Mal’tsev Correspondence and Bi-Interpretability

We study the famous Mal’tsev correspondence between nilpotent k-groups G and nilpotent Lie k-algebras L over a field k of characteristic zero from the model-theoretic, algebro-geometric, and algorithmic viewpoints. It is proved that, in this case, a group G and the corresponding Lie algebra L(G) are bi-interpretable by equations in each other. This gives a much more precise description of the correspondence, which implies that, in addition to the classical categorical properties, the group G and the algebra L(G) share many more algebraic, algorithmic, and model-theoretic properties.