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A non-associative Baker-Campbell-Hausdorff formula

Authors: J. Mostovoy, J. M. Pérez-Izquierdo and I. P. Shestakov
Journal: Proc. Amer. Math. Soc. 145 (2017), 5109-5122
MSC (2010): Primary 17A50, 20N05
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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J. Mostovoy
Affiliation: Departamento de Matemáticas, CINVESTAV-IPN, Apartado Postal 14–740, 07000 México D.F., Mexico

J. M. Pérez-Izquierdo
Affiliation: Departamento de Matemáticas y Computación, Universidad de La Rioja, 26004, Logroño, Spain

I. P. Shestakov
Affiliation: Instituto de Matemática e Estatística, Universidade de São Paulo, Caixa Postal 66281, São Paulo, SP 05311-970, Brazil

Keywords: Baker-Campbell-Hausdorff formula, primitive elements, Sabinin algebras, Magnus expansion
Received by editor(s): May 11, 2016
Received by editor(s) in revised form: January 9, 2017
Published electronically: June 16, 2017
Additional Notes: The authors acknowledge the support by the Spanish Ministerio de Ciencia e Innovación (MTM2013-45588-C3-3-P) and Brazilian Coordenação de Aperfeiçoamento de Pessoal de Nível Superior (CAPES/DGPU)
The first author was also supported by the CONACYT grant 168093-F
The third author also acknowledges support by FAPESP, processo 2014/09310-5 and CNPq, processos 303916/2014-1 and 456698/2014-0.
Communicated by: Kailash Misra
Article copyright: © Copyright 2017 American Mathematical Society

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