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Journal of Lie Theory 17 (2007), No. 3, 525--538
Copyright Heldermann Verlag 2007

The Baker-Campbell-Hausdorff Formula in the Free Metabelian Lie Algebra

Vitaliy Kurlin
Department of Mathematics, University of Liverpool, Liverpool L69 7ZL, England


The classical Baker-Campbell-Hausdorff formula gives a recursive way to compute the Hausdorff series $H=\ln(e^Xe^Y)$ for non-commuting $X,Y$. Formally $H$ lives in the graded completion of the free Lie algebra $L$ generated by $X,Y$. We present a closed explicit formula for $H=\ln(e^Xe^Y)$ in a linear basis of the graded completion of the free metabelian Lie algebra $L/[[L,L],[L,L]]$.

Keywords: Lie algebra, metabelian Lie algebra, Hausdorff series, Baker-Campbell-Hausdorff formula, metabelian BCH formula, Zassenhaus formula, Kashiwara-Vergne conjecture.

MSC: 17B01

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