A comodule-bialgebra structure for word-series substitution and mould composition

Abstract

An internal coproduct is described, which is compatible with Hoffman's quasishuffle product. Hoffman's quasi-shuffle Hopf algebra, with deconcatenation coproduct, is a comodule-Hopf algebra over the bialgebra thus defined. The relation with Ecalle's mould calculus, i.e., mould composition and contracting arborification is precised.1 Note that the comould C is chosen to be a monoid antimorphism in [15]. KURUSCH EBRAHIMI-FARD, FRÉDÉRIC FAUVET, AND DOMINIQUE MANCHONThe algebra k Ω is the dual of the coalgebra k Ω of noncommutative polynomials with variables in Ω, endowed with the deconcatenation coproduct:Moreover, a mould gives rise, by linear extension, to a unique linear form on k Ω . The identification of the vector space of moulds with k Ω is achieved through the map:which associates to each mould M its corresponding word series W M ∈ k Ω . It is wellknown [26] that H Ω = (k Ω , ∐ ∐ -, ∆) is a commutative Hopf algebra, where ∐ ∐ -is Hoffman's quasi-shuffle product, recursively defined by ω∐∐ -1 = 1∐∐ω = ω (here 1 stands for the empty word) and:. Here a and b are letters in Ω and ω ′ , ω ′′ are words in Ω * . The notation [a + b] stands for the internal sum of the two letters in the commutative semigroup Ω. The Hopf algebra H Ω is (Ω ⊔ {0})-graded by the weight defined by ||1|| := 0 and: ||ω|| := [ω 1 + · · · + ω ℓ ] ∈ Ω for a word ω = ω 1 · · · ω ℓ ∈ Ω * of length |ω| := ℓ. The mould product × of [15] is hence obtained by dualizing the deconcatenation coproduct ∆, and thus reflects the noncommutative concatenation product in k Ω , namely:

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

“…The key point of this work, which allows for the comparison envisaged, is that these two bialgebra structures together form a comodule bialgebra. This is an intricate structure of two interacting bialgebras which recently has appeared in numerical analysis [6], local dynamical systems [10], and stochastic integration and renormalization [5]. Precisely, in Theorem 5.1.1 we establish that the gap-insertion bialgebra is an unshuffle-type bialgebra object in the symmetric monoidal category of comodules for the block-substitution bialgebra.…”

“…In particular, m n = P ∈SP(n) ψ(P ) and equations ( 12) and ( 13) can be interpreted as algebraic and operadic lifts of the classical moment-cumulant formula, respectively formula (10).…”

Quasi‐shuffle algebras and renormalisation of rough differential equations

“…Generally, the process of (contracting) arborification is given by a surjective Hopf algebra morphism from the ‐decorated Butcher–Connes–Kreimer Hopf algebra onto the (quasi) shuffle Hopf algebra () defined over the alphabet . See for details. In the shuffle case, the arborification morphism is defined by and , that is, …”

“…If f is invertible we may assume that f 1 = 1, such that ψ f • ψ f −1 = ψ t = id. Hoffman's exponential (16) and logarithm (17) follow from the regular exponential and logarithm. Recall that Lemma 2 states that the map Ψ v = (v ⊗ id) • Φ associated to a character v of H A + is a Hopf algebra automorphism of H A BCK .…”

Ecalle's paralogarithmic resurgence monomials and effective synthesis

“…A mould can alternatively be seen as a linear function from the free vector space spanned by sequences/words to the algebra A; this point of view can be quite fruitful (see e.g. [9,16]), although we shall stick here to Ecalle's definitions and notations.…”