The Akivis algebra of a homogeneous loop

Abstract

A (local) Lie loop is a real analytic manifold M with a base point e and three analytic functions ( x , y ) → x ° y , x y , x / y : M × M → M (respectively, U × U → M for an open neighbourhood U of e in M ) such that the following conditions are satisfied: (i) x ° e = e ° x = e , (ii) x ° ( x y ) = y , and (iii) ( x / y )° y = x for all x , y e M (respectively, U ). If the multiplication ° is associative, then M is a (local) Lie group. The tangent vector space L ( M ) in e is equipped with an anticommutative bilinear operation ( X , Y ) →[ X , Y ] and a trilinear operation ( X , Y , Z ) →〈 X , Y , Z 〉. These are defined as follows: Let B be a convex symmetric open neighbourhood of 0 in L ( M ) such that the exponential function maps B diffeomorphically onto an open neighbourhood V of e in M and transport the operation ° into L ( M ) by defining X ° Y = (exp| B ) −1 ((exp X )° (exp Y )) for X and Y in a neighbourhood C of 0 in B such that (exp C ) ° (exp C ) ⊂ V . Similarly, we transport / and \.