Lie’s fundamental theorems for local analytical loops.

*(English)*Zbl 0596.22002Lie’s fundamental theorems assert that associated with each local Lie group is a Lie algebra (which is the tangent object at the origin) and conversely every Lie algebra determines a local Lie group for which it is the tangent object. The main purpose of this paper is to extend these results as far as possible to local analytical loops (which fail to satisfy the associative law). Lie algebras are replaced by what the authors call Akivis algebras; vector spaces equipped with a bilinear anticommutative multiplication [.,.] and a trilinear ternary multiplication \(<.,.,.>\); the two are related by an identity in which the ternary operation is used to measure how far the operation [.,.] deviates from satisfying the Jacobi identity.

The authors show how to associate to any local analytical loop an Akivis algebra (the operations can be defined directly from the multiplication in a way analogous to what is done in Lie theory). Conversely, it is shown that given any Akivis algebra, there exist many inequivalent local analytical loops having that algebra as tangent object (the authors show that one of these loop multiplications may be defined by polynomials of degree three). In the case that the loop is power associative or alternative, the Campbell-Hausdorff formula may be introduced, and the authors consider how their results specialize in these cases.

The authors show how to associate to any local analytical loop an Akivis algebra (the operations can be defined directly from the multiplication in a way analogous to what is done in Lie theory). Conversely, it is shown that given any Akivis algebra, there exist many inequivalent local analytical loops having that algebra as tangent object (the authors show that one of these loop multiplications may be defined by polynomials of degree three). In the case that the loop is power associative or alternative, the Campbell-Hausdorff formula may be introduced, and the authors consider how their results specialize in these cases.

Reviewer: J.D.Lawson