Search

After providing a short review on the recently introduced notion of post-group by Bai, Guo, Sheng and Tang, we exhibit post-group counterparts of important post-Lie algebras in the literature, including the infinite-dimensional post-Lie algebra of Lie group integrators. The notion of free post-group is examined, and a group isomorphism between the two group structures associated to a free post-group is explicitly constructed.

The goal of this paper is to establish some facts concerning the problem of zeroing the output of an input-output system that does not have relative degree. The approach taken is to work with systems that have Chen-Fliess series representations. The main result is that a class of generating series called primely nullable series provides the building blocks for solving this problem using shuffle algebra. This is achieved by viewing the latter as the symmetric algebra over the commutative polynomials in Lyndon words in order to show that it is a unique factorization domain. Next, the focus turns to factoring generating series in the shuffle algebra into its irreducible elements. A specific algorithm based on the Chen-Fox-Lyndon factorization of words is given.

We give series solutions to single insertion place propagator-type systems of Dyson--Schwinger equations using binary tubings of rooted trees. These solutions are combinatorially transparent in the sense that each tubing has a straightforward contribution. The Dyson--Schwinger equations solved here are more general than those previously solved by chord diagram techniques, including systems and non-integer values of the insertion parameter $s$. We remark on interesting combinatorial connections and properties., 59 pages, more references and other edits from comments received

The goal of the paper is multi-fold. The first of which is to derive an explicit formula to compute the generating series of a closed-loop system when a plant, given in a Chen--Fliess series description is in multiplicative output feedback connection with another system given in Chen--Fliess series description. Further, the objective extends in showing that the multiplicative dynamic output feedback connection has a natural interpretation as a transformation group acting on the plant. A computational framework for computing the generating series for multiplicative dynamic output feedback is devised utilizing the dual Hopf algebras corresponding to the shuffle group and the multiplicative feedback group., arXiv admin note: text overlap with arXiv:2207.08300

The goal of this paper is to establish some facts concerning the problem of zeroing the output of an input-output system that does not have relative degree. The approach taken is to work with systems that have a Chen-Fliess series representation. The main result is that a class of generating series called primely nullable series provides the building blocks for solving this problem using the shuffle algebra. It is shown that the shuffle algebra on the set of generating polynomials is a unique factorization domain so that any polynomial can be uniquely factored modulo a permutation into its irreducible elements for the purpose of identifying nullable factors. This is achieved using the fact that this shuffle algebra is isomorphic to the symmetric algebra over the vector space spanned by Lyndon words. A specific algorithm for factoring generating polynomials into its irreducible factors is presented based on the Chen-Fox-Lyndon factorization of words.

We extend Foissy's work on finite-dimensional comodules over the Butcher--Connes--Kreimer Hopf algebra of non-planar rooted trees to the Munthe-Kaas--Wright Hopf algebra of planar rooted trees. This includes the description of its endomorphisms as well as the recursive construction of its primitive elements. The findings are applied in the context of rough paths, where we describe an isomorphism between planarly branched and geometric rough paths. We also explore the geometric embedding for planar regularity structures as well as the link between the concept of bialgebras in cointeraction, the Guin--Oudom construction and the notion of translation on rough paths.

Recent works have explored relations between classical and quantum statistical physics on the one hand and Voiculescu's theory of free probability on the other. Motivated by these results, the present work focuses on the notion of effective action, which is closely related to the large deviation rate function in classical probability and one-particle irreducible correlation functions in quantum field theories. The central aim is to understand how it can be defined and studied in free probability. In this respect, we introduce a suitable diagrammatic formalism.

