The Magnus expansion and post-Lie algebras

Curry, Charles; Ebrahimi‐Fard, Kurusch; Owren, Brynjulf

doi:10.1090/mcom/3541

Cited by 15 publications

(13 citation statements)

References 28 publications

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“…In [17] it was underlined the relevance of the pLMe in the theory of the Lie group integrators and in [35] it was proven that on a post-Lie algebra, in analogy to what happens on every pre-Lie algebra, the pLMe provides an isomorphism between the group of formal flows and the BCH-group defined on h, generalizing the analog well known result proven in [1], see also [18,4].…”

mentioning

confidence: 71%

Crossed morphisms, (integration of) post-Lie algebras and the post-Lie Magnus expansion

Mencattini¹,

Quesney²

2020

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This letter is divided in two parts. In the first one it will be shown that the datum of a post-Lie product is equivalent to the one of an invertible crossed morphism between two Lie algebras. Moreover it will be argued that the integration of such a crossed morphism yields the post-Lie Magnus expansion associated to the original post-Lie algebra. The second part is devoted to present two combinatorial methods to compute the coefficients of this remarkable formal series. Both methods are based on special tubings on planar trees. Contents 1. Introduction 1.1. Conventions 2. Crossed morphisms 2.1. Crossed morphisms of Lie algebras 2.2. Crossed morphism of Lie group type objects 3. Universal enveloping algebras and the post-Lie Magnus expansion 3.1. Integration of post-Lie algebras 3.2. The post-Lie Magnus expansion 4. Computing the Post-Lie Magnus expansion 4.1. Planar trees and forests 4.2. Definition of PSB 4.3. The free post-Lie algebra 4.4. The universal enveloping algebra of the free post-Lie algebra 4.5. Nested tubings 4.6. Post-Lie Magnus expansion in terms of nested tubings

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confidence: 71%

Crossed morphisms, (integration of) post-Lie algebras and the post-Lie Magnus expansion

Mencattini¹,

Quesney²

2020

Preprint

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“…On the other hand, research on post-Lie and D-algebras is more recent. See for instance [2,7,9,11,25,26,27].…”

Section: Introductionmentioning

confidence: 99%

Algebraic aspects of connections: from torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials

Al-Kaabi,

Ebrahimi-Fard,

Manchon

et al. 2022

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Understanding the algebraic structure underlying a manifold with a general affine connection is a natural problem. In this context, A. V. Gavrilov introduced the notion of framed Lie algebra, consisting of a Lie bracket (the usual Jacobi bracket of vector fields) and a magmatic product without any compatibility relations between them. In this work we will show that an affine connection with curvature and torsion always gives rise to a post-Lie algebra as well as a D-algebra. The notions of torsion and curvature together with Gavrilov's special polynomials and double exponential are revisited in this post-Lie algebraic framework. We unfold the relations between the post-Lie Magnus expansion, the Grossman-Larson product and the K-map, α-map and β-map, three particular functions introduced by Gavrilov with the aim of understanding the geometric and algebraic properties of the double-exponential, which can be understood as a geometric variant of the Baker-Campbell-Hausdorff formula. We propose a partial answer to a conjecture by Gavrilov, by showing that a particular class of geometrically special polynomials is generated by torsion and curvature. This approach unlocks many possibilities for further research such as numerical integrators and rough paths on Riemannian manifolds.

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“…Recently, the deformation and cohomology theories of relative Rota-Baxter operators on both Lie and associative algebras were studied in [14,35,39]. Post-Lie algebras, as natural generalizations of pre-Lie algebras [8], were introduced by Vallette in [37], and have important applications in geometric numerical integration and mathematical physics [5,9,12,13,17,29]. In particular, a relative Rota-Baxter operator of nonzero weights on Lie algebras induces a post-Lie algebra.…”

Section: Introductionmentioning

confidence: 99%

Relative Rota-Baxter operators of nonzero weights on $3$-Lie algebras and 3-post-Lie algebras

Hou¹,

Sheng²,

Zhou³

2022

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In this paper, first we introduce the notion of a relative Rota-Baxter operator of nonzero weights on a 3-Lie algebra with respect to an action on another 3-Lie algebra, which can be characterized by graphs of the semidirect product 3-Lie algebra constructed from the action. Then we introduce a new algebraic structure, which is called a 3-post-Lie algebra. A 3-post-Lie algebra consists of a 3-Lie algebra structure and a ternary operation such that some compatibility conditions are satisfied. We show that a relative Rota-Baxter operator of nonzero weights induces a 3-post-Lie algebra naturally. Thus 3-post-Lie algebras can be viewed as the underlying algebraic structures of relative Rota-Baxter operators of nonzero weights on 3-Lie algebras. Moreover, a 3-post-Lie algebra also gives rise to a new 3-Lie algebra, which is called the subadjacent 3-Lie algebra, and an action on the original 3-Lie algebra. Next we construct an L ∞ -algebra from an action of 3-Lie algebras whose Maurer-Cartan elements are relative Rota-Baxter operators of nonzero weights. Consequently, we obtain the twisted L ∞ -algebra that controls deformations of a given relative Rota-Baxter operator of nonzero weights on 3-Lie algebras. Finally, we construct a cohomology theory for a relative Rota-Baxter operator of nonzero weights on 3-Lie algebras and use the second cohomology group to classify infinitesimal deformations.

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The Magnus expansion and post-Lie algebras

Abstract: We relate the classical and post-Lie Magnus expansions. Intertwining algebraic and geometric arguments allows to placing the classical Magnus expansion in the context of Lie group integrators.

Cited by 15 publications

References 28 publications

Crossed morphisms, (integration of) post-Lie algebras and the post-Lie Magnus expansion

Crossed morphisms, (integration of) post-Lie algebras and the post-Lie Magnus expansion

Algebraic aspects of connections: from torsion, curvature, and post-Lie algebras to Gavrilov's double exponential and special polynomials

Relative Rota-Baxter operators of nonzero weights on $3$-Lie algebras and 3-post-Lie algebras

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