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 forminfinite-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, also known as the controlled character group of the Butcher–Connes–Kreimer Hopf algebra.
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DOI : 10.4171/aihpd/90
Mots-clés : Real analytic, infinite-dimensional Lie group, (combinatorial) Hopf algebra, Silva space, weighted sequence space, inductive limit of Banach spaces, regularity of Lie groups, Faà di Bruno Hopf algebra, Butcher–Connes–Kreimer Hopf algebra, (tame) Butcher group
@article{AIHPD_2020__7_3_395_0, author = {Dahmen, Rafael and Schmeding, Alexander}, title = {Lie groups of controlled characters of combinatorial {Hopf} algebras}, journal = {Annales de l{\textquoteright}Institut Henri Poincar\'e D}, pages = {395--456}, volume = {7}, number = {3}, year = {2020}, doi = {10.4171/aihpd/90}, mrnumber = {4152619}, zbl = {1456.22007}, language = {en}, url = {http://archive.numdam.org/articles/10.4171/aihpd/90/} }
TY - JOUR AU - Dahmen, Rafael AU - Schmeding, Alexander TI - Lie groups of controlled characters of combinatorial Hopf algebras JO - Annales de l’Institut Henri Poincaré D PY - 2020 SP - 395 EP - 456 VL - 7 IS - 3 UR - http://archive.numdam.org/articles/10.4171/aihpd/90/ DO - 10.4171/aihpd/90 LA - en ID - AIHPD_2020__7_3_395_0 ER -
%0 Journal Article %A Dahmen, Rafael %A Schmeding, Alexander %T Lie groups of controlled characters of combinatorial Hopf algebras %J Annales de l’Institut Henri Poincaré D %D 2020 %P 395-456 %V 7 %N 3 %U http://archive.numdam.org/articles/10.4171/aihpd/90/ %R 10.4171/aihpd/90 %G en %F AIHPD_2020__7_3_395_0
Dahmen, Rafael; Schmeding, Alexander. Lie groups of controlled characters of combinatorial Hopf algebras. Annales de l’Institut Henri Poincaré D, Tome 7 (2020) no. 3, pp. 395-456. doi : 10.4171/aihpd/90. http://archive.numdam.org/articles/10.4171/aihpd/90/
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