Gian-Carlo Rota mentioned in one of his last articles the problem of developing a theory around the notion of integration algebras, which should be dual to the one of differential algebras. This idea has been developed historically along three lines: using properties of iterated integrals and generalisations thereof, as in the theory of rough paths; using the algebraic structures underlying chronological calculus such as shuffle and pre-Lie products, as they appear in theoretical physics and control theory; and, more generally, using a particular operator identity which came to be known as Rota-Baxter relation. The recent developments along each of these lines of research and their various application domains are not always known to other communities in mathematics and related fields. The general aim of this survey is therefore to present a modern and unified perspective on these approaches, featuring among others non-commutative Rota-Baxter algebras and related Hopf and Lie algebraic structures such as descent algebras, algebras of simplices as well as shuffle, pre- and post-Lie algebras and their enveloping algebras. Accordingly, two viewpoints are proposed. A geometric one, that can be derived using combinatorial properties of Euclidean simplices and the duality between integrands and integration domains in iterated integrals. An algebraic one, provided by the axiomatisation of integral calculus in terms of Rota-Baxter algebra.

We consider the bialgebra of hypergraphs, a generalization of Schmitt's Hopf algebra of graphs, and show it has a cointeracting bialgebra. So one has a double bialgebra in the sense of L. Foissy, who recently proved there is then a unique double bialgebra morphism to the double bialgebra structure on the polynomial ring ${\mathbb Q}[x]$. We show the polynomial associated to a hypergraph is the hypergraph chromatic polynomial. Moreover hypergraphs occurs in quartets: there is a dual, a complement, and a dual complement hypergraph. These correspondences are involutions and give rise to three other double bialgebras, and three more chromatic polynomials. In all we give eight quartets of bialgebras which includes recent bialgebras of M. Aguiar and F. Ardila, and by L. Foissy., 29 pages. 1. Title is changed. Previously: "Twelve bialgebras of bialgebras, ..." 2. An inaccuracy concerning the cointeraction has been corrected. The coproduct of the bialgebra in the last section has been corrected. 3. Two more quartets of bialgebras are added. The last quartet comes from a recent extraction-contraction bialgebra introduced by L. Foissy

The role of coalgebras as well as algebraic groups in non-commutative probability has long been advocated by the school of von Waldenfels and Sch\"urmann. Another algebraic approach was introduced more recently, based on shuffle and pre-Lie calculus, and resulting in another construction of groups of characters encoding the behaviour of states. Comparing the two, the first approach, recast recently in a general categorical language by Manzel and Sch\"urmann, can be seen as largely driven by the theory of universal products, whereas the second construction builds on Hopf algebras and a suitable algebraization of the combinatorics of noncrossing set partitions. Although both address the same phenomena, moving between the two viewpoints is not obvious. We present here an attempt to unify the two approaches by making explicit the Hopf algebraic connections between them. Our presentation, although relying largely on classical ideas as well as results closely related to Manzel and Sch\"urmann's aforementioned work, is nevertheless original on several points and fills a gap in the free probability literature. In particular, we systematically use the language and techniques of algebraic groups together with shuffle group techniques to prove that two notions of algebraic groups naturally associated with free, respectively Boolean and monotone, probability theories identify. We also obtain explicit formulas for various Hopf algebraic structures and detail arguments that had been left implicit in the literature., Comment: 24 pages

We study a particular group law on formal power series in non-commuting parameters induced by their interpretation as linear forms on a suitable non-commutative and non- cocommutative graded connected word Hopf algebra. This group law is left-linear and is therefore associated to a pre-Lie structure on formal power series. We study these structures and show how they can be used to recast in a group theoretic form various identities and transformations on formal power series that have been central in the context of non-commutative probability theory, in particular in Voiculescu?s theory of free probability.

This article approaches the counting of subgraphs, in terms of signature-type functionals defined over combinatorial Hopf algebras of graphs. Well-known algebraic identities that arise in the context of counting subgraphs are then captured by their character property and a type of "Chen's identity". While different notions of subgraphs (and homomorphisms) correspond to different combinatorial Hopf algebras on graphs, we will show that they are all isomorphic to a polynomial Hopf algebra. In addition, the isomorphy between the Hopf algebras can be realized by maps that respect the counting operations., Comment: 43 pages, 1 figure

International audience; We identify the Baker-Campbell-Hausdorff recursion driven by a weight$\lambda=1$ Rota-Baxter operator with the Magnus expansion relativeto the post-Lie structure naturally associated to the correspondingRota-Baxter algebra. Post-Lie Magnus expansion and BCH-recursionare reviewed before the proof of the main result.

Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free infinite divisibility via the Boolean Bercovici--Pata bijection. On the other hand, in recent years the concept of infinitesimal non-commutative probability has been developed, together with the notion of infinitesimal cumulants which can be useful in the context of combinatorial questions. In this paper, we show that the known relations among free, Boolean and monotone cumulants still hold in the infinitesimal framework. Our approach is based on the use of Grassmann algebra. Formulas involving infinitesimal cumulants can be obtained by applying a formal derivation to known formulas. The relations between the various types of cumulants turn out to be captured via the shuffle algebra approach to moment-cumulant relations in non-commutative probability theory. In this formulation, (free, Boolean and monotone) cumulants are represented as elements of the Lie algebra of infinitesimal characters over a particular combinatorial Hopf algebra. The latter consists of the graded connected double tensor algebra defined over a non-commutative probability space and is neither commutative nor cocommutative. In this note it is shown how the shuffle algebra approach naturally extends to the notion of infinitesimal non-commutative probability space. The basic step consists in replacing the base field as target space of linear Hopf algebra maps by the Grassmann algebra over the base field. We also consider the infinitesimal analog of the Boolean Bercovici--Pata map., Comment: Typos corrected and reference [Has11] added

International audience; This chapter is divided into two parts. The first is largely expository and builds on Karandikar's axiomatisation of Itô calculus for matrix-valued semimartin-gales. Its aim is to unfold in detail the algebraic structures implied for iterated Itô and Stratonovich integrals. These constructions generalise the classical rules of Chen calculus for deterministic scalar-valued iterated integrals. The second part develops the stochastic analog of what is commonly called chronological calculus in control theory. We obtain in particular a pre-Lie Magnus formula for the logarithm of the Itô stochastic exponential of matrix-valued semimartingales.

Hairer’s regularity structures transformed the solution theory of singular stochastic partial differential equations. The notions of positive and negative renormalisation are central and the intricate interplay between these two renormalisation procedures is captured through the combination of cointeracting bialgebras and an algebraic Birkhoff-type decomposition of bialgebra morphisms. This work revisits the latter by defining Bogoliubov-type recursions similar to Connes and Kreimer’s formulation of BPHZ renormalisation. We then apply our approach to the renormalisation problem for SPDEs as well as the proposal for resonance based numerical schemes for certain partial differential equations in numerical analysis introduced in a recent work by the first author.

This chapter is divided into two parts. The first is largely expository and builds on Karandikar's axiomatisation of It{\^o} calculus for matrix-valued semimartin-gales. Its aim is to unfold in detail the algebraic structures implied for iterated It{\^o} and Stratonovich integrals. These constructions generalise the classical rules of Chen calculus for deterministic scalar-valued iterated integrals. The second part develops the stochastic analog of what is commonly called chronological calculus in control theory. We obtain in particular a pre-Lie Magnus formula for the logarithm of the It{\^o} stochastic exponential of matrix-valued semimartingales.

International audience

This chapter is divided into two parts. The first is largely expository and builds on Karandikar's axiomatisation of It{��} calculus for matrix-valued semimartin-gales. Its aim is to unfold in detail the algebraic structures implied for iterated It{��} and Stratonovich integrals. These constructions generalise the classical rules of Chen calculus for deterministic scalar-valued iterated integrals. The second part develops the stochastic analog of what is commonly called chronological calculus in control theory. We obtain in particular a pre-Lie Magnus formula for the logarithm of the It{��} stochastic exponential of matrix-valued semimartingales.

