2006/03/01 by Hans Munthe–Kaas, Munthe-Kaas, H. Z., William M. D. Wright +1 · 5 citations
Computer Science · Mathematics · #16W25 #22E60 #37M99 #65L05 #81R60 #Advanced Topics in Algebra #Commutative Algebra (math.AC) #FOS: Mathematics #Matrix Theory and Algorithms #Numerical Analysis (math.NA) #Numerical methods for differential equations
paper · pdf · doi:10.48550/arxiv.math/0603023
openalex publication_date 2006/03/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01
A commutative but not cocommutative graded Hopf algebra \Hn, based on ordered rooted trees, is studied. This Hopf algebra generalizes the Hopf algebraic structure of unordered rooted trees \Hc, developed by Butcher in his study of Runge--Kutta methods and later rediscovered by Connes and Moscovici in the context of non-commutative geometry and by Kreimer where it is used to describe renormalization in quantum field theory. It is shown that \Hn is naturally obtained from a universal object in a category of non-commutative derivations, and in particular, it forms a foundation for the study of numerical integrators based on non-commutative Lie group actions on a manifold. Recursive and non-recursive definitions of the coproduct and the antipode are derived. It is also shown that the dual of \Hn is a Hopf algebra of Grossman and Larson. \Hn contains two well-known Hopf algebras as special cases: The Hopf algebra \Hc of Butcher--Connes--Kreimer is identified as a proper subalgebra of \Hn using the image of a tree symmetrization operator. The Hopf algebra \Hf of the Free Associative Algebra is obtained from \Hn by a quotient construction.