Associative and non-associative automorphisms of Newtonian space-time

Abstract

In this article, an algebraic study of the transformations

$$(x,t) \mapsto \left( {Rx + \sum\limits_{i = 0}^m {A_i t^i /i!,t + t'} } \right)$$

of Newtonian space-time onto itself is presented. The study is carried out within the framework of Eilenberg and Maclane's co-homology theory of group prolongations, a generalisation of the theory of group extensions (Eilenberg & Maclane, 1947a-d). The ‘loops’ or non-associative groups involving ‘m’ from 0 to 5 are placed in classes of decreasing associativity described in the text. We also discuss the physical applications of the cup-product of co-chains and the vector bundle structure of Newtonian space-time.