Hairer's regularity structures transformed the solution theory of singular stochastic partial differential equations. The notions of positive and negative renormalisation are central and the intricate interplay between these two renormalisation procedures is captured through the combination of cointeracting bialgebras and an algebraic Birkhoff-type decomposition of bialgebra morphisms. This work revisits the latter by defining Bogoliubov-type recursions similar to Connes and Kreimer's formulation of BPHZ renormalisation. We then apply our approach to the renormalisation problem for SPDEs., Comment: 28 pages

We explore the algebraic properties of a generalized version of the iterated-sums signature, inspired by previous work of F.~Kir\'aly and H.~Oberhauser. In particular, we show how to recover the character property of the associated linear map over the tensor algebra by considering a deformed quasi-shuffle product of words on the latter. We introduce three non-linear transformations on iterated-sums signatures, close in spirit to Machine Learning applications, and show some of their properties., Comment: 17 pages

We survey and extend results on a recently defined character on the quasi-shuffle algebra. This character, termed iterated-sums signature, appears in the context of time series analysis and originates from a problem in dynamic time warping. Algebraically, it relates to (multidimensional) quasisymmetric functions as well as (deformed) quasi-shuffle algebras.

Driven by the need for principled extraction of features from time series, we introduce the iterated-sums signature over any commutative semiring. The case of the tropical semiring is a central, and our motivating, example, as it leads to features of (real-valued) time series that are not easily available using existing signature-type objects.

Wick polynomials and Wick products are studied in the context of non-commutative probability theory. It is shown that free, boolean and conditionally free Wick polynomials can be defined and related through the action of the group of characters over a particular Hopf algebra. These results generalize our previous developments of a Hopf algebraic approach to cumulants and Wick products in classical probability theory.

A detailed expose of the Hopf algebra approach to interconnected input-output systems in nonlinear control theory is presented. The focus is on input-output systems that can been represented in terms of Chen-Fliess functional expansions or Fliess operators. This provides a starting point for a discrete-time version of this theory. In particular, the notion of a discrete-time Fliess operator is given and a class of parallel interconnections is described., 43 pages, review article

We explain the notion of a post-Lie algebra and outline its role in the theory of Lie group integrators. This article will not be available due to copyright restrictions (c) 2018 by Springer

In these notes we review and further explore the Lie enveloping algebra of a post-Lie algebra. From a Hopf algebra point of view, one of the central results, which will be recalled in detail, is the existence of a second Hopf algebra structure. By comparing group-like elements in suitable completions of these two Hopf algebras, we derive a particular map which we dub post-Lie Magnus expansion. These results are then considered in the case of Semenov-Tian-Shansky's double Lie algebra, where a post-Lie algebra is defined in terms of solutions of modified classical Yang-Baxter equation. In this context, we prove a factorization theorem for group-like elements. An explicit exponential solution of the corresponding Lie bracket flow is presented, which is based on the aforementioned post-Lie Magnus expansion., Comment: 49 pages, no-figures, review article

We consider multiple polylogarithms in a single variable at non-positive integers. Defining a connected graded Hopf algebra, we apply Connes' and Kreimer's algebraic Birkhoff decomposition to renormalize multiple polylogarithms at non-positive integer arguments, which satisfy the shuffle relation. The q-analogue of this result is as well presented, and compared to the classical case., Comment: some typos fixed, references updated

We explain the notion of a post-Lie algebra and outline its role in the theory of Lie group integrators.

Poincare's center problem asks for conditions under which a planar polynomial system of ordinary differential equations has a center. It is well understood that the Abel equation naturally describes the problem in a convenient coordinate system. In 1989, Devlin described an algebraic approach for constructing sufficient conditions for a center using a linear recursion for the generating series of the solution to the Abel equation. Subsequent work by the authors linked this recursion to feedback structures in control theory and combinatorial Hopf algebras, but only for the lowest degree case. The present work introduces what turns out to be the nontrivial multivariable generalization of this connection between the center problem, feedback control, and combinatorial Hopf algebras. Once the picture is completed, it is possible to provide generalizations of some known identities involving the Abel generating series. A linear recursion for the antipode of this new Hopf algebra is also developed using coderivations. Finally, the results are used to further explore what is called the composition condition for the center problem., Comment: final version

The goal of the paper is to give a systematic way to numerically evaluate the generating function of a periodic multiple polylogarithm using a Chen-Fliess series with a rational generating series. The idea is to realize the corresponding Chen-Fliess series as a bilinear dynamical system. A standard form for such a realization is given. The method is also generalized to the case where the multiple polylogarithm has non-periodic components. This allows one, for instance, to numerically validate the Hoffman conjecture. Finally, a setting in terms of dendriform algebras is provided.

