Pre-Lie Groups in Abstract Differential Geometry

We study groups with “differential structure” in the framework of Abstract Differential Geometry, an abstraction of the classical differential geometry of manifolds, via sheaf-theoretic methods, without ordinary calculus; the basic tool is the notion of a differential triad. First, we consider pre-Lie groups, i.e., semi-topological groups with compatible differential triads and we prove that such groups have “left-invariant vector fields” and “left-invariant derivations”, behaving like the classical ones. Next, for every pre-Lie group, we define an appropriate Lie algebra and prove the existence of a naturally associated adjoint representation of the initial group into the latter.