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Journal of Lie Theory 25 (2015), No. 4, 1139--1165
Copyright Heldermann Verlag 2015

On the Lie Enveloping Algebra of a Post-Lie Algebra

Kurusch Ebrahimi-Fard
ICMAT -- UAM, Calle Nicolás Cabrera 13-15, Campus de Cantoblanco, 28049 Madrid, Spain

Alexander Lundervold
Dept. of Computing Mathematics and Physics, Bergen University College, Postboks 7030, 5020 Bergen, Norway

Hans Z. Munthe-Kaas
Dept. of Mathematics, University of Bergen, Postbox 7800, 5020 Bergen, Norway


\def\g{{\frak g}} We consider pairs of Lie algebras $\g$ and $\overline{\g}$, defined over a common vector space, where the Lie brackets of $\g$ and $\overline{\g}$ are related via a post-Lie algebra structure. The latter can be extended to the Lie enveloping algebra ${\cal U}(\g)$. This permits us to define another associative product on ${\cal U}(\g)$, which gives rise to a Hopf algebra isomorphism between ${\cal U}(\overline{\g})$ and a new Hopf algebra assembled from ${\cal U}(\g)$ with the new product.\endgraf For the free post-Lie algebra these constructions provide a refined understanding of a fundamental Hopf algebra appearing in the theory of numerical integration methods for differential equations on manifolds. In the pre-Lie setting, the algebraic point of view developed here also provides a concise way to develop Butcher's order theory for Runge-Kutta methods.

Keywords: Rooted trees, combinatorial Hopf algebras, post-Lie algebras, universal enveloping algebras, numerical Lie group integration, geometric numerical integration, Butcher's order theory.

MSC: 65L, 53C, 16T

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