The Pre-Lie Structure of the Time-Ordered Exponential

Ebrahimi‐Fard, Kurusch; Patras, Frédéric

doi:10.1007/s11005-014-0703-4

Cited by 18 publications

(23 citation statements)

References 41 publications

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“…It implies that the bracket [a, b] := a b − b a satisfies the Jacobi identity. For several reasons, largely owing to the general theory of integration encoded by Rota-Baxter algebras (see [18]), pre-Lie algebras play a key role, e.g. in the understanding of recursive equations such as Bogoliubov's counterterm formula in perturbative quantum field theory.…”

Section: Shuffle Algebrasmentioning

confidence: 99%

Cumulants, free cumulants and half-shuffles

2015

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Free cumulants were introduced as the proper analogue of classical cumulants in the theory of free probability. There is a mix of similarities and differences, when one considers the two families of cumulants. Whereas the combinatorics of classical cumulants is well expressed in terms of set partitions, that of free cumulants is described and often introduced in terms of non-crossing set partitions. The formal series approach to classical and free cumulants also largely differs. The purpose of this study is to put forward a different approach to these phenomena. Namely, we show that cumulants, whether classical or free, can be understood in terms of the algebra and combinatorics underlying commutative as well as non-commutative (half-)shuffles and (half-) unshuffles. As a corollary, cumulants and free cumulants can be characterized through linear fixed point equations. We study the exponential solutions of these linear fixed point equations, which display well the commutative, respectively non-commutative, character of classical and free cumulants.

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Section: Shuffle Algebrasmentioning

confidence: 99%

Cumulants, free cumulants and half-shuffles

2015

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“…The corresponding endomorphism of a pre-Lie algebra satisfying suitable completion properties appeared in the context of enveloping algebras of pre-Lie algebras in [1], together with its inverse, denoted W . The Hopf and group-theoretical properties of and its links with the Mielnik-Plebanskii-Strichartz continuous Baker-Campbell-Hausdorff formula have been investigated in [9,14]. We refer the reader to [20,24] for more details, including its range of applications and mathematical properties.…”

Section: Pre-lie Algebra and Cumulantsmentioning

confidence: 99%

Cumulant–Cumulant Relations in Free Probability Theory from Magnus’ Expansion

et al. 2021

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Relations between moments and cumulants play a central role in both classical and non-commutative probability theory. The latter allows for several distinct families of cumulants corresponding to different types of independences: free, Boolean and monotone. Relations among those cumulants have been studied recently. In this work, we focus on the problem of expressing with a closed formula multivariate monotone cumulants in terms of free and Boolean cumulants. In the process, we introduce various constructions and statistics on non-crossing partitions. Our approach is based on a pre-Lie algebra structure on cumulant functionals. Relations among cumulants are described in terms of the pre-Lie Magnus expansion combined with results on the continuous Baker–Campbell–Hausdorff formula due to A. Murua.

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“…From the general theory of shuffle algebras, it is known that the

BCH

formula and the pre‐Lie Magnus expansion are closely related (see for more details). In classical Lie theory, the

BCH

formula allows to define a universal group law at the Lie algebra level.…”

Section: Universal Shuffle Group Lawsmentioning

confidence: 99%

“…The central observation in reference is that monotone, free and boolean moment‐cumulant relations can be described in terms of three different exponential‐type maps associated, respectively, to the convolution product and the two half‐products defined on the dual

H^{*}

. Here, the notion of exponential‐type map generalises the one of time‐ordered exponential in physics . Logarithm‐type maps corresponding to these exponential‐type maps can be defined, and shuffle algebra identities permit to express monotone, free and boolean cumulants in terms of each other using the Magnus expansion familiar in the context of pre‐Lie algebras.…”

Section: Introductionmentioning

confidence: 99%

Shuffle group laws: applications in free probability

Ebrahimi‐Fard

Patras

2019

Proc. London Math. Soc.

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Commutative shuffle products are known to be intimately related to universal formulas for products, exponentials and logarithms in group theory as well as in the theory of free Lie algebras, such as, for instance, the Baker–Campbell–Hausdorff formula or the analytic expression of a Lie group law in exponential coordinates in the neighbourhood of the identity. Non‐commutative shuffle products happen to have similar properties with respect to pre‐Lie algebras. However, the situation is more complex since in the non‐commutative framework three exponential‐type maps and corresponding logarithms are naturally defined. This results in several new formal group laws together with new operations, for example, a new notion of adjoint action particularly well fitted to the new theory. These developments are largely motivated by various constructions in non‐commutative probability theory. The second part of the article is devoted to exploring and deepening this perspective. We illustrate our approach by revisiting universal products from a group‐theoretical viewpoint, including additive convolution in monotone, free and boolean probability, as well as the Bercovici–Pata bijection and subordination products.

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The Pre-Lie Structure of the Time-Ordered Exponential

Cited by 18 publications

References 41 publications

Cumulants, free cumulants and half-shuffles

Cumulants, free cumulants and half-shuffles

Cumulant–Cumulant Relations in Free Probability Theory from Magnus’ Expansion

Shuffle group laws: applications in free probability

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