Gyrogroups are generalized groups modelled on the Einstein groupoid of all relativistically admissible velocities with their Einstein’s velocity addition as a binary operation. Einstein’s gyrogroup fails to form a group since it is nonassociative. The breakdown of associativity in the Einstein addition does not result in loss of mathematical regularity owing to the presence of the relativistic effect known as the Thomas precession which, by abstraction, becomes an automorphism called the Thomas gyration. The Thomas gyration turns out to be the missing link that gives rise to analogies shared by gyrogroups and groups. In particular, it gives rise to the gyroassociative and the gyrocommuttive laws that Einstein’s addition possesses, in full analogy with the associative and the commutative laws that vector addition possesses in a vector space. The existence of striking analogies shared by gyrogroups

Abstract. In this paper we consider an algebraic generalization of symmetric spaces of noncompact type to a more general class of symmetric structures equipped with midpoints. These symmetric structures are shown to have close relationships to and even categorical equivalences with a variety of other algebraic structures: transversal twisted subgroups of involutive groups, a special class of loops called B-loops, and gyrocommutative gyrogroups. 1.

Central extensions of gyrocommutative gyrogroups (K-loops) are studied in order to clarify the status of a cocycle equation introduced by Smith and Ungar. A sufficient and necessary conditions under which a central invariant extension is a gyrocommutative gyrogroup are formulated in terms of a 2-cochain f(x, y). In particular, it is shown that for central invariant extensions of gyrocommutative gyrogroups defined by Cartan decompositions of simple Lie algebras, the corresponding f(x, y) satisfies the cocycle equation, provided an extension is a gyrocommutative gyrogroup. 1. Introduction. There has been a renewal of an interest in loop theory in recent years, concerning a special non-associative loop structure called a gyrocommutative gyrogroup, known also under the name of a K-loop. It began with a paper by A. Ungar [15], who pointed it out that the addition law of relativistic velocities