Séminaire Lotharingien de Combinatoire, B93c (2025), 37 pp.
Adrián Celestino and Frédéric Patras
A forest formula for pre-Lie exponentials, Magnus operator and cumulant-cumulant relations
Abstract.
Pre-Lie algebras and their enveloping algebras have broad applications in various fields, including geometry, analysis, numerical analysis, deformation theory, operad theory, quantum field theory, combinatorial Hopf algebras, and non-commutative probability. In this work, we introduce a novel method for computing iterated coproducts on these algebras using combinatorial forest formulas. We demonstrate the effectiveness of our approach by investigating two fundamental maps in pre-Lie algebras: the pre-Lie exponential or Agrachev--Gamkrelidze operator and its inverse, the pre-Lie logarithm or Magnus operator. We apply our method to the examples of the free pre-Lie algebra and a pre-Lie algebra of words based on free probability, highlighting meaningful combinatorial properties on rooted trees. Specifically, we can obtain combinatorial formulas that link various types of cumulants in non-commutative probability.
Received: July 1, 2025.
Revised: November 6, 2025.
Accepted: November 6, 2025.
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