A Non-crossing Word Cooperad for Free Homotopy Probability Theory

Drummond-Cole, Gabriel C.

doi:10.1007/978-3-319-72299-3_5

Cited by 3 publications

(2 citation statements)

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“…We shall see in Proposition 5.2.5 that (11) and the sum formula for m n find a more natural expression in terms of the bialgebra interaction we introduce in this paper. We can thus interpret Equation ( 11) as an algebraic lift of (9).…”

Section: Moments and Cumulantsmentioning

confidence: 99%

“…A different operadic viewpoint on free moment-cumulant relations was taken by Drummond-Cole [8,9], who gave an operadic interpretation of Speicher's multiplicativity [40], and exhibited the free moment-cumulant relations in terms of convolution of maps from a cooperad to an endomorphism operad. Again, the operads and cooperads considered are not directly on noncrossing partitions, but rather on auxiliary classes of decorated trees.…”

Section: Introductionmentioning

confidence: 99%

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Operads of (noncrossing) partitions, interacting bialgebras, and moment-cumulant relations

Ebrahimi-Fard,

Foissy,

Kock

et al. 2019

Preprint

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We establish and explore a relationship between two approaches to moment-cumulant relations in free probability theory: on one side the main approach, due to Speicher, given in terms of Möbius inversion on the lattice of noncrossing partitions, and on the other side the more recent non-commutative shuffle-algebra approach, where the moment-cumulant relations take the form of certain exponential-logarithm relations. We achieve this by exhibiting two operad structures on (noncrossing) partitions, different in nature: one is an ordinary, non-symmetric operad whose composition law is given by insertion into gaps between elements, the other is a coloured, symmetric operad with composition law expressing refinement of blocks. We show that these operad structures interact so as to make the corresponding incidence bialgebra of the former a comodule bialgebra for the latter. Furthermore, this interaction is compatible with the shuffle structure and thus unveils how the two approaches are intertwined. Moreover, the constructions and results are general enough to extend to ordinary set partitions.

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Section: Moments and Cumulantsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%