Extensions and tangent prolongations of differentiable loops

Abstract

The tangent Akivis algebra and Sabinin algebra of degree 3 of a differentiable loop is the tangential object determined by the third-order Taylor polynomial of the multiplication function of the loop. It is endowed with a bilinear skew-symmetric and a trilinear operation defined by the infinitesimal commutator and associator of the loop. The aim of our work is to study tangent algebras of degree 3 of abelian extensions of differentiable loops, which are affine extensions of the tangent algebras of the loop by abelian algebras. This class of loop extensions has previously been studied in terms of computational complexity and in terms of universal algebra. We apply the obtained results to the determination of tangent algebras of degree 3 of tangent prolongation of differentiable loops.