The planar Hopf algebra of noncommutative multi-indices

Abstract

We construct the planar Linares--Otto--Tempelmayr Hopf algebra, thereby filling the missing planar noncommutative multi-index corner in the square relating the LOT, Butcher--Connes--Kreimer, and Munthe-Kaas--Wright Hopf algebras. Starting from the free associative algebra on a weighted alphabet $\mathbb Z_{\ge -1}\times A$, we define an insertion-type product yielding a post-Lie structure on the Lie algebra generated by the linear span $V(A)$ of weight $-1$ monomials whose proper left prefixes all have nonnegative weight, and the Guin--Oudom construction then produces the planar LOT Hopf algebra. We introduce a planar tree fertility map from decorated planar rooted trees to monomials in $V(A)$, prove that it is a linear isomorphism, and obtain a natural Hopf algebra isomorphism with the Munthe-Kaas--Wright Hopf algebra. We further derive an explicit coproduct formula in terms of left-admissible cuts, establish the extraction-contraction coproduct, and construct a word symmetrization operator compatible with the classical tree symmetrization operator.