Over NIP fields, infinite dimensional alternating n-linear spaces are strictly n-dependent and NSOP1, proved via a new composition lemma for NIP relations and connected-component analysis.
A two-sorted theory of nilpotent Lie algebras
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We prove the existence of a model companion of the two-sorted theory of $c$-nilpotent Lie algebras over a field satisfying a given theory of fields. We describe a language in which it admits relative quantifier elimination up to the field sort. Using a new criterion which does not rely on a stationary independence relation, we prove that if the field is NSOP$_1$, then the model companion is NSOP$_4$. We also prove that if the field is algebraically closed, then the model companion is $c$-NIP.
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On n-dependent groups and fields III. Multilinear forms and invariant connected components
Over NIP fields, infinite dimensional alternating n-linear spaces are strictly n-dependent and NSOP1, proved via a new composition lemma for NIP relations and connected-component analysis.