We describe in this work all solutions to the problem of renormalizing multiple zeta values at arguments of any sign in a quasi-shuffle compatible way. As a corollary we clarify the relation between different renormalizations at non-positive values appearing in the recent literature., Comment: Abbreviated version of a manuscript, 8 pages

The notion of trees plays an important role in Butcher's B-series. More recently, a refined understanding of algebraic and combinatorial structures underlying the Magnus expansion has emerged thanks to the use of rooted trees. We follow these ideas by further developing the observation that the logarithm of the solution of a lihear first-order finite-difference equation can be written in terms of the Magnus expansion taking place in a pre-Lie algebra. By using basic combinatorics on planar reduced trees we derive a closed formula for the Magnus expansion in the context of free tridendriform algebra. The tridendriform algebra structure on word quasi-symmetric functions permits us to derive a discrete analogue of the Mielnik-Plebanski-Strichartz formula for this logarithm., To appear in Discrete and Continuous Dynamical Systems series A (AIMS). Partly supported by Agence Nationale de la Recherche, projet CARMA ANR-12-BS01-0017, and Ministerio de Econom{\i}a y Competitividad.MTM2011-23050

Gian-Carlo Rota suggested in one of his last articles the problem of developing a theory around the notion of integration algebras, complementary to the already existing theory of differential algebras. This idea was mainly motivated by Rota's deep appreciation for Kuo-Tsai Chen's seminal work on iterated integrals. As a starting point for such a theory of integration algebras Rota proposed to consider a particular operator identity first introduced by the mathematician Glen Baxter. Later it was coined Rota-Baxter identity. In this article we briefly recall basic properties of Rota--Baxter algebras, and present a concise review of recent work with a particular emphasis of noncommutative aspects.

International audience; Hopf algebra structures on rooted trees are by now a well-studied object, especially in the context of combinatorics. They are essentially characterized by the coproduct map. In this work we define yet another Hopf algebra H by introducing a new coproduct on a (commutative) algebra of rooted forests, considering each tree of the forest (which must contain at least one edge) as a Feyman-like graph without loops. The primitive part of the graded dual is endowed with a pre-Lie product defined in terms of insertion of a tree inside another. We establish a surprising link between the Hopf algebra H obtained this way and the well-known Connes-Kreimer Hopf algebra of rooted trees by means of a natural H-bicomodule structure on the latter. This enables us to recover results in the field of numerical methods for differential equations due to Chartier, Hairer and Vilmart as well as Murua.

Hopf algebra structures on rooted trees are by now a well-studied object, especially in the context of combinatorics. In this work we consider a Hopf algebra H by introducing a coproduct on a (commutative) algebra of rooted forests, considering each tree of the forest (which must contain at least one edge) as a Feynman-like graph without loops. The primitive part of the graded dual is endowed with a pre-Lie product defined in terms of insertion of a tree inside another. We establish a surprising link between the Hopf algebra H obtained this way and the well-known Connes-Kreimer Hopf algebra of rooted trees by means of a natural H-bicomodule structure on the latter. This enables us to recover recent results in the field of numerical methods for differential equations due to Chartier, Hairer and Vilmart as well as Murua., Comment: Error in antipode formula (section 7) corrected. Erratum submitted

