Post-Lie Algebras, Factorization Theorems and Isospectral Flows

Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 267)


In these notes we review and further explore the Lie enveloping algebra of a post-Lie algebra. From a Hopf algebra point of view, one of the central results, which will be recalled in detail, is the existence of second Hopf algebra structure. By comparing group-like elements in suitable completions of these two Hopf algebras, we derive a particular map which we dub post-Lie Magnus expansion. These results are then considered in the case of Semenov-Tian-Shansky’s double Lie algebra, where a post-Lie algebra is defined in terms of solutions of modified classical Yang–Baxter equation. In this context, we prove a factorization theorem for group-like elements. An explicit exponential solution of the corresponding Lie bracket flow is presented, which is based on the aforementioned post-Lie Magnus expansion.


Post-Lie algebra Universal enveloping algebra Hopf algebra Magnus expansion Classical r-matrices Classical Yang–Baxter equation Factorization theorems Isospectral flow 

MSC Classification

16T05 16T10 16T25 16T30 17D25 



The first author acknowledges support from the Spanish government under the project MTM2013-46553-C3-2-P.


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Authors and Affiliations

  1. 1.Department of Mathematical SciencesNorwegian University of Science and Technology – NTNUTrondheimNorway
  2. 2.Instituto de Ciências Matemáticas e de Computação, Universidade de São Paulo (USP)São CarlosBrazil

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