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Co-Addition for free Non-Associative Algebras and the Hausdorff Series

Generalizations of the series exp and log to noncommutative non-associative and other types of algebras were considered by M.Lazard, and recently by V.Drensky and L.Gerritzen. There is a unique power series exp(x) in one non-associative variable x such that exp(x)exp(x)=exp(2x), exp'(0)=1. We call the unique series H=H(x,y) in two non-associative variables satisfying exp(H)=exp(x)exp(y) the non-associative Hausdorff series, and we show that the homogeneous components of H are primitive elements with respect to the co-addition for non-associative variables. We describe the space of primitive elements for the co-addition in non-associative variables using Taylor expansion and a projector onto the algebra A_0 of constants for the partial derivations. By a theorem of Kurosh, A_0 is a free algebra. We describe a procedure to construct a free algebra basis consisting of primitive elements.