On a Pre-Lie Algebra Defined by Insertion of Rooted Trees

Saidi, Anouar

doi:10.1007/s11005-010-0377-5

Cited by 5 publications

(8 citation statements)

References 7 publications

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“…The Jacobi identity for the bracket ., . is a consequence of the pre-Lie property of the Lie derivative [43,34,44], extended straightforwardly to aromatic trees. Define for v P V a node of γ, L v τ γ " i τ pγ vÑ 1 q.…”

Section: Remark 33 the Vertical Homotopy Operator Is Linked To The Co...mentioning

confidence: 99%

“…and the insertion product [43,34,44], that coincides with the Lie derivative L τ γ. We mention that the dual of the grafting product is also called Lie derivative (though it differs from L τ γ) in [19] (see also [20, Sec.…”

Section: Remark 33 the Vertical Homotopy Operator Is Linked To The Co...mentioning

confidence: 99%

“…The Lie derivative is also the pre-Lie version of the substitution law on B-series. It appears under the name pre-Lie insertion product in [43,34,44] for the study of the freeness of the pre-Lie insertion algebra. In the calculus of variations, the Lie-derivative defines symmetries, that are perturbations that leave the input unchanged at first order.…”

Section: Introductionmentioning

confidence: 99%

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The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators

Laurent

McLachlan

Munthe-Kaas

et al. 2023

Forum of Mathematics, Sigma

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Aromatic B-series were introduced as an extension of standard Butcher-series for the study of volume-preserving integrators. It was proven with their help that the only volume-preserving B-series method is the exact flow of the differential equation. The question was raised whether there exists a volume-preserving integrator that can be expanded as an aromatic B-series. In this work, we introduce a new algebraic tool, called the aromatic bicomplex, similar to the variational bicomplex in variational calculus. We prove the exactness of this bicomplex and use it to describe explicitly the key object in the study of volume-preserving integrators: the aromatic forms of vanishing divergence. The analysis provides us with a handful of new tools to study aromatic B-series, gives insights on the process of integration by parts of trees, and allows to describe explicitly the aromatic B-series of a volume-preserving integrator. In particular, we conclude that an aromatic Runge–Kutta method cannot preserve volume.

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Section: Remark 33 the Vertical Homotopy Operator Is Linked To The Co...mentioning

confidence: 99%

Section: Remark 33 the Vertical Homotopy Operator Is Linked To The Co...mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators

Laurent

McLachlan

Munthe-Kaas

et al. 2023

Forum of Mathematics, Sigma

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“…By the same work as in section 1, we show that F is the free graded P-algebra over V . Special case: if for any n ∈ N * , dim V n = 1 then: -A right pre-Lie algebra if λ = 1 [19], [1] [20].…”

Section: Deformed Nap Algebramentioning

confidence: 99%

The Pre-Lie Operad as a Deformation of Nap

Saidi

2013

J. Algebra Appl.

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“…The Lie derivative is also the pre-Lie version of the substitution law on B-series. It appears under the name pre-Lie insertion product in [44,34,45] for the study of the freeness of the pre-Lie insertion algebra. In the calculus of variations, the Lie-derivative defines symmetries, that are perturbations that leave the input unchanged at first order.…”

mentioning

confidence: 99%

The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators

Laurent

2024

JCD

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The aromatic bicomplex is an algebraic tool based on aromatic Butcher trees and used in particular for the explicit description of volumepreserving affine-equivariant numerical integrators. The present work defines new tools inspired from variational calculus such as the Lie derivative, different concepts of symmetries, and Noether's theory in the context of aromatic forests. The approach allows to draw a correspondence between aromatic volume-preserving methods and symmetries on the Euler-Lagrange complex, to write Noether's theorem in the aromatic context, and to describe the aromatic B-series of volume-preserving methods explicitly with the Lie derivative.

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On a Pre-Lie Algebra Defined by Insertion of Rooted Trees

Cited by 5 publications

References 7 publications

The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators

The aromatic bicomplex for the description of divergence-free aromatic forms and volume-preserving integrators

The Pre-Lie Operad as a Deformation of Nap

The Lie derivative and Noether's theorem on the aromatic bicomplex for the study of volume-preserving numerical integrators

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