There are different ways of building a Lie algebra of a group. One approach, due to A. I. Malcev, allows to introduce the structure of a Lie algebra on the (divisible nilpotent) group itself. Conversely, one can build a group on a (nilpotent) Lie algebra, using Baker-Campbell-Hausdorf formula.
As a result, we have a "hybrid" object, one author called them "groupalgebras" or "algebragroups". We observe that this approach works in a far wider setting of arbitrary nilpotent algebras. We also have new applications in the case of the classical Malcev correspondence.
(Joint work with A. Olshanskii)