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Hopf co-addition for free magma algebras and the non-associative Hausdorff series

2003; Elsevier BV; Volume: 265; Issue: 1 Linguagem: Inglês

10.1016/s0021-8693(03)00157-1

1090-266X

Lothar Gerritzen, Ralf Holtkamp,

Homotopy and Cohomology in Algebraic Topology

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(2 x ), 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 H n 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.