Zimmermann type cancellation in the free Faà di Bruno algebra

Abstract

Haiman and Schmitt showed that one can use the antipode S F of the colored Faà di Bruno Hopf algebra F to compute the (compositional) inverse of a multivariable formal power series. It is shown that the antipode S H of an algebraically free analogue H of F may be used to invert non-commutative power series. Whereas F is the incidence Hopf algebra of the colored partitions of finite colored sets, H is the incidence Hopf algebra of the colored interval partitions of finite totally ordered colored sets. Haiman and Schmitt showed that the monomials in the geometric series for S F are labeled by trees. By contrast, the noncommuting monomials of S H are labeled by colored planar trees. The order of the factors in each summand is determined by the breadth first ordering on the vertices of the planar tree. Finally there is a parallel to Haiman and Schmitt's reduced tree formula for the antipode, in which one uses reduced planar trees and the depth first ordering on the vertices. The reduced planar tree formula is proved by recursion, and again by an unusual cancellation technique. The one variable case of H has also been considered by Brouder, Frabetti, and Krattenthaler, who point out its relation to Foissy's free analogue of the Connes-Kreimer Hopf algebra.

Quantum field theory meets Hopf algebra

“…The symmetric algebra S(C) = ∞ n=0 S n (C) can be equipped with the structure of a bialgebra over C. The product of the bialgebra S(C) is the symmetric product (denoted by juxtaposition) and its coproduct Δ is defined on S 1 (C) ∼ = C by Δa = Δ a and extended to S(C) by algebra morphism: Δ1 = 1 ⊗ 1 and Δ(uv) = u (1) (2) . The elements of S n (C) are said to be of degree n. The counit ε of S(C) is defined to be equal to ε on S 1 (C) ∼ = C and extended to S(C) by algebra morphism: ε(1) = 1 and ε(uv) = ε(u)ε(v).…”

A Faà di Bruno Hopf algebra for a group of Fliess operators with applications to feedback

“…For example, a ∅ ∈ A 0 but a ∅ ∈σ(R). It is also clear that the coproduct terms computed above do not satisfy (2). Despite this fact, the following theorem still holds.…”

“…The treatment is based on [1], [2], [6], [25]. The starting point is a systematic statement of what it means for a set A to be an associative R-algebra.…”

“…In either case, if f 1 = 0 then f has a compositional inverse, f −1 , and therefore, the corresponding set of functions forms a group under composition. In the special case where f 1 = 1, the coordinate functions a n : f → f n , n ≥ 1 on the corresponding subgroup form a graded connected Hopf algebra, a so called Faà di Bruno Hopf algebra [2], [3], [6], [20]. The antipode of this Hopf algebra acts on each coordinate function to produce a polynomial expression for the coordinates of the compositional inverse.…”

Hopf algebras and the logarithm of the $S$-transform in free probability

“…This follows immediately when we combine the formal series expansion which defines log χ μ (Equation ( Case |w| = 2. There exists a unique chain in NC (2), namely Γ = (0 2 , 1 2 ). Thus for a word of length 2, w = (i 1 , i 2 ), the formula (4.8) simply amounts to (4.9)…”