Lie groups of controlled characters of combinatorial Hopf algebras

TL;DR

This paper constructs infinite-dimensional Lie groups from controlled characters of combinatorial Hopf algebras, with applications in physics, numerical analysis, and control theory, and establishes their regularity and algebraic structure.

Contribution

It introduces a general framework for Lie groups of controlled characters in combinatorial Hopf algebras, extending previous constructions like the Butcher group.

Findings

Controlled characters form infinite-dimensional Lie groups under certain growth conditions.

The Lie algebra of these groups is explicitly identified.

Regularity of the Lie groups is established in the sense of Milnor.

Abstract

In this article groups of controlled characters of a combinatorial Hopf algebra are considered from the perspective of infinite-dimensional Lie theory. A character is controlled in our sense if it satisfies certain growth bounds, e.g.\ exponential growth. We study these characters for combinatorial Hopf algebras. Following Loday and Ronco, a combinatorial Hopf algebra is a graded and connected Hopf algebra which is a polynomial algebra with an explicit choice of basis (usually identified with combinatorial objects such as trees, graphs, etc.). If the growth bounds and the Hopf algebra are compatible we prove that the controlled characters form infinite-dimensional Lie groups. Further, we identify the Lie algebra and establish regularity results (in the sense of Milnor) for these Lie groups. The general construction principle exhibited here enables to treat a broad class of examples from physics, numerical analysis and control theory. Groups of controlled characters appear in renormalisation of quantum field theories, numerical analysis and control theory in the guise of groups of locally convergent power series. The results presented here, generalise the construction of the (tame) Butcher group, aka the controlled character group of the Butcher-Connes-Kreimer Hopf algebra.