Lie's fundamental theorems for local analytical loops

Karl Hofmann, Karl Strambach
1986 Pacific Journal of Mathematics  
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
more » ... ess 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.
doi:10.2140/pjm.1986.123.301 fatcat:fdpai7nukjei3id4bkrkwgyv5e