Wilson conjecture for omega-categorical Lie algebras, the case 4-Engel characteristic 3

Abstract

We continue our study of the Wilson conjecture for $\omega$-categorical Lie algebras and prove that $\omega$-categorical $4$-Engel Lie algebras of characteristic $3$ are nilpotent. We develop a set of tools to adapt in the definable context some classical methods for studying Engel Lie algebras (Higgins, Kostrikin, Zelmanov, Vaughan-Lee, Traustason and others). We solve the case at hand by starting a systematic study of Lie algebras for which there is a $k$ such that the principal ideal generated by any element is nilpotent of class $<k$ (which we call $k$-strong Lie algebras). We use computer algebra to check basic cases of a conjectural arithmetical property of those, namely that $x^{k-1}y^{k-1} = (-1)^{k-1}y^{k-1}x^{k-1}$ is an identity for Lie elements of the enveloping algebra. The solution is given by reducing the problem to $k$-strong Lie algebras generated by particularly well behaved sandwiches in the sense of Kostrikin.