Post-symmetric braces and integration of post-Lie algebras

Mencattini, Igor; Quesney, Alexandre; Silva, Pryscilla

doi:10.1016/j.jalgebra.2020.03.018

Cited by 14 publications

(20 citation statements)

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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].…”

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confidence: 71%

“…Since then, they appeared unexpectedly in almost every area of the modern mathematics, from differential geometry, [34,41,4] to combinatorics [5,14,15], from mathematical physics, see [24], to numerical analysis [9,28], see [10,33,25] for comprehensive reviews. In spite post-Lie algebras have been introduced much more recently by Vallette, see [43], and independently by Lundervold and Munthe-Kaas [32], since then they have been deeply studied, both from point of view of pure, see for example [3,11,22,35] and of applied mathematics [20,36,16], see also [23,26].…”

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confidence: 99%

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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%

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

Mencattini¹,

Quesney²

2020

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“…Remark 2.2. Similar axioms in the definition of a post-Hopf algebra also appeared in the definition of D-algebras [33,34] and D-bialgebras [31] with motivations from the studies of numerical Lie group integrators and the algebraic structure on the universal enveloping algebra of a post-Lie algebra.…”

Section: Post-hopf Algebrasmentioning

confidence: 85%

Post-Hopf algebras, relative Rota-Baxter operators and solutions of the Yang-Baxter equation

Li¹,

Sheng²,

Tang³

2022

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In this paper, first we introduce the notion of a post-Hopf algebra, which gives rise to a post-Lie algebra on the space of primitive elements and there is naturally a post-Hopf algebra structure on the universal enveloping algebra of a post-Lie algebra. A novel property is that a cocommutative post-Hopf algebra gives rise to a generalized Grossman-Larsson product, which leads to a subadjacent Hopf algebra and can be used to construct solutions of the Yang-Baxter equation. Then we introduce the notion of relative Rota-Baxter operators on Hopf algebras. A cocommutative post-Hopf algebra gives rise to a relative Rota-Baxter operator on its subadjacent Hopf algebra. Conversely, a relative Rota-Baxter operator also induces a post-Hopf algebra. Then we show that relative Rota-Baxter operators give rise to matched pairs of Hopf algebras. Consequently, post-Hopf algebras and relative Rota-Baxter operators give solutions of the Yang-Baxter equation in certain cocommutative Hopf algebras. Finally we characterize relative Rota-Baxter operators on Hopf algebras using relative Rota-Baxter operators on the Lie algebra of primitive elements, graphs and module bialgebra structures.

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“…Theorem 14. The post-Lie Magnus expansion χ, described in (33), coincides with the weighted BCH-recursion χ λ recursively given by (26), with weight λ = 1.…”

Section: Post-lie Magnus Expansion and Bch-recursionmentioning

confidence: 99%

“…We close this introduction by noting that the Magnus expansion, in its various forms (classical [24,27], pre-Lie [1,10,16] and post-Lie [17,18,19,26]), has been studied in applied mathematics, control theory, physics and chemistry. See reference [6] for details on the classical Magnus expansion in applied mathematics.…”

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confidence: 99%