### Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 11 (2015), 093, 16 pages      arXiv:1505.02436      https://doi.org/10.3842/SIGMA.2015.093

### Post-Lie Algebras and Isospectral Flows

Kurusch Ebrahimi-Fard a, Alexander Lundervold b, Igor Mencattini c and Hans Z. Munthe-Kaas d
a) ICMAT, C/ Nicolás Cabrera 13-15, 28049 Madrid, Spain
b) Department of Computing, Mathematics and Physics, Faculty of Engineering, Bergen University College, Postbox 7030, N-5020 Bergen, Norway
c) Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo, Campus de São Carlos, Caixa Postal 668, 13560-970 São Carlos, SP, Brazil
d) Department of Mathematics, University of Bergen, Postbox 7803, N-5020 Bergen, Norway

Received August 13, 2015, in final form November 16, 2015; Published online November 20, 2015

Abstract
In this paper we explore the Lie enveloping algebra of a post-Lie algebra derived from a classical $R$-matrix. An explicit exponential solution of the corresponding Lie bracket flow is presented. It is based on the solution of a post-Lie Magnus-type differential equation.

Key words: isospectral flow equation; $R$-matrix; Magnus expansion; post-Lie algebra.

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