Cohomology in algebra, geometry, physicsand statistics

Exponential maps, ODE-s and a Hopf algebroid

Hopf algebroids over noncommutative base are generalizations of convolution algebras of groupoids. In a series of works with collaborators we have constructed a completed Hopf algebroid of differential operators on a Lie group and related examples of Drinfeld-Xu twists. Comparing the twist formulas in an arbitrary coordinate chart with the twist formulas in the normal coordinates (i.e. given by the exponential map) I have observed a new formulas for passage between the coordinates, inverse to the passage given by solving flow ODEs. It is interesting to compare these formulas with other series for ODEs including the ones coming from Lie integrators, Runge-Kutta integrators and the related considerations around Butcher group. As the latter combinatorics is parallel to the combinatorics of renormalization of QFTs, I expect that our approach could be eventually useful there as well.