Recurrence relations and convergence theory of the generalized polar decomposition on Lie groups

The subject matter of this paper is the analysis of some issues related to generalized polar decompositions on Lie groups. This decomposition, depending on an involutive automorphism $\sigma$, is equivalent to a factorization of $z\in G$, $G$ being a Lie group, as $z=xy$ with $\sigma(x)=x^{-1}$ and $\sigma(y)=y$, and was recently discussed by Munthe-Kaas, Quispel and Zanna together with its many applications to numerical analysis. It turns out that, contrary to $X(t) = \log(x)$, an analysis of $Y(t) = \log(y)$ is a very complicated task. In this paper we derive the series expansion for $Y(t)=\log(y)$, obtaining an explicit recurrence relation that completely defines the function $Y(t)$ in terms of projections on a Lie triple system ${\mathfrak p}_\sigma$ and a subalgebra $\mathfrak{k}_{\sigma}$ of the Lie algebra ${\mathfrak g}$, and obtain bounds on its region of analyticity. The results presented in this paper have direct application, among others, to linear algebra, integration of differential equations and approximation of the exponential.