A central piece of classical
Lie theory is the fact that with each local Lie group, a Lie algebra is associated as
tangent object at the origin, and that, conversely and more importantly, every Lie
algebra determines a local Lie group whose tangent algebra it is. Up to equivalence of
local groups, this correspondence is bijective.
Attempts at the development of a Lie theory for analytical loops have not been
entirely satisfactory in this direction, since they relied more or less on certain
associativity assumptions. Here we associate with an arbitrary local analytical loop a
unique tangent algebra with a ternary multiplication in addition to the standard
binary one, and we call this algebra an Akivis algebra. Our main objective is to show
that, conversely, for every Akivis algebra there exist many inequivalent local
analytical loops with the given Akivis algebra as tangent algebra. We shall give a
good idea about the degree of non-uniqueness. It is curious to note that, on account
of this non-uniqueness, the construction is more elementary than in the case of
analytical groups.
