Abstract

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 suficient 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.

Authors