Tangent prolongation of ${\mathcal{C}}^r$-differentiable loops

Abstract

The aim of our paper is to generalize the tangent prolongation of Lie groups to non-associative multiplications and to examine how the weak associative and weak inverse properties are transferred to the multiplication defined on the tangent bundle. We obtain that the tangent prolongation of a ${\mathcal{C}}^r$-differentiable loop ($r\ge 1$) is a ${\mathcal{C}}^{r-1}$-differentiable loop that acquires the classical weak inverse and weak associative properties of the initial loop.