Prime Lie algebras satisfying the standard Lie identity of degree 5

Abstract

For every commutative differential algebra one can define the Lie algebra of special derivations. It is known for years that not every Lie algebra can be embedded to the Lie algebra of special derivations of some differential algebra. More precisely, the Lie algebra of special derivations of a commutative algebra always satisfy the standard Lie identity of degree $5$. The problem of existence of such embedding is a long-standing problem (see \cite{YuPSt5,PoinsotAAM,Poinsot}), which is closely related to the Lie algebra of vector fields on the affine line (see \cite{YuPSt5}). It was solved by Razmyslov in \cite{YuPSt5} for simple Lie algebras satisfying this identity (see also \cite[Th. 16]{Poinsot}). We extend this result to prime (and semiprime) Lie algebras over a field of zero characteristic satisfying the standard Lie identity of degree~$5$. As an application, we prove that for any semiprime Lie algebra the standard identity $St_5$ implies all other identities of the Lie algebra of polynomial vector fields on the affine line. We also generalize some previous results about primeness of the Lie algebra of special derivations of a prime differential algebra to the case of non-unitary differential algebra.