A non-associative Baker-Campbell-Hausdorff formula

Mostovoy, J.; Pérez-Izquierdo, J.; Shestakov, I.

doi:10.1090/proc/13684

A non-associative Baker-Campbell-Hausdorff formula
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by J. Mostovoy, J. M. Pérez-Izquierdo and I. P. Shestakov

Proc. Amer. Math. Soc. 145 (2017), 5109-5122

DOI: https://doi.org/10.1090/proc/13684

Published electronically: June 16, 2017

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Abstract:

We address the problem of constructing the non-associative version of the Dynkin form of the Baker-Campbell-Hausdorff formula; that is, expressing $\log (\exp (x)\exp (y))$, where $x$ and $y$ are non-associative variables, in terms of the Shestakov-Umirbaev primitive operations. In particular, we obtain a recursive expression for the Magnus expansion of the Baker-Campbell-Hausdorff series and an explicit formula in degrees smaller than 5. Our main tool is a non-associative version of the Dynkin-Specht-Wever Lemma. A construction of Bernouilli numbers in terms of binary trees is also recovered.

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