p$-nilpotent, then $L/L'^p+\gamma_3(L)\cong H/H'^p+\gamma_3(H)$.