This paper introduces a new Lie-theoretic approach to the computation of counterterms in perturbative renormalization. Contrary to the usual approach, the devised version of the Bogoliubov recursion does not follow a linear induction on the number of loops. It is well-behaved with respect to the Connes-Kreimer approach: that is, the recursion takes place inside the group of Hopf algebra characters with values in regularized Feynman amplitudes. (Paradigmatically, we use dimensional regularization in the minimal subtraction scheme, although our procedure is generalizable to other schemes.) The new method is related to Zassenhaus' approach to the Baker-Campbell-Hausdorff formula for computing products of exponentials. The decomposition of counterterms is parametrized by a family of Lie idempotents known as the Zassenhaus idempotents. It is shown, inter alia, that the corresponding Feynman rules generate the same algebra as the graded components of the Connes-Kreimer beta-function. This further extends previous work of ours (together with Jose M. Gracia-Bondia) on the connection between Lie idempotents and renormalization procedures, where we constructed the Connes-Kreimer beta-function by means of the classical Dynkin idempotent., 11 pages, improved version, to appear in Lie Theory and Its Applicatons in Physics VII ed. V.K. Dobrev et al., Heron Press, Sofia, 2008

We describe various combinatorial aspects of the Birkhoff-Connes-Kreimer factorization in perturbative renormalisation. The analog of Bogoliubov's preparation map on the Lie algebra of Feynman graphs is identified with the pre-Lie Magnus expansion. Our results apply to any connected filtered Hopf algebra, based on the pro-nilpotency of the Lie algebra of infinitesimal characters., improved version, 20 pages, CIRM 2006 workshop "Renormalization and Galois Theory", Org. F. Fauvet, J.-P. Ramis

We study multiple zeta values and their generalizations from the point of view of Rota--Baxter algebras. We obtain a general framework for this purpose and derive relations on multiple zeta values from relations in Rota--Baxter algebras.

Coherent unit actions on a binary, quadratic operad were introduced by Loday and were shown by him to give Hopf algebra structures on the free algebras when the operad is also regular with a splitting of associativity. Working with such operads, we characterize coherent unit actions in terms of linear equations of the generators of the operads. We then use these equations to give all possible operad relations that allow such coherent unit actions. We further show that coherent unit actions are preserved under taking products and thus yield Hopf algebras on the free object of the product operads when the factor operads have coherent unit actions. On the other hand, coherent unit actions are never preserved under taking the dual in the operadic sense except for the operad of associative algebras.

In this brief note we would like to report on an observation concerning the relation between Rota-Baxter operators and Loday-type algebras, i.e. dendriform di- and trialgebras. It is shown that associative algebras equipped with a Rota-Baxter operator of arbitrary weight always give such dendriform structures., 13 pages, no figures, short remark at the end of Sect. 4 added, minor typos corrected

In this note we comment on part of a recent article by B. Schroer and H.-W. Wiesbrock. Therein they calculate some new modular structure for the U(1)-current-algebra (Weyl-algebra). We point out that their findings are true in a more general setting. The split-property allows an extension to doubly-localized algebras., Comment: 13 pages, corrected version

Boolean, free and monotone cumulants as well as relations among them, have proven to be important in the study of non-commutative probability theory. Quite notably, Boolean cumulants were successfully used to study free infinite divisibility via the Boolean Bercovici-Pata bijection. On the other hand, in recent years the concept of infinitesimal non-commutative probability has been developed, together with the notion of infinitesimal cumulants which can be useful in the context of combinatorial questions. In this paper, we show that the known relations among free, Boolean and monotone cumulants still hold in the infinitesimal framework. Our approach is based on the use of algebra of Grassmann numbers. Formulas involving infinitesimal cumulants can be obtained by applying a formal derivation to known formulas. The relations between the various types of cumulants turn out to be captured via the shuffle algebra approach to moment-cumulant relations in non-commutative probability theory. In this formulation, (free, Boolean and monotone) cumulants are represented as elements of the Lie algebra of infinitesimal characters over a particular combinatorial Hopf algebra. The latter consists of the graded connected double tensor algebra defined over a non-commutative probability space and is neither commutative nor cocommutative. In this note it is shown how the shuffle algebra approach naturally extends to the notion of infinitesimal non-commutative probability space. The basic step consists in replacing the base field as target space of linear Hopf algebra maps by the algebra of Grassmann numbers defined over the base field. We also consider the infinitesimal analog of the Boolean Bercovici-Pata map., DOCUMENTA MATHEMATICA, Vol 26 (2021), p. 1145-1185

