Hopf co-addition for free magma algebras and the non-associative Hausdorff series
✍ Scribed by Lothar Gerritzen; Ralf Holtkamp
- Publisher
- Elsevier Science
- Year
- 2003
- Tongue
- English
- Weight
- 194 KB
- Volume
- 265
- Category
- Article
- ISSN
- 0021-8693
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✦ Synopsis
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 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.