Pre-Lie Groups in Abstract Differential Geometry

Abstract

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.