Noen uttrykk for løsningen på en ordinær differensiallikning utledes ved hjelp av en metode som innebærer Taylor-rekkeutvikling av vektorfeltet som definerer differensiallikningen. En generalisering av et kjent resultat utledes ved hjelp av denne fremgangsmåten. Følger av denne generaliseringen drøftes, og det vises hvordan disse gir ytterligere innsikt i løsningen på en beslektet mengde partielle differensiallikninger. To fysiske anvendelser av resultatene presenteres. Den første er en metode for å evaluere Feynman-integraler, og den andre er en metode for å finne svømmebanen til mikrober. Resultatene i denne oppgaven kan anvendes i mange fysiske fagfelt. Some expressions for the solution of an ordinary differential equation (ODE) are derived. The method used involves Taylor expanding the vector field defining the ODE. A generalization of a known result is derived taking this approach. Consequences of this generalization are elaborated upon and are shown to produce further insights into the solution of a related set of partial differential equations. Two physical applications of the results are presented. The first is a method for evaluating Feynman integrals, the second is a method for finding the path of swimming microbes. The results of this thesis have applications in many fields of Physics.

The notion of cumulants plays a significant role in the combinatorial study of noncommutative probability theory. In this thesis, we study several problems associated with the notions of cumulants for free, Boolean and monotone independence via a unified Hopf algebraic framework for non-commutative probability. In this framework, developed by Ebrahimi-Fard and Patras [EFP15, EFP16, EFP18], the relations between moments and the different brands of cumulants are described by three different exponentials that relate the group of characters and the Lie algebra of infinitesimal characters on a particular word Hopf algebra H. The first question that we address is to show how the algebraic central limit theorems for free, Boolean and monotone independence enter into the shuffle-algebraic picture for non-commutative probability. We analyze how the different notions of additive convolutions are described in the shuffle framework and prove combinatorial formulas for the powers of the half-shuffle products. Next, we focus on extending the shuffle framework in order to consider the extension of non-commutative probability called infinitesimal non-commutative probability. We show how infinitesimal cumulants also correspond to infinitesimal characters on H and describe the shuffle-algebraic equations for the infinitesimal moment-cumulant relations. We also prove that the combinatorial relations between infinitesimal cumulants follow via the extended shuffle-algebraic framework. Afterwards, we concentrate on the problem of writing the multivariate monotone cumulants of random variables in terms of their moments. The starting point to obtain this formula is to compute a logarithm of a certain character on H. Then by investigating a connection of H with a Hopf algebra of decorated rooted trees, we compute the logarithm instead in the Hopf algebra of trees, which yields a new combinatorial formula from moments to monotone cumulants in terms of Schröder trees. Thereafter, we turn to the problem, left open in [AHLV15], of expressing multivariate monotone cumulants in terms of the free and Boolean cumulants. Our approach relies on a pre-Lie algebraic relation between the three infinitesimal characters associated to the free, Boolean and monotone cumulants. This relation is known as the pre-Lie Magnus expansion and is defined in terms of iterations of a pre-Lie product. Our problem is then transformed into finding a combinatorial formula in terms of non-crossing partitions of the iterated pre-Lie products and identifying and describing the coefficients that govern the transition from free and Boolean to monotone cumulants. The coefficients obtained in the solution of the previous problem hint at a deeper relation between non-commutative probability, combinatorics of rooted trees and free pre-Lie algebras. Our last question is to have a more systematic understanding of the previous relation. In the process, we develop a concrete and effective method of computations on a specific class of pre-Lie algebras, where iterations of the pre-Lie product can be computed in terms of forest formulas for iterated coproducts. We illustrate our method by retrieving the formulas between monotone and free (Boolean) cumulants, as well as the computation of the pre-Lie Magnus expansion on the generator of the free pre-Lie algebra of rooted trees.

Following Manin's approach to renormalization in the theory of computation, we investigate Dyson-Schwinger equations on Hopf algebras, operads and properads of flow charts, as a way of encoding self-similarity structures in the theory of algorithms computing primitive and partial recursive functions and in the halting problem.