Non-isomorphic restricted Lie algebras with isomorphic restricted enveloping algebras

Abstract

Let $p$ be a prime. For every field $F$ of characteristic $p$ we exhibit pairs of non-isomorphic finite-dimensional $p$-nilpotent restricted Lie algebras $L$ and $H$ over $F$ whose restricted enveloping algebras $u(L)$ and $u(H)$ are isomorphic as $F$-algebras. Such pairs exist in every dimension at least $p+5$, with $\dim L^{\prime }=p$ and $\dim H^{\prime }=p+1$. Thus, the restricted isomorphism problem has a negative answer over every field of positive characteristic, even for $p$-nilpotent algebras over perfect fields.