Abstract
Abstract The tangent space of a Lie loop, a non-associative Lie group, carries the structure of a Sabinin algebra, an algebraic concept generalizing Lie algebras. Alternatively a Sabinin algebra can be interpreted as the universal local covariant of a flat affine connection. Combining this interpretation of Sabinin algebras with a version of the parallel transport equation we prove that every Sabinin algebra structure determines and is determined by the one-sided linear terms in the Baker-Campbell-Hausdorff formula for the multiplication in the underlying loop. Based on this resultwe prove that every abstract Sabinin algebra over ℝ is the tangent algebra of a global Lie loop.
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