A formal Lie correspondence

Abstract

We establish an equivalence between categories of 'formally nilpotent' Lie algebras and exponential groups in characteristic zero. It extends the equivalences of Mal'cev, Lazard, Quillen and Warfield, and applies to groups under composition of generalized formal series or automorphisms of algebras of generalized formal series. We obtain first-order transfer results from finite dimensional nilpotent objects to formally nilpotent ones. We give applications to solving equations over groups, to the theory of nilpotent exponential groups as per Miasnikov-Remeslennikov, and to definability problems in certain groups of formal series.