Post-Lie Algebras, Factorization Theorems and Isospectral-Flows

@article{EbrahimiFard2017PostLieAF,
  title={Post-Lie Algebras, Factorization Theorems and Isospectral-Flows},
  author={Kurusch Ebrahimi-Fard and Igor Mencattini},
  journal={arXiv: Mathematical Physics},
  year={2017}
}
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 a 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… 
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8 Citations

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References

SHOWING 1-10 OF 88 REFERENCES

Post-Lie algebras and factorization theorems

On Post-Lie Algebras, Lie–Butcher Series and Moving Frames

This paper investigates algebras arising from flat connections with constant torsion, leading to the definition of post- Lie–Butcher series, a generalization of pre-Lie algebrae, and develops new formulas for computations in free post-LieAlgebrAs and D-algebra, and shows that Lie– butcher series are related to invariants of curves described by moving frames.

The Hopf Algebra of Fliess Operators and Its Dual Pre-lie Algebra

We study the Hopf algebra H of Fliess operators coming from Control Theory in the one-dimensional case. We prove that it admits a graded, finite-dimensional, connected grading. Dually, the vector…

Left-symmetric algebras, or pre-Lie algebras in geometry and physics

In this survey article we discuss the origin, theory and applications of left-symmetric algebras (LSAs in short) in geometry in physics. Recently Connes, Kreimer and Kontsevich have introduced LSAs…

Affine actions on Lie groups and post-Lie algebra structures

On the Lie enveloping algebra of a post-Lie algebra

We consider pairs of Lie algebras $g$ and $\bar{g}$, defined over a common vector space, where the Lie brackets of $g$ and $\bar{g}$ are related via a post-Lie algebra structure. The latter can be…

Faà di Bruno Hopf Algebras, Dyson–Schwinger Equations, and Lie–Butcher Series

Numerical analysis of time-integration algorithms has been applying advanced algebraic techniques for more than fourty years. An explicit description of the group of characters in the…

On the Lie envelopping algebra of a pre-Lie algebra

We construct an associative product on the symmetric module S(L) of any pre-Lie algebra L. Then we proove that in the case of rooted trees our construction is dual to that of Connes and Kreimer. We…

On the Structure of Hopf Algebras

induced by the product M x M e M. The structure theorem of Hopf concerning such algebras has been generalized by Borel, Leray, and others. This paper gives a comprehensive treatment of Hopf algebras…

Birkhoff Type Decompositions and the Baker–Campbell–Hausdorff Recursion

We describe a unification of several apparently unrelated factorizations arising from quantum field theory, vertex operator algebras, combinatorics and numerical methods in differential equations.…
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